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

SchurVeronese -- syzygy data for Veronese embeddings of P^1 and P^2
*******************************************************************

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

The authors of this package used a combination of high-throughput and
high-performance computing and sparse numerical linear algebra to compute the
syzygies of $\mathbb{P}^{2}$ under the $d$-fold Veronese embedding for various
values of $d$. See the paper ``Conjectures and Computations about Veronese
Syzygies'' by Bruce, Erman, Goldstein and Yang, which we refer to as [BEGY]
(see arXiv:1711.03513 (see http://arxiv.org/abs/1711.03513 )) throughout the
documentation for this package. In addition, much of the data generated from
these computations (graded Betti numbers, multigraded Betti numbers, Schur
functor decompositions, etc.) is currently available online via syzygydata.com.
The goal of this package is to make this data more accessible and easy to use
by providing a way to access it via Macaulay2.

Most functions have been implemented with three parameters $(d,n,b)$, where the
goal is to compute the syzygies of the pushforward of the line bundle
$\mathcal{O}(b)$ under the $d$-fold embedding.  However, we have produced data
for $n=1$ and $n=2$, for $b$ between $0$ and $d$ and for a limited range of
values of $d$.  Other inputs will produce an error message.  Our hope is that
as we (or others) are able to compute new data, we will be able to update the
package.

One of the main functions is totalBettiTally, which
produces the standard graded Betti table of the corresponding standard graded
Veronese module. Other main functions refine the data in the Betti table by
providing the multigraded Betti number or the Schur functor decompositions, or
by computing statistics related to the Betti table (e.g., the BoijSoederberg
coefficients) or related to the SchurFunctor decomposition.

A number of functions in this package produce individual entries of a Betti
table. There are two common notations for referring to Betti numbers in the
literature, and it will be useful to reference these notations throughout the
documentation, similar to how they are referenced in the corresponding paper.
For a graded module $M$ we will write $\beta_{i,j}(M)$ for $\dim Tor_i(M,k)_j$
and we write $K_{p,q}(M)$ for the vector space $Tor_p(M,k)_{p+q}$.

Authors
=======

  * Juliette Bruce (see https://juliettebruce.github.io ) <jebruce2@wisc.edu>
  * Daniel Erman (see http://www.math.wisc.edu/~derman/ )
    <derman@math.wisc.edu>
  * Steve Goldstein (see  ) <sgoldstein@wisc.edu>
  * Jay Yang (see http://www-users.math.umn.edu/~jkyang/ ) <jkyang@umn.edu>

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

Version 1.1 of this package was accepted for publication in volume 11 (see
https://msp.org/jsag/2021/11-1/ ) of The Journal of Software for Algebra and
Geometry (see https://msp.org/jsag/ ) on 5 May 2021, in the article The
Schur–Veronese package in Macaulay2 (see
https://msp.org/jsag/2021/11-1/p09.xhtml ) (DOI: 10.2140/jsag.2021.11.83 (see
https://doi.org/10.2140/jsag.2021.11.83 )). That version can be obtained from
the journal (see https://msp.org/jsag/2021/11-1/jsag-v11-n1-x09-SchurVeronese.
zip ).

Version
=======

This documentation describes version 1.1 of SchurVeronese, released 13 May
2019.

Citation
========

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

+-----------------------------------------------------------------------------+
|@misc{SchurVeroneseSource,                                                   |
|  title = {{SchurVeronese: Data for Veronese embeddings of projective space. |
|  author = {Juliette Bruce and Daniel Erman and Steve Goldstein and Jay Yang |
|  howpublished = {A \emph{Macaulay2} package available at                    |
|    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}} |
|}                                                                            |
|                                                                             |
|@article{SchurVeroneseArticle,                                               |
|  title = {{The Schur–Veronese package in \emph{Macaulay2}}},                |
|  author = {Juliette Bruce and Daniel Erman and Steve Goldstein and Jay Yang |
|  journal = {The Journal of Software for Algebra and Geometry},              |
|  volume = {11},                                                             |
|  year = {2021},                                                             |
|}                                                                            |
|-----------------------------------------------------------------------------|
| Version~1.1}},                                                              |
|},                                                                           |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|},                                                                           |
+-----------------------------------------------------------------------------+

