This is PieriMaps.info, produced by Macaulay2, version 1.26.06

PieriMaps -- Pieri inclusions
*****************************

Description
===========

For mathematical background of this package and some examples of use, see:
Steven V Sam, Computing inclusions of Schur modules, arXiv:0810.4666
Some other references:
Andrzej Daszkiewicz, On the Invariant Ideals of the Symmetric Algebra $S.(V
\oplus \wedge^2 V)$, J. Algebra 125, 1989, 444-473.
David Eisenbud, Gunnar Fl\o ystad, and Jerzy Weyman, The existence of pure free
resolutions, arXiv:0709.1529.
William Fulton, Young Tableaux: With Applications to Representation Theory and
Geometry, London Math. Society Student Texts 35, 1997.
Jerzy Weyman, Cohomology of Vector Bundles and Syzygies, Cambridge University
Press, 2002.

Let $V$ be a vector space over $K$. If $K$ has characteristic 0, then given the
partition $\mu$ and the partition $\mu'$ obtained from $\mu$ by removing a
single box, there is a unique (up to nonzero scalar) $GL(V)$-equivariant
inclusion $S_\mu V \to V \otimes S_{\mu'} V$, where $S_\mu$ refers to the
irreducible representation of $GL(V)$ with highest weight $\mu$. This can be
extended uniquely to a map of $P = \mathrm{Sym}(V)$-modules $P \otimes S_\mu V
\to P \otimes S_{\mu'} V$. The purpose of this package is to write down matrix
representatives for these maps. The main function for doing so is pieri.
Here is an example of the use of the package PieriMaps, which is also
designed to check whether the maps are being constructed correctly (important,
since it is notoriously difficult to get the signs and coefficients right.) We
will construct by hand the free resolution in 3 variables corresponding to the
degree sequence (0,2,3,6). Let's start with the packaged code:
+-----------------------------------------------------------------------------+
|i1 : R = QQ[a,b,c];                                                          |
+-----------------------------------------------------------------------------+
|i2 : f = pureFree({0,2,3,6}, R)                                              |
|                                                                             |
|o2 = | 12a2 0    0    6ab 0   0   6ac   0    0   2b2 0   0   2bc  0    0     |
|     | 0    12a2 0    0   6ab 0   0     6ac  0   0   2b2 0   0    2bc  0     |
|     | 0    0    12a2 0   0   6ab 0     0    6ac 0   0   2b2 0    0    2bc   |
|     | 0    0    0    0   6a2 0   -12a2 0    0   0   8ab 0   -8ab 4ac  0     |
|     | 0    0    0    0   0   6a2 0     -3a2 0   0   0   8ab 0    -2ab 4ac   |
|     | 0    0    0    0   0   0   0     0    0   0   0   2a2 0    -a2  0     |
|     ------------------------------------------------------------------------|
|     2c2   0    0   0   0   0    0   0    0   0    0   0   |                 |
|     0     2c2  0   0   0   0    0   0    0   0    0   0   |                 |
|     0     0    2c2 0   0   0    0   0    0   0    0   0   |                 |
|     -16ac 0    0   6b2 0   4bc  0   2c2  0   0    0   0   |                 |
|     0     -4ac 0   0   6b2 -b2  4bc -2bc 2c2 0    0   0   |                 |
|     2a2   0    0   0   6ab -2ab 2ac -2ac 0   12b2 6bc 2c2 |                 |
|                                                                             |
|             6      27                                                       |
|o2 : Matrix R  <-- R                                                         |
+-----------------------------------------------------------------------------+
|i3 : betti res coker f                                                       |
|                                                                             |
|            0  1  2 3                                                        |
|o3 = total: 6 27 24 3                                                        |
|         0: 6  .  . .                                                        |
|         1: . 27 24 .                                                        |
|         2: .  .  . .                                                        |
|         3: .  .  . 3                                                        |
|                                                                             |
|o3 : BettiTally                                                              |
+-----------------------------------------------------------------------------+
By the general theory from Eisenbud-Fl\o ystad-Weyman, each of these free
modules should be essentially Schur functors corresponding to the following
partitions.
+------------------------------+
|i4 : needsPackage "SchurRings"|
|                              |
|o4 = SchurRings               |
|                              |
|o4 : Package                  |
+------------------------------+
|i5 : schurRing(s,3)           |
|                              |
|o5 = schurRing (QQ, s, 3)     |
|                              |
|o5 : SchurRing                |
+------------------------------+
|i6 : dim s_{2,2}              |
|                              |
|o6 = 6                        |
+------------------------------+
|i7 : dim s_{4,2}              |
|                              |
|o7 = 27                       |
+------------------------------+
|i8 : dim s_{4,3}              |
|                              |
|o8 = 24                       |
+------------------------------+
|i9 : dim s_{4,3,3}            |
|                              |
|o9 = 3                        |
+------------------------------+
This package also provides a routine schurRank, for computing
this dimension:
+---------------------------+
|i10 : schurRank(3, {2,2})  |
|                           |
|o10 = 6                    |
+---------------------------+
|i11 : schurRank(3, {4,2})  |
|                           |
|o11 = 27                   |
+---------------------------+
|i12 : schurRank(3, {4,3})  |
|                           |
|o12 = 24                   |
+---------------------------+
|i13 : schurRank(3, {4,3,3})|
|                           |
|o13 = 3                    |
+---------------------------+
We now use pieri, to construct each of the maps of the resolution
separately.
+-----------------------------------------------------------------------------+
|i14 : f1 = pieri({4,2,0},{1,1}, R)                                           |
|                                                                             |
|o14 = | 12a2 0    0    6ab 0   0   6ac   0    0   2b2 0   0   2bc  0    0    |
|      | 0    12a2 0    0   6ab 0   0     6ac  0   0   2b2 0   0    2bc  0    |
|      | 0    0    12a2 0   0   6ab 0     0    6ac 0   0   2b2 0    0    2bc  |
|      | 0    0    0    0   6a2 0   -12a2 0    0   0   8ab 0   -8ab 4ac  0    |
|      | 0    0    0    0   0   6a2 0     -3a2 0   0   0   8ab 0    -2ab 4ac  |
|      | 0    0    0    0   0   0   0     0    0   0   0   2a2 0    -a2  0    |
|      -----------------------------------------------------------------------|
|      2c2   0    0   0   0   0    0   0    0   0    0   0   |                |
|      0     2c2  0   0   0   0    0   0    0   0    0   0   |                |
|      0     0    2c2 0   0   0    0   0    0   0    0   0   |                |
|      -16ac 0    0   6b2 0   4bc  0   2c2  0   0    0   0   |                |
|      0     -4ac 0   0   6b2 -b2  4bc -2bc 2c2 0    0   0   |                |
|      2a2   0    0   0   6ab -2ab 2ac -2ac 0   12b2 6bc 2c2 |                |
|                                                                             |
|              6      27                                                      |
|o14 : Matrix R  <-- R                                                        |
+-----------------------------------------------------------------------------+
|i15 : f2 = pieri({4,3,0},{2}, R)                                             |
|                                                                             |
|o15 = | 3b  c   0   0   0     0     0     0     0     0     0     0     0    |
|      | 0   2b  2c  0   0     0     0     0     0     0     0     0     0    |
|      | 0   0   b   3c  0     0     0     0     0     0     0     0     0    |
|      | -6a 0   0   0   3/2b  c     0     0     -3/2c 0     0     0     0    |
|      | 0   -4a 0   0   0     b     2c    0     0     -c    0     0     0    |
|      | 0   0   -2a 0   0     0     1/2b  3c    0     0     -1/2c 0     0    |
|      | 0   -2a 0   0   0     -1/2b 0     0     3b    1/2c  0     0     0    |
|      | 0   0   -4a 0   0     0     -b    0     0     2b    c     0     0    |
|      | 0   0   0   -6a 0     0     0     -3/2b 0     0     b     3/2c  0    |
|      | 0   0   0   0   -9/2a 0     0     0     0     0     0     0     c    |
|      | 0   0   0   0   0     -3a   0     0     0     0     0     0     0    |
|      | 0   0   0   0   0     0     -3/2a 0     0     0     0     0     0    |
|      | 0   0   0   0   0     -3/2a 0     0     -9/2a 0     0     0     -b   |
|      | 0   0   0   0   0     0     -3a   0     0     -3a   0     0     0    |
|      | 0   0   0   0   0     0     0     -9/2a 0     0     -3/2a 0     0    |
|      | 0   0   0   0   0     0     0     0     0     -3/2a 0     0     0    |
|      | 0   0   0   0   0     0     0     0     0     0     -3a   0     0    |
|      | 0   0   0   0   0     0     0     0     0     0     0     -9/2a 0    |
|      | 0   0   0   0   0     0     0     0     0     0     0     0     -4/3a|
|      | 0   0   0   0   0     0     0     0     0     0     0     0     0    |
|      | 0   0   0   0   0     0     0     0     0     0     0     0     0    |
|      | 0   0   0   0   0     0     0     0     0     0     0     0     0    |
|      | 0   0   0   0   0     0     0     0     0     0     0     0     0    |
|      | 0   0   0   0   0     0     0     0     0     0     0     0     0    |
|      | 0   0   0   0   0     0     0     0     0     0     0     0     0    |
|      | 0   0   0   0   0     0     0     0     0     0     0     0     0    |
|      | 0   0   0   0   0     0     0     0     0     0     0     0     0    |
|      -----------------------------------------------------------------------|
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      2c  0   -c    0     0     0     0     0     0     0   0     |          |
|      0   3c  0     -1/2c 0     0     0     0     0     0   0     |          |
|      0   0   1/2c  0     0     0     0     0     0     0   0     |          |
|      -2b 0   b     c     0     0     0     0     0     0   0     |          |
|      0   -3b 0     1/2b  3/2c  0     0     0     0     0   0     |          |
|      0   0   -1/2b 0     0     0     0     0     0     0   0     |          |
|      0   0   0     -b    0     0     0     0     0     0   0     |          |
|      0   0   0     0     -3/2b 0     0     0     0     0   0     |          |
|      0   0   0     0     0     2c    0     0     0     0   0     |          |
|      -a  0   1/3a  0     0     -1/2b 3c    -1/2c 0     0   0     |          |
|      -2a 0   0     0     0     -3b   0     c     0     0   0     |          |
|      0   -3a 0     -a    0     0     -9/2b 0     3/2c  0   0     |          |
|      0   0   0     -2a   0     0     0     -2b   0     0   0     |          |
|      0   0   0     0     -3a   0     0     0     -3b   0   0     |          |
|      0   0   0     0     0     -1/4a 0     0     0     3c  0     |          |
|      0   0   0     0     0     0     -3/2a 1/6a  0     -6b 3/2c  |          |
|      0   0   0     0     0     0     0     0     -3/2a 0   -9/2b |          |
|                                                                             |
|              27      24                                                     |
|o15 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
|i16 : f3 = pieri({4,3,3},{3,3,3}, R)                                         |
|                                                                             |
|o16 = | 6c3       0         0         |                                      |
|      | -18bc2    0         0         |                                      |
|      | 18b2c     0         0         |                                      |
|      | -6b3      0         0         |                                      |
|      | 0         6c3       0         |                                      |
|      | 72/5ac2   -72/5bc2  18/5c3    |                                      |
|      | -144/5abc 54/5b2c   -36/5bc2  |                                      |
|      | 72/5ab2   -12/5b3   18/5b2c   |                                      |
|      | -72/5ac2  -18/5bc2  12/5c3    |                                      |
|      | 144/5abc  36/5b2c   -54/5bc2  |                                      |
|      | -72/5ab2  -18/5b3   72/5b2c   |                                      |
|      | 0         0         -6b3      |                                      |
|      | 0         27ac2     0         |                                      |
|      | -108/5a2c -162/5abc 108/5ac2  |                                      |
|      | -36/5a2b  36/5ab2   -54/5abc  |                                      |
|      | -432/5a2c -108/5abc 162/5ac2  |                                      |
|      | 216/5a2b  54/5ab2   -216/5abc |                                      |
|      | 0         0         18ab2     |                                      |
|      | 0         18a2c     0         |                                      |
|      | -24/5a3   -36/5a2b  54/5a2c   |                                      |
|      | -216/5a3  -54/5a2b  216/5a2c  |                                      |
|      | 0         0         -18a2b    |                                      |
|      | 0         3/2a3     0         |                                      |
|      | 0         0         6a3       |                                      |
|                                                                             |
|              24      3                                                      |
|o16 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
Fix the degrees (i.e. make sure that the target of f2 is the source of f1,
etc). Otherwise the test of exactness below would fail.
+-----------------------------------------------------------------------------+
|i17 : f1                                                                     |
|                                                                             |
|o17 = | 12a2 0    0    6ab 0   0   6ac   0    0   2b2 0   0   2bc  0    0    |
|      | 0    12a2 0    0   6ab 0   0     6ac  0   0   2b2 0   0    2bc  0    |
|      | 0    0    12a2 0   0   6ab 0     0    6ac 0   0   2b2 0    0    2bc  |
|      | 0    0    0    0   6a2 0   -12a2 0    0   0   8ab 0   -8ab 4ac  0    |
|      | 0    0    0    0   0   6a2 0     -3a2 0   0   0   8ab 0    -2ab 4ac  |
|      | 0    0    0    0   0   0   0     0    0   0   0   2a2 0    -a2  0    |
|      -----------------------------------------------------------------------|
|      2c2   0    0   0   0   0    0   0    0   0    0   0   |                |
|      0     2c2  0   0   0   0    0   0    0   0    0   0   |                |
|      0     0    2c2 0   0   0    0   0    0   0    0   0   |                |
|      -16ac 0    0   6b2 0   4bc  0   2c2  0   0    0   0   |                |
|      0     -4ac 0   0   6b2 -b2  4bc -2bc 2c2 0    0   0   |                |
|      2a2   0    0   0   6ab -2ab 2ac -2ac 0   12b2 6bc 2c2 |                |
|                                                                             |
|              6      27                                                      |
|o17 : Matrix R  <-- R                                                        |
+-----------------------------------------------------------------------------+
|i18 : f2 = map(source f1,,f2)                                                |
|                                                                             |
|o18 = {2} | 3b  c   0   0   0     0     0     0     0     0     0     0      |
|      {2} | 0   2b  2c  0   0     0     0     0     0     0     0     0      |
|      {2} | 0   0   b   3c  0     0     0     0     0     0     0     0      |
|      {2} | -6a 0   0   0   3/2b  c     0     0     -3/2c 0     0     0      |
|      {2} | 0   -4a 0   0   0     b     2c    0     0     -c    0     0      |
|      {2} | 0   0   -2a 0   0     0     1/2b  3c    0     0     -1/2c 0      |
|      {2} | 0   -2a 0   0   0     -1/2b 0     0     3b    1/2c  0     0      |
|      {2} | 0   0   -4a 0   0     0     -b    0     0     2b    c     0      |
|      {2} | 0   0   0   -6a 0     0     0     -3/2b 0     0     b     3/2c   |
|      {2} | 0   0   0   0   -9/2a 0     0     0     0     0     0     0      |
|      {2} | 0   0   0   0   0     -3a   0     0     0     0     0     0      |
|      {2} | 0   0   0   0   0     0     -3/2a 0     0     0     0     0      |
|      {2} | 0   0   0   0   0     -3/2a 0     0     -9/2a 0     0     0      |
|      {2} | 0   0   0   0   0     0     -3a   0     0     -3a   0     0      |
|      {2} | 0   0   0   0   0     0     0     -9/2a 0     0     -3/2a 0      |
|      {2} | 0   0   0   0   0     0     0     0     0     -3/2a 0     0      |
|      {2} | 0   0   0   0   0     0     0     0     0     0     -3a   0      |
|      {2} | 0   0   0   0   0     0     0     0     0     0     0     -9/2a  |
|      {2} | 0   0   0   0   0     0     0     0     0     0     0     0      |
|      {2} | 0   0   0   0   0     0     0     0     0     0     0     0      |
|      {2} | 0   0   0   0   0     0     0     0     0     0     0     0      |
|      {2} | 0   0   0   0   0     0     0     0     0     0     0     0      |
|      {2} | 0   0   0   0   0     0     0     0     0     0     0     0      |
|      {2} | 0   0   0   0   0     0     0     0     0     0     0     0      |
|      {2} | 0   0   0   0   0     0     0     0     0     0     0     0      |
|      {2} | 0   0   0   0   0     0     0     0     0     0     0     0      |
|      {2} | 0   0   0   0   0     0     0     0     0     0     0     0      |
|      -----------------------------------------------------------------------|
|      0     0   0   0     0     0     0     0     0     0     0   0     |    |
|      0     0   0   0     0     0     0     0     0     0     0   0     |    |
|      0     0   0   0     0     0     0     0     0     0     0   0     |    |
|      0     0   0   0     0     0     0     0     0     0     0   0     |    |
|      0     0   0   0     0     0     0     0     0     0     0   0     |    |
|      0     0   0   0     0     0     0     0     0     0     0   0     |    |
|      0     0   0   0     0     0     0     0     0     0     0   0     |    |
|      0     0   0   0     0     0     0     0     0     0     0   0     |    |
|      0     0   0   0     0     0     0     0     0     0     0   0     |    |
|      c     0   0   0     0     0     0     0     0     0     0   0     |    |
|      0     2c  0   -c    0     0     0     0     0     0     0   0     |    |
|      0     0   3c  0     -1/2c 0     0     0     0     0     0   0     |    |
|      -b    0   0   1/2c  0     0     0     0     0     0     0   0     |    |
|      0     -2b 0   b     c     0     0     0     0     0     0   0     |    |
|      0     0   -3b 0     1/2b  3/2c  0     0     0     0     0   0     |    |
|      0     0   0   -1/2b 0     0     0     0     0     0     0   0     |    |
|      0     0   0   0     -b    0     0     0     0     0     0   0     |    |
|      0     0   0   0     0     -3/2b 0     0     0     0     0   0     |    |
|      -4/3a 0   0   0     0     0     2c    0     0     0     0   0     |    |
|      0     -a  0   1/3a  0     0     -1/2b 3c    -1/2c 0     0   0     |    |
|      0     -2a 0   0     0     0     -3b   0     c     0     0   0     |    |
|      0     0   -3a 0     -a    0     0     -9/2b 0     3/2c  0   0     |    |
|      0     0   0   0     -2a   0     0     0     -2b   0     0   0     |    |
|      0     0   0   0     0     -3a   0     0     0     -3b   0   0     |    |
|      0     0   0   0     0     0     -1/4a 0     0     0     3c  0     |    |
|      0     0   0   0     0     0     0     -3/2a 1/6a  0     -6b 3/2c  |    |
|      0     0   0   0     0     0     0     0     0     -3/2a 0   -9/2b |    |
|                                                                             |
|              27      24                                                     |
|o18 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
|i19 : f3 = map(source f2,,f3)                                                |
|                                                                             |
|o19 = {3} | 6c3       0         0         |                                  |
|      {3} | -18bc2    0         0         |                                  |
|      {3} | 18b2c     0         0         |                                  |
|      {3} | -6b3      0         0         |                                  |
|      {3} | 0         6c3       0         |                                  |
|      {3} | 72/5ac2   -72/5bc2  18/5c3    |                                  |
|      {3} | -144/5abc 54/5b2c   -36/5bc2  |                                  |
|      {3} | 72/5ab2   -12/5b3   18/5b2c   |                                  |
|      {3} | -72/5ac2  -18/5bc2  12/5c3    |                                  |
|      {3} | 144/5abc  36/5b2c   -54/5bc2  |                                  |
|      {3} | -72/5ab2  -18/5b3   72/5b2c   |                                  |
|      {3} | 0         0         -6b3      |                                  |
|      {3} | 0         27ac2     0         |                                  |
|      {3} | -108/5a2c -162/5abc 108/5ac2  |                                  |
|      {3} | -36/5a2b  36/5ab2   -54/5abc  |                                  |
|      {3} | -432/5a2c -108/5abc 162/5ac2  |                                  |
|      {3} | 216/5a2b  54/5ab2   -216/5abc |                                  |
|      {3} | 0         0         18ab2     |                                  |
|      {3} | 0         18a2c     0         |                                  |
|      {3} | -24/5a3   -36/5a2b  54/5a2c   |                                  |
|      {3} | -216/5a3  -54/5a2b  216/5a2c  |                                  |
|      {3} | 0         0         -18a2b    |                                  |
|      {3} | 0         3/2a3     0         |                                  |
|      {3} | 0         0         6a3       |                                  |
|                                                                             |
|              24      3                                                      |
|o19 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
|i20 : f1 * f2                                                                |
|                                                                             |
|o20 = 0                                                                      |
|                                                                             |
|              6      24                                                      |
|o20 : Matrix R  <-- R                                                        |
+-----------------------------------------------------------------------------+
|i21 : f2 * f3                                                                |
|                                                                             |
|o21 = 0                                                                      |
|                                                                             |
|              27      3                                                      |
|o21 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
Check that the complex is exact.
+------------------------+
|i22 : ker f1 == image f2|
|                        |
|o22 = true              |
+------------------------+
|i23 : ker f2 == image f3|
|                        |
|o23 = true              |
+------------------------+
Looks great! Now let's try it modulo some prime numbers and see if we get
exactness.
+-----------------------------------------------------------------------------+
|i24 : p = 32003                                                              |
|                                                                             |
|o24 = 32003                                                                  |
+-----------------------------------------------------------------------------+
|i25 : R = ZZ/p[a,b,c];                                                       |
+-----------------------------------------------------------------------------+
|i26 : f1 = pieri({4,2,0},{1,1},R)                                            |
|                                                                             |
|o26 = | 7012a2+6250ab-5379b2+12089ac-8371bc-3669c2                           |
|      | 3630a2-5863ab-8195b2+3219ac-14676bc+10443c2                          |
|      | -12738a2-14389ab-3669b2+14830ac+10443bc+4994c2                       |
|      | 1962a2+704ab+42b2                                                    |
|      | 3a2                                                                  |
|      | 0                                                                    |
|      -----------------------------------------------------------------------|
|      9770a2-3164ab+14689b2+12630ac+661bc+10959c2                            |
|      -6841a2-2950ab-13606b2-14090ac-15537bc+8572c2                          |
|      15976a2-1642ab-2725b2-14788ac+7543bc-1774c2                            |
|      3793a2+6938ab+14609b2+9266ac-12554bc+1946c2                            |
|      5403a2+4637ab-2058ac+1946bc+86c2                                       |
|      a2-12863ab                                                             |
|      -----------------------------------------------------------------------|
|      1642a2+15650ab+932b2+4677ac-13229bc+10676c2                            |
|      5880a2+12908ab+12962b2-12390ac+2856bc+4469c2                           |
|      -285a2+6636ab-9146b2-10575ac-3456bc-2216c2                             |
|      3554a2+8755ab-5997b2-15690ac+6585bc-1155c2                             |
|      12831a2-7437ab+b2-15850ac-1155bc+5350c2                                |
|      102a2-9646ab                                                           |
|      -----------------------------------------------------------------------|
|      -1128a2-6948ab+12866b2+4136ac+9742bc-3952c2                            |
|      -15652a2+9995ab-13630b2+4919ac+3717bc+6340c2                           |
|      3004a2-6111ab-10090b2-12960ac-556bc+4552c2                             |
|      1723a2+2521ab+11439b2+7047ac-12864bc-3101c2                            |
|      7431a2-12074ab+b2-13792ac-3101bc+5264c2                                |
|      101a2+3217ab                                                           |
|      -----------------------------------------------------------------------|
|      7230a2-7752ab-6045b2-4786ac+6938bc-8543c2                              |
|      12526a2-7587ab+12804b2+13972ac+8090bc-1035c2                           |
|      -15894a2+3623ab+12687b2+11930ac-5873bc-3927c2                          |
|      1985a2-8539ab+14392b2-11485ac+12244bc-6993c2                           |
|      -3363a2+10655ab+b2-9676ac-6993bc+5092c2                                |
|      99a2-3060ab                                                            |
|      -----------------------------------------------------------------------|
|      10499a2+15367ab-14668b2+2246ac+2607bc+4192c2                           |
|      -9328a2+14587ab+6412b2+3856ac-14242bc+1359c2                           |
|      10229a2-12217ab-11216b2+131ac-1421bc+7570c2                            |
|      10095a2+15220ab-14490b2+1986ac+5349bc-10885c2                          |
|      -14145a2+1381ab+b2-5560ac-10885bc+4920c2                               |
|      97a2-9337ab                                                            |
|      -----------------------------------------------------------------------|
|      1047a2-9148ab-14489b2-12525ac-4085bc+2166c2                            |
|      7697a2+1003ab-1637b2+4780ac+391bc+13521c2                              |
|      -15962a2+9478ab+14126b2+15628ac+12799bc+7040c2                         |
|      -5950a2+9792ab-11201b2+15457ac-1546bc-14777c2                          |
|      7088a2-7893ab+b2-1444ac-14777bc+4748c2                                 |
|      95a2-15614ab                                                           |
|      -----------------------------------------------------------------------|
|      -11780a2-7783ab+15644b2-12502ac+9471bc+15795c2                         |
|      -6257a2-7148ab+11266b2-12116ac+13638bc+3415c2                          |
|      13379a2-9728ab-8883b2-6095ac+4751bc-5518c2                             |
|      -14147a2+7180ab-7744b2-3075ac-8441bc+13334c2                           |
|      -3670a2+14836ab+b2+2672ac+13334bc+4576c2                               |
|      93a2+10112ab                                                           |
|      -----------------------------------------------------------------------|
|      3034a2-13972ab+4905b2-4278ac-5652bc-10955c2                            |
|      -3240a2-9891ab-2428b2+461ac-6617bc-1500c2                              |
|      -1071a2+1855ab-8264b2-12598ac+1894bc-1475c2                            |
|      -1335a2+12896ab-3702b2+10400ac-15336bc+9442c2                          |
|      -14377a2+5564ab+b2+6788ac+9442bc+4404c2                                |
|      91a2+3835ab                                                            |
|      -----------------------------------------------------------------------|
|      -831a2-13029ab+6536b2+734ac-2467bc-6439c2                              |
|      10974a2-15649ab-3280b2+1186ac+2373bc+9234c2                            |
|      -10599a2+8862ab-8287b2-601ac+14687bc+1563c2                            |
|      14886a2-3252ab+10396b2-7748ac+9856bc+5550c2                            |
|      8269a2-3518ab+b2+10906ac+5550bc+4232c2                                 |
|      89a2-2463ab                                                            |
|      -----------------------------------------------------------------------|
|      -12775a2-2850ab-13594b2+4415ac-8872bc+82c2                             |
|      -9498a2+4038ab-12853b2+11380ac-7464bc-2812c2                           |
|      -6720a2-158ab-3727b2-13068ac+4744bc+997c2                              |
|      -4792a2-15924ab-5194b2-15584ac+6009bc+1659c2                           |
|      -5848a2-7444ab+b2+15112ac+1659bc+4060c2                                |
|      87a2-9502ab                                                            |
|      -----------------------------------------------------------------------|
|      -15504a2+13987ab+14845b2-3188ac+14858bc-13007c2                        |
|      10549a2+8441ab-15477b2+8136ac-11717bc-5829c2                           |
|      -13699a2-12105ab-8768b2+14289ac+4845bc+5278c2                          |
|      14817a2+6672ab+10707b2-15387ac-5252bc-2188c2                           |
|      15830a2+8648ab+b2-10655ac-2188bc+3889c2                                |
|      85a2+1314ab                                                            |
|      -----------------------------------------------------------------------|
|      4524a2+12312ab-14403b2+7223ac-12012bc+3211c2                           |
|      -6731a2-15509ab-7762b2-6386ac+9930bc+1264c2                            |
|      -12893a2+13347ab+3656b2-6061ac-15138bc+8000c2                          |
|      2048a2-15003ab+58b2-5828ac+9823bc+10672c2                              |
|      -15463a2+4694ab+42b2-801ac+10674bc-2188c2                              |
|      1902a2+5587ab+6b2+ac                                                   |
|      -----------------------------------------------------------------------|
|      -874a2+8323ab+7624b2-3641ac+7499bc+6453c2                              |
|      -2574a2-1514ab-11310b2+9477ac+8166bc-8357c2                            |
|      11944a2+3335ab+2767b2+4436ac+704bc-15849c2                             |
|      5768a2-11230ab-11596b2-3289ac-13091bc+1170c2                           |
|      14598a2+2902ab+1380b2-13881ac+1458bc-9464c2                            |
|      5137a2-3348ab+816b2+144ac+27bc+c2                                      |
|      -----------------------------------------------------------------------|
|      8798a2+6400ab-5840b2-1845ac-5373bc+3153c2                              |
|      5490a2-3378ab-15485b2-10675ac-10730bc+8660c2                           |
|      7941a2+2999ab-5200b2+11546ac-13300bc+4886c2                            |
|      312a2-8445ab-3198b2-14681ac-14607bc-15411c2                            |
|      -7665a2-10070ab+1339b2-11917ac-15125bc-3555c2                          |
|      3318a2-3010ab+810b2+143ac+27bc+c2                                      |
|      -----------------------------------------------------------------------|
|      2831a2-3560ab+571b2-13064ac+1640bc-3083c2                              |
|      -13668a2-12329ab-13022b2-4900ac+3246bc-15294c2                         |
|      -8960a2-6361ab+2810b2-2056ac+11836bc+7296c2                            |
|      13799a2+5358ab+2103b2-847ac-14917bc+2457c2                             |
|      -3911a2+7994ab+1219b2-9746ac+2737bc-7160c2                             |
|      -1896a2+12949ab+792b2+140ac+27bc+c2                                    |
|      -----------------------------------------------------------------------|
|      -10208a2-10208ab+4359b2+7863ac+12227bc-1636c2                          |
|      -6975a2-5478ab+525b2+4489ac-3792bc+2838c2                              |
|      -9436a2-12004ab-6659b2-1756ac+13227bc-12080c2                          |
|      10799a2-14805ab-5853b2+5504ac+8783bc+567c2                             |
|      1753a2+9159ab+1108b2-11225ac+841bc-12195c2                             |
|      -6399a2+6913ab+774b2+137ac+27bc+c2                                     |
|      -----------------------------------------------------------------------|
|      15978a2+15978ab-3205b2-15047ac+11129bc-1579c2                          |
|      13751a2-10331ab+10561b2+12886ac+11403bc+12979c2                        |
|      12020a2+6580ab+13037b2-5038ac-8432bc+4988c2                            |
|      -12240a2-2272ab+2215b2+3435ac+3661bc-12579c2                           |
|      137a2+8742ab+1006b2-10819ac-12311bc+9131c2                             |
|      -10245a2-7910ab+756b2+134ac+27bc+c2                                    |
|      -----------------------------------------------------------------------|
|      8a2+8ab-6399b2-9590ac+2115bc-5865c2                                    |
|      -15989a2+3346ab+2178b2+12477ac-5945bc-3518c2                           |
|      -14213a2-12550ab+5406b2-7599ac+5172bc+13573c2                          |
|      -9477a2+252ab-4635b2+3027ac+11793bc+15836c2                            |
|      -2787a2-4434ab+913b2-14623ac-15905bc-11076c2                           |
|      -13488a2-10036ab+738b2+131ac+27bc+c2                                   |
|      -----------------------------------------------------------------------|
|      4045a2+4045ab+809b2-2113ac+1696bc+1963c2                               |
|      -1304a2+12277ab-3177b2+973ac+7104bc-15817c2                            |
|      -4410a2-7310ab-14593b2+12296ac-7956bc-10499c2                          |
|      -15500a2+12511ab-5323b2-1496ac+4892bc-10046c2                          |
|      8205a2-1470ab+829b2+15722ac-9790bc-12374c2                             |
|      15821a2-394ab+720b2+128ac+27bc+c2                                      |
|      -----------------------------------------------------------------------|
|      15219a2+15219ab+15845b2-11552ac-1366bc-2180c2                          |
|      4293a2-1686ab+6688b2-10945ac-10702bc+1327c2                            |
|      4900a2+4550ab+10451b2+5419ac+1789bc-6152c2                             |
|      -10585a2+213ab-7314b2-3964ac-4016bc-15700c2                            |
|      -5979a2-11500ab+754b2+924ac-15450bc+1997c2                             |
|      13622a2+1758ab+702b2+125ac+27bc+c2                                     |
|      -----------------------------------------------------------------------|
|      -2422a2-2422ab-6885b2+7401ac+4968bc-5617c2                             |
|      516a2-8522ab-3268b2+14073ac+8136bc-9874c2                              |
|      -6986a2-6472ab-10454b2+11582ac-7971bc-5982c2                           |
|      8679a2-941ab+3008b2-2798ac+11988bc-13214c2                             |
|      2493a2+14054ab+688b2+3806ac-12970bc-2882c2                             |
|      11864a2-2309ab+684b2+122ac+27bc+c2                                     |
|      -----------------------------------------------------------------------|
|      6487a2+6487ab+7698b2-6463ac+10096bc+983c2                              |
|      1235a2+15775ab-1860b2-10278ac-6994bc+15653c2                           |
|      -9492a2+14649ab+5162b2+2860ac-12427bc+11060c2                          |
|      -3302a2-15821ab-4033b2+10151ac-13273bc-4590c2                          |
|      -12215a2-4361ab+582b2+7846ac-4366bc-11113c2                            |
|      9642a2+11901ab+626b2+112ac+26bc+c2                                     |
|      -----------------------------------------------------------------------|
|      -12758a2-12758ab+3849b2-15384ac+5255bc-6838c2                          |
|      -11535a2+14459ab+12641b2-534ac-8196bc+6473c2                           |
|      12756a2-4746ab+9071b2-3909ac-380bc-2164c2                              |
|      13224a2-2459ab+12661b2+6309ac+15482bc-12673c2                          |
|      -4479a2+12477ab+401b2-13706ac-12495bc-1648c2                           |
|      6331a2+12532ab+494b2+89ac+23bc+c2                                      |
|      -----------------------------------------------------------------------|
|      3849ac-7137bc+4496c2                                                   |
|      3849a2-4713ab+13571b2-5152ac-1772bc-5389c2                             |
|      -930a2-14501ab-3597b2-15426ac+13417bc-2977c2                           |
|      -12411a2-13177ab+6020b2-7509ac+14376bc+4715c2                          |
|      -11925a2-12616ab+135b2+5609ac+4813bc-9068c2                            |
|      1785a2-3459ab+266b2+49ac+17bc+c2                                       |
|      -----------------------------------------------------------------------|
|      -7344bc-4176c2                                0               |        |
|      4717ab-9757ac+2927bc+11966c2                  0               |        |
|      13571a2+13571ab-10087b2-10087ac-8418bc+4717c2 0               |        |
|      4717a2-2627ab+7344b2+7259ac+8259bc+15083c2    -10c2           |        |
|      2448a2-11272ab+25b2-8765ac+15125bc+3025c2     0               |        |
|      275a2+5961ab+110b2+21ac+11bc+c2               25b2+5ac+5bc+c2 |        |
|                                                                             |
|              6      27                                                      |
|o26 : Matrix R  <-- R                                                        |
+-----------------------------------------------------------------------------+
|i27 : betti res coker f1                                                     |
|                                                                             |
|             0  1  2 3                                                       |
|o27 = total: 6 27 24 3                                                       |
|          0: 6  .  . .                                                       |
|          1: . 27 24 .                                                       |
|          2: .  .  . .                                                       |
|          3: .  .  . 3                                                       |
|                                                                             |
|o27 : BettiTally                                                             |
+-----------------------------------------------------------------------------+
|i28 : f2 = pieri({4,3,0},{2},R)                                              |
|                                                                             |
|o28 = | 6986b+4510c           -1451b-7220c         13064b+9527c              |
|      | 9020b-10741c          -14440b+3016c        -12949b-15317c            |
|      | 10631b+2953c          1508b+10808c         8343b+684c                |
|      | -13972a+14166b+3105c  2902a-9397b-5451c    5875a-5254b+4348c         |
|      | 13963a+923b+2263c     -3123a-12640b-10200c -6105a-4110b+15487c       |
|      | 10741a+10088b-320c    -3016a+9732b+14476c  15317a-12189b+420c        |
|      | -9020a+11176b+3043c   14440a-8058b-7439c   12949a-14861b+15883c      |
|      | -10521a-8004b-15157c  -6032a+14786b+14710c -1369a-8099b+10532c       |
|      | -5906a-3207b+8650c    10387a-12207b+8371c  -1368a+7689b+1079c        |
|      | -10495a-11636c        -3812a-11660c        15762a-3966c              |
|      | -2769a+2034c          5917a+2034c          12330a+1147c              |
|      | 1739a+15880c          2807a+15880c         4564a+13355c              |
|      | -10840a+11636b-13565c 8524a+11660b-13565c  -464a+3966b+11577c        |
|      | -14780a-2034b+11327c  -13758a-2034b+11327c 9839a-1147b-4667c         |
|      | 10581a-15880b-270c    -6807a-15880b-270c   7676a-13355b-10233c       |
|      | -9129a+13565b         -9686a+13565b        -15646a-11577b            |
|      | 13468a-11327b         -12127a-11327b       407a+4667b                |
|      | 6053a+270b            6890a+270b           -3237a+10233b             |
|      | 4847a-24c             4879a-24c            5288a-1392c               |
|      | -7163a+6b+2034c       -7163a+6b+2034c      11569a+348b-2296c         |
|      | -6907a+36b-12204c     -6907a+36b-12204c    7702a+2088b+13764c        |
|      | 15134a+6102b          15134a+6102b         9647a-6879b               |
|      | 9349a-7595b           9349a-7595b          9334a+4475b               |
|      | 540a                  540a                 -11537a                   |
|      | 3a                    3a                   174a                      |
|      | 0                     0                    a                         |
|      | 0                     0                    0                         |
|      -----------------------------------------------------------------------|
|      -15183b-7919c       4969b-6623c          15163b-9429c                  |
|      -15838b-4094c       -13246b+10515c       13145b-2015c                  |
|      -2047b-12069c       -10744b-5499c        14994b+30c                    |
|      -1637a+7830b-12120c -9938a+8975b+14045c  1677a-13275b-6277c            |
|      -327a-10408b+12432c -5511a+13349b+6604c  5713a+7996b-3832c             |
|      4094a+11292b-3503c  -10515a+11413b+7642c 2015a+15497b+8124c            |
|      15838a+8628b-15635c 13246a+7935b-12479c  -13145a-7455b+907c            |
|      8188a+10885b-5888c  10973a-8736b+10767c  4030a+4637b+4339c             |
|      -7865a+13337b+7255c 10998a+12255b+10478c -60a+7193b-12886c             |
|      8513a+14347c        5078a+11453c         7822a-12082c                  |
|      -779a-299c          -8044a-4319c         8015a-8243c                   |
|      -1873a+13436c       -2236a-2363c         -14488a+13760c                |
|      10476a-14347b-648c  -1934a-11453b+1094c  9193a+12082b-15937c           |
|      -5945a+299b+9117c   6396a+4319b+11574c   -2415a+8243b+247c             |
|      2501a-13436b-10053c 4315a+2363b-9603c    -13948a-13760b-9333c          |
|      14902a+648b         5434a-1094b          -2721a+15937b                 |
|      -14339a-9117b       -298a-11574b         -13017a-247b                  |
|      10238a+10053b       569a+9603b           6655a+9333b                   |
|      2206a-1376c         -4603a-1336c         -5226a-1312c                  |
|      -15636a+344b-3652c  -3539a+334b-7042c    4100a+328b-9076c              |
|      1595a+2064b-10103c  2131a+2004b+10237c   8114a+1968b-9562c             |
|      -11110a-10947b      -14998a+10886b       1871a+4784b                   |
|      13769a-11797b       8855a+11529b         -494a-12879b                  |
|      -11897a             -12797a              -13337a                       |
|      172a                167a                 164a                          |
|      a                   a                    a                             |
|      0                   0                    0                             |
|      -----------------------------------------------------------------------|
|      -15488b-2190c         -6016b-15512c        -10796b-3322c               |
|      -4380b+3603c          979b+8643c           -6644b+13540c               |
|      -14200b+3911c         -11680b+13264c       6770b+594c                  |
|      -1027a+13161b+1107c   12032a+12462b+8085c  -10411a-8151b+73c           |
|      8760a-6030b+5120c     -1958a+13732b+1140c  13288a+11081b+15412c        |
|      -3603a+3224b-8085c    -8643a+188b+1322c    -13540a-146b+14592c         |
|      4380a-11259b+3888c    -979a+4428b-194c     6644a+474b-7998c            |
|      -7206a+9104b+401c     14717a-1152b-7492c   4923a+303b+6218c            |
|      -7822a+12344b+6148c   5475a-6280b-1448c    -1188a+13369b-8693c         |
|      -7480a-6710c          -5383a-4866c         -7550a-6982c                |
|      -13913a-13787c        -9193a+11052c        -1240a+2268c                |
|      -9672a-2120c          -564a+14003c         438a-1877c                  |
|      -1547a+6710b+241c     -5536a+4866b-14378c  -1641a+6982b+4212c          |
|      -10669a+13787b-11080c 36a-11052b+9596c     -15142a-2268b-1731c         |
|      -12777a+2120b-9063c   14874a-14003b-8793c  12126a+1877b-8523c          |
|      -11664a-241b          582a+14378b          -8009a-4212b                |
|      -1203a+11080b         -9527a-9596b         13349a+1731b                |
|      13559a+9063b          4344a+8793b          -5924a+8523b                |
|      -1721a-1288c          6488a-1264c          -12026a-1240c               |
|      12147a+322b-11110c    -11401a+316b-13144c  -2538a+310b-15178c          |
|      13305a+1932b+2642c    -14299a+1896b+14846c -10692a+1860b-4953c         |
|      -13263a-1318b         3606a-7420b          -11528a-13522b              |
|      -9843a-5284b          12811a+2311b         3462a+9906b                 |
|      -13877a               -14417a              -14957a                     |
|      161a                  158a                 155a                        |
|      a                     a                    a                           |
|      0                     0                    0                           |
|      -----------------------------------------------------------------------|
|      -5877b-6258c         -8813b-14176c        9247b+1060c                  |
|      -12516b+12885c       3651b-8309c          2120b+10717c                 |
|      -9559b+5151c         11847b-9902c         -10643b+15903c               |
|      11754a-5269b-13526c  -14377a+10506b-6944c 13509a+1070b-4108c           |
|      -6971a-4888b+6606c   -7302a-12365b+14458c -4240a+14466b-7935c          |
|      -12885a-6457b+8767c  8309a+3644b+324c     -10717a+5242b+6182c          |
|      12516a-12283b+15786c -3651a+11342b+59c    -2120a-2088b-1550c           |
|      6233a+12052b+3081c   -15385a-7052b+12629c 10569a+15319b+7046c          |
|      -10302a+5928b+11394c -12199a+1309b-9825c  197a-13808b+9463c            |
|      15807a-13490c        485a+7187c           -3210a-5592c                 |
|      14664a-8136c         5092a+11835c         -11395a-9899c                |
|      -12632a+14246c       -10932a-1634c        -15726a+14484c               |
|      13421a+13490b-7995c  -13194a-7187b+13013c -13415a+5592b+9233c          |
|      8032a+8136b-13058c   9785a-11835b+7618c   9851a+9899b-3711c            |
|      -12082a-14246b-8253c -4899a+1634b-7983c   -9125a-14484b-7713c          |
|      -15355a+7995b        -177a-13013b         4650a-9233b                  |
|      -9243a+13058b        -5884a-7618b         10865a+3711b                 |
|      -2179a+8253b         -2528a+7983b         3614a+7713b                  |
|      7319a-1216c          1085a-1192c          7456a-1168c                  |
|      6733a+304b+14791c    -15589a+298b+12757c  -3462a+292b+10723c           |
|      -7877a+1824b+7251c   -5858a+1788b-12548c  -8567a+1752b-344c            |
|      5341a+12379b         -9793a+6277b         7084a+175b                   |
|      -5887a-14502b        -15236a-6907b        7422a+688b                   |
|      -15497a              15966a               15426a                       |
|      152a                 149a                 146a                         |
|      a                    a                    a                            |
|      0                    0                    0                            |
|      -----------------------------------------------------------------------|
|      8346b+14184c         -9595b+9348c         -13631b+10629c               |
|      -3635b+3555c         -13307b+15643c       -10745b+3953c                |
|      -14224b+9547c        -8180b+3540c         -14025b-10748c               |
|      15311a+7734b-3965c   -12813a+9990b-9574c  -4741a+2846b+13893c          |
|      7270a-8118b+15295c   -5389a-15886b+2496c  -10513a+4749b+11970c         |
|      -3555a+11902b-6621c  -15643a+1686b+83c    -3953a+11113b-5917c          |
|      3635a-4247b+155c     13307a-4681b+2124c   10745a-4661b-15762c          |
|      -7110a+8819b-14645c  717a+5124b-523c      -7906a+10735b+15973c         |
|      12909a-15674b-15693c -7080a+7567b+10712c  -10507a-9064b+15358c         |
|      8801a-643c           2033a-10155c         -8538a+8235c                 |
|      -7649a-5644c         15655a+15019c        -14247a-9613c                |
|      -3703a-1484c         -5058a+13376c        -1336a-1110c                 |
|      -7367a+643b-6563c    10762a+10155b+14319c 4307a-8235b+12905c           |
|      -8336a+5644b-15074c  9143a-15019b+14327c  -4109a+9613b+15512c          |
|      2879a+1484b-7443c    9053a-13376b-6634c   12940a+1110b-2619c           |
|      -465a+6563b          -6372a-14319b        15283a-12905b                |
|      11932a+15074b        1569a-14327b         -15916a-15512b               |
|      15076a+7443b         -133a+6634b          -14071a+2619b                |
|      11525a-1144c         13540a-1072c         -10980a-7952c                |
|      -5658a+286b+8690c    3719a+268b+2616c     -4829a+1988b+8796c           |
|      -13233a+1716b+11854c -11654a+1608b-15708c 15806a+11928b+11062c         |
|      -7908a-5924b         13138a+7857b         9845a-5489b-c                |
|      -1855a+8295b         3349a-587b           979a+9879b                   |
|      14886a               13268a               5238a+2b                     |
|      143a                 134a                 994a-2c                      |
|      a                    a                    14a+4b                       |
|      0                    0                    a                            |
|      -----------------------------------------------------------------------|
|      13944b+8113c         -12487b-10199c       11493b-3230c                 |
|      -15777b-3330c        11605b+15016c        -6460b+10930c                |
|      -1665b-9464c         7508b-2013c          5465b+9843c                  |
|      4115a+3655b-13627c   -7029a+11202b-9870c  9017a-13905b-10840c          |
|      -449a-96b-10941c     8793a-4578b+5692c    12920a+12644b-2954c          |
|      3330a+1715b+9216c    -15016a-1300b-13233c -10930a+10268b-5069c         |
|      15777a-4893b-7101c   -11605a+12873b-5446c 6460a+8643b-9990c            |
|      6660a+169b+6374c     1971a+12819b-4774c   10143a+3510b+14230c          |
|      -13075a-15829b-1522c 4026a+3036b-13463c   12317a+13207b+10645c         |
|      -10965a-3751c        -1603a+15007c        9712a-3181c                  |
|      288a-2428c           13734a+7603c         -5929a+4719c                 |
|      -5145a-3628c         3900a-15438c         1199a-5707c                  |
|      -8446a+3751b-10743c  -9009a-15007b-13513c 6591a+3181b-7312c            |
|      313a+2428b+14400c    8473a-7603b+6775c    -1668a-4719b+6030c           |
|      -12164a+3628b-8838c  -1412a+15438b-12058c 7589a+5707b+3309c            |
|      -10700a+10743b       -15665a+13513b       -2033a+7312b                 |
|      12881a-14400b        14322a-6775b         -10687a-6030b                |
|      4566a+8838b          8386a+12058b         68a-3309b                    |
|      15669a-7533c         1326a-581c           14909a+11467c                |
|      4795a+9884b-8050c    -4631a+8146b+11934c  -5256a+5134b+3257c           |
|      -8089a-4702b+15121c  -12580a-15130b-8618c 9905a-1199b+11729c           |
|      14031a+8735b-16c     -10726a+4564b-15c    -3338a+10320b-13c            |
|      3203a+1761b          -13550a-14767b       -12060a+8545b                |
|      -14327a+32b          -7887a+30b           -6618a+26b                   |
|      4942a-30c            4073a-28c            2567a-24c                    |
|      98a+60b-c            85a+56b-c            61a+48b-c                    |
|      16a+3b               15a+3b               13a+3b                       |
|      -----------------------------------------------------------------------|
|      -12152b+5475c         -7500b-3750c          3601b-14201c               |
|      10950b-4201c          -7500b-12651c         3601b-14201c               |
|      13901b+9024c          9676b+7975c           8901b+14976c               |
|      -7699a-11453b-530c    15000a-7001b-1228c    -7202a+6075b+12181c        |
|      10103a-1952b-5791c    15000a+9022b-1078c    -7202a-8095b-2134c         |
|      4201a+421b-6692c      12651a+13839b+2171c   14201a+11919b+9122c        |
|      -10950a-1868b-12264c  7500a+15989b-3786c    -3601a+11500b-7098c        |
|      8402a+14108b-605c     -6701a-10819b-13931c  -3601a+11776b+12273c       |
|      13955a+10893b+6492c   -15950a-3533b+171c    2051a-3357b-13273c         |
|      2356a-10975c          -11000a+10754c        13778a-13697c              |
|      5856a+9441c           4937a-6991c           -7718a+6289c               |
|      -1263a+9943c          -9514a-7889c          -3754a+14309c              |
|      7194a+10975b-15119c   -12280a-10754b+15534c -7037a+13697b+1965c        |
|      7052a-9441b-3531c     3688a+6991b+2007c     3077a-6289b+8197c          |
|      -12603a-9943b+1032c   4086a+7889b-13825c    14708a-14309b-5059c        |
|      4789a+15119b          11358a-15534b         -10709a-1965b              |
|      1815a+3531b           9790a-2007b           -4816a-8197b               |
|      12527a-1032b          -513a+13825b          7816a+5059b                |
|      -6702a-11912c         -3671a-6136c          7595a-2632c                |
|      5653a+2978b+7658c     14319a+1534b-5087c    12202a+658b+8068c          |
|      -11206a-14135b-14437c 7926a+9204b-1781c     -10219a+3948b+15442c       |
|      11355a-8660b-11c      -11123a-15036b-9c     -10603a-7682b-7c           |
|      7062a-3129b           -4014a+3562b          15609a+1119b               |
|      -2064a+22b            -4353a+18b            10118a+14b                 |
|      1489a-20c             767a-16c              329a-12c                   |
|      41a+40b-c             25a+32b-c             13a+24b-c                  |
|      11a+3b                9a+3b                 7a+3b                      |
|      -----------------------------------------------------------------------|
|      0                  0             0    |                                |
|      0                  0             0    |                                |
|      8901c              0             0    |                                |
|      8901b+4661c        0             0    |                                |
|      -4240b-8732c       0             0    |                                |
|      2077b-12513c       0             0    |                                |
|      -15682b+8520c      0             0    |                                |
|      -2077b+3408c       0             0    |                                |
|      14201a-7189b+1704c 0             0    |                                |
|      5300a+639c         0             0    |                                |
|      12720a-5184c       0             0    |                                |
|      -6231a-8527c       -748c         0    |                                |
|      1060a-639b-13516c  0             0    |                                |
|      424a+5184b-4170c   -374c         0    |                                |
|      -4900a+8527b-1337c 748b-187c     0    |                                |
|      6443a+13516b       0             0    |                                |
|      -10224a+4170b      374b          0    |                                |
|      -5112a+1337b       187b          0    |                                |
|      -852a-824c         -136c         0    |                                |
|      -14238a+206b+1696c 34b+184c      0    |                                |
|      213a+1236b-10236c  204b-1116c    0    |                                |
|      14782a+5133b-5c    1309a+561b-3c -c   |                                |
|      8340a-11531b       748a+2232b    0    |                                |
|      2674a+10b          374a+6b       2b   |                                |
|      103a-8c            17a-4c        -c   |                                |
|      5a+16b-c           a+8b-c        2b-c |                                |
|      5a+3b              3a+3b         a+3b |                                |
|                                                                             |
|              27      24                                                     |
|o28 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
|i29 : f3 = pieri({4,3,3},{3,3,3},R)                                          |
|                                                                             |
|o29 = | -20c3         -15c3         0      |                                 |
|      | 60bc2         45bc2         0      |                                 |
|      | -60b2c        -45b2c        0      |                                 |
|      | 20b3          15b3          0      |                                 |
|      | 20c3          20c3          0      |                                 |
|      | -48ac2-48bc2  -36ac2-48bc2  3c3    |                                 |
|      | 96abc+36b2c   72abc+36b2c   -6bc2  |                                 |
|      | -48ab2-8b3    -36ab2-8b3    3b2c   |                                 |
|      | 48ac2-12bc2   36ac2-12bc2   2c3    |                                 |
|      | -96abc+24b2c  -72abc+24b2c  -9bc2  |                                 |
|      | 48ab2-12b3    36ab2-12b3    12b2c  |                                 |
|      | 0             0             -5b3   |                                 |
|      | 90ac2         90ac2         0      |                                 |
|      | 72a2c-108abc  54a2c-108abc  18ac2  |                                 |
|      | 24a2b+24ab2   18a2b+24ab2   -9abc  |                                 |
|      | 288a2c-72abc  216a2c-72abc  27ac2  |                                 |
|      | -144a2b+36ab2 -108a2b+36ab2 -36abc |                                 |
|      | 0             0             15ab2  |                                 |
|      | 60a2c         60a2c         0      |                                 |
|      | 16a3-24a2b    12a3-24a2b    9a2c   |                                 |
|      | 144a3-36a2b   108a3-36a2b   36a2c  |                                 |
|      | 0             0             -15a2b |                                 |
|      | 5a3           5a3           0      |                                 |
|      | 0             0             5a3    |                                 |
|                                                                             |
|              24      3                                                      |
|o29 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
|i30 : f2 = map(source f1,,f2)                                                |
|                                                                             |
|o30 = {2} | 6986b+4510c           -1451b-7220c         13064b+9527c          |
|      {2} | 9020b-10741c          -14440b+3016c        -12949b-15317c        |
|      {2} | 10631b+2953c          1508b+10808c         8343b+684c            |
|      {2} | -13972a+14166b+3105c  2902a-9397b-5451c    5875a-5254b+4348c     |
|      {2} | 13963a+923b+2263c     -3123a-12640b-10200c -6105a-4110b+15487c   |
|      {2} | 10741a+10088b-320c    -3016a+9732b+14476c  15317a-12189b+420c    |
|      {2} | -9020a+11176b+3043c   14440a-8058b-7439c   12949a-14861b+15883c  |
|      {2} | -10521a-8004b-15157c  -6032a+14786b+14710c -1369a-8099b+10532c   |
|      {2} | -5906a-3207b+8650c    10387a-12207b+8371c  -1368a+7689b+1079c    |
|      {2} | -10495a-11636c        -3812a-11660c        15762a-3966c          |
|      {2} | -2769a+2034c          5917a+2034c          12330a+1147c          |
|      {2} | 1739a+15880c          2807a+15880c         4564a+13355c          |
|      {2} | -10840a+11636b-13565c 8524a+11660b-13565c  -464a+3966b+11577c    |
|      {2} | -14780a-2034b+11327c  -13758a-2034b+11327c 9839a-1147b-4667c     |
|      {2} | 10581a-15880b-270c    -6807a-15880b-270c   7676a-13355b-10233c   |
|      {2} | -9129a+13565b         -9686a+13565b        -15646a-11577b        |
|      {2} | 13468a-11327b         -12127a-11327b       407a+4667b            |
|      {2} | 6053a+270b            6890a+270b           -3237a+10233b         |
|      {2} | 4847a-24c             4879a-24c            5288a-1392c           |
|      {2} | -7163a+6b+2034c       -7163a+6b+2034c      11569a+348b-2296c     |
|      {2} | -6907a+36b-12204c     -6907a+36b-12204c    7702a+2088b+13764c    |
|      {2} | 15134a+6102b          15134a+6102b         9647a-6879b           |
|      {2} | 9349a-7595b           9349a-7595b          9334a+4475b           |
|      {2} | 540a                  540a                 -11537a               |
|      {2} | 3a                    3a                   174a                  |
|      {2} | 0                     0                    a                     |
|      {2} | 0                     0                    0                     |
|      -----------------------------------------------------------------------|
|      -15183b-7919c       4969b-6623c          15163b-9429c                  |
|      -15838b-4094c       -13246b+10515c       13145b-2015c                  |
|      -2047b-12069c       -10744b-5499c        14994b+30c                    |
|      -1637a+7830b-12120c -9938a+8975b+14045c  1677a-13275b-6277c            |
|      -327a-10408b+12432c -5511a+13349b+6604c  5713a+7996b-3832c             |
|      4094a+11292b-3503c  -10515a+11413b+7642c 2015a+15497b+8124c            |
|      15838a+8628b-15635c 13246a+7935b-12479c  -13145a-7455b+907c            |
|      8188a+10885b-5888c  10973a-8736b+10767c  4030a+4637b+4339c             |
|      -7865a+13337b+7255c 10998a+12255b+10478c -60a+7193b-12886c             |
|      8513a+14347c        5078a+11453c         7822a-12082c                  |
|      -779a-299c          -8044a-4319c         8015a-8243c                   |
|      -1873a+13436c       -2236a-2363c         -14488a+13760c                |
|      10476a-14347b-648c  -1934a-11453b+1094c  9193a+12082b-15937c           |
|      -5945a+299b+9117c   6396a+4319b+11574c   -2415a+8243b+247c             |
|      2501a-13436b-10053c 4315a+2363b-9603c    -13948a-13760b-9333c          |
|      14902a+648b         5434a-1094b          -2721a+15937b                 |
|      -14339a-9117b       -298a-11574b         -13017a-247b                  |
|      10238a+10053b       569a+9603b           6655a+9333b                   |
|      2206a-1376c         -4603a-1336c         -5226a-1312c                  |
|      -15636a+344b-3652c  -3539a+334b-7042c    4100a+328b-9076c              |
|      1595a+2064b-10103c  2131a+2004b+10237c   8114a+1968b-9562c             |
|      -11110a-10947b      -14998a+10886b       1871a+4784b                   |
|      13769a-11797b       8855a+11529b         -494a-12879b                  |
|      -11897a             -12797a              -13337a                       |
|      172a                167a                 164a                          |
|      a                   a                    a                             |
|      0                   0                    0                             |
|      -----------------------------------------------------------------------|
|      -15488b-2190c         -6016b-15512c        -10796b-3322c               |
|      -4380b+3603c          979b+8643c           -6644b+13540c               |
|      -14200b+3911c         -11680b+13264c       6770b+594c                  |
|      -1027a+13161b+1107c   12032a+12462b+8085c  -10411a-8151b+73c           |
|      8760a-6030b+5120c     -1958a+13732b+1140c  13288a+11081b+15412c        |
|      -3603a+3224b-8085c    -8643a+188b+1322c    -13540a-146b+14592c         |
|      4380a-11259b+3888c    -979a+4428b-194c     6644a+474b-7998c            |
|      -7206a+9104b+401c     14717a-1152b-7492c   4923a+303b+6218c            |
|      -7822a+12344b+6148c   5475a-6280b-1448c    -1188a+13369b-8693c         |
|      -7480a-6710c          -5383a-4866c         -7550a-6982c                |
|      -13913a-13787c        -9193a+11052c        -1240a+2268c                |
|      -9672a-2120c          -564a+14003c         438a-1877c                  |
|      -1547a+6710b+241c     -5536a+4866b-14378c  -1641a+6982b+4212c          |
|      -10669a+13787b-11080c 36a-11052b+9596c     -15142a-2268b-1731c         |
|      -12777a+2120b-9063c   14874a-14003b-8793c  12126a+1877b-8523c          |
|      -11664a-241b          582a+14378b          -8009a-4212b                |
|      -1203a+11080b         -9527a-9596b         13349a+1731b                |
|      13559a+9063b          4344a+8793b          -5924a+8523b                |
|      -1721a-1288c          6488a-1264c          -12026a-1240c               |
|      12147a+322b-11110c    -11401a+316b-13144c  -2538a+310b-15178c          |
|      13305a+1932b+2642c    -14299a+1896b+14846c -10692a+1860b-4953c         |
|      -13263a-1318b         3606a-7420b          -11528a-13522b              |
|      -9843a-5284b          12811a+2311b         3462a+9906b                 |
|      -13877a               -14417a              -14957a                     |
|      161a                  158a                 155a                        |
|      a                     a                    a                           |
|      0                     0                    0                           |
|      -----------------------------------------------------------------------|
|      -5877b-6258c         -8813b-14176c        9247b+1060c                  |
|      -12516b+12885c       3651b-8309c          2120b+10717c                 |
|      -9559b+5151c         11847b-9902c         -10643b+15903c               |
|      11754a-5269b-13526c  -14377a+10506b-6944c 13509a+1070b-4108c           |
|      -6971a-4888b+6606c   -7302a-12365b+14458c -4240a+14466b-7935c          |
|      -12885a-6457b+8767c  8309a+3644b+324c     -10717a+5242b+6182c          |
|      12516a-12283b+15786c -3651a+11342b+59c    -2120a-2088b-1550c           |
|      6233a+12052b+3081c   -15385a-7052b+12629c 10569a+15319b+7046c          |
|      -10302a+5928b+11394c -12199a+1309b-9825c  197a-13808b+9463c            |
|      15807a-13490c        485a+7187c           -3210a-5592c                 |
|      14664a-8136c         5092a+11835c         -11395a-9899c                |
|      -12632a+14246c       -10932a-1634c        -15726a+14484c               |
|      13421a+13490b-7995c  -13194a-7187b+13013c -13415a+5592b+9233c          |
|      8032a+8136b-13058c   9785a-11835b+7618c   9851a+9899b-3711c            |
|      -12082a-14246b-8253c -4899a+1634b-7983c   -9125a-14484b-7713c          |
|      -15355a+7995b        -177a-13013b         4650a-9233b                  |
|      -9243a+13058b        -5884a-7618b         10865a+3711b                 |
|      -2179a+8253b         -2528a+7983b         3614a+7713b                  |
|      7319a-1216c          1085a-1192c          7456a-1168c                  |
|      6733a+304b+14791c    -15589a+298b+12757c  -3462a+292b+10723c           |
|      -7877a+1824b+7251c   -5858a+1788b-12548c  -8567a+1752b-344c            |
|      5341a+12379b         -9793a+6277b         7084a+175b                   |
|      -5887a-14502b        -15236a-6907b        7422a+688b                   |
|      -15497a              15966a               15426a                       |
|      152a                 149a                 146a                         |
|      a                    a                    a                            |
|      0                    0                    0                            |
|      -----------------------------------------------------------------------|
|      8346b+14184c         -9595b+9348c         -13631b+10629c               |
|      -3635b+3555c         -13307b+15643c       -10745b+3953c                |
|      -14224b+9547c        -8180b+3540c         -14025b-10748c               |
|      15311a+7734b-3965c   -12813a+9990b-9574c  -4741a+2846b+13893c          |
|      7270a-8118b+15295c   -5389a-15886b+2496c  -10513a+4749b+11970c         |
|      -3555a+11902b-6621c  -15643a+1686b+83c    -3953a+11113b-5917c          |
|      3635a-4247b+155c     13307a-4681b+2124c   10745a-4661b-15762c          |
|      -7110a+8819b-14645c  717a+5124b-523c      -7906a+10735b+15973c         |
|      12909a-15674b-15693c -7080a+7567b+10712c  -10507a-9064b+15358c         |
|      8801a-643c           2033a-10155c         -8538a+8235c                 |
|      -7649a-5644c         15655a+15019c        -14247a-9613c                |
|      -3703a-1484c         -5058a+13376c        -1336a-1110c                 |
|      -7367a+643b-6563c    10762a+10155b+14319c 4307a-8235b+12905c           |
|      -8336a+5644b-15074c  9143a-15019b+14327c  -4109a+9613b+15512c          |
|      2879a+1484b-7443c    9053a-13376b-6634c   12940a+1110b-2619c           |
|      -465a+6563b          -6372a-14319b        15283a-12905b                |
|      11932a+15074b        1569a-14327b         -15916a-15512b               |
|      15076a+7443b         -133a+6634b          -14071a+2619b                |
|      11525a-1144c         13540a-1072c         -10980a-7952c                |
|      -5658a+286b+8690c    3719a+268b+2616c     -4829a+1988b+8796c           |
|      -13233a+1716b+11854c -11654a+1608b-15708c 15806a+11928b+11062c         |
|      -7908a-5924b         13138a+7857b         9845a-5489b-c                |
|      -1855a+8295b         3349a-587b           979a+9879b                   |
|      14886a               13268a               5238a+2b                     |
|      143a                 134a                 994a-2c                      |
|      a                    a                    14a+4b                       |
|      0                    0                    a                            |
|      -----------------------------------------------------------------------|
|      13944b+8113c         -12487b-10199c       11493b-3230c                 |
|      -15777b-3330c        11605b+15016c        -6460b+10930c                |
|      -1665b-9464c         7508b-2013c          5465b+9843c                  |
|      4115a+3655b-13627c   -7029a+11202b-9870c  9017a-13905b-10840c          |
|      -449a-96b-10941c     8793a-4578b+5692c    12920a+12644b-2954c          |
|      3330a+1715b+9216c    -15016a-1300b-13233c -10930a+10268b-5069c         |
|      15777a-4893b-7101c   -11605a+12873b-5446c 6460a+8643b-9990c            |
|      6660a+169b+6374c     1971a+12819b-4774c   10143a+3510b+14230c          |
|      -13075a-15829b-1522c 4026a+3036b-13463c   12317a+13207b+10645c         |
|      -10965a-3751c        -1603a+15007c        9712a-3181c                  |
|      288a-2428c           13734a+7603c         -5929a+4719c                 |
|      -5145a-3628c         3900a-15438c         1199a-5707c                  |
|      -8446a+3751b-10743c  -9009a-15007b-13513c 6591a+3181b-7312c            |
|      313a+2428b+14400c    8473a-7603b+6775c    -1668a-4719b+6030c           |
|      -12164a+3628b-8838c  -1412a+15438b-12058c 7589a+5707b+3309c            |
|      -10700a+10743b       -15665a+13513b       -2033a+7312b                 |
|      12881a-14400b        14322a-6775b         -10687a-6030b                |
|      4566a+8838b          8386a+12058b         68a-3309b                    |
|      15669a-7533c         1326a-581c           14909a+11467c                |
|      4795a+9884b-8050c    -4631a+8146b+11934c  -5256a+5134b+3257c           |
|      -8089a-4702b+15121c  -12580a-15130b-8618c 9905a-1199b+11729c           |
|      14031a+8735b-16c     -10726a+4564b-15c    -3338a+10320b-13c            |
|      3203a+1761b          -13550a-14767b       -12060a+8545b                |
|      -14327a+32b          -7887a+30b           -6618a+26b                   |
|      4942a-30c            4073a-28c            2567a-24c                    |
|      98a+60b-c            85a+56b-c            61a+48b-c                    |
|      16a+3b               15a+3b               13a+3b                       |
|      -----------------------------------------------------------------------|
|      -12152b+5475c         -7500b-3750c          3601b-14201c               |
|      10950b-4201c          -7500b-12651c         3601b-14201c               |
|      13901b+9024c          9676b+7975c           8901b+14976c               |
|      -7699a-11453b-530c    15000a-7001b-1228c    -7202a+6075b+12181c        |
|      10103a-1952b-5791c    15000a+9022b-1078c    -7202a-8095b-2134c         |
|      4201a+421b-6692c      12651a+13839b+2171c   14201a+11919b+9122c        |
|      -10950a-1868b-12264c  7500a+15989b-3786c    -3601a+11500b-7098c        |
|      8402a+14108b-605c     -6701a-10819b-13931c  -3601a+11776b+12273c       |
|      13955a+10893b+6492c   -15950a-3533b+171c    2051a-3357b-13273c         |
|      2356a-10975c          -11000a+10754c        13778a-13697c              |
|      5856a+9441c           4937a-6991c           -7718a+6289c               |
|      -1263a+9943c          -9514a-7889c          -3754a+14309c              |
|      7194a+10975b-15119c   -12280a-10754b+15534c -7037a+13697b+1965c        |
|      7052a-9441b-3531c     3688a+6991b+2007c     3077a-6289b+8197c          |
|      -12603a-9943b+1032c   4086a+7889b-13825c    14708a-14309b-5059c        |
|      4789a+15119b          11358a-15534b         -10709a-1965b              |
|      1815a+3531b           9790a-2007b           -4816a-8197b               |
|      12527a-1032b          -513a+13825b          7816a+5059b                |
|      -6702a-11912c         -3671a-6136c          7595a-2632c                |
|      5653a+2978b+7658c     14319a+1534b-5087c    12202a+658b+8068c          |
|      -11206a-14135b-14437c 7926a+9204b-1781c     -10219a+3948b+15442c       |
|      11355a-8660b-11c      -11123a-15036b-9c     -10603a-7682b-7c           |
|      7062a-3129b           -4014a+3562b          15609a+1119b               |
|      -2064a+22b            -4353a+18b            10118a+14b                 |
|      1489a-20c             767a-16c              329a-12c                   |
|      41a+40b-c             25a+32b-c             13a+24b-c                  |
|      11a+3b                9a+3b                 7a+3b                      |
|      -----------------------------------------------------------------------|
|      0                  0             0    |                                |
|      0                  0             0    |                                |
|      8901c              0             0    |                                |
|      8901b+4661c        0             0    |                                |
|      -4240b-8732c       0             0    |                                |
|      2077b-12513c       0             0    |                                |
|      -15682b+8520c      0             0    |                                |
|      -2077b+3408c       0             0    |                                |
|      14201a-7189b+1704c 0             0    |                                |
|      5300a+639c         0             0    |                                |
|      12720a-5184c       0             0    |                                |
|      -6231a-8527c       -748c         0    |                                |
|      1060a-639b-13516c  0             0    |                                |
|      424a+5184b-4170c   -374c         0    |                                |
|      -4900a+8527b-1337c 748b-187c     0    |                                |
|      6443a+13516b       0             0    |                                |
|      -10224a+4170b      374b          0    |                                |
|      -5112a+1337b       187b          0    |                                |
|      -852a-824c         -136c         0    |                                |
|      -14238a+206b+1696c 34b+184c      0    |                                |