Exports
=======

  * Functions and commands
      * bsCoeffs -- a list of the Boij-Soederberg coefficients
        of a Veronese embedding
      * dominantWeightsBetti -- a hash table
        containing the dominant Schur functors of a Veronese embedding
      * lexWeightsBetti -- a hash table containing the
        lex-leading weight Schur functors of a Veronese embedding
      * makeBettiTally -- converts a hash table
        representing a Betti table to a Betti tally
      * multiBetti -- a hash table containing the
        multigraded Betti numbers of a Veronese embedding
      * numDistinctRepsBetti -- a hash table
        containing the number of distinct Schur functors of a Veronese
        embedding
      * numRepsBetti -- a hash table containing the number
        of Schur functors of a Veronese embedding
      * schurBetti -- a hash table for Schur module
        decomposition of Veronese Betti tables
      * totalBetti -- a hash table containing the graded
        Betti numbers of a Veronese embedding
      * totalBettiTally -- a Betti tally containing the
        graded Betti numbers of a Veronese embedding
  * Methods
      * "bsCoeffs(ZZ,ZZ,ZZ)" -- see bsCoeffs, -- a list of the
        Boij-Soederberg coefficients of a Veronese embedding
      * "dominantWeightsBetti(ZZ,ZZ,ZZ)" -- see dominantWeightsBetti,
        -- a hash table containing the dominant Schur
        functors of a Veronese embedding
      * "lexWeightsBetti(ZZ,ZZ,ZZ)" -- see lexWeightsBetti,
        -- a hash table containing the lex-leading weight
        Schur functors of a Veronese embedding
      * "makeBettiTally(HashTable)" -- see makeBettiTally,
        -- converts a hash table representing a Betti table to
        a Betti tally
      * "multiBetti(ZZ,ZZ,ZZ)" -- see multiBetti, -- a hash
        table containing the multigraded Betti numbers of a Veronese embedding
      * "numDistinctRepsBetti(ZZ,ZZ,ZZ)" -- see numDistinctRepsBetti,
        -- a hash table containing the number of distinct
        Schur functors of a Veronese embedding
      * "numRepsBetti(ZZ,ZZ,ZZ)" -- see numRepsBetti, -- a
        hash table containing the number of Schur functors of a Veronese
        embedding
      * "schurBetti(ZZ,ZZ,ZZ)" -- see schurBetti, -- a hash
        table for Schur module decomposition of Veronese Betti tables
      * "totalBetti(ZZ,ZZ,ZZ)" -- see totalBetti, -- a hash
        table containing the graded Betti numbers of a Veronese embedding
      * "totalBettiTally(ZZ,ZZ,ZZ)" -- see totalBettiTally,
        -- a Betti tally containing the graded Betti numbers
        of a Veronese embedding

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

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

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

The source of this document is in SchurVeronese/doc.m2:42:0.

bsCoeffs -- a list of the Boij-Soederberg coefficients of a Veronese embedding
******************************************************************************

  * Usage:
        bsCoeffs(d,n,b)
  * Inputs:
      * d, an integer (Macaulay2Doc),
      * n, an integer (Macaulay2Doc),
      * b, an integer (Macaulay2Doc),
  * Outputs:
      * a list (Macaulay2Doc),

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

This function returns a list of the Boij-Soederberg coefficients for the
decomposition of the Betti table of $\mathcal{O}(b)$ on $\mathbb{P}^{n}$ under
the $d$-fold Veronese embedding. See Section 6.3 of [BEGY].  Of course, these
coefficients and the corresponding pure diagrams are easily computed by
applying decompose(BettiTally) (BoijSoederberg),
from the BoijSoederberg package to
the corresponding totalBettiTally.