|      213a+1236b-10236c  204b-1116c    0    |                                |
|      14782a+5133b-5c    1309a+561b-3c -c   |                                |
|      8340a-11531b       748a+2232b    0    |                                |
|      2674a+10b          374a+6b       2b   |                                |
|      103a-8c            17a-4c        -c   |                                |
|      5a+16b-c           a+8b-c        2b-c |                                |
|      5a+3b              3a+3b         a+3b |                                |
|                                                                             |
|              27      24                                                     |
|o30 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
|i31 : f3 = map(source f2,,f3)                                                |
|                                                                             |
|o31 = {3} | -20c3         -15c3         0      |                             |
|      {3} | 60bc2         45bc2         0      |                             |
|      {3} | -60b2c        -45b2c        0      |                             |
|      {3} | 20b3          15b3          0      |                             |
|      {3} | 20c3          20c3          0      |                             |
|      {3} | -48ac2-48bc2  -36ac2-48bc2  3c3    |                             |
|      {3} | 96abc+36b2c   72abc+36b2c   -6bc2  |                             |
|      {3} | -48ab2-8b3    -36ab2-8b3    3b2c   |                             |
|      {3} | 48ac2-12bc2   36ac2-12bc2   2c3    |                             |
|      {3} | -96abc+24b2c  -72abc+24b2c  -9bc2  |                             |
|      {3} | 48ab2-12b3    36ab2-12b3    12b2c  |                             |
|      {3} | 0             0             -5b3   |                             |
|      {3} | 90ac2         90ac2         0      |                             |
|      {3} | 72a2c-108abc  54a2c-108abc  18ac2  |                             |
|      {3} | 24a2b+24ab2   18a2b+24ab2   -9abc  |                             |
|      {3} | 288a2c-72abc  216a2c-72abc  27ac2  |                             |
|      {3} | -144a2b+36ab2 -108a2b+36ab2 -36abc |                             |
|      {3} | 0             0             15ab2  |                             |
|      {3} | 60a2c         60a2c         0      |                             |
|      {3} | 16a3-24a2b    12a3-24a2b    9a2c   |                             |
|      {3} | 144a3-36a2b   108a3-36a2b   36a2c  |                             |
|      {3} | 0             0             -15a2b |                             |
|      {3} | 5a3           5a3           0      |                             |
|      {3} | 0             0             5a3    |                             |
|                                                                             |
|              24      3                                                      |
|o31 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
|i32 : f1 * f2                                                                |
|                                                                             |
|o32 = | 3253a3+5047a2b-1601ab2-12426b3+6717a2c+6959abc-4464b2c-5683ac2-3930bc|
|      | 267a3+9095a2b-93ab2-14086b3-11447a2c-14714abc-2149b2c-10471ac2+11716b|
|      | 11576a3-10838a2b-12579ab2+2734b3-10134a2c+10361abc+8410b2c-4926ac2+11|
|      | 5577a3+971a2b-1520ab2-8124b3-4824a2c-9075abc-15841b2c+11537ac2-12867b|
|      | -299a3+4752a2b+10383ab2+11115b3+4920a2c-2145abc-1583b2c+212ac2-5152bc|
|      | 6023a3-1770a2b-15405ab2+9867b3+15602a2c+5903abc-1371b2c+12587ac2+6444|
|      -----------------------------------------------------------------------|
|      2-6379c3     -10844a3+10340a2b+4202ab2+15438b3+4976a2c-2293abc-13300b2c|
|      c2+12911c3   -8459a3-14651a2b-7616ab2+7859b3+2835a2c+9117abc+3093b2c-15|
|      220bc2-826c3 6483a3-6927a2b+8397ab2+10240b3+8686a2c+4987abc+13256b2c+58|
|      c2-8831c3    15419a3+15273a2b+7039ab2-3395b3+5236a2c+15383abc-4577b2c+4|
|      2+3108c3     -1568a3+13761a2b-5619ab2-7923b3-14571a2c-7211abc+13872b2c+|
|      bc2+863c3    11655a3-12102a2b-7669ab2+10011b3+6370a2c-15951abc-1371b2c+|
|      -----------------------------------------------------------------------|
|      -11659ac2+15014bc2+4334c3                                              |
|      798ac2-9663bc2+15851c3                                                 |
|      24ac2+12168bc2+11899c3                                                 |
|      216ac2+6510bc2+3985c3                                                  |
|      3274ac2-14865bc2+14248c3                                               |
|      2941ac2+6444bc2+863c3                                                  |
|      -----------------------------------------------------------------------|
|      -12893a3+11254a2b+9293ab2                                              |
|      -13273a3-9154a2b-2421ab2-                                              |
|      -8742a3-10835a2b-12394ab2                                              |
|      1057a3+15620a2b+10655ab2-                                              |
|      -2510a3+12559a2b-8149ab2+                                              |
|      9381a3-10179a2b+11313ab2-                                              |
|      -----------------------------------------------------------------------|
|      +6315b3+6513a2c+8291abc+5745b2c+1999ac2+7067bc2-15927c3                |
|      11012b3-11023a2c-1747abc+4939b2c-1254ac2+2368bc2-15065c3               |
|      +14008b3+7277a2c+15288abc-15126b2c-14357ac2-11433bc2-14903c3           |
|      12440b3-12125a2c-8473abc-13417b2c+10782ac2-10863bc2+7756c3             |
|      2209b3-7337a2c-15633abc+13837b2c+7241ac2-11889bc2-12698c3              |
|      8922b3+4063a2c+13041abc-6571b2c+15768ac2-13383bc2-4824c3               |
|      -----------------------------------------------------------------------|
|      10432a3-6305a2b-8548ab2-683b3+8196a2c-2128abc-8250b2c+12467ac2-6360bc2+|
|      13831a3+7064a2b-11476ab2-11340b3+1496a2c-11410abc-307b2c+14972ac2+7752b|
|      7519a3+11323a2b+8321ab2-13808b3+15193a2c-443abc-14102b2c+15854ac2-15195|
|      -7764a3+15894a2b+2935ab2-11060b3+9707a2c-2953abc+15212b2c-3471ac2-3332b|
|      11581a3-708a2b+8334ab2-4803b3+7982a2c+15687abc-243b2c-5403ac2-9747bc2+6|
|      -5351a3+544a2b-10469ab2+5548b3-3586a2c+15343abc-4634b2c-1317ac2+14347bc|
|      -----------------------------------------------------------------------|
|      11673c3     14229a3-5657a2b                                            |
|      c2+13183c3  8899a3-7400a2b+                                            |
|      bc2+12224c3 2407a3-6580a2b+                                            |
|      c2-15634c3  4425a3-11834a2b                                            |
|      7c3         9232a3-15000a2b                                            |
|      2+15936c3   7646a3+11764a2b                                            |
|      -----------------------------------------------------------------------|
|      -731ab2-11568b3-7671a2c-13273abc-8756b2c+1366ac2-5118bc2-1302c3        |
|      1534ab2-4742b3+537a2c+8801abc-3226b2c+9844ac2+3689bc2-6585c3           |
|      568ab2-5261b3-5882a2c-15147abc-6891b2c+5926ac2+3281bc2+6489c3          |
|      -10836ab2+10920b3-6937a2c-2897abc-7641b2c+14895ac2-7544bc2-12339c3     |
|      -14636ab2+6196b3+16a2c-5212abc+9336b2c+3599ac2-4423bc2+15238c3         |
|      +10992ab2+9435b3-11569a2c-143abc+4812b2c+9342ac2+3768bc2+3830c3        |
|      -----------------------------------------------------------------------|
|      7671a3+4827a2b+879ab2-14812b3+14268a2c+11388abc+6628b2c-10653ac2+9294bc|
|      1163a3-14544a2b-925ab2-12106b3+11335a2c+15325abc-14257b2c-8917ac2-14630|
|      -2226a3+4183a2b+10664ab2-9775b3+6707a2c-1362abc-7235b2c-15270ac2-11936b|
|      -6935a3+8207a2b-5528ab2+1688b3+2610a2c+15332abc+14151b2c+11188ac2+2890b|
|      15556a3-11286a2b+7548ab2-3982b3-367a2c-2865abc+4109b2c+1307ac2-14871bc2|
|      14355a3+9489a2b-985ab2-1570b3-8034a2c+8065abc-4337b2c-1728ac2-2193bc2+2|
|      -----------------------------------------------------------------------|
|      2-7463c3   12906a3+12736a2b+7065ab2                                    |
|      bc2+5399c3 -4058a3-9133a2b+7088ab2-                                    |
|      c2+3947c3  -10475a3+6079a2b-6788ab2                                    |
|      c2+7298c3  -13796a3-9895a2b-9911ab2                                    |
|      -9516c3    13371a3+9285a2b-2923ab2-                                    |
|      967c3      7519a3-12292a2b-12788ab2                                    |
|      -----------------------------------------------------------------------|
|      +15342b3+11045a2c-14954abc-8387b2c-355ac2-3169bc2-9756c3               |
|      7990b3-10773a2c+1218abc+6606b2c+2722ac2+7937bc2+7435c3                 |
|      -15739b3-2385a2c+2010abc+11769b2c+4234ac2-3177bc2-5692c3               |
|      +10102b3-11040a2c+9420abc-135b2c-1329ac2-15303bc2-8515c3               |
|      1106b3-2079a2c-9751abc-12049b2c+5151ac2-14552bc2-10484c3               |
|      -11267b3-10778a2c-15745abc+4928b2c+4703ac2-7740bc2+2104c3              |
|      -----------------------------------------------------------------------|
|      5650a3+5484a2b-7668ab2+10288b3+3029a2c+670abc-11707b2c+14062ac2+4403bc2|
|      -1210a3-2881a2b+3550ab2+12803b3+945a2c+11551abc+12875b2c-1289ac2+3636bc|
|      10599a3-12691a2b-8232ab2+14005b3+3182a2c-14925abc+15614b2c-4736ac2-1463|
|      4235a3-6922a2b+6616ab2+10735b3-3296a2c+3614abc-2633b2c-3554ac2-7006bc2+|
|      -15506a3-11539a2b+4278ab2+15942b3+2475a2c-11680abc+7820b2c-14465ac2-734|
|      -8649a3-14167a2b-2321ab2+14939b3+8326a2c-15416abc+604b2c+15028ac2-12873|
|      -----------------------------------------------------------------------|
|      +13213c3    -2353a3-15525a2b+13178ab2+8936b3+5483a2c-12960abc-11958b2c+|
|      2-1472c3    5807a3+487a2b-7366ab2-8686b3-1836a2c-4182abc-9678b2c+4991ac|
|      4bc2-7958c3 -9966a3+2858a2b-4076ab2-2761b3-4878a2c-12583abc+5833b2c-255|
|      2873c3      8957a3+11679a2b-15648ab2-2245b3+3892a2c-14493abc+8661b2c-66|
|      7bc2+1602c3 -15113a3+3397a2b+8533ab2+11354b3+11895a2c+2805abc-5733b2c+5|
|      bc2+1241c3  -13746a3+3192a2b-13762ab2+15634b3-12432a2c-8571abc+14694b2c|
|      -----------------------------------------------------------------------|
|      12671ac2+15570bc2+2596c3                                               |
|      2+12477bc2-9719c3                                                      |
|      0ac2-9414bc2+14942c3                                                   |
|      22ac2+20bc2+10764c3                                                    |
|      237ac2+7282bc2+5180c3                                                  |
|      -15517ac2+14411bc2+378c3                                               |
|      -----------------------------------------------------------------------|
|      -13868a3-4297a2b+2974ab2-6824b3-6079a2c-9352abc-9913b2c+49ac2+12922bc2+|
|      15301a3+12293a2b-14636ab2-12572b3+14418a2c+9548abc+14285b2c+7799ac2+271|
|      -15968a3+5103a2b-15086ab2-14693b3-8694a2c+5856abc+10687b2c+8471ac2+1079|
|      -9344a3+15556a2b+7793ab2+6852b3-8467a2c+10046abc-2755b2c+12556ac2+13704|
|      241a3+10771a2b-4401ab2+4454b3+8570a2c-14136abc-1276b2c-4367ac2-7818bc2-|
|      -15064a3-9475a2b-15076ab2-6590b3+12675a2c-6167abc+15195b2c+11304ac2+101|
|      -----------------------------------------------------------------------|
|      10004c3      -3290a3-13425a2b-4342ab2+7276b3-6779a2c+                  |
|      4bc2+13919c3 6190a3-5185a2b+7550ab2+12249b3+11643a2c-                  |
|      7bc2+12958c3 -14630a3+63a2b-10355ab2+12513b3-1086a2c-                  |
|      bc2-5092c3   11510a3-1757a2b-13134ab2-2031b3-6080a2c-                  |
|      9127c3       -6042a3-9830a2b+6158ab2-3094b3-14827a2c+                  |
|      06bc2-485c3  -9123a3+191a2b-11126ab2-15398b3-14699a2c                  |
|      -----------------------------------------------------------------------|
|      2901abc-6364b2c+8832ac2-10473bc2-10420c3                               |
|      12201abc-10924b2c-4203ac2+15873bc2+3674c3                              |
|      15127abc+3205b2c-2029ac2+5637bc2+13024c3                               |
|      80abc-2931b2c+1075ac2+2276bc2-12480c3                                  |
|      11833abc+9897b2c+5373ac2-5717bc2+14146c3                               |
|      -9472abc+2197b2c+3717ac2+6217bc2-1348c3                                |
|      -----------------------------------------------------------------------|
|      -8012a3+4342a2b-3093ab2-1300b3-7823a2c-8326abc+5125b2c-7611ac2-3781bc2-|
|      2144a3-10955a2b+8130ab2+14568b3-10249a2c-9343abc-10688b2c+3489ac2-1212b|
|      -3559a3+14046a2b+15269ab2+7582b3+10327a2c+11878abc-6114b2c+12224ac2-614|
|      9309a3-4576a2b+298ab2-4786b3-13038a2c-6878abc-14214b2c-14461ac2-117bc2+|
|      -12075a3-5922a2b+6244ab2-4842b3-12375a2c-4435abc+3871b2c-5036ac2-2895bc|
|      1423a3-5688a2b+12657ab2-15545b3-8719a2c-3591abc+2186b2c+9826ac2+4768bc2|
|      -----------------------------------------------------------------------|
|      15124c3     -12193a3+8854a2b+2142ab2+14078b3+2728a2c-9601abc-12831b2c+2|
|      c2+6997c3   -323a3-7291a2b-13103ab2+12554b3+5771a2c+7057abc+15111b2c-56|
|      7bc2+9397c3 8094a3-3742a2b-13325ab2-2931b3-14207a2c+15902abc+3966b2c-11|
|      10903c3     -1435a3+9142a2b+10638ab2-3574b3+13837a2c+12959abc+10780b2c+|
|      2-9339c3    -9513a3+4720a2b-6329ab2+1728b3+13726a2c-6042abc+12552b2c-74|
|      -2213c3     4047a3-6300a2b-2637ab2+3796b3+997a2c+4811abc+5169b2c-10303a|
|      -----------------------------------------------------------------------|
|      514ac2-3367bc2-4049c3                                                  |
|      22ac2-13413bc2-4512c3                                                  |
|      733ac2+9698bc2+9857c3                                                  |
|      13503ac2-3238bc2-3017c3                                                |
|      40ac2-14165bc2-2035c3                                                  |
|      c2-11575bc2-3117c3                                                     |
|      -----------------------------------------------------------------------|
|      13323a3-1873a2b+5179ab2+15531b3-2685a2c-549abc+3779b2c+5846ac2+14768bc2|
|      11894a3+12256a2b-4976ab2+15861b3-4991a2c+2617abc-10062b2c+9698ac2+4312b|
|      -1179a3+13369a2b+6846ab2+1067b3+4241a2c+14428abc+8485b2c+10751ac2-2797b|
|      2277a3+387a2b-2835ab2+4249b3+7407a2c-12988abc-14540b2c-11889ac2-5823bc2|
|      -991a3+15960a2b+4943ab2+2131b3-14798a2c-11947abc-13658b2c+15863ac2-7111|
|      291a3+12162a2b-13458ab2+124b3-9892a2c+8079abc-4451b2c-15530ac2+8043bc2-|
|      -----------------------------------------------------------------------|
|      -8771c3    13479a3-2673a2b+8186ab2-7869b3-6472a2c+4045abc+9313b2c-10778|
|      c2+348c3   1191a3+14474a2b+3119ab2+3625b3-2603a2c+4966abc+11921b2c+1251|
|      c2-7585c3  -14967a3-4339a2b+9699ab2-2730b3+2157a2c-13109abc+11927b2c+40|
|      -1840c3    -6819a3+12142a2b-2172ab2+3448b3-2571a2c-2685abc+13957b2c-151|
|      bc2-6084c3 -5032a3+5168a2b+3912ab2+4394b3-4291a2c-2358abc-388b2c-3471ac|
|      6471c3     -12663a3+10566a2b+8887ab2+10773b3-975a2c-1605abc+6359b2c-102|
|      -----------------------------------------------------------------------|
|      ac2-9874bc2+14937c3                                                    |
|      9ac2-14658bc2-1402c3                                                   |
|      85ac2+12262bc2-15188c3                                                 |
|      88ac2+2238bc2-3094c3                                                   |
|      2-13714bc2-4572c3                                                      |
|      98ac2+686bc2-7207c3                                                    |
|      -----------------------------------------------------------------------|
|      13168a3+786a2b+13215ab2+5939b3+8856a2c-12310abc-9093b2c-4497ac2+15683bc|
|      805a3-14218a2b+11609ab2-403b3-9697a2c+4951abc-1903b2c+15458ac2-4805bc2-|
|      1194a3+5762a2b+12765ab2+3415b3+2489a2c+4864abc+2150b2c-9933ac2-1342bc2-|
|      3864a3+2859a2b+11409ab2-12452b3-731a2c-10818abc+14326b2c-7744ac2+14185b|
|      -3087a3+14005a2b-8485ab2-4328b3+1747a2c-2169abc-12619b2c-6276ac2+2294bc|
|      13527a3+14393a2b-10837ab2-3763b3+7557a2c+3547abc+13651b2c+11653ac2+3742|
|      -----------------------------------------------------------------------|
|      2-9242c3   2206a3+5191a2b+3792ab2+12204b3-2642a2c-11609abc-3522b2c+6597|
|      6518c3     -15326a3-3718a2b+14495ab2+15256b3-7099a2c-4043abc-13729b2c-6|
|      7123c3     7514a3+436a2b-14241ab2+14798b3+6510a2c-12122abc-2128b2c+1247|
|      c2-6792c3  13822a3-7511a2b+3419ab2-13285b3+2645a2c+2104abc-2951b2c+443a|
|      2-2364c3   10294a3-11777a2b-691ab2+12756b3+14535a2c-14905abc-10428b2c+3|
|      bc2+5053c3 -13160a3+13348a2b-11663ab2+6603b3+5647a2c+1762abc-351b2c-497|
|      -----------------------------------------------------------------------|
|      ac2-8682bc2-7612c3                                                     |
|      174ac2-11643bc2-12072c3                                                |
|      8ac2-6741bc2-9805c3                                                    |
|      c2+11334bc2-12747c3                                                    |
|      316ac2+12116bc2+12228c3                                                |
|      1ac2+3851bc2-2592c3                                                    |
|      -----------------------------------------------------------------------|
|      1490a3-7028a2b-867ab2+15524b3-15341a2c                                 |
|      -1064a3+9823a2b-5852ab2-8089b3+8359a2c                                 |
|      6548a3-6654a2b-5503ab2+15986b3+136a2c-                                 |
|      -7792a3-11541a2b+437ab2+2978b3-7578a2c                                 |
|      15813a3-9657a2b-6303ab2-6330b3+9805a2c                                 |
|      1953a3+858a2b+15816ab2+4986b3+9192a2c-                                 |
|      -----------------------------------------------------------------------|
|      +10883abc+5463b2c-810ac2+10379bc2+7684c3                               |
|      +10828abc+10341b2c-10325ac2-12190bc2-15790c3                           |
|      14927abc+4275b2c+8500ac2-5614bc2-13485c3                               |
|      +3393abc-8394b2c+12776ac2-4378bc2-11814c3                              |
|      +14185abc-5099b2c+2659ac2-714bc2+13689c3                               |
|      5886abc-7448b2c+5621ac2-4641bc2+3751c3                                 |
|      -----------------------------------------------------------------------|
|      -14618a3+5980a2b-8488ab2+9007b3+7300a2c-4212abc-679b2c-10439ac2+13602bc|
|      -10529a3-12894a2b+1116ab2-7308b3+13622a2c+15225abc+11997b2c+11361ac2+12|
|      -13229a3+9124a2b-2252ab2-9649b3-12812a2c+12094abc+4646b2c+10714ac2+7719|
|      15887a3+14099a2b+6519ab2+14350b3-6307a2c-13167abc+4278b2c+2762ac2+7172b|
|      -4201a3-15945a2b+9210ab2+2519b3-15231a2c-14362abc-2581b2c+14925ac2-6448|
|      2504a3-160a2b-487ab2-615b3+940a2c+11163abc+1273b2c+8254ac2+10632bc2+107|
|      -----------------------------------------------------------------------|
|      2+10226c3      10049a3-14286a2b                                        |
|      823bc2-12909c3 -5069a3+5393a2b-                                        |
|      bc2+11052c3    14871a3+4636a2b-                                        |
|      c2-11151c3     12876a3-4698a2b-                                        |
|      bc2-15150c3    -11577a3-5736a2b                                        |
|      81c3           10625a3-14342a2b                                        |
|      -----------------------------------------------------------------------|
|      +11005ab2-13930b3-1058a2c+9559abc-3698b2c+8326ac2-5257bc2-7889c3       |
|      8502ab2-1087b3-3595a2c+2865abc+4407b2c+4457ac2-15699bc2+15492c3        |
|      11213ab2-8339b3+10251a2c+14831abc+13739b2c+6731ac2+15350bc2+298c3      |
|      13580ab2-9883b3-2577a2c-35abc+1265b2c+13376ac2-2844bc2+9988c3          |
|      -9028ab2+1370b3+12545a2c+14902abc-2735b2c-13701ac2+11920bc2-10797c3    |
|      +2064ab2-11826b3-5008a2c+1660abc-14605b2c+4693ac2+11824bc2+7155c3      |
|      -----------------------------------------------------------------------|
|      -13610a3+4417a2b-5385ab2-1174b3-7698a2c+5290abc-8615b2c-3373ac2-178bc2-|
|      6546a3-10112a2b+4114ab2+9977b3-3918a2c-5632abc+5684b2c-13949ac2+6130bc2|
|      -10040a3-13438a2b-1613ab2-7103b3+8760a2c-5258abc-11467b2c+10352ac2+926b|
|      7298a3+13745a2b+474ab2-7454b3+13945a2c+8819abc-2848b2c+2002ac2-14025bc2|
|      13822a3+10660a2b+1494ab2+4950b3+1046a2c-3780abc+3631b2c-9126ac2-14046bc|
|      -14049a3+2458a2b-15382ab2+2871b3+15070a2c+10050abc+8655b2c+6431ac2+1235|
|      -----------------------------------------------------------------------|
|      10789c3     2535a3-473a2b+9322ab2+7574b3+1243a2c-3242abc-3058b2c+4676ac|
|      +6655c3     9239a3-14867a2b-11414ab2+3128b3+15678a2c+1410abc-330b2c-731|
|      c2-15613c3  7124a3-13944a2b-7560ab2+2728b3-6872a2c+14114abc-7155b2c+142|
|      -2394c3     -4869a3-2380a2b+15526ab2+4547b3-9399a2c+9740abc+10443b2c+14|
|      2+1470c3    -2959a3+10334a2b+14997ab2+11856b3+416a2c-777abc-12168b2c+64|
|      4bc2-8007c3 -9794a3-14999a2b+230ab2-2656b3+1294a2c+11721abc-14178b2c+52|
|      -----------------------------------------------------------------------|
|      2-12956bc2+9434c3                                                      |
|      4ac2-1741bc2-15645c3                                                   |
|      90ac2+6585bc2+14317c3                                                  |
|      865ac2-8765bc2-2036c3                                                  |
|      00ac2-11550bc2+4668c3                                                  |
|      83ac2+10044bc2-14885c3                                                 |
|      -----------------------------------------------------------------------|
|      14677a3+1508a2b-5369ab2+15800b3-5266a2c-6770abc-                       |
|      6816a3-3106a2b+14665ab2-14894b3-7378a2c+2323abc+                       |
|      12892a3+11448a2b+1861ab2-3997b3+3432a2c-8070abc+                       |
|      -2857a3+14361a2b+3505ab2+5812b3+3174a2c+4496abc+                       |
|      -4180a3-1058a2b+3822ab2-14641b3+15496a2c+3527abc                       |
|      -13752a3-7719a2b-2773ab2+13012b3+5821a2c+5315abc                       |
|      -----------------------------------------------------------------------|
|      6318b2c-10093ac2-4920bc2+15992c3                                       |
|      4675b2c+15268ac2-10292bc2+8404c3                                       |
|      6380b2c-102ac2-14388bc2-13468c3                                        |
|      14777b2c-9399ac2-7061bc2+3990c3                                        |
|      -12509b2c-5926ac2-3472bc2+2266c3                                       |
|      -7909b2c+11529ac2-7783bc2-1637c3                                       |
|      -----------------------------------------------------------------------|
|      6487a2b+6487ab2+7698b3+2422a2c+3657abc+2707b2c-11250ac2-12515bc2+5297c3|
|      8933a2b+6349ab2-6721b3-4365a2c-12064abc+11162b2c+836ac2-4416bc2+3297c3 |
|      -11352a2b-14353ab2-2032b3-5655a2c+11413abc+6542b2c+13931ac2+8078bc2+424|
|      3879a2b-10172ab2+8007b3-985a2c+11878abc-893b2c+3038ac2+2170bc2-6584c3  |
|      -4062a2b+2410ab2+852b3+6984a2c-3105abc+4412b2c-650ac2-4214bc2+8925c3   |
|      13212a2b+5008ab2+1283b3-13919a2c+47abc-977b2c-191ac2-48bc2-3c3         |
|      -----------------------------------------------------------------------|
|          |                                                                  |
|          |                                                                  |
|      2c3 |                                                                  |
|          |                                                                  |
|          |                                                                  |
|          |                                                                  |
|                                                                             |
|              6      24                                                      |
|o32 : Matrix R  <-- R                                                        |
+-----------------------------------------------------------------------------+
|i33 : f2 * f3                                                                |
|                                                                             |
|o33 = {2} | 14508a3b-14796a2b2-14848ab3+10239b4+7254a3c-7562a2bc-10217ab2c-25|
|      {2} | 14508a3b+9541a2b2+129ab3+15632b4-7150a3c+15121a2bc+7611ab2c+11536|
|      {2} | -3575a3b+13774a2b2+6885ab3+6345b4+11931a3c-4949a2bc+11480ab2c+227|
|      {2} | 2987a4-7699a3b+2383a2b2+5177ab3-5162b4-1084a3c-13260a2bc-7417ab2c|
|      {2} | 2987a4+15689a3b+15764a2b2+9374ab3-9621b4-11663a3c-9755a2bc-5526ab|
|      {2} | 7150a4-9968a3b-1048a2b2-10926ab3-11410b4-7027a3c+11968a2bc+7672ab|
|      {2} | -14508a4+14106a3b-11183a2b2-4259ab3+1032b4-14162a3c-11707a2bc+110|
|      {2} | 14300a4-4594a3b+12052a2b2-5495ab3-8490b4-14138a3c-3732a2bc+13017a|
|      {2} | 8141a4-6785a3b+3472a2b2+9302ab3+13245b4-2135a3c-11369a2bc+5459ab2|
|      {2} | 15864a4-14070a3b-12959a2b2+15486ab3+1950a3c-1690a2bc+759ab2c+1560|
|      {2} | -8304a4+15940a3b+6098a2b2-3140ab3-12998a3c-14486a2bc-8077ab2c-123|
|      {2} | 11266a4-6163a3b-5292a2b2+2227ab3+8816a3c-780a2bc+14029ab2c-15714b|
|      {2} | 6306a4+4669a3b-12276a2b2-8216ab3-15603b4+9626a3c-13344a2bc-9366ab|
|      {2} | -9952a4-7556a3b+10336a2b2+2842ab3+12395b4+13948a3c+4029a2bc+1158a|
|      {2} | 7124a4+3996a3b+220a2b2+10677ab3+15714b4-13801a3c-145a2bc-2138ab2c|
|      {2} | 15761a4-4227a3b+2576a2b2-10434ab3-9914b4+11262a3c+14621a2bc+4624a|
|      {2} | 7187a4-4198a3b+6927a2b2+10863ab3-14156b4-4773a3c-11433a2bc-7530ab|
|      {2} | -2809a4+360a3b+7156a2b2-13655ab3+2917b4+8860a3c-2858a2bc-4940ab2c|
|      {2} | 10848a4+280a3b+4917a2b2+11199ab3-864a3c+4816a2bc+4151ab2c-3104b3c|
|      {2} | 2006a4+15518a3b-5296a2b2-14370ab3+776b4-3060a3c-10440a2bc-6243ab2|
|      {2} | -594a4+13032a3b+1372a2b2-15001ab3+4656b4+12648a3c-2937a2bc-12486a|
|      {2} | -2096a4-2597a3b+15546a2b2+12441ab3-10883b4+2466a3c+13424a2bc-8708|
|      {2} | 11008a4+10234a3b+11173a2b2+2780ab3-11529b4-12838a3c+9146a2bc-1012|
|      {2} | 13085a4+11905a3b+14836a2b2+6962ab3+1589a3c-11323a2bc+10434ab2c+13|
|      {2} | -4273a4+15149a3b+10037a2b2+388ab3-15864a3c-12599a2bc-2052ab2c+259|
|      {2} | 2277a4-8964a3b-6204a2b2+2112ab3-1412a3c+12720a2bc-4356ab2c-258a2c|
|      {2} | 1167a4-2109a3b-48a2b2+108ab3+5268a3c-108a2bc-216ab2c             |
|      -----------------------------------------------------------------------|
|      02b3c+9788a2c2-1925abc2-13846b2c2+1549ac3-13087bc3+1361c4              |
|      b3c+11205a2c2-1439abc2+4342b2c2+10491ac3-10090bc3+9081c4               |
|      4b3c-9132a2c2+9277abc2-7047b2c2-9782ac3-11609bc3-9025c4                |
|      -9214b3c-7578a2c2+4043abc2+9576b2c2-3373ac3-2448bc3-5221c4             |
|      2c+10235b3c+3360a2c2+4006abc2-5362b2c2-9398ac3-12455bc3-9189c4         |
|      2c-145b3c+7963a2c2-4997abc2-8012b2c2-6480ac3+9974bc3-775c4             |
|      66ab2c+7700b3c-6852a2c2+11615abc2+10578b2c2-173ac3-10678bc3+9590c4     |
|      b2c-2524b3c+12612a2c2+15584abc2-10604b2c2+2097ac3+9018bc3+6432c4       |
|      c+15864b3c+5943a2c2-8595abc2-7993b2c2-8945ac3-1800bc3+4557c4           |
|      3b3c-15489a2c2-25abc2-7330b2c2-13658ac3-3877bc3+13738c4                |
|      95b3c+7932a2c2+11637abc2+7656b2c2-12923ac3+10443bc3+952c4              |
|      3c+1882a2c2-7763abc2+8335b2c2-3546ac3-5186bc3-12827c4                  |
|      2c-14759b3c+7172a2c2-15356abc2-9866b2c2+10551ac3+14806bc3+5153c4       |
|      b2c+6500b3c-10591a2c2-6379abc2+5347b2c2-3988ac3+15521bc3+4940c4        |
|      -11252b3c-10275a2c2+2543abc2-1183b2c2+11762ac3+2836bc3+5358c4          |
|      b2c+13743b3c-7647a2c2+5143abc2+3459b2c2+3233ac3-5153bc3                |
|      2c-15790b3c+3229a2c2+4501abc2+15530b2c2+12707ac3-4940bc3               |
|      +6369b3c-15024a2c2-4037abc2+9991b2c2-13671ac3-5358bc3                  |
|      -10348a2c2-4986abc2+7968b2c2-477ac3+12410bc3+5763c4                    |
|      c+5048b3c-2178a2c2-8414abc2+9674b2c2-14370ac3+10315bc3+10498c4         |
|      b2c+9814b3c+7970a2c2-9011abc2+735b2c2-3286ac3-14453bc3+778c4           |
|      ab2c-9675b3c+15989a2c2-6086abc2+10725b2c2+5417ac3-329bc3               |
|      3ab2c-6697b3c-9065a2c2+15116abc2+11617b2c2-9880ac3-1556bc3             |
|      863a2c2-12021abc2-10716ac3                                             |
|      8a2c2-7392abc2+3280ac3                                                 |
|      2+12abc2+20ac3                                                         |
|                                                                             |
|      -----------------------------------------------------------------------|
|      10881a3b+1494a2b2-13287ab3-9855b4-10561a3c+10362a2bc+12397ab2c+9032b3c-|
|      10881a3b+11746a2b2+68ab3-1187b4+10639a3c-12731a2bc+15771ab2c-2214b3c+13|
|      -10682a3b-15824a2b2+12588ab3-15423b4-15054a3c-1833a2bc-15183ab2c-4254b3|
|      10241a4-6954a3b+1421a2b2+4831ab3-12309b4+11714a3c-12120a2bc+10754ab2c+4|
|      10241a4+10587a3b+14602a2b2+4143ab3+10416b4+6785a3c+6650a2bc-5488ab2c+14|
|      -10639a4+12830a3b+2080a2b2-8862ab3-3864b4-14262a3c-5439a2bc-14944ab2c-5|
|      -10881a4+13990a3b+227a2b2-6402ab3-10105b4+1145a3c+6579a2bc-14329ab2c-90|
|      10725a4-10838a3b+2278a2b2-7638ab3+1091b4+3416a3c+10708a2bc+13800ab2c+14|
|      -1895a4-15856a3b-4796a2b2-10438ab3+10566b4+7705a3c+3744a2bc-4515ab2c-10|
|      11898a4+11453a3b+12634a2b2+4924ab3-11202a3c-8059a2bc-3456ab2c+7874b3c+1|
|      -6228a4-12211a3b-5307a2b2+755ab3-1916a3c+10655a2bc+7462ab2c-10900b3c-85|
|      -7552a4+14241a3b-8888a2b2+11592ab3-13268a3c-12871a2bc+14109ab2c+13115b3|
|      -11272a4+4502a3b+4817a2b2-11969ab3-7874b4+3119a3c+1191a2bc+9720ab2c+159|
|      -7464a4+8518a3b+10004a2b2+9960ab3+10900b4+3763a3c-4859a2bc+4852ab2c+771|
|      5343a4+7059a3b-13471a2b2+2546ab3-13115b4+8390a3c-6248a2bc-1475ab2c-1297|
|      3820a4-14941a3b-472a2b2-12276ab3-13154b4+274a3c-13663a2bc-399ab2c+103b3|
|      13391a4+13926a3b+4401a2b2-5184ab3-574b4-357a3c+2191a2bc+14044ab2c-9791b|
|      5894a4-10757a3b-4054a2b2+8881ab3-15345b4+2532a3c+15790a2bc-8929ab2c-151|
|      8136a4-14116a3b+5747a2b2+169ab3-5339a3c-7452a2bc-13402ab2c+3776b3c-7136|
|      -14497a4+7967a3b+6258a2b2+20ab3-944b4-13279a3c+45a2bc-15940ab2c-3475b3c|
|      15556a4-9562a3b-12102a2b2+10323ab3-5664b4+16a3c-2271a2bc+13141ab2c-4360|
|      8065a4-15098a3b-10464a2b2-364ab3+11849b4+4155a3c+9070a2bc-15320ab2c+151|
|      9191a4+6079a3b+12875a2b2+12548ab3+15453b4-15709a3c-12136a2bc+4664ab2c-3|
|      -13721a4+15999a3b+3341a2b2+2441ab3-5766a3c+11499a2bc-2606ab2c+1300a2c2+|
|      12818a4-4202a3b+10145a2b2-472ab3+10432a3c-13811a2bc+558ab2c+4866a2c2-74|
|      1709a4-6980a3b-4212a2b2+2107ab3-8446a3c+8364a2bc-4341ab2c-186a2c2+12abc|
|      879a4-1695a3b+60a2b2+108ab3+4116a3c-324a2bc-216ab2c                    |
|      -----------------------------------------------------------------------|
|      6543a2c2-12070abc2-9191b2c2-7729ac3+2131bc3-8092c4                     |
|      94a2c2+10755abc2-8813b2c2+15849ac3-10230bc3-12621c4                    |
|      c+499a2c2+9713abc2+12596b2c2+15453ac3+7438bc3+5740c4                   |
|      579b3c+10834a2c2-4125abc2-8270b2c2+10582ac3-9868bc3+10304c4            |
|      437b3c+15224a2c2-14511abc2+504b2c2-10496ac3-14855bc3+2126c4            |
|      450b3c+9622a2c2-7653abc2+12323b2c2+1612ac3+3289bc3-2375c4              |
|      28b3c+4699a2c2-14581abc2-14343b2c2-2419ac3-3228bc3-7198c4              |
|      40b3c+424a2c2-1629abc2-10408b2c2-9050ac3+4372bc3-5347c4                |
|      85b3c+253a2c2+15404abc2-721b2c2+7218ac3-15388bc3+15804c4               |
|      3780a2c2-9556abc2-2814b2c2+679ac3+11008bc3-12439c4                     |
|      37a2c2-5500abc2-7142b2c2+7115ac3+11936bc3+11122c4                      |
|      c+3233a2c2-9117abc2-15361b2c2+775ac3+12638bc3+2567c4                   |
|      68b3c+12874a2c2+8347abc2-11111b2c2+2590ac3-11566bc3+1334c4             |
|      6b3c-445a2c2-9376abc2-2145b2c2+9854ac3-4539bc3-2431c4                  |
|      b3c-15457a2c2+450abc2-12487b2c2-9059ac3-8508bc3+4008c4                 |
|      c-8197a2c2+8185abc2-7998b2c2-10409ac3-1334bc3                          |
|      3c+6867a2c2+2655abc2-6583b2c2-15962ac3+2431bc3                         |
|      b3c+7912a2c2-1658abc2+5941b2c2-15409ac3-4008bc3                        |
|      a2c2-12437abc2-12912b2c2-9109ac3+12770bc3+5643c4                       |
|      +1933a2c2-15198abc2+7147b2c2-8964ac3+11838bc3-11335c4                  |
|      b3c+3728a2c2+14488abc2+14637b2c2+2880ac3+8772bc3+3764c4                |
|      52b3c-14089a2c2-13424abc2+15204b2c2-14922ac3-1822bc3                   |
|      578b3c+7469a2c2-14503abc2-2470b2c2+4862ac3-7528bc3                     |
|      11882abc2-8016ac3                                                      |
|      37abc2+3295ac3                                                         |
|      2+20ac3                                                                |
|                                                                             |
|      -----------------------------------------------------------------------|
|      12380ab3-14232b4-1870a2bc+11713ab2c-1092b3c-935a2c2-4764abc2-6821b2c2+3|
|      -891ab3-10600b4-1870a2bc+2890ab2c-6836b3c+14965a2c2+6459abc2+1139b2c2-3|
|      -14034ab3-10791b4-14022a2bc+3582ab2c-4394b3c+2854a2c2+9008abc2-3810b2c2|
|      1740a2b2+11907ab3-5350b4+3740a3c-2547a2bc+8155ab2c-8020b3c-12095a2c2-68|
|      1376a2b2-13161ab3-8324b4+3740a3c-13146a2bc-15741ab2c-3503b3c-3633a2c2-1|
|      -3087a2b2+15587ab3+5793b4-14965a3c+5360a2bc+8533ab2c+13382b3c-3748a2c2+|
|      12100a2b2+12233ab3+10440b4+1870a3c+15107a2bc-10021ab2c-2877b3c-8133a2c2|
|      -8718a2b2-195ab3-12589b4+2073a3c-12384a2bc-5073ab2c+4702b3c-2873a2c2-15|
|      11006a3b+4563a2b2+5102ab3+5034b4-5708a3c-4561a2bc+1992ab2c+13559b3c+333|
|      -15494a3b-14335a2b2-15953ab3+12972a3c-3047a2bc+5962ab2c-4043b3c-15707a2|
|      1218a3b+7074a2b2-7031ab3-9394a3c-414a2bc+8301ab2c-14511b3c+5009a2c2-116|
|      -2544a3b-14018a2b2+14624ab3+3251a3c-471a2bc+5540ab2c-8414b3c-9857a2c2+1|
|      -15900a3b+12441a2b2-13222ab3+4043b4-11819a3c-11477a2bc+9721ab2c+10201b3|
|      -6360a3b-6771a2b2+7972ab3+14511b4+15952a3c+15054a2bc+11133ab2c+3394b3c+|
|      9494a3b-14070a2b2+3218ab3+8414b4-9792a3c-9651a2bc+13762ab2c-15839b3c+80|
|      -636a3b-9214a2b2-9579ab3+14162b4+4725a3c-8174a2bc+13335ab2c+14990b3c+12|
|      -6655a3b+1166a2b2+15240ab3+13448b4+10743a3c-13440a2bc-3708ab2c+7808b3c-|
|      12674a3b+12968a2b2-3699ab3-6562b4-11268a3c+3280a2bc+5082ab2c-5837b3c-71|
|      12780a3b-386a2b2-5277ab3-15643a3c-562a2bc+12471ab2c+5840b3c+4759a2c2+49|
|      -10451a3b+14079a2b2-1689ab3-1460b4-7911a3c-1790a2bc-3519ab2c+14915b3c-6|
|      -3195a3b+2023a2b2-7153ab3-8760b4-8523a3c-14936a2bc-11978ab2c-3139b3c+15|
|      2291a3b+947a2b2-8632ab3-875b4-1775a3c+13742a2bc+2225ab2c-10942b3c+6931a|
|      2912a3b-7935a2b2-4944ab3-3440b4+13750a3c+7280a2bc+10275ab2c-11945b3c-13|
|      -8097a3b-3411a2b2-12734ab3+5041a3c+13435a2bc-12772ab2c+12010a2c2-422abc|
|      -1545a3b+6502a2b2-730ab3-13261a3c+4561a2bc+1902ab2c+7258a2c2-1308abc2-8|
|      -65a3b+675a2b2+715ab3+688a3c-2019a2bc-2052ab2c+2619a2c2+1641abc2-22ac3 |
|      5a4-60a3b+150a2b2+45ab3+333a3c-414a2bc-108ab2c+432a2c2+81abc2          |
|      -----------------------------------------------------------------------|
|      279ac3-2650bc3-2928c4                                 |                |
|      54ac3-15427bc3-10968c4                                |                |
|      +210ac3-11303bc3+1278c4                               |                |
|      75abc2+5391b2c2-12001ac3-5041bc3+13318c4              |                |
|      4949abc2+1064b2c2-14746ac3-12021bc3-12675c4           |                |
|      7808abc2+14620b2c2+5270ac3+15806bc3-10450c4           |                |
|      +14624abc2-13791b2c2+12376ac3+5199bc3-13275c4         |                |
|      971abc2-2017b2c2-7398ac3-15618bc3-6550c4              |                |
|      6a2c2+2901abc2+6364b2c2-10849ac3+4892bc3+7962c4       |                |
|      c2-7684abc2+7640b2c2+12326ac3+1655bc3+13796c4         |                |
|      68abc2+15161b2c2+2330ac3-4069bc3+11810c4              |                |
|      409abc2-9602b2c2+4215ac3+12518bc3+5523c4              |                |
|      c-15798a2c2-1168abc2+15358b2c2-8022ac3-7293bc3-7384c4 |                |
|      6135a2c2+11489abc2-3739b2c2+1100ac3+12177bc3-2721c4   |                |
|      22a2c2-11801abc2-6681b2c2+8406ac3-4880bc3-13042c4     |                |
|      443a2c2-15526abc2-6503b2c2+7822ac3+7384bc3            |                |
|      8007a2c2-12230abc2+8016b2c2-12353ac3+2721bc3          |                |
|      24a2c2+8268abc2-643b2c2+8117ac3+13042bc3              |                |
|      36abc2+13907b2c2-6393ac3-13331bc3-6416c4              |                |
|      970a2c2+13461abc2+12975b2c2-3023ac3-849bc3+6422c4     |                |
|      699a2c2+7734abc2-6034b2c2+469ac3-7481bc3-6589c4       |                |
|      2c2-6710abc2-7494b2c2+14128ac3-12692bc3               |                |
|      252a2c2-10993abc2+2207b2c2+5442ac3+13178bc3           |                |
|      2-5919ac3                                             |                |
|      ac3                                                   |                |
|                                                            |                |
|                                                            |                |
|                                                                             |
|              27      3                                                      |
|o33 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
|i34 : ker f1 == image f2                                                     |
|                                                                             |
|o34 = false                                                                  |
+-----------------------------------------------------------------------------+
|i35 : ker f2 == image f3                                                     |
|                                                                             |
|o35 = false                                                                  |
+-----------------------------------------------------------------------------+
These do not piece together well. The reason is that pieri,
changes the bases of the free modules in a way which is not invertible (over
ZZ) when the ground field has positive characteristic.

Version 2.0 overhaul.  This release extends the original PieriMaps package in
three respects:
  * Multiple basis conventions. Every map can be expressed in the package's
    original convention (default Convention => "Row"), the column-form Filling
    basis from SchurFunctors, (Convention =>
    "Filling"), or the divided-power Weyl-module basis (Convention => "Weyl").
    See Convention.
  * Both directions of every map. In addition to the inclusion direction (see
    pieri, pieriColumn, lrMap), the
    projection direction is provided via dualPieri,
    dualPieriColumn, and dualLR.  Each pair
    satisfies $\mathrm{dual} \circ \mathrm{incl} = \mathrm{Id}$, the canonical
    $GL$-equivariant projection given by Schur's lemma.
  * Verifications. See verifyEquivariant, checks
    $GL_n$-equivariance of any matrix between Schur modules via the
    Chevalley-generator commutativity test; see verifyWellDefined,
    checks compatibility with the Garnir relations of the
    chosen convention.
Examples.

The single-box Pieri inclusion $S_{(2,1)} V \to V \otimes S_{(1,1)} V$ in three
variables.  Each row of symbolicForm displays a source standard tableau on the
left and its image as a sum of (target tableau, coefficient) pairs on the
right:
+-------------------------------+
|i36 : R = QQ[a,b,c];           |
+-------------------------------+
|i37 : M = pieri({2,1}, {1}, R);|
|                               |
|              3      8         |
|o37 : Matrix R  <-- R          |
+-------------------------------+
|i38 : symbolicForm M           |
|                               |
|      +-----+----------+       |
|      |+-+-+|       +-+|       |
|o38 = ||0|0||2a  *  |0||       |
|      ||1| ||       |1||       |
|      |+-+-+|       +-+|       |
|      +-----+----------+       |
|      |+-+-+|       +-+|       |
|      ||0|0||2a  *  |0||       |
|      ||2| ||       |2||       |
|      |+-+-+|       +-+|       |
|      +-----+----------+       |
|      |+-+-+|      +-+ |       |
|      ||0|1||b  *  |0| |       |
|      ||1| ||      |1| |       |
|      |+-+-+|      +-+ |       |
|      +-----+----------+       |
|      |+-+-+|      +-+ |       |
|      ||0|1||b  *  |0| |       |
|      ||2| ||      |2| |       |
|      |+-+-+|      +-+ |       |
|      |     |      +-+ |       |
|      |     |a  *  |1| |       |
|      |     |      |2| |       |
|      |     |      +-+ |       |
|      +-----+----------+       |
|      |+-+-+|      +-+ |       |
|      ||0|2||c  *  |0| |       |
|      ||1| ||      |1| |       |
|      |+-+-+|      +-+ |       |
|      |     |       +-+|       |
|      |     |-a  *  |1||       |
|      |     |       |2||       |
|      |     |       +-+|       |
|      +-----+----------+       |
|      |+-+-+|      +-+ |       |
|      ||0|2||c  *  |0| |       |
|      ||2| ||      |2| |       |
|      |+-+-+|      +-+ |       |
|      +-----+----------+       |
|      |+-+-+|       +-+|       |
|      ||1|1||2b  *  |1||       |
|      ||2| ||       |2||       |
|      |+-+-+|       +-+|       |
|      +-----+----------+       |
|      |+-+-+|      +-+ |       |
|      ||1|2||c  *  |1| |       |
|      ||2| ||      |2| |       |
|      |+-+-+|      +-+ |       |
|      +-----+----------+       |
+-------------------------------+
The same map in the column-form (Filling) convention is the literal change of
basis.  Both matrices are equivariant injections and have the same rank:
+-----------------------------------------------------------+
|i39 : R = QQ[a,b,c];                                       |
+-----------------------------------------------------------+
|i40 : Mrow = pieri({2,1}, {1}, R, Convention => "Row");    |
|                                                           |
|              3      8                                     |
|o40 : Matrix R  <-- R                                      |
+-----------------------------------------------------------+
|i41 : Mfil = pieri({2,1}, {1}, R, Convention => "Filling");|
|                                                           |
|              3      8                                     |
|o41 : Matrix R  <-- R                                      |
+-----------------------------------------------------------+
|i42 : (rank Mrow, rank Mfil)                               |
|                                                           |
|o42 = (3, 3)                                               |
|                                                           |
|o42 : Sequence                                             |
+-----------------------------------------------------------+
The projection dualPieri, left-inverts pieri.
Verified by $\mathrm{dualPieri} \circ \mathrm{applyPieri}(T_\mu) = T_\mu$ on
the highest-weight vector:
+-----------------------------------------------------------------------------+
|i43 : R = QQ[a,b,c];                                                         |
+-----------------------------------------------------------------------------+
|i44 : Tmu = (standardTableaux(3, {2,1}))#0;                                  |
+-----------------------------------------------------------------------------+
|i45 : img = applyPieri(({2,1}, {1}), Tmu, R);                                |
+-----------------------------------------------------------------------------+
|i46 : back = flatten for term in img list applyDualPieri(({2,1}, {1}),       |
|-----------------------------------------------------------------------------|
|term#0, term#1, 3);                                                          |
+-----------------------------------------------------------------------------+
|i47 : back                                                                   |
|                                                                             |
|o47 = {{1, {{0, 0}, {1}}}}                                                   |
|                                                                             |
|o47 : List                                                                   |
+-----------------------------------------------------------------------------+
When the Littlewood-Richardson coefficient $c^\lambda_{\mu,\nu}$ exceeds 1,
distinct LR tableaux $Q$ produce $c$ linearly independent inclusions $\Psi_Q :
S_\lambda V \hookrightarrow S_\nu V \otimes S_\mu V$:
+-----------------------------------------------------------------------------+
|i48 : Qs = lrTableaux({3,2,1}, {2,1}, {2,1});  -- c^{(3,2,1)}_{(2,1),(2,1)} =|
|-----------------------------------------------------------------------------|
|2                                                                            |
+-----------------------------------------------------------------------------+
|i49 : M1 = lrMap(({3,2,1}, {2,1}, {2,1}), Qs#0, 3);                          |
|                                                                             |
|               64       8                                                    |
|o49 : Matrix QQ   <-- QQ                                                     |
+-----------------------------------------------------------------------------+
|i50 : M2 = lrMap(({3,2,1}, {2,1}, {2,1}), Qs#1, 3);                          |
|                                                                             |
|               64       8                                                    |
|o50 : Matrix QQ   <-- QQ                                                     |
+-----------------------------------------------------------------------------+
|i51 : rank(M1 | M2) == 16    -- 2 * dim S_{(3,2,1)} -- linearly independent  |
|                                                                             |
|o51 = true                                                                   |
+-----------------------------------------------------------------------------+
Each dualLR, projects onto exactly its own $Q$-summand and
annihilates the others:
+--------------------------------------------------+
|i52 : Qs = lrTableaux({3,2,1}, {2,1}, {2,1});     |
+--------------------------------------------------+
|i53 : M0 = lrMap(({3,2,1},{2,1},{2,1}), Qs#0, 3); |
|                                                  |
|               64       8                         |
|o53 : Matrix QQ   <-- QQ                          |
+--------------------------------------------------+
|i54 : M1 = lrMap(({3,2,1},{2,1},{2,1}), Qs#1, 3); |
|                                                  |
|               64       8                         |
|o54 : Matrix QQ   <-- QQ                          |
+--------------------------------------------------+
|i55 : N0 = dualLR(({3,2,1},{2,1},{2,1}), Qs#0, 3);|
|                                                  |
|               8       64                         |
|o55 : Matrix QQ  <-- QQ                           |
+--------------------------------------------------+
|i56 : N0 * M0 == id_(QQ^(numColumns M0))          |
|                                                  |
|o56 = true                                        |
+--------------------------------------------------+
|i57 : N0 * M1 == 0                                |
|                                                  |
|o57 = true                                        |
+--------------------------------------------------+
See verifyEquivariant, checks Chevalley-generator
commutativity for any matrix between Schur reps -- a direct $GL_n$-equivariance
test:
+--------------------------------------------------------------------+
|i58 : R = QQ[a,b,c];                                                |
+--------------------------------------------------------------------+
|i59 : M = pieri({3,2,1}, {2}, R);   -- inclusion at the interior row|
|                                                                    |
|              6      8                                              |
|o59 : Matrix R  <-- R                                               |
+--------------------------------------------------------------------+
|i60 : verifyEquivariant(M, {3,2,1}, {2}, R)                         |
|                                                                    |
|o60 = true                                                          |
+--------------------------------------------------------------------+
And verifyWellDefined, confirms that an LR map factors
through the Garnir relations of the chosen convention:
+-------------------------------------------------------+
|i61 : Qs = lrTableaux({3,2,1}, {2,1}, {2,1});          |
+-------------------------------------------------------+
|i62 : verifyWellDefined(({3,2,1},{2,1},{2,1}), Qs#0, 3,|
|      Convention => "Filling", Verbose => false)       |
|                                                       |
|o62 = true                                             |
+-------------------------------------------------------+
|i63 :                                                  |
+-------------------------------------------------------+

See also
========

  * pieri -- computes a matrix representation for a Pieri
    inclusion of representations of a general linear group
  * pieriColumn -- native column-form (vertical-strip)
    Pieri inclusion
  * dualPieri -- GL-equivariant projection Sym^d V ⊗ S_lambda
    V --> S_mu V
  * dualPieriColumn -- GL-equivariant projection
    wedge^d V ⊗ S_lambda V --> S_mu V
  * lrMap -- GL(V)-equivariant Littlewood-Richardson inclusion
  * dualLR -- GL-equivariant projection S_nu V ⊗ S_mu V -->
    S_lambda V at a chosen LR tableau
  * applyLR -- apply Psi_Q to a single tableau of shape lambda
  * applyPieri -- apply pieri to a single source tableau
    T_mu
  * applyDualPieri -- apply dualPieri to a single basis
    pair (poly, T_lambda)
  * Convention -- tableau basis convention used by
    PieriMaps' overhauled functions
  * verifyEquivariant -- rigorously verify
    GL_n-equivariance of a Schur-rep matrix
  * verifyWellDefined -- verify an LR map respects
    the chosen convention's straightening relations
  * symbolicForm -- print a Schur-rep matrix as a
    basis-labeled symbolic map
  * pureFree -- computes a GL(V)-equivariant map whose
    resolution is pure, or the reduction mod p of such a map
  * schurRank -- computes the dimension of the irreducible
    GL(QQ^n) representation associated to a partition

Authors
=======

  * Steven V Sam (see http://math.mit.edu/~ssam/ ) <ssam@math.mit.edu>
  * Keller VandeBogert (see https://sites.nd.edu/kellerv/ ) <keller.v@uky.edu>

Certification a gold star
=========================

Version 1.0 of this package was accepted for publication in volume 1 (see
https://msp.org/jsag/2009/1-1/ ) of The Journal of Software for Algebra and
Geometry: Macaulay2 (see https://msp.org/jsag/ ) on 2009-06-27, in the article
Computing inclusions of Schur modules (see
https://msp.org/jsag/2009/1-1/p02.xhtml ) (DOI: 10.2140/jsag.2009.1.5 (see
https://doi.org/10.2140/jsag.2009.1.5 )). That version can be obtained from the
journal (see https://msp.org/jsag/2009/1-1/jsag-v1-n1-x02-code.zip ).