+-----------------------------------------------------------------+
|i1 : totalBettiTally(3,2,0)                                      |
|                                                                 |
|            0  1   2   3   4   5  6 7                            |
|o1 = total: 1 27 105 189 189 105 27 1                            |
|         0: 1  .   .   .   .   .  . .                            |
|         1: . 27 105 189 189 105 27 .                            |
|         2: .  .   .   .   .   .  . 1                            |
|                                                                 |
|o1 : BettiTally                                                  |
+-----------------------------------------------------------------+
|i2 : bsCoeffs(3,2,0)                                             |
|                                                                 |
|o2 = {45360}                                                     |
|                                                                 |
|o2 : List                                                        |
+-----------------------------------------------------------------+
|i3 : totalBettiTally(4,2,2)                                      |
|                                                                 |
|            0  1   2   3    4    5    6    7    8    9  10  11 12|
|o3 = total: 6 62 276 715 1275 2628 4488 4950 3630 1804 588 114 10|
|         0: 6 62 276 660  825  252    .    .    .    .   .   .  .|
|         1: .  .   .  55  450 2376 4488 4950 3630 1804 588 114 10|
|         2: .  .   .   .    .    .    .    .    .    .   .   .  .|
|                                                                 |
|o3 : BettiTally                                                  |
+-----------------------------------------------------------------+
|i4 : bsCoeffs(4,2,2)                                             |
|                                                                 |
|o4 = {1219276800, 4746470400, 1219276800, 479001600}             |
|                                                                 |
|o4 : List                                                        |
+-----------------------------------------------------------------+

Ways to use bsCoeffs:
=====================

  * "bsCoeffs(ZZ,ZZ,ZZ)"

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

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

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

The source of this document is in SchurVeronese/doc.m2:373:0.

dominantWeightsBetti -- a hash table containing the dominant Schur functors of a Veronese embedding
***************************************************************************************************

  * Usage:
        dominantWeightsBetti(d,n,b)
  * Inputs:
      * d, an integer (Macaulay2Doc),
      * n, an integer (Macaulay2Doc),
      * b, an integer (Macaulay2Doc),
  * Outputs:
      * a hash table (Macaulay2Doc),

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

This function returns a hash table $H$ whose keys are pairs $(p,q)$ such that
the corresponding value $H#(p,q)$ is a list of the dominant weights appearing
in the Schur functor decomposition of $K_{p,q}(\mathbb{P}^n, d;b)$. The Schur
functors are recorded via the corresponding partitions. See Section 1.3 of
[BEGY].

+-------------------------------------+
|i1 : totalBettiTally(3,2,0)          |
|                                     |
|            0  1   2   3   4   5  6 7|
|o1 = total: 1 27 105 189 189 105 27 1|
|         0: 1  .   .   .   .   .  . .|
|         1: . 27 105 189 189 105 27 .|
|         2: .  .   .   .   .   .  . 1|
|                                     |
|o1 : BettiTally                      |
+-------------------------------------+
|i2 : D = dominantWeightsBetti(3,2,0);|
+-------------------------------------+
|i3 : D#(1,1)                         |
|                                     |
|o3 = {{4, 2, 0}}                     |
|                                     |
|o3 : List                            |
+-------------------------------------+
|i4 : D#(5,1)                         |
|                                     |
|o4 = {{9, 5, 4}, {8, 7, 3}}          |
|                                     |
|o4 : List                            |
+-------------------------------------+
|i5 : D#(7,2)                         |
|                                     |
|o5 = {{9, 9, 9}}                     |
|                                     |
|o5 : List                            |
+-------------------------------------+

Ways to use dominantWeightsBetti:
=================================

  * "dominantWeightsBetti(ZZ,ZZ,ZZ)"

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

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

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

The source of this document is in SchurVeronese/doc.m2:242:0.

lexWeightsBetti -- a hash table containing the lex-leading weight Schur functors of a Veronese embedding
********************************************************************************************************

  * Usage:
        lexWeightsBetti(d,n,b)
  * Inputs:
      * d, an integer (Macaulay2Doc),
      * n, an integer (Macaulay2Doc),
      * b, an integer (Macaulay2Doc),
  * Outputs:
      * a hash table (Macaulay2Doc),

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

This function returns a hash table $H$ whose keys are pairs $(p,q)$ such that
the corresponding value $H#(p,q)$ is a list of the lexicographically maximal
weights of the Schur functors appearing in the Schur functor decomposition of
$K_{p,q}(\mathbb{P}^n, d;b)$. See Section 1.3 of [BEGY].