Version
=======

This documentation describes version 2.0 of PieriMaps, released May 1, 2026.

Citation
========

If you have used this package in your research, please cite it as follows:

+----------------------------------------------------------------------------+
|@article{Sam2009,                                                           |
|    AUTHOR = {Sam, Steven V},                                               |
|    TITLE = {Computing inclusions of {S}chur modules},                      |
|    JOURNAL = {The Journal of Software for Algebra and Geometry: Macaulay2},|
|    VOLUME = {1},                                                           |
|    YEAR = {2009},                                                          |
|    PAGES = {5--10},                                                        |
|    DOI = {10.2140/jsag.2009.1.5},                                          |
|    URL = {https://msp.org/jsag/2009/1-1/p02.xhtml}                         |
|}                                                                           |
+----------------------------------------------------------------------------+

Exports
=======

  * Functions and commands
      * applyDualLR -- apply dualLR to a single basis pair
        (T_nu, T_mu)
      * applyDualPieri -- apply dualPieri to a single
        basis pair (poly, T_lambda)
      * applyDualPieriColumn -- apply
        dualPieriColumn to a single basis pair (wedge_poly, T_lambda)
      * applyLR -- apply Psi_Q to a single tableau of shape
        lambda
      * applyPieri -- apply pieri to a single source tableau
        T_mu
      * applyPieriColumn -- apply pieriColumn to a
        single source tableau T_mu
      * displayLRImage -- pretty-print the result of
        applyLR
      * dualLR -- GL-equivariant projection S_nu V ⊗ S_mu V -->
        S_lambda V at a chosen LR tableau
      * dualPieri -- GL-equivariant projection Sym^d V ⊗
        S_lambda V --> S_mu V
      * dualPieriColumn -- GL-equivariant projection
        wedge^d V ⊗ S_lambda V --> S_mu V
      * fillingToPM -- convert a SchurFunctors column-form
        Filling to PieriMaps row-form tableaux
      * fillingToPMMatrix -- QQ matrix of the
        Filling-to-PM change of basis on a Schur module
      * fillingToWeyl -- convert a Filling to a sum of
        WeylFillings (compose pmToWeyl with fillingToPM)
      * lrMap -- GL(V)-equivariant Littlewood-Richardson
        inclusion
      * lrTableaux -- enumerate Littlewood-Richardson
        tableaux
      * pieri -- computes a matrix representation for a Pieri
        inclusion of representations of a general linear group
      * pieriColumn -- native column-form (vertical-strip)
        Pieri inclusion
      * pmToFilling -- convert a PieriMaps row-form tableau
        to SchurFunctors column-form Fillings
      * pmToFillingMatrix -- QQ matrix of the
        PM-to-Filling change of basis on a Schur module
      * pmToWeyl -- convert a PieriMaps row-form tableau to a
        SchurFunctors WeylFilling
      * pureFree -- computes a GL(V)-equivariant map whose
        resolution is pure, or the reduction mod p of such a map
      * schurRank -- computes the dimension of the
        irreducible GL(QQ^n) representation associated to a partition
      * standardTableaux -- list all standard tableaux
        of a certain shape with bounded labels
      * straighten -- computes straightening of a tableau
      * symbolicForm -- print a Schur-rep matrix as a
        basis-labeled symbolic map
      * verifyEquivariant -- rigorously verify
        GL_n-equivariance of a Schur-rep matrix
      * verifyWellDefined -- verify an LR map
        respects the chosen convention's straightening relations
      * weylToFilling -- convert a WeylFilling to a
        SchurFunctors Filling (compose pmToFilling with weylToPM)
      * weylToPM -- convert a SchurFunctors WeylFilling to a
        PieriMaps row-form tableau
  * Methods
      * "applyDualLR(Sequence,List,List,ZZ)" -- see applyDualLR,
        -- apply dualLR to a single basis pair (T_nu, T_mu)
      * "applyDualPieri(Sequence,RingElement,BasicList,ZZ)" -- see
        applyDualPieri, -- apply dualPieri to a single basis
        pair (poly, T_lambda)
      * "applyDualPieriColumn(Sequence,RingElement,BasicList,ZZ)" -- see
        applyDualPieriColumn, -- apply dualPieriColumn to
        a single basis pair (wedge_poly, T_lambda)
      * "applyLR(Sequence,List,BasicList,ZZ)" -- see applyLR, --
        apply Psi_Q to a single tableau of shape lambda
      * "applyPieri(Sequence,BasicList,PolynomialRing)" -- see
        applyPieri, -- apply pieri to a single source tableau T_mu
      * "applyPieriColumn(Sequence,BasicList,PolynomialRing)" -- see
        applyPieriColumn, -- apply pieriColumn to a single
        source tableau T_mu
      * "displayLRImage(List)" -- see displayLRImage, --
        pretty-print the result of applyLR
      * "dualLR(Sequence,List,PolynomialRing)" -- see dualLR, --
        GL-equivariant projection S_nu V ⊗ S_mu V --> S_lambda V at a chosen LR
        tableau
      * "dualLR(Sequence,List,ZZ)" -- see dualLR, --
        GL-equivariant projection S_nu V ⊗ S_mu V --> S_lambda V at a chosen LR
        tableau
      * "dualPieri(List,List,PolynomialRing)" -- see dualPieri,
        -- GL-equivariant projection Sym^d V ⊗ S_lambda V --> S_mu V
      * "dualPieriColumn(List,List,PolynomialRing)" -- see
        dualPieriColumn, -- GL-equivariant projection wedge^d
        V ⊗ S_lambda V --> S_mu V
      * "lrMap(Sequence,List,PolynomialRing)" -- see lrMap, --
        GL(V)-equivariant Littlewood-Richardson inclusion
      * "lrMap(Sequence,List,ZZ)" -- see lrMap, --
        GL(V)-equivariant Littlewood-Richardson inclusion
      * "lrTableaux(List,List,List)" -- see lrTableaux, --
        enumerate Littlewood-Richardson tableaux
      * "pieri(List,List,PolynomialRing)" -- see pieri, --
        computes a matrix representation for a Pieri inclusion of
        representations of a general linear group
      * "pieriColumn(List,List,PolynomialRing)" -- see pieriColumn,
        -- native column-form (vertical-strip) Pieri inclusion
      * "pureFree(List,PolynomialRing)" -- see pureFree, --
        computes a GL(V)-equivariant map whose resolution is pure, or the
        reduction mod p of such a map
      * "schurRank(ZZ,List)" -- see schurRank, -- computes the
        dimension of the irreducible GL(QQ^n) representation associated to a
        partition
      * "standardTableaux(ZZ,List)" -- see standardTableaux,
        -- list all standard tableaux of a certain shape with
        bounded labels
      * "straighten(List)" -- see straighten, -- computes
        straightening of a tableau
      * "straighten(List,MutableHashTable)" -- see straighten,
        -- computes straightening of a tableau
      * "symbolicForm(Matrix)" -- see symbolicForm, --
        print a Schur-rep matrix as a basis-labeled symbolic map
      * "verifyEquivariant(Matrix,List,List,PolynomialRing)" -- see
        verifyEquivariant, -- rigorously verify
        GL_n-equivariance of a Schur-rep matrix
      * "verifyEquivariant(Matrix,Sequence,ZZ)" -- see verifyEquivariant,
        -- rigorously verify GL_n-equivariance of a
        Schur-rep matrix
      * "verifyWellDefined(List,List,PolynomialRing)" -- see
        verifyWellDefined, -- verify an LR map respects the
        chosen convention's straightening relations
      * "verifyWellDefined(Sequence,List,ZZ)" -- see verifyWellDefined,
        -- verify an LR map respects the chosen convention's
        straightening relations
  * Symbols
      * Convention -- tableau basis convention used by
        PieriMaps' overhauled functions
      * "Direction" -- see verifyEquivariant, --
        rigorously verify GL_n-equivariance of a Schur-rep matrix

For the programmer
==================

The object PieriMaps, is a package (Macaulay2Doc),
defined in PieriMaps.m2, with auxiliary files in PieriMaps/.

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:214:0.

applyDualLR -- apply dualLR to a single basis pair (T_nu, T_mu)
***************************************************************

  * Usage:
        applyDualLR((lambda, mu, nu), Q, {T_nu, T_mu}, n)
        applyDualLR(..., Convention => "Row" | "Filling" | "Weyl")
  * Inputs:
      * (lambda, mu, nu), a sequence (Macaulay2Doc),
      * Q, a list (Macaulay2Doc), an LR tableau
      * {T_nu, T_mu}, a list (Macaulay2Doc), a 2-element list
        containing the source basis pair (a tableau of shape $\nu$ and a
        tableau of shape $\mu$).  Each may be a List or a Filling, interpreted
        in the natural form of the convention (see applyPieri),
        independently.
      * n, an integer (Macaulay2Doc),
  * Optional inputs (Macaulay2Doc),:
      * Convention => a string (Macaulay2Doc), default value
        "Row"
  * Outputs:
      * a list (Macaulay2Doc), list of pairs {coefficient,
        T_lambda} representing the image in $S_\lambda V$

Description
===========

Note: the {T_nu, T_mu} argument must be a List (not a Sequence), since M2
flattens nested Sequences in function-call arg lists but does not flatten
Lists.  Non-standard inputs are straightened first.
+-----------------------------------------------------+
|i1 : shapes = ({2,1}, {1}, {1,1});                   |
+-----------------------------------------------------+
|i2 : Q = (lrTableaux shapes)#0;                      |
+-----------------------------------------------------+
|i3 : applyDualLR(shapes, Q, {{{0},{1}}, {{0},{}}}, 3)|
|                                                     |
|       1                                             |
|o3 = {{-, {{0, 0}, {1}}}}                            |
|       3                                             |
|                                                     |
|o3 : List                                            |
+-----------------------------------------------------+

See also
========

  * dualLR -- GL-equivariant projection S_nu V ⊗ S_mu V -->
    S_lambda V at a chosen LR tableau
  * lrMap -- GL(V)-equivariant Littlewood-Richardson inclusion
  * applyLR -- apply Psi_Q to a single tableau of shape lambda
  * applyDualPieri -- apply dualPieri to a single basis
    pair (poly, T_lambda)
  * Convention -- tableau basis convention used by
    PieriMaps' overhauled functions

Ways to use applyDualLR:
========================

  * "applyDualLR(Sequence,List,List,ZZ)"

For the programmer
==================

The object applyDualLR, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:1162:0.

applyDualPieri -- apply dualPieri to a single basis pair (poly, T_lambda)
*************************************************************************

  * Usage:
        applyDualPieri((mu, boxes), poly, T, n)
        applyDualPieri(..., Convention => "Row" | "Filling" | "Weyl")
  * Inputs:
      * (mu, boxes), a sequence (Macaulay2Doc), the same data
        passed to dualPieri,
      * poly, a ring element (Macaulay2Doc), a homogeneous
        polynomial of degree $d = \#\mathrm{boxes}$ in the polynomial ring P
        (an element of $\mathrm{Sym}^d V$)
      * T, a basic list (Macaulay2Doc), a tableau of shape
        $\lambda$.  Accepted as either a List or a Filling.  In "Row"/"Weyl"
        convention a List is the PM row form; in "Filling" convention a List is
        column form (auto-wrapped).  A Filling is always treated as column
        form.  Non-standard inputs are straightened first using the
        convention's relations.
      * n, an integer (Macaulay2Doc), $\dim V$
  * Optional inputs (Macaulay2Doc),:
      * Convention => a string (Macaulay2Doc), default value
        "Row", , basis for source $S_\lambda$ and target $S_\mu$.
  * Outputs:
      * a list (Macaulay2Doc), a list of pairs {coefficient, T_mu}
        representing the image as a sum $\sum c_i \cdot T_\mu^{(i)}$ in $S_\mu
        V$.  Zero coefficients are dropped.

Description
===========

Convenience wrapper for dualPieri,: evaluates the projection
on a single source basis vector $\mathrm{poly} \otimes T_\lambda$ and returns a
sparse representation of its image.  Useful for hand-checks and interactive
exploration when the full dualPieri, matrix is too large.

Input flexibility: T may be in either tableau type.  For Convention => "Row", a
Filling input is converted via fillingToPM, before
straightening; for Convention => "Filling", a List input is converted via
pmToFilling.  Non-standard tableaux straighten first.
+----------------------------------------------------------+
|i1 : P = QQ[a,b,c];                                       |
+----------------------------------------------------------+
|i2 : Tlam = {{0,0},{1}};   -- standard PM tableau of {2,1}|
+----------------------------------------------------------+
|i3 : applyDualPieri(({3,1}, {1}), a, Tlam, 3)             |
|                                                          |
|       1                                                  |
|o3 = {{-, {{0, 0, 0}, {1}}}}                              |
|       3                                                  |
|                                                          |
|o3 : List                                                 |
+----------------------------------------------------------+
+--------------------------------------------------------------+
|i4 : -- Non-standard input is straightened first (row-Garnir).|
|     P = QQ[a,b,c,d];                                         |
+--------------------------------------------------------------+
|i5 : applyDualPieri(({3,2,1}, {3}), a, {{1,0},{2}}, 4)        |
|                                                              |
|o5 = {}                                                       |
|                                                              |
|o5 : List                                                     |
+--------------------------------------------------------------+
|i6 :                                                          |
+--------------------------------------------------------------+

See also
========

  * dualPieri -- GL-equivariant projection Sym^d V ⊗ S_lambda
    V --> S_mu V
  * pieri -- computes a matrix representation for a Pieri
    inclusion of representations of a general linear group
  * applyLR -- apply Psi_Q to a single tableau of shape lambda
  * applyDualLR -- apply dualLR to a single basis pair
    (T_nu, T_mu)
  * Convention -- tableau basis convention used by
    PieriMaps' overhauled functions

Ways to use applyDualPieri:
===========================

  * "applyDualPieri(Sequence,RingElement,BasicList,ZZ)"

For the programmer
==================

The object applyDualPieri, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:1068:0.

applyDualPieriColumn -- apply dualPieriColumn to a single basis pair (wedge_poly, T_lambda)
*******************************************************************************************

  * Usage:
        applyDualPieriColumn((mu, cols), poly, T, n)
        applyDualPieriColumn(..., Convention => "Row" | "Filling" | "Weyl")
  * Inputs:
      * (mu, cols), a sequence (Macaulay2Doc), the data passed
        to dualPieriColumn,
      * poly, a ring element (Macaulay2Doc), a homogeneous
        degree-$d$ wedge polynomial
      * T, a basic list (Macaulay2Doc), a tableau of shape
        $\lambda$.  May be a List or a Filling (interpreted in the natural form
        of the convention; see applyPieri).
      * n, an integer (Macaulay2Doc), $\dim V$
  * Optional inputs (Macaulay2Doc),:
      * Convention => a string (Macaulay2Doc), default value
        "Filling"
  * Outputs:
      * a list (Macaulay2Doc), a list of pairs {coefficient, T_mu}
        representing the image in $S_\mu V$

Description
===========

+-----------------------------------------------------------------------------+
|i1 : needsPackage "SchurFunctors";                                           |
+-----------------------------------------------------------------------------+
|i2 : P = QQ[e_0..e_2, SkewCommutative => true];                              |
+-----------------------------------------------------------------------------+
|i3 : sfStdTab = value (SchurFunctors#"private                                |
|-----------------------------------------------------------------------------|
|dictionary"#"standardTableaux");                                             |
+-----------------------------------------------------------------------------+
|i4 : T = (sfStdTab(3, {1,1}))#0;                                             |
+-----------------------------------------------------------------------------+
|i5 : applyDualPieriColumn(({2,1}, {1}), e_0, T, 3)                           |
|                                                                             |
|o5 = {}                                                                      |
|                                                                             |
|o5 : List                                                                    |
+-----------------------------------------------------------------------------+

See also
========

  * dualPieriColumn -- GL-equivariant projection
    wedge^d V ⊗ S_lambda V --> S_mu V
  * pieriColumn -- native column-form (vertical-strip)
    Pieri inclusion
  * applyDualPieri -- apply dualPieri to a single basis
    pair (poly, T_lambda)
  * Convention -- tableau basis convention used by
    PieriMaps' overhauled functions

Ways to use applyDualPieriColumn:
=================================

  * "applyDualPieriColumn(Sequence,RingElement,BasicList,ZZ)"

For the programmer
==================

The object applyDualPieriColumn, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:1391:0.

applyLR -- apply Psi_Q to a single tableau of shape lambda
**********************************************************

  * Usage:
        applyLR((lambda, mu, nu), Q, T, n)
        applyLR(..., Convention => "Row" | "Filling" | "Weyl")
  * Inputs:
      * (lambda, mu, nu), a sequence (Macaulay2Doc), three
        partitions specifying the LR map
      * Q, a list (Macaulay2Doc), an LR tableau as returned by
        lrTableaux,
      * T, a basic list (Macaulay2Doc), a tableau of shape
        $\lambda$.  May be supplied as a List or as a Filling.  In "Row"/"Weyl"
        convention a List is interpreted as PM row form; in "Filling"
        convention a List is interpreted as column form (and wrapped
        automatically). A Filling is always treated as column form. T need not
        be standard; it is straightened internally using the convention's
        straightening relations.
      * n, an integer (Macaulay2Doc), the dimension of $V$
  * Optional inputs (Macaulay2Doc),:
      * Convention => a string (Macaulay2Doc), default value
        "Row", , basis convention used to interpret T and label the output
        $T_\nu$, $T_\mu$.
  * Outputs:
      * a list (Macaulay2Doc), a list of triples {c, T_nu, T_mu}
        representing the image as the sum $\sum c_i \cdot T_\nu^{(i)} \otimes
        T_\mu^{(i)}$ in $S_\nu V \otimes S_\mu V$

Description
===========

Computes the image of a single basis vector (or, more generally, of any
tableau, which is first straightened into the SSYT basis of $S_\lambda V$)
under the LR inclusion $\Psi_Q$.  The output is intended for human reading or
interactive exploration when the full lrMap, matrix is too large
to display.  Use displayLRImage, for pretty-printing.

When Convention => "Filling" is chosen, the input T must be a SchurFunctors
Filling, and the output uses Fillings on both target sides.  Straightening uses
SchurFunctors' column-Garnir relations.  When Convention => "Row" or "Weyl", T
is a list of rows and PieriMaps' row-Garnir straightening is applied.
+-----------------------------------------------------------------------------+
|i1 : Qs = lrTableaux({3,2,1}, {2,1}, {2,1});                                 |
+-----------------------------------------------------------------------------+
|i2 : T = {{0,0,0},{1,1},{2}};   -- highest-weight vector of S_(3,2,1) QQ^3   |
+-----------------------------------------------------------------------------+
|i3 : applyLR(({3,2,1}, {2,1}, {2,1}), Qs#0, T, 3)                            |
|                                                                             |
|       27                                                                    |
|o3 = {{--, {{0, 0}, {1}}, {{0, 1}, {2}, {}}}, {9, {{0, 0}, {1}}, {{0, 2},    |
|        4                                                                    |
|     ------------------------------------------------------------------------|
|                   63                                        9               |
|     {1}, {}}}, {- --, {{0, 0}, {2}}, {{0, 1}, {1}, {}}}, {- -, {{0, 1},     |
|                    4                                        2               |
|     ------------------------------------------------------------------------|
|                                27                                        9  |
|     {1}}, {{0, 0}, {2}, {}}}, {--, {{0, 1}, {2}}, {{0, 0}, {1}, {}}}, {- -, |
|                                 4                                        4  |
|     ------------------------------------------------------------------------|
|     {{0, 2}, {1}}, {{0, 0}, {1}, {}}}}                                      |
|                                                                             |
|o3 : List                                                                    |
+-----------------------------------------------------------------------------+
|i4 : displayLRImage applyLR(({3,2,1}, {2,1}, {2,1}), Qs#0, T, 3)             |
|                                                                             |
|o4 =   +  27/4 . [1 1 ] ⊗ [1 2 ]                                             |
|                  2        3                                                 |
|       +  9 . [1 1 ] ⊗ [1 3 ]                                                |
|               2        2                                                    |
|       -  63/4 . [1 1 ] ⊗ [1 2 ]                                             |
|                  3        2                                                 |
|       -  9/2 . [1 2 ] ⊗ [1 1 ]                                              |
|                 2        3                                                  |
|       +  27/4 . [1 2 ] ⊗ [1 1 ]                                             |
|                  3        2                                                 |
|       -  9/4 . [1 3 ] ⊗ [1 1 ]                                              |
|                 2        2                                                  |
+-----------------------------------------------------------------------------+
+-----------------------------------------------------------------------------+
|i5 : -- Non-standard input: row-Garnir straightening before applying.        |
|     Tnonstd = {{1,0},{2}};   -- row 0 not weakly-increasing                 |
+-----------------------------------------------------------------------------+
|i6 : applyLR(({2,1}, {1}, {1,1}), (lrTableaux({2,1},{1},{1,1}))#0, Tnonstd,  |
|                                                                             |
|       3                           3                                         |
|o6 = {{-, {{0}, {2}}, {{1}, {}}}, {-, {{1}, {2}}, {{0}, {}}}}                |
|       2                           2                                         |
|                                                                             |
|o6 : List                                                                    |
|-----------------------------------------------------------------------------|
|3)                                                                           |
+-----------------------------------------------------------------------------+
|i7 :                                                                         |
+-----------------------------------------------------------------------------+

See also
========

  * lrMap -- GL(V)-equivariant Littlewood-Richardson inclusion
  * lrTableaux -- enumerate Littlewood-Richardson tableaux
  * displayLRImage -- pretty-print the result of applyLR
  * verifyWellDefined -- verify an LR map respects
    the chosen convention's straightening relations

Ways to use applyLR:
====================

  * "applyLR(Sequence,List,BasicList,ZZ)"

For the programmer
==================

The object applyLR, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:619:0.

applyPieri -- apply pieri to a single source tableau T_mu
*********************************************************

  * Usage:
        applyPieri((mu, boxes), T, P)
        applyPieri(..., Convention => "Row" | "Filling" | "Weyl")
  * Inputs:
      * (mu, boxes), a sequence (Macaulay2Doc), the same data
        passed to pieri,
      * T, a basic list (Macaulay2Doc), a tableau of shape
        $\mu$.  May be a List or a Filling.  In "Row"/"Weyl" convention a List
        is the PM row form; in "Filling" convention a List is column form
        (auto-wrapped).  A Filling is always treated as column form.
      * P, a polynomial ring (Macaulay2Doc), whose
        generators carry $\mathrm{Sym}^d V$
  * Optional inputs (Macaulay2Doc),:
      * Convention => a string (Macaulay2Doc), default value
        "Row"
  * Outputs:
      * a list (Macaulay2Doc), a list of pairs {poly, T_lambda}
        where each poly is a homogeneous degree-$d$ form and $T_\lambda$ is a
        standard tableau of $\lambda$; the image is $\sum \mathrm{poly}_i
        \otimes T_{\lambda,i}$ in $\mathrm{Sym}^d V \otimes S_\lambda V$

Description
===========

Convenience point-evaluator for pieri,: applies the inclusion map
to a single source basis vector T and returns its image as a sparse
representation.  Non-standard inputs are straightened first.
+------------------------------------------+
|i1 : P = QQ[a,b,c];                       |
+------------------------------------------+
|i2 : T = (standardTableaux(3, {3,2,1}))#0;|
+------------------------------------------+
|i3 : applyPieri(({3,2,1}, {1}), T, P)     |
|                                          |
|o3 = {{3a, {{0, 0}, {1, 1}, {2}}}}        |
|                                          |
|o3 : List                                 |
+------------------------------------------+

See also
========

  * pieri -- computes a matrix representation for a Pieri
    inclusion of representations of a general linear group
  * applyDualPieri -- apply dualPieri to a single basis
    pair (poly, T_lambda)
  * applyLR -- apply Psi_Q to a single tableau of shape lambda
  * applyPieriColumn -- apply pieriColumn to a single
    source tableau T_mu
  * Convention -- tableau basis convention used by
    PieriMaps' overhauled functions

Ways to use applyPieri:
=======================

  * "applyPieri(Sequence,BasicList,PolynomialRing)"

For the programmer
==================

The object applyPieri, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:1287:0.

applyPieriColumn -- apply pieriColumn to a single source tableau T_mu
*********************************************************************

  * Usage:
        applyPieriColumn((mu, cols), T, P)
        applyPieriColumn(..., Convention => "Row" | "Filling" | "Weyl")
  * Inputs:
      * (mu, cols), a sequence (Macaulay2Doc), the same data
        passed to pieriColumn,
      * T, a basic list (Macaulay2Doc), a tableau of shape
        $\mu$.  May be a List or a Filling (interpreted in the natural form of
        the convention; see applyPieri).
      * P, a polynomial ring (Macaulay2Doc), a
        SkewCommutative ring whose generators carry $\wedge^d V$
  * Optional inputs (Macaulay2Doc),:
      * Convention => a string (Macaulay2Doc), default value
        "Filling"
  * Outputs:
      * a list (Macaulay2Doc), a list of pairs {wedge_poly,
        T_lambda} representing the image in $\wedge^d V \otimes S_\lambda V$

Description
===========

Column-form analogue of applyPieri.
+-----------------------------------------------------------------------------+
|i1 : needsPackage "SchurFunctors";                                           |
+-----------------------------------------------------------------------------+
|i2 : P = QQ[e_0..e_2, SkewCommutative => true];                              |
+-----------------------------------------------------------------------------+
|i3 : sfStdTab = value (SchurFunctors#"private                                |
|-----------------------------------------------------------------------------|
|dictionary"#"standardTableaux");                                             |
+-----------------------------------------------------------------------------+
|i4 : F = (sfStdTab(3, {2,1}))#0;                                             |
+-----------------------------------------------------------------------------+
|i5 : applyPieriColumn(({2,1}, {1}), F, P)                                    |
|                                                                             |
|           +-+-+         +-+-+                                               |
|o5 = {{e , |0|0|}, {-e , |0|1|}}                                             |
|        1  +-+-+      0  +-+-+                                               |
|                                                                             |
|o5 : List                                                                    |
+-----------------------------------------------------------------------------+

See also
========

  * pieriColumn -- native column-form (vertical-strip)
    Pieri inclusion
  * applyPieri -- apply pieri to a single source tableau
    T_mu
  * dualPieriColumn -- GL-equivariant projection
    wedge^d V ⊗ S_lambda V --> S_mu V
  * Convention -- tableau basis convention used by
    PieriMaps' overhauled functions

Ways to use applyPieriColumn:
=============================

  * "applyPieriColumn(Sequence,BasicList,PolynomialRing)"

For the programmer
==================

The object applyPieriColumn, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:1319:0.

Convention -- tableau basis convention used by PieriMaps' overhauled functions
******************************************************************************

Description
===========

An option accepted by pieri, lrMap, applyLR,
see pureFree, and verifyWellDefined.
Possible values:
  * "Row" (default): the original PieriMaps row-form basis.  Source/target
    Schur modules are indexed by SSYTs as lists of weakly-increasing rows.
  * "Filling": the SchurFunctors column-form basis.  Source/target are indexed
    by Fillings, with column-Garnir straightening relations.  The matrix is
    obtained by post-hoc basis change with the equivariant iso
    pmToFilling.
  * "Weyl": the divided-power row-form basis.  Source/target are indexed by
    WeylFillings.  Numerically the same data as the Row matrix; it is just that
    the bases are interpreted on the Weyl (divided-power) side rather than the
    Schur side.
Use verifyWellDefined, to verify that the
chosen-convention map respects the appropriate straightening relations.
+--------------------------------------------------+
|i1 : P = QQ[a,b,c];                               |
+--------------------------------------------------+
|i2 : pieri({2,1}, {1}, P, Convention => "Row")    |
|                                                  |
|o2 = | 2a 0  b 0 c  0 0  0 |                      |
|     | 0  2a 0 b 0  c 0  0 |                      |
|     | 0  0  0 a -a 0 2b c |                      |
|                                                  |
|             3      8                             |
|o2 : Matrix P  <-- P                              |
+--------------------------------------------------+
|i3 : pieri({2,1}, {1}, P, Convention => "Filling")|
|                                                  |
|o3 = | 3/2a 3/2b c     0    1/2c 0    0    0    | |
|     | 0    0    1/2b  3/2a b    3/2c 0    0    | |
|     | 0    0    -1/2a 0    1/2a 0    3/2b 3/2c | |
|                                                  |
|             3      8                             |
|o3 : Matrix P  <-- P                              |
+--------------------------------------------------+

See also
========

  * pmToFilling -- convert a PieriMaps row-form tableau to
    SchurFunctors column-form Fillings
  * fillingToPM -- convert a SchurFunctors column-form
    Filling to PieriMaps row-form tableaux
  * pmToWeyl -- convert a PieriMaps row-form tableau to a
    SchurFunctors WeylFilling
  * weylToPM -- convert a SchurFunctors WeylFilling to a
    PieriMaps row-form tableau
  * verifyWellDefined -- verify an LR map respects
    the chosen convention's straightening relations

Functions with optional argument named Convention:
==================================================

  * "applyDualLR(...,Convention=>...)" -- see applyDualLR,
    -- apply dualLR to a single basis pair (T_nu, T_mu)
  * "applyDualPieri(...,Convention=>...)" -- see applyDualPieri,
    -- apply dualPieri to a single basis pair (poly, T_lambda)
  * "applyDualPieriColumn(...,Convention=>...)" -- see
    applyDualPieriColumn, -- apply dualPieriColumn to a
    single basis pair (wedge_poly, T_lambda)
  * "applyLR(...,Convention=>...)" -- see applyLR, -- apply
    Psi_Q to a single tableau of shape lambda
  * "applyPieri(...,Convention=>...)" -- see applyPieri, --
    apply pieri to a single source tableau T_mu
  * "applyPieriColumn(...,Convention=>...)" -- see applyPieriColumn,
    -- apply pieriColumn to a single source tableau T_mu
  * "dualLR(...,Convention=>...)" -- see dualLR, --
    GL-equivariant projection S_nu V ⊗ S_mu V --> S_lambda V at a chosen LR
    tableau
  * "dualPieri(...,Convention=>...)" -- see dualPieri, --
    GL-equivariant projection Sym^d V ⊗ S_lambda V --> S_mu V
  * "dualPieriColumn(...,Convention=>...)" -- see dualPieriColumn,
    -- GL-equivariant projection wedge^d V ⊗ S_lambda V -->
    S_mu V
  * "lrMap(...,Convention=>...)" -- see lrMap, --
    GL(V)-equivariant Littlewood-Richardson inclusion
  * "pieri(...,Convention=>...)" -- see pieri, -- computes a
    matrix representation for a Pieri inclusion of representations of a general
    linear group
  * "pieriColumn(...,Convention=>...)" -- see pieriColumn,
    -- native column-form (vertical-strip) Pieri inclusion
  * "pureFree(...,Convention=>...)" -- see pureFree, --
    computes a GL(V)-equivariant map whose resolution is pure, or the reduction
    mod p of such a map
  * "verifyEquivariant(...,Convention=>...)" -- see verifyEquivariant,
    -- rigorously verify GL_n-equivariance of a Schur-rep
    matrix
  * "verifyWellDefined(...,Convention=>...)" -- see verifyWellDefined,
    -- verify an LR map respects the chosen convention's
    straightening relations

For the programmer
==================

The object Convention, is a symbol (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:659:0.

displayLRImage -- pretty-print the result of applyLR
****************************************************

  * Usage:
        displayLRImage(image)
  * Inputs:
      * image, a list (Macaulay2Doc), as returned by applyLR,
  * Outputs:
      * a net (Macaulay2Doc), a stacked display of the symbolic sum
        $\sum c_i \cdot T_\nu^{(i)} \otimes T_\mu^{(i)}$

Description
===========

+----------------------------------------+
|i1 : shapes = ({2,1}, {1}, {1,1});      |
+----------------------------------------+
|i2 : Q = (lrTableaux shapes)#0;         |
+----------------------------------------+
|i3 : T = (standardTableaux(3, {2,1}))#0;|
+----------------------------------------+
|i4 : img = applyLR(shapes, Q, T, 3);    |
+----------------------------------------+
|i5 : displayLRImage img                 |
|                                        |
|o5 =   +  3 . [1 ] ⊗ [1 ]               |
|               2                        |
+----------------------------------------+

See also
========

  * applyLR -- apply Psi_Q to a single tableau of shape lambda
  * lrMap -- GL(V)-equivariant Littlewood-Richardson inclusion

Ways to use displayLRImage:
===========================

  * "displayLRImage(List)"

For the programmer
==================

The object displayLRImage, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:640:0.

dualLR -- GL-equivariant projection S_nu V ⊗ S_mu V --> S_lambda V at a chosen LR tableau
*****************************************************************************************

  * Usage:
        dualLR((lambda, mu, nu), Q, n)
        dualLR((lambda, mu, nu), Q, P)
        dualLR(..., Convention => "Row" | "Filling" | "Weyl")
  * Inputs:
      * (lambda, mu, nu), a sequence (Macaulay2Doc), three
        partitions with $|\lambda| = |\mu| + |\nu|$ and $S_\lambda V$ appearing
        in $S_\nu V \otimes S_\mu V$ (i.e. $c^\lambda_{\mu,\nu} > 0$)
      * Q, a list (Macaulay2Doc), an LR tableau of shape
        $\lambda/\mu$ with content $\nu$ (as returned by lrTableaux)
        singling out one of the $c^\lambda_{\mu,\nu}$ copies of
        $S_\lambda V$
      * n, an integer (Macaulay2Doc), or pass a polynomial ring whose
        numgens is used
  * Optional inputs (Macaulay2Doc),:
      * Convention => a string (Macaulay2Doc), default value
        "Row", , basis convention for the three Schur factors.
  * Outputs:
      * a matrix (Macaulay2Doc), a $\mathbb{Q}$-matrix of size
        $\dim S_\lambda V \times (\dim S_\nu V \cdot \dim S_\mu V)$, the
        $GL_n$-equivariant projection onto Q's specific copy of $S_\lambda V$
        inside $S_\nu V \otimes S_\mu V$

Description
===========

Returns the $GL_n$-equivariant projection onto the LR-summand of $S_\nu V
\otimes S_\mu V$ indexed by Q (unique up to scalar by Schur's lemma).  When the
LR coefficient $c^\lambda_{\mu,\nu}$ exceeds 1, distinct LR tableaux Q give
different projections, each onto its own copy of $S_\lambda V$;
$\mathrm{dualLR}_Q \circ \mathrm{lrMap}_{Q'} = 0$ when $Q \neq Q'$, and
$\mathrm{dualLR}_Q \circ \mathrm{lrMap}_Q = \mathrm{Id}$.

Construction: stack the LR inclusions lrMap, for every $(\lambda',
Q')$ with $c^{\lambda'}_{\mu,\nu} > 0$ into a square invertible matrix on
$S_\nu V \otimes S_\mu V$ (square by the LR/Pieri rule) and read off the rows
of its inverse corresponding to $(\lambda, Q)$.
+-----------------------------------------------------------------------------+
|i1 : shapes = ({2,1}, {1}, {1,1});                                           |
+-----------------------------------------------------------------------------+
|i2 : Q = (lrTableaux shapes)#0;                                              |
+-----------------------------------------------------------------------------+
|i3 : M = lrMap(shapes, Q, 3);   -- inclusion S_(2,1) V -> S_(1,1) V ⊗ S_(1) V|
|                                                                             |
|              9       8                                                      |
|o3 : Matrix QQ  <-- QQ                                                       |
+-----------------------------------------------------------------------------+
|i4 : N = dualLR(shapes, Q, 3);  -- projection back onto Q's copy             |
|                                                                             |
|              8       9                                                      |
|o4 : Matrix QQ  <-- QQ                                                       |
+-----------------------------------------------------------------------------+
|i5 : N * M == id_(QQ^(numColumns M))                                         |
|                                                                             |
|o5 = true                                                                    |
+-----------------------------------------------------------------------------+
+---------------------------------------------------------------------+
|i6 : Qs = lrTableaux({3,2,1}, {2,1}, {2,1});  -- multiplicity-2 case |
+---------------------------------------------------------------------+
|i7 : M0 = lrMap(({3,2,1},{2,1},{2,1}), Qs#0, 3);                     |
|                                                                     |
|              64       8                                             |
|o7 : Matrix QQ   <-- QQ                                              |
+---------------------------------------------------------------------+
|i8 : M1 = lrMap(({3,2,1},{2,1},{2,1}), Qs#1, 3);                     |
|                                                                     |
|              64       8                                             |
|o8 : Matrix QQ   <-- QQ                                              |
+---------------------------------------------------------------------+
|i9 : N0 = dualLR(({3,2,1},{2,1},{2,1}), Qs#0, 3);                    |
|                                                                     |
|              8       64                                             |
|o9 : Matrix QQ  <-- QQ                                               |
+---------------------------------------------------------------------+
|i10 : N0 * M0 == id_(QQ^(numColumns M0))   -- recovers Q_0's copy    |
|                                                                     |
|o10 = true                                                           |
+---------------------------------------------------------------------+
|i11 : N0 * M1 == 0                          -- annihilates Q_1's copy|
|                                                                     |
|o11 = true                                                           |
+---------------------------------------------------------------------+

See also
========

  * lrMap -- GL(V)-equivariant Littlewood-Richardson inclusion
  * lrTableaux -- enumerate Littlewood-Richardson tableaux
  * applyDualLR -- apply dualLR to a single basis pair
    (T_nu, T_mu)
  * dualPieri -- GL-equivariant projection Sym^d V ⊗ S_lambda
    V --> S_mu V
  * verifyEquivariant -- rigorously verify
    GL_n-equivariance of a Schur-rep matrix
  * Convention -- tableau basis convention used by
    PieriMaps' overhauled functions

Ways to use dualLR:
===================

  * "dualLR(Sequence,List,PolynomialRing)"
  * "dualLR(Sequence,List,ZZ)"

For the programmer
==================

The object dualLR, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:1133:0.

dualPieri -- GL-equivariant projection Sym^d V ⊗ S_lambda V --> S_mu V
**********************************************************************

  * Usage:
        dualPieri(mu, boxes, P)
        dualPieri(mu, boxes, P, Convention => "Row" | "Filling" | "Weyl")
  * Inputs:
      * mu, a list (Macaulay2Doc), a partition (the target shape):
        $\mu = \lambda + (\text{horizontal $d$-strip})$ where $d =
        \#\mathrm{boxes}$.
      * boxes, a list (Macaulay2Doc), a list of 1-indexed row
        indices recording the horizontal strip $\mu/\lambda$.  Same convention
        as pieri.
      * P, a polynomial ring (Macaulay2Doc), a polynomial
        ring whose generators carry the symmetric factor $\mathrm{Sym}^d V$ on
        the source side.
  * Optional inputs (Macaulay2Doc),:
      * Convention => a string (Macaulay2Doc), default value
        "Row", , basis convention for both source and target Schur factors.
        "Row" (default), "Filling", or "Weyl".
  * Outputs:
      * a matrix (Macaulay2Doc), a $K$-matrix of size $\dim S_\mu
        V \times \bigl(\dim S_\lambda V \cdot \binom{n + d - 1}{d}\bigr)$.
        Columns are indexed by pairs $(T_\lambda, \alpha)$ (a standard tableau
        of $\lambda$ paired with a degree-$d$ monomial of P), in lex order with
        $T_\lambda$ outer and $\alpha$ inner.

Description
===========

Returns the $GL_n$-equivariant projection $\mathrm{Sym}^d V \otimes S_\lambda V
\to S_\mu V$ (unique up to scalar by Schur's lemma).  By Pieri's rule,
$\mathrm{Sym}^d V \otimes S_\lambda V$ decomposes as a direct sum of $S_{\mu'}$
over all $\mu' = \lambda + (\text{horizontal $d$-strip})$; this function
returns the projection onto the summand selected by $\mu$.

The construction stacks the inclusion matrices pieri, for every
addable horizontal $d$-strip into a square invertible matrix, then reads off
the rows of its inverse corresponding to $S_\mu$.  This realizes
$\mathrm{dualPieri} \circ \mathrm{pieri} = \mathrm{Id}$ on $S_\mu V$ exactly.

When Convention => "Filling", the source $S_\lambda$ factor and target $S_\mu$
are expressed in the SchurFunctors column-form basis; "Weyl" uses the
divided-power Weyl-module basis (matrix data is identical to "Row").
+-------------------------------------------------+
|i1 : P = QQ[a,b,c];                              |
+-------------------------------------------------+
|i2 : N = dualPieri({3,2,1}, {1}, P);             |
|                                                 |
|              8       9                          |
|o2 : Matrix QQ  <-- QQ                           |
+-------------------------------------------------+
|i3 : M = pieri({3,2,1}, {1}, P);                 |
|                                                 |
|             3      8                            |
|o3 : Matrix P  <-- P                             |
+-------------------------------------------------+
|i4 : numRows N == numColumns M and numRows M == 3|
|                                                 |
|o4 = true                                        |
+-------------------------------------------------+
+-----------------------------------------------------------------------------+
|i5 : P = QQ[a,b,c,d];  -- interior-row removal mu={3,2,1}, k=2,              |
|-----------------------------------------------------------------------------|
|lambda={3,1,1}                                                               |
+-----------------------------------------------------------------------------+
|i6 : N = dualPieri({3,2,1}, {2}, P);                                         |
|                                                                             |
|              64       144                                                   |
|o6 : Matrix QQ   <-- QQ                                                      |
+-----------------------------------------------------------------------------+
|i7 : numRows N, numColumns N                                                 |
|                                                                             |
|o7 = (64, 144)                                                               |
|                                                                             |
|o7 : Sequence                                                                |
+-----------------------------------------------------------------------------+

Performance. See dualPieri, caches the stacked block matrix used
to invert the GL_n decomposition, keyed by $(\lambda, d, n, K,
\mathrm{Convention})$.  The first call at a given key builds every Pieri
inclusion $S_{\mu'}\to S_\lambda \otimes \mathrm{Sym}^d V$ for $\mu'$ ranging
over all addable horizontal $d$-strips and stacks them into one square matrix
$A$; subsequent calls (for sister summands $\mu$ of the same ambient tensor
product) reuse $A$ and only run a single back-solve, so they cost less than the
corresponding pieri, inclusion.  The selected rows of $A^{-1}$ are
computed via solve(transpose A, E) (E a small standard-basis block) rather than
by forming the full inverse.  boxes is normalized to canonical (sorted
ascending) order; pass the same canonical order to pieri, for the
round-trip $\mathrm{dualPieri} \circ \mathrm{pieri} = \mathrm{Id}$.

See also
========

  * pieri -- computes a matrix representation for a Pieri
    inclusion of representations of a general linear group
  * applyDualPieri -- apply dualPieri to a single basis
    pair (poly, T_lambda)
  * dualLR -- GL-equivariant projection S_nu V ⊗ S_mu V -->
    S_lambda V at a chosen LR tableau
  * verifyEquivariant -- rigorously verify
    GL_n-equivariance of a Schur-rep matrix
  * Convention -- tableau basis convention used by
    PieriMaps' overhauled functions

Ways to use dualPieri:
======================

  * "dualPieri(List,List,PolynomialRing)"

For the programmer
==================

The object dualPieri, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:1016:0.

dualPieriColumn -- GL-equivariant projection wedge^d V ⊗ S_lambda V --> S_mu V
******************************************************************************

  * Usage:
        dualPieriColumn(mu, cols, P)
        dualPieriColumn(..., Convention => "Row" | "Filling" | "Weyl")
  * Inputs:
      * mu, a list (Macaulay2Doc), the target shape: $\mu = \lambda
        + (\text{vertical $d$-strip})$ with $d = \#\mathrm{cols}$
      * cols, a list (Macaulay2Doc), a list of column indices
        recording the strip $\mu/\lambda$
      * P, a polynomial ring (Macaulay2Doc), a
        SkewCommutative ring ($\wedge^d V$ is encoded there)
  * Optional inputs (Macaulay2Doc),:
      * Convention => a string (Macaulay2Doc), default value
        "Filling"
  * Outputs:
      * a matrix (Macaulay2Doc), a $K$-matrix of size $\dim S_\mu
        V \times \bigl(\dim S_\lambda V \cdot \binom{n}{d}\bigr)$.  Columns
        indexed by $(T_\lambda, \alpha)$ pairs ($\alpha$ an exterior monomial).

Description
===========

Column-form analogue of dualPieri,: the $GL_n$-equivariant
projection from $\wedge^d V \otimes S_\lambda V$ onto the chosen $S_\mu V$
summand (unique up to scalar).  Construction stacks pieriColumn,
inclusions for every addable vertical $d$-strip and inverts.
+-----------------------------------------------+
|i1 : P = QQ[e_0..e_2, SkewCommutative => true];|
+-----------------------------------------------+
|i2 : M = pieriColumn({2,1}, {1}, P);           |
|                                               |
|             6      8                          |
|o2 : Matrix P  <-- P                           |
+-----------------------------------------------+
|i3 : N = dualPieriColumn({2,1}, {1}, P);       |
|                                               |
|              8       18                       |
|o3 : Matrix QQ  <-- QQ                         |
+-----------------------------------------------+
|i4 : numRows N, numColumns N                   |
|                                               |
|o4 = (8, 18)                                   |
|                                               |
|o4 : Sequence                                  |
+-----------------------------------------------+

See also
========

  * pieriColumn -- native column-form (vertical-strip)
    Pieri inclusion
  * dualPieri -- GL-equivariant projection Sym^d V ⊗ S_lambda
    V --> S_mu V
  * applyDualPieriColumn -- apply dualPieriColumn
    to a single basis pair (wedge_poly, T_lambda)
  * Convention -- tableau basis convention used by
    PieriMaps' overhauled functions

Ways to use dualPieriColumn:
============================

  * "dualPieriColumn(List,List,PolynomialRing)"

For the programmer
==================

The object dualPieriColumn, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:1360:0.

fillingToPM -- convert a SchurFunctors column-form Filling to PieriMaps row-form tableaux
*****************************************************************************************

  * Usage:
        fillingToPM F
  * Inputs:
      * F, Filling, a SchurFunctors column-form Filling
  * Outputs:
      * a hash table (Macaulay2Doc), the image of F under the
        antisymmetrize-then-symmetrize equivariant map $\wedge^{\lambda'} V \to
        \mathrm{Sym}^\lambda V$, expressed as a sum of standard PM-form
        tableaux with rational coefficients (after PieriMaps straightening).

Description
===========

Inverse (up to a uniform scalar) of pmToFilling.  Each
column of F is wedge-expanded into $V^{\otimes |\lambda|}$ via the column
antisymmetrizer (signed sum over permutations, with normalization $1/\prod
\lambda'_j!$), reordered by rows, and projected to symmetric (sort) per row.
The result is straightened into the standard PM basis.
+--------------------------------------+
|i1 : needsPackage "SchurFunctors";    |
+--------------------------------------+
|i2 : F = new Filling from {{0,2},{1}};|
+--------------------------------------+
|i3 : fillingToPM F                    |
|                                      |
|o3 = HashTable{{{0, 1}, {2}} => 1}    |
|                                1     |
|               {{0, 2}, {1}} => -     |
|                                2     |
|                                      |
|o3 : HashTable                        |
+--------------------------------------+

See also
========

  * pmToFilling -- convert a PieriMaps row-form tableau to
    SchurFunctors column-form Fillings
  * weylToFilling -- convert a WeylFilling to a
    SchurFunctors Filling (compose pmToFilling with weylToPM)
  * fillingToPMMatrix -- QQ matrix of the
    Filling-to-PM change of basis on a Schur module

For the programmer
==================

The object fillingToPM, is a function closure (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:728:0.

fillingToPMMatrix -- QQ matrix of the Filling-to-PM change of basis on a Schur module
*************************************************************************************

  * Usage:
        fillingToPMMatrix(mu, n)
  * Inputs:
      * mu, a list (Macaulay2Doc), a partition
      * n, an integer (Macaulay2Doc),
  * Outputs:
      * a matrix (Macaulay2Doc), a $\mathbb{Q}$ matrix; rows = PM
        tableaux of $\mu$, cols = Fillings of $\mu^t$.  Entry $(i, j)$ =
        coefficient of PM tableau $i$ in fillingToPM(filling_j).

Description
===========

Inverse direction of pmToFillingMatrix,; same
equivariant iso, transposed convention.
+---------------------------------------------+
|i1 : symbolicForm fillingToPMMatrix({2,1}, 3)|
|                                             |
|     +-----+-----------+                     |
|     |+-+-+|3     +-+-+|                     |
|o1 = ||0|0||-  *  |0|0||                     |
|     ||1| ||4     |1| ||                     |
|     |+-+-+|      +-+-+|                     |
|     +-----+-----------+                     |
|     |+-+-+|3     +-+-+|                     |
|     ||0|1||-  *  |0|1||                     |
|     ||1| ||2     |1| ||                     |
|     |+-+-+|      +-+-+|                     |
|     +-----+-----------+                     |
|     |+-+-+|1     +-+-+|                     |
|     ||0|2||-  *  |0|1||                     |
|     ||1| ||2     |2| ||                     |
|     |+-+-+|      +-+-+|                     |
|     |     |      +-+-+|                     |
|     |     |1  *  |0|2||                     |
|     |     |      |1| ||                     |
|     |     |      +-+-+|                     |
|     +-----+-----------+                     |
|     |+-+-+|3     +-+-+|                     |
|     ||0|0||-  *  |0|0||                     |
|     ||2| ||4     |2| ||                     |
|     |+-+-+|      +-+-+|                     |
|     +-----+-----------+                     |
|     |+-+-+|      +-+-+|                     |
|     ||0|1||1  *  |0|1||                     |
|     ||2| ||      |2| ||                     |
|     |+-+-+|      +-+-+|                     |
|     |     |1     +-+-+|                     |
|     |     |-  *  |0|2||                     |
|     |     |2     |1| ||                     |
|     |     |      +-+-+|                     |
|     +-----+-----------+                     |
|     |+-+-+|3     +-+-+|                     |
|     ||0|2||-  *  |0|2||                     |
|     ||2| ||2     |2| ||                     |
|     |+-+-+|      +-+-+|                     |
|     +-----+-----------+                     |
|     |+-+-+|3     +-+-+|                     |
|     ||1|1||-  *  |1|1||                     |
|     ||2| ||4     |2| ||                     |
|     |+-+-+|      +-+-+|                     |
|     +-----+-----------+                     |
|     |+-+-+|3     +-+-+|                     |
|     ||1|2||-  *  |1|2||                     |
|     ||2| ||2     |2| ||                     |
|     |+-+-+|      +-+-+|                     |
|     +-----+-----------+                     |
+---------------------------------------------+

See also
========

  * pmToFillingMatrix -- QQ matrix of the
    PM-to-Filling change of basis on a Schur module
  * fillingToPM -- convert a SchurFunctors column-form
    Filling to PieriMaps row-form tableaux
  * symbolicForm -- print a Schur-rep matrix as a
    basis-labeled symbolic map

For the programmer
==================

The object fillingToPMMatrix, is a function closure (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:848:0.

fillingToWeyl -- convert a Filling to a sum of WeylFillings (compose pmToWeyl with fillingToPM)
***********************************************************************************************

  * Usage:
        fillingToWeyl F
  * Inputs:
      * F, Filling
  * Outputs:
      * a hash table (Macaulay2Doc), a HashTable {WeylFilling
        => coeff}

Description
===========

+--------------------------------------+
|i1 : needsPackage "SchurFunctors";    |
+--------------------------------------+
|i2 : F = new Filling from {{0,2},{1}};|
+--------------------------------------+
|i3 : fillingToWeyl F                  |
|                                      |
|               +-+-+                  |
|o3 = HashTable{|0|1| => 1}            |
|               |2| |                  |
|               +-+-+                  |
|               +-+-+    1             |
|               |0|2| => -             |
|               |1| |    2             |
|               +-+-+                  |
|                                      |
|o3 : HashTable                        |
+--------------------------------------+

See also
========

  * pmToWeyl -- convert a PieriMaps row-form tableau to a
    SchurFunctors WeylFilling
  * fillingToPM -- convert a SchurFunctors column-form
    Filling to PieriMaps row-form tableaux
  * weylToFilling -- convert a WeylFilling to a
    SchurFunctors Filling (compose pmToFilling with weylToPM)

For the programmer
==================

The object fillingToWeyl, is a function closure (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:792:0.

lrMap -- GL(V)-equivariant Littlewood-Richardson inclusion
**********************************************************

  * Usage:
        lrMap((lambda, mu, nu), Q, n)
        lrMap((lambda, mu, nu), Q, P)
        lrMap(..., Convention => "Row" | "Filling" | "Weyl")
  * Inputs:
      * (lambda, mu, nu), a sequence (Macaulay2Doc), three
        partitions with $|\lambda| = |\mu| + |\nu|$ and $\mu$ contained in
        $\lambda$
      * Q, a list (Macaulay2Doc), an LR tableau of shape
        $\lambda/\mu$ with content $\nu$, as returned by lrTableaux,
      * n, an integer (Macaulay2Doc), the dimension of $V$ (or pass a
        PolynomialRing (Macaulay2Doc), whose numgens is
        used)
  * Optional inputs (Macaulay2Doc),:
      * Convention => a string (Macaulay2Doc), default value
        "Row", , basis convention for both the source $S_\lambda V$ and the
        target $S_\nu V \otimes S_\mu V$.  "Row" (default), "Filling", or
        "Weyl".
  * Outputs:
      * a matrix (Macaulay2Doc), a matrix over $\mathbb{Q}$ whose
        columns index a basis of $S_\lambda V$ and rows index a basis of $S_\nu
        V \otimes S_\mu V$

Description
===========

Builds the $GL(V)$-equivariant inclusion $\Psi_Q \colon S_\lambda V \to S_\nu V
\otimes S_\mu V$ associated to the LR tableau Q.  The map is constructed by
iterating row-Pieri inclusions $S_{\lambda^{(a)}} V \to \mathrm{Sym}^{\nu_a} V
\otimes S_{\lambda^{(a-1)}} V$ in the order $a = r, r-1, \ldots, 1$, then
projecting the $\mathrm{Sym}$ tensor side onto $S_\nu V$ via straightening.

The columns of the output are indexed by standardTableaux(n, lambda) (or the
corresponding Filling / WeylFilling basis if a non-default Convention is
chosen).  The rows are indexed by pairs $(T_\nu, T_\mu)$ in lex order: row $i
\cdot \#S_\mu + j$ corresponds to $(T_\nu = N\nu\#i, T_\mu = N\mu\#j)$ where
Nnu = standardTableaux(n, nu) and Nmu = standardTableaux(n, mu padded to length
#lambda).  When Convention => "Filling" is used, the equivariant change of
basis pmToFillingMatrix, is applied on each side, and
the rank is preserved.
+-----------------------------------------------------------------------------+
|i1 : shapes = ({2,1}, {1}, {1,1});                  -- a multiplicity-1      |
|-----------------------------------------------------------------------------|
|example                                                                      |
+-----------------------------------------------------------------------------+
|i2 : Q = (lrTableaux shapes)#0;                                              |
+-----------------------------------------------------------------------------+
|i3 : Mrow = lrMap(shapes, Q, 3, Convention => "Row")                         |
|                                                                             |
|o3 = | 3 0 0   0   0    0   0 0   |                                          |
|     | 0 0 3/2 0   0    0   0 0   |                                          |
|     | 0 0 0   0   3/2  0   0 0   |                                          |
|     | 0 3 0   0   0    0   0 0   |                                          |
|     | 0 0 0   3/2 0    0   0 0   |                                          |
|     | 0 0 0   0   0    3/2 0 0   |                                          |
|     | 0 0 0   3/2 -3/2 0   0 0   |                                          |
|     | 0 0 0   0   0    0   3 0   |                                          |
|     | 0 0 0   0   0    0   0 3/2 |                                          |
|                                                                             |
|              9       8                                                      |
|o3 : Matrix QQ  <-- QQ                                                       |
+-----------------------------------------------------------------------------+
+------------------------------------------------------------+
|i4 : shapes = ({2,1}, {1}, {1,1});                          |
+------------------------------------------------------------+
|i5 : Q = (lrTableaux shapes)#0;                             |
+------------------------------------------------------------+
|i6 : Mfilling = lrMap(shapes, Q, 3, Convention => "Filling")|
|                                                            |
|o6 = | 3 0 0  0 0 0 0 0 |                                   |
|     | 0 3 0  0 0 0 0 0 |                                   |
|     | 0 0 2  0 1 0 0 0 |                                   |
|     | 0 0 0  3 0 0 0 0 |                                   |
|     | 0 0 1  0 2 0 0 0 |                                   |
|     | 0 0 0  0 0 3 0 0 |                                   |
|     | 0 0 -1 0 1 0 0 0 |                                   |
|     | 0 0 0  0 0 0 3 0 |                                   |
|     | 0 0 0  0 0 0 0 3 |                                   |
|                                                            |
|              9       8                                     |
|o6 : Matrix QQ  <-- QQ                                      |
+------------------------------------------------------------+
+------------------------------------------------------+
|i7 : shapes = ({2,1}, {1}, {1,1});                    |
+------------------------------------------------------+
|i8 : Q = (lrTableaux shapes)#0;                       |
+------------------------------------------------------+
|i9 : Mweyl = lrMap(shapes, Q, 3, Convention => "Weyl")|
|                                                      |
|o9 = | 3 0 0   0   0    0   0 0   |                   |
|     | 0 0 3/2 0   0    0   0 0   |                   |
|     | 0 0 0   0   3/2  0   0 0   |                   |
|     | 0 3 0   0   0    0   0 0   |                   |
|     | 0 0 0   3/2 0    0   0 0   |                   |
|     | 0 0 0   0   0    3/2 0 0   |                   |
|     | 0 0 0   3/2 -3/2 0   0 0   |                   |
|     | 0 0 0   0   0    0   3 0   |                   |
|     | 0 0 0   0   0    0   0 3/2 |                   |
|                                                      |
|              9       8                               |
|o9 : Matrix QQ  <-- QQ                                |
+------------------------------------------------------+
All three are GL-equivariant injections of $S_\lambda V$ into $S_\nu V \otimes
S_\mu V$.  The Row and Weyl matrices have the same raw entries (only the
source/target bases are reinterpreted on the divided-power side); the Filling
matrix is obtained by post-hoc basis change.  All three have the same rank.
+--------------------------------------------------------------+
|i10 : shapes = ({2,1}, {1}, {1,1});                           |
+--------------------------------------------------------------+
|i11 : Q = (lrTableaux shapes)#0;                              |
+--------------------------------------------------------------+
|i12 : Mrow = lrMap(shapes, Q, 3, Convention => "Row");        |
|                                                              |
|               9       8                                      |
|o12 : Matrix QQ  <-- QQ                                       |
+--------------------------------------------------------------+
|i13 : Mfilling = lrMap(shapes, Q, 3, Convention => "Filling");|
|                                                              |
|               9       8                                      |
|o13 : Matrix QQ  <-- QQ                                       |
+--------------------------------------------------------------+
|i14 : Mweyl = lrMap(shapes, Q, 3, Convention => "Weyl");      |
|                                                              |
|               9       8                                      |
|o14 : Matrix QQ  <-- QQ                                       |
+--------------------------------------------------------------+
|i15 : (rank Mrow, rank Mfilling, rank Mweyl)                  |
|                                                              |
|o15 = (8, 8, 8)                                               |
|                                                              |
|o15 : Sequence                                                |
+--------------------------------------------------------------+
When $c^\lambda_{\mu,\nu} > 1$, distinct LR tableaux $Q$ give linearly
independent inclusions; the joint matrix has rank $c^\lambda_{\mu,\nu} \cdot
\dim S_\lambda V$.  The canonical multiplicity-2 example is
$c^{(3,2,1)}_{(2,1),(2,1)} = 2$:
+----------------------------------------------------------------------------+
|i16 : Qs = lrTableaux({3,2,1}, {2,1}, {2,1});  -- two distinct LR tableaux  |
+----------------------------------------------------------------------------+
|i17 : M1 = lrMap(({3,2,1}, {2,1}, {2,1}), Qs#0, 3);                         |
|                                                                            |
|               64       8                                                   |
|o17 : Matrix QQ   <-- QQ                                                    |
+----------------------------------------------------------------------------+
|i18 : M2 = lrMap(({3,2,1}, {2,1}, {2,1}), Qs#1, 3);                         |
|                                                                            |
|               64       8                                                   |
|o18 : Matrix QQ   <-- QQ                                                    |
+----------------------------------------------------------------------------+
|i19 : rank M1, rank M2          -- each = dim S_(3,2,1) = 8                 |
|                                                                            |
|o19 = (8, 8)                                                                |
|                                                                            |
|o19 : Sequence                                                              |
+----------------------------------------------------------------------------+
|i20 : rank(M1 | M2) == 16       -- 2 * 8: M1 and M2 are linearly independent|
|                                                                            |
|o20 = true                                                                  |
+----------------------------------------------------------------------------+
Use symbolicForm, to read off the action of $\Psi_Q$ on
each standard tableau of $S_\lambda V$, as a sum over basis pairs of $S_\nu V
\otimes S_\mu V$:
+--------------------------------------+
|i21 : shapes = ({2,1}, {1}, {1,1});   |
+--------------------------------------+
|i22 : Q = (lrTableaux shapes)#0;      |
+--------------------------------------+
|i23 : symbolicForm lrMap(shapes, Q, 3)|
|                                      |
|      +-----+-------------------+     |
|      |+-+-+|      +-+     +-+  |     |
|o23 = ||0|0||3  *  |0|  ⊗  |0|  |     |
|      ||1| ||      |1|     +-+  |     |
|      |+-+-+|      +-+          |     |
|      +-----+-------------------+     |
|      |+-+-+|      +-+     +-+  |     |
|      ||0|0||3  *  |0|  ⊗  |0|  |     |
|      ||2| ||      |2|     +-+  |     |
|      |+-+-+|      +-+          |     |
|      +-----+-------------------+     |
|      |+-+-+|3     +-+     +-+  |     |
|      ||0|1||-  *  |0|  ⊗  |1|  |     |
|      ||1| ||2     |1|     +-+  |     |
|      |+-+-+|      +-+          |     |
|      +-----+-------------------+     |
|      |+-+-+|3     +-+     +-+  |     |
|      ||0|1||-  *  |0|  ⊗  |1|  |     |
|      ||2| ||2     |2|     +-+  |     |
|      |+-+-+|      +-+          |     |
|      |     |3     +-+     +-+  |     |
|      |     |-  *  |1|  ⊗  |0|  |     |
|      |     |2     |2|     +-+  |     |
|      |     |      +-+          |     |
|      +-----+-------------------+     |
|      |+-+-+|3     +-+     +-+  |     |
|      ||0|2||-  *  |0|  ⊗  |2|  |     |
|      ||1| ||2     |1|     +-+  |     |
|      |+-+-+|      +-+          |     |
|      |     |  3     +-+     +-+|     |
|      |     |- -  *  |1|  ⊗  |0||     |
|      |     |  2     |2|     +-+|     |
|      |     |        +-+        |     |
|      +-----+-------------------+     |
|      |+-+-+|3     +-+     +-+  |     |
|      ||0|2||-  *  |0|  ⊗  |2|  |     |
|      ||2| ||2     |2|     +-+  |     |
|      |+-+-+|      +-+          |     |
|      +-----+-------------------+     |
|      |+-+-+|      +-+     +-+  |     |
|      ||1|1||3  *  |1|  ⊗  |1|  |     |
|      ||2| ||      |2|     +-+  |     |
|      |+-+-+|      +-+          |     |
|      +-----+-------------------+     |
|      |+-+-+|3     +-+     +-+  |     |
|      ||1|2||-  *  |1|  ⊗  |2|  |     |
|      ||2| ||2     |2|     +-+  |     |
|      |+-+-+|      +-+          |     |
|      +-----+-------------------+     |
+--------------------------------------+