+-------------------------------------+
|i1 : totalBettiTally(3,2,0)          |
|                                     |
|            0  1   2   3   4   5  6 7|
|o1 = total: 1 27 105 189 189 105 27 1|
|         0: 1  .   .   .   .   .  . .|
|         1: . 27 105 189 189 105 27 .|
|         2: .  .   .   .   .   .  . 1|
|                                     |
|o1 : BettiTally                      |
+-------------------------------------+
|i2 : lexWeightsBetti(3,2,0)          |
|                                     |
|o2 = HashTable{(0, 0) => {0, 0, 0}}  |
|               (0, 1) => {}          |
|               (0, 2) => {}          |
|               (1, 0) => {}          |
|               (1, 1) => {4, 2, 0}   |
|               (1, 2) => {}          |
|               (2, 0) => {}          |
|               (2, 1) => {6, 2, 1}   |
|               (2, 2) => {}          |
|               (3, 0) => {}          |
|               (3, 1) => {7, 4, 1}   |
|               (3, 2) => {}          |
|               (4, 0) => {}          |
|               (4, 1) => {8, 5, 2}   |
|               (4, 2) => {}          |
|               (5, 0) => {}          |
|               (5, 1) => {9, 5, 4}   |
|               (5, 2) => {}          |
|               (6, 0) => {}          |
|               (6, 1) => {9, 7, 5}   |
|               (6, 2) => {}          |
|               (7, 0) => {}          |
|               (7, 1) => {}          |
|               (7, 2) => {9, 9, 9}   |
|                                     |
|o2 : HashTable                       |
+-------------------------------------+

Ways to use lexWeightsBetti:
============================

  * "lexWeightsBetti(ZZ,ZZ,ZZ)"

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

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

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

The source of this document is in SchurVeronese/doc.m2:270:0.

makeBettiTally -- converts a hash table representing a Betti table to a Betti tally
***********************************************************************************

  * Usage:
        makeBettiTally(H)
  * Inputs:
      * H, a hash table (Macaulay2Doc),
  * Outputs:
      * a Betti tally (Macaulay2Doc),

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

Given a hash table $H$ whose keys are pairs of integers $(p,q)$ this function
presents the data in the Betti tally format. For instance, combining this with
the totalBetti, function reproduces the standard Betti table.
By contrast, combining this with the numRepsBetti function produces a table
where the entry in position $(p,q)$ is the number of Schur functors in the
representation corresponding to that Betti table entry.

+---------------------------------------+
|i1 : H = totalBetti(3,2,0);            |
+---------------------------------------+
|i2 : makeBettiTally H                  |
|                                       |
|            0  1   2   3   4   5  6 7  |
|o2 = total: 1 27 105 189 189 105 27 1  |
|         0: 1  .   .   .   .   .  . .  |
|         1: . 27 105 189 189 105 27 .  |
|         2: .  .   .   .   .   .  . 1  |
|                                       |
|o2 : BettiTally                        |
+---------------------------------------+
|i3 : makeBettiTally numRepsBetti(3,2,0)|
|                                       |
|            0 1 2 3 4 5 6 7            |
|o3 = total: 1 1 4 7 7 4 1 1            |
|         0: 1 . . . . . . .            |
|         1: . 1 4 7 7 4 1 .            |
|         2: . . . . . . . 1            |
|                                       |
|o3 : BettiTally                        |
+---------------------------------------+

Ways to use makeBettiTally:
===========================

  * "makeBettiTally(HashTable)"

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

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

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

The source of this document is in SchurVeronese/doc.m2:71:0.

multiBetti -- a hash table containing the multigraded Betti numbers of a Veronese embedding
*******************************************************************************************

  * Usage:
        multiBetti(d,n,b)
  * Inputs:
      * d, an integer (Macaulay2Doc),
      * n, an integer (Macaulay2Doc),
      * b, an integer (Macaulay2Doc),
  * Outputs:
      * a hash table (Macaulay2Doc),

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

This function returns a hash table $H$ containing the multigraded Betti numbers
for $\mathcal{O}(b)$ on $\mathbb{P}^{n}$ under the embedding by $d$-fold
Veronese embedding given by $\mathcal{O}(d)$. The keys of the returned hash
table $H$ are pairs $(p,q)$ where $H#(p,q)$ gives the the multigraded Betti
decomposition of $K_{p,q}(\mathbb{P}^n, d;b)$. We record the multigraded Betti
numbers via a multigraded Hilbert series. See Section 1.1 of [BEGY] for a more
precise description of the multigraded Betti numbers of a Veronese embedding.