See also
========

  * lrTableaux -- enumerate Littlewood-Richardson tableaux
  * pieri -- computes a matrix representation for a Pieri
    inclusion of representations of a general linear group
  * straighten -- computes straightening of a tableau
  * applyLR -- apply Psi_Q to a single tableau of shape lambda
  * verifyWellDefined -- verify an LR map respects
    the chosen convention's straightening relations
  * symbolicForm -- print a Schur-rep matrix as a
    basis-labeled symbolic map

Ways to use lrMap:
==================

  * "lrMap(Sequence,List,PolynomialRing)"
  * "lrMap(Sequence,List,ZZ)"

For the programmer
==================

The object lrMap, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:565:0.

lrTableaux -- enumerate Littlewood-Richardson tableaux
******************************************************

  * Usage:
        lrTableaux(lambda, mu, nu)
  * Inputs:
      * lambda, a list (Macaulay2Doc), a partition
      * mu, a list (Macaulay2Doc), a partition contained in lambda
      * nu, a list (Macaulay2Doc), a partition with |nu| = |lambda|
        - |mu|
  * Outputs:
      * a list (Macaulay2Doc), the list of LR fillings of the skew
        shape lambda/mu with content nu

Description
===========

An LR tableau is a semistandard skew tableau (rows weakly increasing, columns
strictly increasing) whose reverse reading word is a lattice word.  Each
filling is returned as a list of rows; row $i$ has length $\lambda_i - \mu_i$
and entries in $\{0, \ldots, \#\nu - 1\}$ (so label 0 corresponds to row 1 of
$\nu$, etc.).  The number of LR tableaux equals the Littlewood-Richardson
coefficient $c^\lambda_{\mu,\nu}$.
+--------------------------------------------------------------------------+
|i1 : #lrTableaux({3,2,1}, {2,1}, {2,1}) -- = c^{(3,2,1)}_{(2,1),(2,1)} = 2|
|                                                                          |
|o1 = 2                                                                    |
+--------------------------------------------------------------------------+
|i2 : #lrTableaux({2,1}, {1}, {1,1})     -- = 1                            |
|                                                                          |
|o2 = 1                                                                    |
+--------------------------------------------------------------------------+

See also
========

  * lrMap -- GL(V)-equivariant Littlewood-Richardson inclusion

Ways to use lrTableaux:
=======================

  * "lrTableaux(List,List,List)"

For the programmer
==================

The object lrTableaux, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:469:0.

pieri -- computes a matrix representation for a Pieri inclusion of representations of a general linear group
************************************************************************************************************

  * Usage:
        pieri(mu, boxes, P)
        pieri(mu, boxes, P, Convention => "Row" | "Filling" | "Weyl")
  * Inputs:
      * mu, a list (Macaulay2Doc), a partition $(\mu_1, \ldots,
        \mu_r)$ where $\mu_i$ is the number of boxes in the $i$th row
      * boxes, a list (Macaulay2Doc), a list of rows from which to
        remove boxes (boxes are always removed from the end of the row). This
        specifies which map of $GL(V)$ representations we want. The row indices
        start from 1 and not 0, and this must specify a horizontal strip in
        $\mu$ (see description below).
      * P, a polynomial ring (Macaulay2Doc), a polynomial
        ring over a field $K$ in $n$ variables
  * Optional inputs (Macaulay2Doc),:
      * Convention => a string (Macaulay2Doc), default value
        "Row", , the tableau basis convention used to label the source $S_\mu
        V$ and target $S_\lambda V$.  One of "Row" (default; PieriMaps row-form
        SSYT basis), "Filling" (SchurFunctors column-form Filling basis on both
        sides), or "Weyl" (data-equivalent to Row, but the bases are
        interpreted as WeylFillings on the divided-power side).  The strip
        factor $(\mathrm{Sym}^k V)$ stays in P regardless.
  * Outputs:
      * a matrix (Macaulay2Doc), If $K$ has characteristic 0,
        then given the partition $\mu$ and the partition $\mu'$ obtained from
        $\mu$ by removing a single box, there is a unique (up to nonzero
        scalar) $GL(V)$-equivariant inclusion $S_\mu V \to V \otimes S_{\mu'}
        V$, where $S_\mu$ refers to the irreducible representation of $GL(V)$
        with highest weight $\mu$. This can be extended uniquely to a map of $P
        = \mathrm{Sym}(V)$-modules $P \otimes S_\mu V \to P \otimes S_{\mu'}
        V$. This method computes the matrix representation for the composition
        of maps that one obtains by iterating this procedure of removing boxes,
        i.e., the final output is a $GL(V)$-equivariant map $P \otimes S_\mu V
        \to P \otimes S_\lambda V$ where $\lambda$ is the partition obtained
        from $\mu$ by deleting a box from row $\mathrm{boxes}_0$, a box from
        row $\mathrm{boxes}_1$, etc. If $K$ has positive characteristic, then
        the corresponding map is calculated over $\mathbb{Q}$, lifted to a
        $\mathbb{Z}$-form of the representation which has the property that the
        map has a torsion-free cokernel over $\mathbb{Z}$, and then the
        coefficients are reduced to $K$.

Description
===========

Convention: the partition $(d)$ represents the $d$th symmetric power, while the
partition $(1, \ldots, 1)$ represents the $d$th exterior power. Using the
notation from the output, $\mu/\lambda$ must be a horizontal strip. Precisely,
this means that $\lambda_i \geq \mu_{i+1}$ for all $i$. If this condition is
not satisfied, the program throws an error because a nonzero equivariant map of
the desired form will not exist.

The Convention option re-expresses the same equivariant map in different
tableau bases on the source $S_\mu V$ and target $S_\lambda V$.  In "Row"
(default), the bases are PM-style SSYT (rows weakly increasing).  In "Filling",
they are SchurFunctors-style column-form Fillings (the matrix is obtained by
post-hoc basis change with the equivariant iso pmToFilling).
In "Weyl", the matrix data is the same as "Row" but is
interpreted on the WeylFilling (divided-power) side.  All three give matrices
of the same rank.
+-----------------------------------------------------------------------------+
|i1 : pieri({3,1}, {1}, QQ[a,b,c]) -- removes the last box from row 1 of the  |
|                                                                             |
|o1 = | 3a 0  b  0  c  0  0  0  0     0 0   0  0  0  0  |                     |
|     | 0  3a 0  b  0  c  0  0  0     0 0   0  0  0  0  |                     |
|     | 0  0  2a 0  0  0  2b 0  c     0 0   0  0  0  0  |                     |
|     | 0  0  0  2a 0  0  0  2b 0     c 0   0  0  0  0  |                     |
|     | 0  0  0  0  2a 0  0  0  b     0 2c  0  0  0  0  |                     |
|     | 0  0  0  0  0  2a 0  0  0     b 0   2c 0  0  0  |                     |
|     | 0  0  0  0  0  0  0  a  -1/2a 0 0   0  3b c  0  |                     |
|     | 0  0  0  0  0  0  0  0  0     a -2a 0  0  2b 2c |                     |
|                                                                             |
|                      8               15                                     |
|o1 : Matrix (QQ[a..c])  <-- (QQ[a..c])                                       |
|-----------------------------------------------------------------------------|
|partition {3,1}                                                              |
+-----------------------------------------------------------------------------+
|i2 : res coker oo -- resolve this map                                        |
|                                                                             |
|               8               15               10               3           |
|o2 = (QQ[a..c])  <-- (QQ[a..c])   <-- (QQ[a..c])   <-- (QQ[a..c])            |
|                                                                             |
|     0               1                2                3                     |
|                                                                             |
|o2 : Complex                                                                 |
+-----------------------------------------------------------------------------+
|i3 : betti oo -- check that the resolution is pure                           |
|                                                                             |
|            0  1  2 3                                                        |
|o3 = total: 8 15 10 3                                                        |
|         0: 8 15  . .                                                        |
|         1: .  . 10 .                                                        |
|         2: .  .  . 3                                                        |
|                                                                             |
|o3 : BettiTally                                                              |
+-----------------------------------------------------------------------------+
+-----------------------------------------------------+
|i4 : P = QQ[a,b,c];                                  |
+-----------------------------------------------------+
|i5 : Mrow = pieri({2,1}, {1}, P, Convention => "Row")|
|                                                     |
|o5 = | 2a 0  b 0 c  0 0  0 |                         |
|     | 0  2a 0 b 0  c 0  0 |                         |
|     | 0  0  0 a -a 0 2b c |                         |
|                                                     |
|             3      8                                |
|o5 : Matrix P  <-- P                                 |
+-----------------------------------------------------+
+---------------------------------------------------------+
|i6 : P = QQ[a,b,c];                                      |
+---------------------------------------------------------+
|i7 : Mfil = pieri({2,1}, {1}, P, Convention => "Filling")|
|                                                         |
|o7 = | 3/2a 3/2b c     0    1/2c 0    0    0    |        |
|     | 0    0    1/2b  3/2a b    3/2c 0    0    |        |
|     | 0    0    -1/2a 0    1/2a 0    3/2b 3/2c |        |
|                                                         |
|             3      8                                    |
|o7 : Matrix P  <-- P                                     |
+---------------------------------------------------------+
+------------------------------------------------------+
|i8 : P = QQ[a,b,c];                                   |
+------------------------------------------------------+
|i9 : Mwey = pieri({2,1}, {1}, P, Convention => "Weyl")|
|                                                      |
|o9 = | 2a 0  b 0 c  0 0  0 |                          |
|     | 0  2a 0 b 0  c 0  0 |                          |
|     | 0  0  0 a -a 0 2b c |                          |
|                                                      |
|             3      8                                 |
|o9 : Matrix P  <-- P                                  |
+------------------------------------------------------+
The Row and Weyl matrices share raw data (only the source/target bases are
reinterpreted on the divided-power side); the Filling matrix is obtained by
post-hoc basis change.  All three have the same rank.
+-----------------------------------------------------------+
|i10 : P = QQ[a,b,c];                                       |
+-----------------------------------------------------------+
|i11 : Mrow = pieri({2,1}, {1}, P, Convention => "Row");    |
|                                                           |
|              3      8                                     |
|o11 : Matrix P  <-- P                                      |
+-----------------------------------------------------------+
|i12 : Mfil = pieri({2,1}, {1}, P, Convention => "Filling");|
|                                                           |
|              3      8                                     |
|o12 : Matrix P  <-- P                                      |
+-----------------------------------------------------------+
|i13 : Mwey = pieri({2,1}, {1}, P, Convention => "Weyl");   |
|                                                           |
|              3      8                                     |
|o13 : Matrix P  <-- P                                      |
+-----------------------------------------------------------+
|i14 : rank Mrow, rank Mfil, rank Mwey                      |
|                                                           |
|o14 = (3, 3, 3)                                            |
|                                                           |
|o14 : Sequence                                             |
+-----------------------------------------------------------+
The Betti table of the cokernel is the same in every convention (the
resolutions are isomorphic up to an equivariant change of basis):
+---------------------------------------------------------------+
|i15 : P = QQ[a,b,c];                                           |
+---------------------------------------------------------------+
|i16 : betti res coker pieri({2,1}, {1}, P, Convention => "Row")|
|                                                               |
|             0 1 2 3                                           |
|o16 = total: 3 8 6 1                                           |
|          0: 3 8 6 .                                           |
|          1: . . . 1                                           |
|                                                               |
|o16 : BettiTally                                               |
+---------------------------------------------------------------+
+-------------------------------------------------------------------+
|i17 : P = QQ[a,b,c];                                               |
+-------------------------------------------------------------------+
|i18 : betti res coker pieri({2,1}, {1}, P, Convention => "Filling")|
|                                                                   |
|             0 1 2 3                                               |
|o18 = total: 3 8 6 1                                               |
|          0: 3 8 6 .                                               |
|          1: . . . 1                                               |
|                                                                   |
|o18 : BettiTally                                                   |
+-------------------------------------------------------------------+

Multi-box horizontal strips: boxes can list more than one row index (with
repeats), and the result is the composition of single-box Pieri inclusions.
The same row index can appear repeatedly when row 1 has enough boxes; entries
become degree-$d$ polynomials.
+-----------------------------------------------------------------------------+
|i19 : P = QQ[a,b,c];                                                         |
+-----------------------------------------------------------------------------+
|i20 : pieri({3,1}, {1,1}, P)         -- removes 2 boxes from row 1; entries  |
|                                                                             |
|o20 = | 6a2 0   4ab 0   4ac  0   2b2 0   2bc  0   2c2  0   0   0   0   |     |
|      | 0   6a2 0   4ab 0    4ac 0   2b2 0    2bc 0    2c2 0   0   0   |     |
|      | 0   0   0   2a2 -2a2 0   0   4ab -2ab 2ac -4ac 0   6b2 4bc 2c2 |     |
|                                                                             |
|              3      15                                                      |
|o20 : Matrix P  <-- P                                                        |
|-----------------------------------------------------------------------------|
|are deg 2                                                                    |
+-----------------------------------------------------------------------------+
+-----------------------------------------------------------------------------+
|i21 : P = QQ[a,b,c];                                                         |
+-----------------------------------------------------------------------------+
|i22 : pieri({3,2,1}, {1,3}, P)       -- removes one box from row 1, then row |
|                                                                             |
|o22 = | 3ac  0      bc  0      c2  0      0      0      |                    |
|      | -3ab 3/2ac  -b2 1/2bc  -bc 1/2c2  0      0      |                    |
|      | 0    -3/2ab 0   -1/2b2 0   -1/2bc 0      0      |                    |
|      | 6a2  0      2ab ac     0   0      bc     1/2c2  |                    |
|      | 0    3/2a2  0   0      ab  1/2ac  -1/2b2 -1/4bc |                    |
|      | 0    0      0   1/2a2  -a2 0      1/2ab  1/4ac  |                    |
|                                                                             |
|              6      8                                                       |
|o22 : Matrix P  <-- P                                                        |
|-----------------------------------------------------------------------------|
|3                                                                            |
+-----------------------------------------------------------------------------+
And symbolicForm, makes the basis-by-basis action of a
multi-box map readable:
+-----------------------------------------+
|i23 : P = QQ[a,b,c];                     |
+-----------------------------------------+
|i24 : symbolicForm pieri({3,1}, {1,1}, P)|
|                                         |
|      +-------+-------------+            |
|      |+-+-+-+|  2     +-+  |            |
|o24 = ||0|0|0||6a   *  |0|  |            |
|      ||1| | ||        |1|  |            |
|      |+-+-+-+|        +-+  |            |
|      +-------+-------------+            |
|      |+-+-+-+|  2     +-+  |            |
|      ||0|0|0||6a   *  |0|  |            |
|      ||2| | ||        |2|  |            |
|      |+-+-+-+|        +-+  |            |
|      +-------+-------------+            |
|      |+-+-+-+|         +-+ |            |
|      ||0|0|1||4a*b  *  |0| |            |
|      ||1| | ||         |1| |            |
|      |+-+-+-+|         +-+ |            |
|      +-------+-------------+            |
|      |+-+-+-+|         +-+ |            |
|      ||0|0|1||4a*b  *  |0| |            |
|      ||2| | ||         |2| |            |
|      |+-+-+-+|         +-+ |            |
|      |       |  2     +-+  |            |
|      |       |2a   *  |1|  |            |
|      |       |        |2|  |            |
|      |       |        +-+  |            |
|      +-------+-------------+            |
|      |+-+-+-+|         +-+ |            |
|      ||0|0|2||4a*c  *  |0| |            |
|      ||1| | ||         |1| |            |
|      |+-+-+-+|         +-+ |            |
|      |       |   2     +-+ |            |
|      |       |-2a   *  |1| |            |
|      |       |         |2| |            |
|      |       |         +-+ |            |
|      +-------+-------------+            |
|      |+-+-+-+|         +-+ |            |
|      ||0|0|2||4a*c  *  |0| |            |
|      ||2| | ||         |2| |            |
|      |+-+-+-+|         +-+ |            |
|      +-------+-------------+            |
|      |+-+-+-+|  2     +-+  |            |
|      ||0|1|1||2b   *  |0|  |            |
|      ||1| | ||        |1|  |            |
|      |+-+-+-+|        +-+  |            |
|      +-------+-------------+            |
|      |+-+-+-+|  2     +-+  |            |
|      ||0|1|1||2b   *  |0|  |            |
|      ||2| | ||        |2|  |            |
|      |+-+-+-+|        +-+  |            |
|      |       |         +-+ |            |
|      |       |4a*b  *  |1| |            |
|      |       |         |2| |            |
|      |       |         +-+ |            |
|      +-------+-------------+            |
|      |+-+-+-+|         +-+ |            |
|      ||0|1|2||2b*c  *  |0| |            |
|      ||1| | ||         |1| |            |
|      |+-+-+-+|         +-+ |            |
|      |       |          +-+|            |
|      |       |-2a*b  *  |1||            |
|      |       |          |2||            |
|      |       |          +-+|            |
|      +-------+-------------+            |
|      |+-+-+-+|         +-+ |            |
|      ||0|1|2||2b*c  *  |0| |            |
|      ||2| | ||         |2| |            |
|      |+-+-+-+|         +-+ |            |
|      |       |         +-+ |            |
|      |       |2a*c  *  |1| |            |
|      |       |         |2| |            |
|      |       |         +-+ |            |
|      +-------+-------------+            |
|      |+-+-+-+|  2     +-+  |            |
|      ||0|2|2||2c   *  |0|  |            |
|      ||1| | ||        |1|  |            |
|      |+-+-+-+|        +-+  |            |
|      |       |          +-+|            |
|      |       |-4a*c  *  |1||            |
|      |       |          |2||            |
|      |       |          +-+|            |
|      +-------+-------------+            |
|      |+-+-+-+|  2     +-+  |            |
|      ||0|2|2||2c   *  |0|  |            |
|      ||2| | ||        |2|  |            |
|      |+-+-+-+|        +-+  |            |
|      +-------+-------------+            |
|      |+-+-+-+|  2     +-+  |            |
|      ||1|1|1||6b   *  |1|  |            |
|      ||2| | ||        |2|  |            |
|      |+-+-+-+|        +-+  |            |
|      +-------+-------------+            |
|      |+-+-+-+|         +-+ |            |
|      ||1|1|2||4b*c  *  |1| |            |
|      ||2| | ||         |2| |            |
|      |+-+-+-+|         +-+ |            |
|      +-------+-------------+            |
|      |+-+-+-+|  2     +-+  |            |
|      ||1|2|2||2c   *  |1|  |            |
|      ||2| | ||        |2|  |            |
|      |+-+-+-+|        +-+  |            |
|      +-------+-------------+            |
+-----------------------------------------+

See also
========

  * pureFree -- computes a GL(V)-equivariant map whose
    resolution is pure, or the reduction mod p of such a map
  * pieriColumn -- native column-form (vertical-strip)
    Pieri inclusion
  * lrMap -- GL(V)-equivariant Littlewood-Richardson inclusion

Ways to use pieri:
==================

  * "pieri(List,List,PolynomialRing)"

For the programmer
==================

The object pieri, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:381:0.

pieriColumn -- native column-form (vertical-strip) Pieri inclusion
******************************************************************

  * Usage:
        pieriColumn(mu, cols, P)
  * Inputs:
      * mu, a list (Macaulay2Doc), a partition
      * cols, a list (Macaulay2Doc), a list of 1-indexed column
        indices.  One box is removed from the bottom of each listed column; the
        cumulative removal must form a vertical strip (no two boxes in the same
        row).
      * P, a polynomial ring (Macaulay2Doc), a
        SkewCommutative $\mathbb{Q}$-algebra in $n$ variables.  Its generators
        carry the wedge factor $\wedge^{|\mathrm{cols}|} V$ on the output side.
  * Optional inputs (Macaulay2Doc),:
      * Convention => ..., default value "Filling"
  * Outputs:
      * a matrix (Macaulay2Doc), a matrix from $S_\mu V$ to
        $\wedge^{|\mathrm{cols}|} V \otimes S_\lambda V$, with rows indexed by
        standard Fillings of $\lambda$ and columns indexed by standard Fillings
        of $\mu$.

Description
===========

The dual of pieri.  Where pieri(mu, rows, P) removes a horizontal
strip and lands in $\mathrm{Sym}^k V$, pieriColumn(mu, cols, P) removes a
vertical strip and lands in $\wedge^k V$.  Both maps are unique (up to scalar)
$GL(V)$-equivariant inclusions $S_\mu V \to (\,?\text{-tensor}\,) \otimes
S_\lambda V$; pieri uses Olver's formula with $c_J = \prod (\mu_{J_q} - \mu_k +
k - J_q)$ on the symmetric ($\mathrm{Sym}$) side, while pieriColumn uses the
dual formula with $c'_J = \prod (\mu'_k - \mu'_{J_q} + J_q - k)$ on the wedge
side, where $\mu'$ is the conjugate partition.  The extra sign in $c'_J$
relative to a literal transpose accounts for the wedge anticommutativity.
+-----------------------------------------------+
|i1 : needsPackage "SchurFunctors";             |
+-----------------------------------------------+
|i2 : P = QQ[e_0..e_3, SkewCommutative => true];|
+-----------------------------------------------+
|i3 : M = pieriColumn({3,2,1}, {1}, P);         |
|                                               |
|             60      64                        |
|o3 : Matrix P   <-- P                          |
+-----------------------------------------------+
|i4 : numRows M, numColumns M                   |
|                                               |
|o4 = (60, 64)                                  |
|                                               |
|o4 : Sequence                                  |
+-----------------------------------------------+
+---------------------------------------------------------+
|i5 : -- Multi-strip: remove a vertical strip of length 2.|
|     P = QQ[e_0..e_3, SkewCommutative => true];          |
+---------------------------------------------------------+
|i6 : M = pieriColumn({3,2,1}, {1,2}, P);                 |
|                                                         |
|             45      64                                  |
|o6 : Matrix P   <-- P                                    |
+---------------------------------------------------------+
|i7 : numRows M, numColumns M                             |
|                                                         |
|o7 = (45, 64)                                            |
|                                                         |
|o7 : Sequence                                            |
+---------------------------------------------------------+
|i8 :                                                     |
+---------------------------------------------------------+

See also
========

  * pieri -- computes a matrix representation for a Pieri
    inclusion of representations of a general linear group
  * fillingToPM -- convert a SchurFunctors column-form
    Filling to PieriMaps row-form tableaux
  * pieri(...,Convention=>...) -- computes a matrix
    representation for a Pieri inclusion of representations of a general linear
    group

Ways to use pieriColumn:
========================

  * "pieriColumn(List,List,PolynomialRing)"

For the programmer
==================

The object pieriColumn, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:899:0.

pmToFilling -- convert a PieriMaps row-form tableau to SchurFunctors column-form Fillings
*****************************************************************************************

  * Usage:
        pmToFilling T
  * Inputs:
      * T, a list (Macaulay2Doc), a PieriMaps row-form tableau
        (list of weakly-increasing rows of the same shape)
  * Outputs:
      * a hash table (Macaulay2Doc), the image of T under the
        symmetrize-then-antisymmetrize equivariant map $\mathrm{Sym}^\lambda V
        \to \wedge^{\lambda'} V$, expressed as a sum of standard Fillings with
        rational coefficients (after SchurFunctors straightening).

Description
===========

PieriMaps' row-form representation $S_\lambda V \subset
\mathrm{Sym}^{\lambda_1} V \otimes \cdots$ and SchurFunctors' column-form
representation $S_\lambda V \subset \wedge^{\lambda'_1} V \otimes \cdots$ are
equivariant isomorphisms (over $\mathbb{Q}$).  This function realizes that iso
explicitly: each PM monomial $T = (m_1, \ldots, m_r)$ is symmetrized into
$V^{\otimes |\lambda|}$ via the row-symmetrizer (with normalization $1/\prod
\lambda_i!$), reordered by columns, and then projected to wedges (sort $+$
sign) per column.  The result is then straightened into the standard Filling
basis via SchurFunctors' straighten.

Round-tripping pmToFilling then fillingToPM, gives a uniform
shape-dependent scalar $c_\lambda \cdot \mathrm{id}$ (the Young symmetrizer
normalization).
+-----------------------------------------------------------------------------+
|i1 : needsPackage "SchurFunctors";                                           |
+-----------------------------------------------------------------------------+
|i2 : pmToFilling {{0,1},{2}}                                                 |
|                                                                             |
|               +-+-+      1                                                  |
|o2 = HashTable{|0|2| => - -}                                                 |
|               |1| |      2                                                  |
|               +-+-+                                                         |
|               +-+-+                                                         |
|               |0|1| => 1                                                    |
|               |2| |                                                         |
|               +-+-+                                                         |
|                                                                             |
|o2 : HashTable                                                               |
+-----------------------------------------------------------------------------+
|i3 : pmToFilling {{0,0,0},{1,1},{2}}  -- highest-weight vector of S_(3,2,1)  |
|                                                                             |
|               +-+-+-+                                                       |
|o3 = HashTable{|0|0|0| => 1}                                                 |
|               |1|1| |                                                       |
|               |2| | |                                                       |
|               +-+-+-+                                                       |
|                                                                             |
|o3 : HashTable                                                               |
|-----------------------------------------------------------------------------|
|Q^3                                                                          |
+-----------------------------------------------------------------------------+

See also
========

  * fillingToPM -- convert a SchurFunctors column-form
    Filling to PieriMaps row-form tableaux
  * pmToWeyl -- convert a PieriMaps row-form tableau to a
    SchurFunctors WeylFilling
  * pmToFillingMatrix -- QQ matrix of the
    PM-to-Filling change of basis on a Schur module
  * Convention -- tableau basis convention used by
    PieriMaps' overhauled functions

For the programmer
==================

The object pmToFilling, is a function closure (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:701:0.

pmToFillingMatrix -- QQ matrix of the PM-to-Filling change of basis on a Schur module
*************************************************************************************

  * Usage:
        pmToFillingMatrix(mu, n)
  * Inputs:
      * mu, a list (Macaulay2Doc), a partition (the Schur shape)
      * n, an integer (Macaulay2Doc), the dimension of $V$
  * Outputs:
      * a matrix (Macaulay2Doc), a $\mathbb{Q}$ matrix; rows
        indexed by Fillings of column heights $\mu^t$, columns indexed by
        PieriMaps' standard tableaux of shape $\mu$.  Entry $(i, j)$ is the
        coefficient of Filling $i$ in pmToFilling(PMtab_j).

Description
===========

This matrix realizes the equivariant change of basis between PieriMaps'
row-form basis of $S_\mu V$ and SchurFunctors' column-form basis.  It is
invertible over $\mathbb{Q}$ (up to a shape-dependent overall scalar) and is
the workhorse behind pieri(...,Convention=>...),/see
lrMap(...,Convention=>...), for the Filling convention.

Source/target basis labels are attached so symbolicForm,
can display the change-of-basis on each PM tableau:
+---------------------------------------------+
|i1 : symbolicForm pmToFillingMatrix({2,1}, 3)|
|                                             |
|     +-----+-------------+                   |
|     |+-+-+|      +-+-+  |                   |
|o1 = ||0|0||1  *  |0|0|  |                   |
|     ||1| ||      |1| |  |                   |
|     |+-+-+|      +-+-+  |                   |
|     +-----+-------------+                   |
|     |+-+-+|      +-+-+  |                   |
|     ||0|0||1  *  |0|0|  |                   |
|     ||2| ||      |2| |  |                   |
|     |+-+-+|      +-+-+  |                   |
|     +-----+-------------+                   |
|     |+-+-+|1     +-+-+  |                   |
|     ||0|1||-  *  |0|1|  |                   |
|     ||1| ||2     |1| |  |                   |
|     |+-+-+|      +-+-+  |                   |
|     +-----+-------------+                   |
|     |+-+-+|  1     +-+-+|                   |
|     ||0|1||- -  *  |0|2||                   |
|     ||2| ||  2     |1| ||                   |
|     |+-+-+|        +-+-+|                   |
|     |     |      +-+-+  |                   |
|     |     |1  *  |0|1|  |                   |
|     |     |      |2| |  |                   |
|     |     |      +-+-+  |                   |
|     +-----+-------------+                   |
|     |+-+-+|      +-+-+  |                   |
|     ||0|2||1  *  |0|2|  |                   |
|     ||1| ||      |1| |  |                   |
|     |+-+-+|      +-+-+  |                   |
|     |     |  1     +-+-+|                   |
|     |     |- -  *  |0|1||                   |
|     |     |  2     |2| ||                   |
|     |     |        +-+-+|                   |
|     +-----+-------------+                   |
|     |+-+-+|1     +-+-+  |                   |
|     ||0|2||-  *  |0|2|  |                   |
|     ||2| ||2     |2| |  |                   |
|     |+-+-+|      +-+-+  |                   |
|     +-----+-------------+                   |
|     |+-+-+|      +-+-+  |                   |
|     ||1|1||1  *  |1|1|  |                   |
|     ||2| ||      |2| |  |                   |
|     |+-+-+|      +-+-+  |                   |
|     +-----+-------------+                   |
|     |+-+-+|1     +-+-+  |                   |
|     ||1|2||-  *  |1|2|  |                   |
|     ||2| ||2     |2| |  |                   |
|     |+-+-+|      +-+-+  |                   |
|     +-----+-------------+                   |
+---------------------------------------------+

See also
========

  * fillingToPMMatrix -- QQ matrix of the
    Filling-to-PM change of basis on a Schur module
  * pmToFilling -- convert a PieriMaps row-form tableau to
    SchurFunctors column-form Fillings
  * symbolicForm -- print a Schur-rep matrix as a
    basis-labeled symbolic map

For the programmer
==================

The object pmToFillingMatrix, is a function closure (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:826:0.

pmToWeyl -- convert a PieriMaps row-form tableau to a SchurFunctors WeylFilling
*******************************************************************************

  * Usage:
        pmToWeyl T
  * Inputs:
      * T, a list (Macaulay2Doc), a PieriMaps row-form tableau
  * Outputs:
      * WeylFilling with the same row-data (a SchurFunctors basis element of
        $K_\lambda V$)

Description
===========

PM tableaux and WeylFillings have the same data layout (rows weakly
increasing).  This is purely a relabel: pmToWeyl wraps a list of rows as a
WeylFilling (interpreting it as a basis element of the divided-power module
$K_\lambda V$), and weylToPM, unwraps.  The conversion is
reversible.Requires the updated SchurFunctors package that exports weyl and
WeylFilling.
+----------------------------------+
|i1 : needsPackage "SchurFunctors";|
+----------------------------------+
|i2 : pmToWeyl {{0,1},{2}}         |
|                                  |
|     +-+-+                        |
|o2 = |0|1|                        |
|     |2| |                        |
|     +-+-+                        |
|                                  |
|o2 : WeylFilling                  |
+----------------------------------+
|i3 : weylToPM pmToWeyl {{0,1},{2}}|
|                                  |
|o3 = {{0, 1}, {2}}                |
|                                  |
|o3 : List                         |
+----------------------------------+

See also
========

  * weylToPM -- convert a SchurFunctors WeylFilling to a
    PieriMaps row-form tableau
  * pmToFilling -- convert a PieriMaps row-form tableau to
    SchurFunctors column-form Fillings
  * weylToFilling -- convert a WeylFilling to a
    SchurFunctors Filling (compose pmToFilling with weylToPM)

For the programmer
==================

The object pmToWeyl, is a function closure (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:749:0.

pureFree -- computes a GL(V)-equivariant map whose resolution is pure, or the reduction mod p of such a map
***********************************************************************************************************

  * Usage:
        pureFree(d, P)
        pureFree(d, P, Convention => "Row" | "Filling" | "Weyl")
  * Inputs:
      * d, a list (Macaulay2Doc), a list of degrees (increasing
        numbers)
      * P, a polynomial ring (Macaulay2Doc), a polynomial
        ring over a field $K$ in $n$ variables
  * Optional inputs (Macaulay2Doc),:
      * Convention => a string (Macaulay2Doc), default value
        "Row", , basis convention used internally by pieri.
        Defaults to "Row"; the resulting Betti table does not depend on the
        choice.
  * Outputs:
      * a matrix (Macaulay2Doc), A map whose cokernel has Betti
        diagram with degree sequence $d$ if $K$ has characteristic 0.  If $K$
        has positive characteristic $p$, then the corresponding map is
        calculated over $\mathbb{Q}$ and is lifted to a $\mathbb{Z}$-form which
        is then reduced mod $p$.

Description
===========

The function translates the data of a degree sequence $d$ for a desired pure
free resolution into the data of a Pieri map according to the formula of
Eisenbud-Fl\o ystad-Weyman and then applies the function pieri.
The Convention option is forwarded to that internal Pieri call so that the
resulting matrix is presented in the user's preferred basis convention; the
Betti table of the resulting resolution is the same in all conventions.
+-----------------------------------------------------------------------------+
|i1 : betti res coker pureFree({0,1,2,4}, QQ[a,b,c]) -- degree sequence       |
|                                                                             |
|            0 1 2 3                                                          |
|o1 = total: 3 8 6 1                                                          |
|         0: 3 8 6 .                                                          |
|         1: . . . 1                                                          |
|                                                                             |
|o1 : BettiTally                                                              |
|-----------------------------------------------------------------------------|
|{0,1,2,4}                                                                    |
+-----------------------------------------------------------------------------+
|i2 : betti res coker pureFree({0,1,2,4}, ZZ/2[a,b,c]) -- same map, but       |
|                                                                             |
|            0 1 2 3                                                          |
|o2 = total: 3 8 6 1                                                          |
|         0: 3 8 6 .                                                          |
|         1: . . . 1                                                          |
|                                                                             |
|o2 : BettiTally                                                              |
|-----------------------------------------------------------------------------|
|reduced mod 2                                                                |
+-----------------------------------------------------------------------------+
|i3 : betti res coker pureFree({0,1,2,4}, GF(4)[a,b,c]) -- can also use non   |
|                                                                             |
|            0 1 2 3                                                          |
|o3 = total: 3 8 6 1                                                          |
|         0: 3 8 6 .                                                          |
|         1: . . . 1                                                          |
|                                                                             |
|o3 : BettiTally                                                              |
|-----------------------------------------------------------------------------|
|prime fields                                                                 |
+-----------------------------------------------------------------------------+
+-----------------------------------------------------------------------------+
|i4 : -- The Betti table is independent of the chosen convention:             |
|     P = QQ[a,b,c];                                                          |
+-----------------------------------------------------------------------------+
|i5 : bettiR = betti res coker pureFree({0,1,2,4}, P, Convention => "Row");   |
+-----------------------------------------------------------------------------+
|i6 : bettiF = betti res coker pureFree({0,1,2,4}, P, Convention =>           |
|-----------------------------------------------------------------------------|
|"Filling");                                                                  |
+-----------------------------------------------------------------------------+
|i7 : bettiR == bettiF                                                        |
|                                                                             |
|o7 = true                                                                    |
+-----------------------------------------------------------------------------+
|i8 :                                                                         |
+-----------------------------------------------------------------------------+

See also
========

  * pieri -- computes a matrix representation for a Pieri
    inclusion of representations of a general linear group

Ways to use pureFree:
=====================

  * "pureFree(List,PolynomialRing)"

For the programmer
==================

The object pureFree, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:420:0.

schurRank -- computes the dimension of the irreducible GL(QQ^n) representation associated to a partition
********************************************************************************************************

  * Usage:
        schurRank(n, mu)
  * Inputs:
      * n, an integer (Macaulay2Doc), the size of the matrix group
        $GL(\mathbb{Q}^n)$
      * mu, a list (Macaulay2Doc), a partition $(\mu_1, \ldots,
        \mu_r)$ where $\mu_i$ is the number of boxes in the $i$th row
  * Outputs:
      * an integer (Macaulay2Doc), The dimension of the irreducible
        $GL(\mathbb{Q}^n)$ representation associated to $\mu$

Description
===========

The dimension is computed using the determinantal formula given by the Weyl
character formula.
+-----------------------------------------+
|i1 : schurRank(5, {4,3}) -- should be 560|
|                                         |
|o1 = 560                                 |
+-----------------------------------------+

See also
========

  * standardTableaux -- list all standard tableaux of
    a certain shape with bounded labels

Ways to use schurRank:
======================

  * "schurRank(ZZ,List)"

For the programmer
==================

The object schurRank, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:443:0.

standardTableaux -- list all standard tableaux of a certain shape with bounded labels
*************************************************************************************

  * Usage:
        standardTableaux(dim, mu)
  * Inputs:
      * dim, an integer (Macaulay2Doc), number of labels to be used
      * mu, a list (Macaulay2Doc), a partition which gives the
        shape
  * Outputs:
      * a list (Macaulay2Doc), list of all standard tableaux of
        shape $\mu$ with labels $0, \ldots, \mathrm{dim}-1$

Description
===========

+-----------------------------------------------------------------------------+
|i1 : standardTableaux(3, {2,2}) -- lists all standard tableaux on the 2x2 squ|
|                                                                             |
|o1 = {{{0, 0}, {1, 1}}, {{0, 0}, {1, 2}}, {{0, 0}, {2, 2}}, {{0, 1}, {1, 2}},|
|     ------------------------------------------------------------------------|
|     {{0, 1}, {2, 2}}, {{1, 1}, {2, 2}}}                                     |
|                                                                             |
|o1 : List                                                                    |
|-----------------------------------------------------------------------------|
|are with entries 0,1,2                                                       |
+-----------------------------------------------------------------------------+

See also
========

  * schurRank -- computes the dimension of the irreducible
    GL(QQ^n) representation associated to a partition

Ways to use standardTableaux:
=============================

  * "standardTableaux(ZZ,List)"

For the programmer
==================

The object standardTableaux, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:232:0.

straighten -- computes straightening of a tableau
*************************************************

  * Usage:
        straighten(t)straighten(t, h)
  * Inputs:
      * t, a list (Macaulay2Doc), a tableau to straighten; a
        tableau looks like {{3,4}, {1,2}} for example, where we list the
        entries from left to right, top to bottom
      * h, a mutable hash table (Macaulay2Doc), where
        the answers should be stored
  * Consequences:
      * If provided, the hashtable h is updated with any calculations which are
        performed as a result of calling this function.

Description
===========

If a hashtable h is provided, then this outputs nothing, it simply just
modifies h. When looking up values, remember that the keys are stored with rows
weakly increasing. If no hashtable is provided, then the user is simply given
the straightening of the tableau in terms of semistandard tableaux. The answer
is in the form a hashtable: each key is a semistandard tableaux, and the value
of the key is the coefficient of that semistandard tableaux used to write the
input t as a linear combination.
+--------------------------------------------------------------+
|i1 : h = new MutableHashTable from {}                         |
|                                                              |
|o1 = MutableHashTable{}                                       |
|                                                              |
|o1 : MutableHashTable                                         |
+--------------------------------------------------------------+
|i2 : straighten({{3,4}, {1,2}}, h)                            |
+--------------------------------------------------------------+
|i3 : h#{{3,4}, {1,2}} -- get the coefficients                 |
|                                                              |
|o3 = HashTable{{{1, 2}, {3, 4}} => 1}                         |
|                                                              |
|o3 : HashTable                                                |
+--------------------------------------------------------------+
|i4 : straighten({{3,4}, {1,2}}) -- just get the answer instead|
|                                                              |
|o4 = HashTable{{{1, 2}, {3, 4}} => 1}                         |
|                                                              |
|o4 : HashTable                                                |
+--------------------------------------------------------------+

Ways to use straighten:
=======================

  * "straighten(List)"
  * "straighten(List,MutableHashTable)"

For the programmer
==================

The object straighten, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:252:0.

symbolicForm -- print a Schur-rep matrix as a basis-labeled symbolic map
************************************************************************

  * Usage:
        symbolicForm M
  * Inputs:
      * M, a matrix (Macaulay2Doc), typically the output of
        pieri, pieriColumn, lrMap,
        dualPieri, dualPieriColumn,
        or dualLR, (which attach source/target basis labels to
        M.cache).  Plain matrices are accepted too: integer indices 1..rank are
        used as labels.
  * Outputs:
      * a net (Macaulay2Doc), a netList; each row shows a source
        basis element i and the nonzero terms of M(i) as a list of
        (target_label, coefficient) pairs

Description
===========

Pretty-prints the matrix as the GL-equivariant map it represents on the labeled
bases.  Each row of the netList corresponds to a source basis element (a
tableau, or a (tableau, monomial) pair, etc.) and lists the image as a sum over
target basis elements with their coefficients.  Rows where the image is
identically zero appear with an empty list.

After symbolicForm M is called once, the rule is cached on M.cache#"rule" so
subsequent calls reuse it.  For interactive exploration, you can also evaluate
the rule on a single basis element directly via (M.cache#"rule")(label).
+------------------------------+
|i1 : P = QQ[a,b,c];           |
+------------------------------+
|i2 : M = pieri({2,1}, {1}, P);|
|                              |
|             3      8         |
|o2 : Matrix P  <-- P          |
+------------------------------+
|i3 : symbolicForm M           |
|                              |
|     +-----+----------+       |
|     |+-+-+|       +-+|       |
|o3 = ||0|0||2a  *  |0||       |
|     ||1| ||       |1||       |
|     |+-+-+|       +-+|       |
|     +-----+----------+       |
|     |+-+-+|       +-+|       |
|     ||0|0||2a  *  |0||       |
|     ||2| ||       |2||       |
|     |+-+-+|       +-+|       |
|     +-----+----------+       |
|     |+-+-+|      +-+ |       |
|     ||0|1||b  *  |0| |       |
|     ||1| ||      |1| |       |
|     |+-+-+|      +-+ |       |
|     +-----+----------+       |
|     |+-+-+|      +-+ |       |
|     ||0|1||b  *  |0| |       |
|     ||2| ||      |2| |       |
|     |+-+-+|      +-+ |       |
|     |     |      +-+ |       |
|     |     |a  *  |1| |       |
|     |     |      |2| |       |
|     |     |      +-+ |       |
|     +-----+----------+       |
|     |+-+-+|      +-+ |       |
|     ||0|2||c  *  |0| |       |
|     ||1| ||      |1| |       |
|     |+-+-+|      +-+ |       |
|     |     |       +-+|       |
|     |     |-a  *  |1||       |
|     |     |       |2||       |
|     |     |       +-+|       |
|     +-----+----------+       |
|     |+-+-+|      +-+ |       |
|     ||0|2||c  *  |0| |       |
|     ||2| ||      |2| |       |
|     |+-+-+|      +-+ |       |
|     +-----+----------+       |
|     |+-+-+|       +-+|       |
|     ||1|1||2b  *  |1||       |
|     ||2| ||       |2||       |
|     |+-+-+|       +-+|       |
|     +-----+----------+       |
|     |+-+-+|      +-+ |       |
|     ||1|2||c  *  |1| |       |
|     ||2| ||      |2| |       |
|     |+-+-+|      +-+ |       |
|     +-----+----------+       |
+------------------------------+
+----------------------------------+
|i4 : shapes = ({2,1}, {1}, {1,1});|
+----------------------------------+
|i5 : Q = (lrTableaux shapes)#0;   |
+----------------------------------+
|i6 : N = lrMap(shapes, Q, 3);     |
|                                  |
|              9       8           |
|o6 : Matrix QQ  <-- QQ            |
+----------------------------------+
|i7 : symbolicForm N               |
|                                  |
|     +-----+-------------------+  |
|     |+-+-+|      +-+     +-+  |  |
|o7 = ||0|0||3  *  |0|  ⊗  |0|  |  |
|     ||1| ||      |1|     +-+  |  |
|     |+-+-+|      +-+          |  |
|     +-----+-------------------+  |
|     |+-+-+|      +-+     +-+  |  |
|     ||0|0||3  *  |0|  ⊗  |0|  |  |
|     ||2| ||      |2|     +-+  |  |
|     |+-+-+|      +-+          |  |
|     +-----+-------------------+  |
|     |+-+-+|3     +-+     +-+  |  |
|     ||0|1||-  *  |0|  ⊗  |1|  |  |
|     ||1| ||2     |1|     +-+  |  |
|     |+-+-+|      +-+          |  |
|     +-----+-------------------+  |
|     |+-+-+|3     +-+     +-+  |  |
|     ||0|1||-  *  |0|  ⊗  |1|  |  |
|     ||2| ||2     |2|     +-+  |  |
|     |+-+-+|      +-+          |  |
|     |     |3     +-+     +-+  |  |
|     |     |-  *  |1|  ⊗  |0|  |  |
|     |     |2     |2|     +-+  |  |
|     |     |      +-+          |  |
|     +-----+-------------------+  |
|     |+-+-+|3     +-+     +-+  |  |
|     ||0|2||-  *  |0|  ⊗  |2|  |  |
|     ||1| ||2     |1|     +-+  |  |
|     |+-+-+|      +-+          |  |
|     |     |  3     +-+     +-+|  |
|     |     |- -  *  |1|  ⊗  |0||  |
|     |     |  2     |2|     +-+|  |
|     |     |        +-+        |  |
|     +-----+-------------------+  |
|     |+-+-+|3     +-+     +-+  |  |
|     ||0|2||-  *  |0|  ⊗  |2|  |  |
|     ||2| ||2     |2|     +-+  |  |
|     |+-+-+|      +-+          |  |
|     +-----+-------------------+  |
|     |+-+-+|      +-+     +-+  |  |
|     ||1|1||3  *  |1|  ⊗  |1|  |  |
|     ||2| ||      |2|     +-+  |  |
|     |+-+-+|      +-+          |  |
|     +-----+-------------------+  |
|     |+-+-+|3     +-+     +-+  |  |
|     ||1|2||-  *  |1|  ⊗  |2|  |  |
|     ||2| ||2     |2|     +-+  |  |
|     |+-+-+|      +-+          |  |
|     +-----+-------------------+  |
+----------------------------------+

See also
========

  * pieri -- computes a matrix representation for a Pieri
    inclusion of representations of a general linear group
  * lrMap -- GL(V)-equivariant Littlewood-Richardson inclusion
  * dualPieri -- GL-equivariant projection Sym^d V ⊗ S_lambda
    V --> S_mu V
  * dualLR -- GL-equivariant projection S_nu V ⊗ S_mu V -->
    S_lambda V at a chosen LR tableau
  * applyLR -- apply Psi_Q to a single tableau of shape lambda
  * applyPieri -- apply pieri to a single source tableau
    T_mu

Ways to use symbolicForm:
=========================

  * "symbolicForm(Matrix)"

For the programmer
==================

The object symbolicForm, is a method function (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:1432:0.

verifyEquivariant -- rigorously verify GL_n-equivariance of a Schur-rep matrix
******************************************************************************

  * Usage:
        verifyEquivariant(M, mu, boxes, P)               -- pieri-type inclusion
        verifyEquivariant(M, shapes, n)                  -- lrMap-type inclusion
        verifyEquivariant(M, mu, boxes, P, Direction => "Dual")  -- dualPieri-type projection
        verifyEquivariant(M, shapes, n, Direction => "Dual")     -- dualLR-type projection
  * Inputs:
      * M, a matrix (Macaulay2Doc), the matrix to test for
        $GL_n$-equivariance
  * Optional inputs (Macaulay2Doc),:
      * Convention => a string (Macaulay2Doc), default value
        "Row", , basis convention used by M's rows and columns.  Default "Row".
      * Verbose => a Boolean value (Macaulay2Doc), default value
        false, , if true print the first failing $(i, j, T)$ when the test
        fails.  Default false.
      * Direction => a string (Macaulay2Doc), default value
        "Inclusion", , "Inclusion" (default) or "Dual" to indicate the
        direction of the map.
  * Outputs:
      * a Boolean value (Macaulay2Doc), true if M commutes with
        every Chevalley generator $E_{i,j}$ of $\mathfrak{gl}_n$ ($i \neq j$);
        false otherwise

Description
===========

Tests whether a matrix M between Schur-functor representations is genuinely
$GL_n$-equivariant by checking that $M \circ \rho_{\mathrm{src}}(E_{i,j}) =
\rho_{\mathrm{tgt}}(E_{i,j}) \circ M$ for every Chevalley generator $E_{i,j}$.
Since the $E_{i,j}$ ($i \neq j$) generate the strictly upper- and strictly
lower-triangular parts of $\mathfrak{gl}_n$, and our matrices commute with the
diagonal automatically (entries are weight-homogeneous), commutativity with all
$E_{i,j}$ is equivalent to full $GL_n$-equivariance.

Action conventions: $E_{i,j}$ acts on $V = K^n$ by $x_l \mapsto \delta_{l,j}\,
x_i$ (i.e. the differential operator $x_i\, \partial/\partial x_j$ on
polynomials), on a Schur module $S_\lambda V$ by replacing each entry equal to
$j$ with $i$ in a tableau (summed over positions, then straightened), and on a
tensor product by Leibniz.

Four overloads cover the maps in this package:
  * verifyEquivariant(M, mu, boxes, P) -- $M \colon S_\mu V \to \mathrm{Sym}^d
    V \otimes S_\lambda V$ (or $\wedge^d V$ for skew-commutative P), the
    inclusion produced by pieri, or pieriColumn.
  * verifyEquivariant(M, (lambda, mu, nu), n) -- $M \colon S_\lambda V \to
    S_\nu V \otimes S_\mu V$, the inclusion produced by lrMap.
  * verifyEquivariant(M, mu, boxes, P, Direction => "Dual") -- $M \colon
    \mathrm{Sym}^d V \otimes S_\lambda V \to S_\mu V$, the projection produced
    by dualPieri.
  * verifyEquivariant(M, (lambda, mu, nu), n, Direction => "Dual") -- $M \colon
    S_\nu V \otimes S_\mu V \to S_\lambda V$, the projection produced by
    dualLR.
+------------------------------------------+
|i1 : P = QQ[a,b,c];                       |
+------------------------------------------+
|i2 : M = pieri({3,2,1}, {2}, P);          |
|                                          |
|             6      8                     |
|o2 : Matrix P  <-- P                      |
+------------------------------------------+
|i3 : verifyEquivariant(M, {3,2,1}, {2}, P)|
|                                          |
|o3 = true                                 |
+------------------------------------------+
+---------------------------------------------------------------+
|i4 : P = QQ[a,b,c];   -- verify the dual Pieri projection      |
+---------------------------------------------------------------+
|i5 : N = dualPieri({3,2,1}, {2}, P);                           |
|                                                               |
|              8       18                                       |
|o5 : Matrix QQ  <-- QQ                                         |
+---------------------------------------------------------------+
|i6 : verifyEquivariant(N, {3,2,1}, {2}, P, Direction => "Dual")|
|                                                               |
|o6 = true                                                      |
+---------------------------------------------------------------+
+-----------------------------------------------------------------------------+
|i7 : Q = (lrTableaux({3,2,1}, {2,1}, {2,1}))#0;  -- multiplicity-2 LR        |
|-----------------------------------------------------------------------------|
|inclusion                                                                    |
+-----------------------------------------------------------------------------+
|i8 : M = lrMap(({3,2,1}, {2,1}, {2,1}), Q, 3);                               |
|                                                                             |
|              64       8                                                     |
|o8 : Matrix QQ   <-- QQ                                                      |
+-----------------------------------------------------------------------------+
|i9 : verifyEquivariant(M, ({3,2,1}, {2,1}, {2,1}), 3)                        |
|                                                                             |
|o9 = true                                                                    |
+-----------------------------------------------------------------------------+

See also
========

  * pieri -- computes a matrix representation for a Pieri
    inclusion of representations of a general linear group
  * lrMap -- GL(V)-equivariant Littlewood-Richardson inclusion
  * dualPieri -- GL-equivariant projection Sym^d V ⊗ S_lambda
    V --> S_mu V
  * dualLR -- GL-equivariant projection S_nu V ⊗ S_mu V -->
    S_lambda V at a chosen LR tableau
  * verifyWellDefined -- verify an LR map respects
    the chosen convention's straightening relations

Ways to use verifyEquivariant:
==============================

  * "verifyEquivariant(Matrix,List,List,PolynomialRing)"
  * "verifyEquivariant(Matrix,Sequence,ZZ)"

For the programmer
==================

The object verifyEquivariant, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:1248:0.

verifyWellDefined -- verify an LR map respects the chosen convention's straightening relations
**********************************************************************************************

  * Usage:
        verifyWellDefined((lambda, mu, nu), Q, n, Convention => ...)
  * Inputs:
      * (lambda, mu, nu), a sequence (Macaulay2Doc), three
        partitions for the LR data
      * Q, a list (Macaulay2Doc), an LR tableau as returned by
        lrTableaux,
      * n, an integer (Macaulay2Doc), the dimension of $V$
  * Optional inputs (Macaulay2Doc),:
      * Convention => a string (Macaulay2Doc), default value
        "Row", , basis convention to check
      * Verbose => a Boolean value (Macaulay2Doc), default value
        true, , whether to print per-test progress (default true)
      * Direction => ..., default value "Inclusion"
  * Outputs:
      * a Boolean value (Macaulay2Doc), true if all generated
        tests pass, false otherwise

Description
===========

For skeptical users.  Generates a few non-standard tableaux of shape lambda in
the chosen basis convention, then for each:
  * (a) computes
  * applyLR(shapes, Q, T, n, Convention => conv)
  *  directly (which uses the convention's straightening internally),
  * (b) independently straightens T into a sum of standard tableaux,
  * (c) computes the same sum applied to the standards,
  * (d) asserts (a) and (c) agree.
If the matrix were silently using the wrong straightening relations (e.g.
PieriMaps row-Garnir on a SchurFunctors column-form Filling input), the check
fails and prints the offending case.
+-----------------------------------------------------------------------------+
|i1 : needsPackage "SchurFunctors";                                           |
+-----------------------------------------------------------------------------+
|i2 : Qs = lrTableaux({3,2,1}, {2,1}, {2,1});                                 |
+-----------------------------------------------------------------------------+
|i3 : verifyWellDefined(({3,2,1},{2,1},{2,1}), Qs#0, 3, Convention =>         |
|  Test #1 (T = Filling{{1, 0, 2}, {0, 1}, {0}}): OK                          |
|  Test #2 (T = Filling{{0, 1, 2}, {1, 0}, {0}}): OK                          |
|  Test #3 (T = Filling{{1, 0, 2}, {0, 1}, {1}}): OK                          |
|verifyWellDefined: 3 test(s) all PASSED for convention Filling.              |
|                                                                             |
|o3 = true                                                                    |
|-----------------------------------------------------------------------------|
|"Filling")                                                                   |
+-----------------------------------------------------------------------------+
|i4 : verifyWellDefined(({3,2,1},{2,1},{2,1}), Qs#0, 3, Convention => "Row")  |
|  Test #1 (T = {{0, 0, 0}, {2, 1}, {2}}): OK                                 |
|  Test #2 (T = {{0, 0, 1}, {2, 1}, {2}}): OK                                 |
|  Test #3 (T = {{0, 0, 2}, {2, 1}, {2}}): OK                                 |
|verifyWellDefined: 3 test(s) all PASSED for convention Row.                  |
|                                                                             |
|o4 = true                                                                    |
+-----------------------------------------------------------------------------+

See also
========

  * lrMap -- GL(V)-equivariant Littlewood-Richardson inclusion
  * applyLR -- apply Psi_Q to a single tableau of shape lambda
  * pmToFilling -- convert a PieriMaps row-form tableau to
    SchurFunctors column-form Fillings
  * Convention -- tableau basis convention used by
    PieriMaps' overhauled functions

Ways to use verifyWellDefined:
==============================

  * "verifyWellDefined(List,List,PolynomialRing)"
  * "verifyWellDefined(Sequence,List,ZZ)"

For the programmer
==================

The object verifyWellDefined, is a method function with options (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:931:0.

weylToFilling -- convert a WeylFilling to a SchurFunctors Filling (compose pmToFilling with weylToPM)
*****************************************************************************************************

  * Usage:
        weylToFilling W
  * Inputs:
      * W, WeylFilling
  * Outputs:
      * a hash table (Macaulay2Doc), image as a sum of
        standard Fillings after straightening

Description
===========

+----------------------------------+
|i1 : needsPackage "SchurFunctors";|
+----------------------------------+
|i2 : W = pmToWeyl {{0,1},{2}};    |
+----------------------------------+
|i3 : weylToFilling W              |
|                                  |
|               +-+-+      1       |
|o3 = HashTable{|0|2| => - -}      |
|               |1| |      2       |
|               +-+-+              |
|               +-+-+              |
|               |0|1| => 1         |
|               |2| |              |
|               +-+-+              |
|                                  |
|o3 : HashTable                    |
+----------------------------------+

See also
========

  * pmToFilling -- convert a PieriMaps row-form tableau to
    SchurFunctors column-form Fillings
  * weylToPM -- convert a SchurFunctors WeylFilling to a
    PieriMaps row-form tableau
  * fillingToWeyl -- convert a Filling to a sum of
    WeylFillings (compose pmToWeyl with fillingToPM)

For the programmer
==================

The object weylToFilling, is a function closure (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:778:0.

weylToPM -- convert a SchurFunctors WeylFilling to a PieriMaps row-form tableau
*******************************************************************************

  * Usage:
        weylToPM W
  * Inputs:
      * W, WeylFilling
  * Outputs:
      * a list (Macaulay2Doc), a list of rows (PM-form tableau)
        with the same data

Description
===========

Inverse of pmToWeyl.  Both WeylFilling and PM-form share the
row-of-multisets data layout, so this is purely an unwrapping.
+----------------------------------+
|i1 : needsPackage "SchurFunctors";|
+----------------------------------+
|i2 : W = pmToWeyl {{0,1},{2}}     |
|                                  |
|     +-+-+                        |
|o2 = |0|1|                        |
|     |2| |                        |
|     +-+-+                        |
|                                  |
|o2 : WeylFilling                  |
+----------------------------------+
|i3 : weylToPM W                   |
|                                  |
|o3 = {{0, 1}, {2}}                |
|                                  |
|o3 : List                         |
+----------------------------------+

See also
========

  * pmToWeyl -- convert a PieriMaps row-form tableau to a
    SchurFunctors WeylFilling
  * fillingToWeyl -- convert a Filling to a sum of
    WeylFillings (compose pmToWeyl with fillingToPM)

For the programmer
==================

The object weylToPM, is a function closure (Macaulay2Doc).

-------------------------------------------------------------------------------

The source of this document is in PieriMaps/doc.m2:764:0.