Note that the output of this function is sometimes enormous and so can take a
long time to print on the screen.

+-----------------------------------------------------------------------------+
|i1 : totalBettiTally(3,2,0)                                                  |
|                                                                             |
|            0  1   2   3   4   5  6 7                                        |
|o1 = total: 1 27 105 189 189 105 27 1                                        |
|         0: 1  .   .   .   .   .  . .                                        |
|         1: . 27 105 189 189 105 27 .                                        |
|         2: .  .   .   .   .   .  . 1                                        |
|                                                                             |
|o1 : BettiTally                                                              |
+-----------------------------------------------------------------------------+
|i2 : multiBetti(3,2,0)                                                       |
|                                                                             |
|o2 = HashTable{(0, 0) => 1                                                   |
|               (0, 1) => 0                                                   |
|               (0, 2) => 0                                                   |
|               (1, 0) => 0                                                   |
|                          4 2                                                |
|               (1, 1) => T T  +                                              |
|                          0 1                                                |
|               (1, 2) => 0                                                   |
|               (2, 0) => 0                                                   |
|                          6 2                                                |
|               (2, 1) => T T T                                               |
|                          0 1 2                                              |
|               (2, 2) => 0                                                   |
|               (3, 0) => 0                                                   |
|                          7 4                                                |
|               (3, 1) => T T T                                               |
|                          0 1 2                                              |
|               (3, 2) => 0                                                   |
|               (4, 0) => 0                                                   |
|                          8 5 2                                              |
|               (4, 1) => T T T                                               |
|                          0 1 2                                              |
|               (4, 2) => 0                                                   |
|               (5, 0) => 0                                                   |
|                          9 5 4                                              |
|               (5, 1) => T T T                                               |
|                          0 1 2                                              |
|               (5, 2) => 0                                                   |
|               (6, 0) => 0                                                   |
|                          9 7 5                                              |
|               (6, 1) => T T T                                               |
|                          0 1 2                                              |
|               (6, 2) => 0                                                   |
|               (7, 0) => 0                                                   |
|               (7, 1) => 0                                                   |
|                          9 9 9                                              |
|               (7, 2) => T T T                                               |
|                          0 1 2                                              |
|                                                                             |
|o2 : HashTable                                                               |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
| 4        4 2    3 3     3 2       3   2    3 3    2 4     2 3       2 2 2   |
|T T T  + T T  + T T  + 2T T T  + 2T T T  + T T  + T T  + 2T T T  + 3T T T  + |
| 0 1 2    0 2    0 1     0 1 2     0 1 2    0 2    0 1     0 1 2     0 1 2   |
|                                                                             |
|                                                                             |
|   6   2    5 4     5 3       5 2 2     5   3    5 4    4 5     4 4       4 3|
|+ T T T  + T T  + 3T T T  + 4T T T  + 3T T T  + T T  + T T  + 4T T T  + 7T T |
|   0 1 2    0 1     0 1 2     0 1 2     0 1 2    0 2    0 1     0 1 2     0 1|
|                                                                             |
|                                                                             |
|    7 3 2     7 2 3    7   4     6 5       6 4 2     6 3 3     6 2 4     6   |
|+ 2T T T  + 2T T T  + T T T  + 2T T T  + 5T T T  + 7T T T  + 5T T T  + 2T T T|
|    0 1 2     0 1 2    0 1 2     0 1 2     0 1 2     0 1 2     0 1 2     0 1 |
|                                                                             |
|                                                                             |
|    8 4 3     8 3 4    8 2 5     7 6 2     7 5 3     7 4 4     7 3 5     7 2 |
|+ 2T T T  + 2T T T  + T T T  + 2T T T  + 5T T T  + 7T T T  + 5T T T  + 2T T T|
|    0 1 2     0 1 2    0 1 2     0 1 2     0 1 2     0 1 2     0 1 2     0 1 |
|                                                                             |
|                                                                             |
|   9 4 5    8 7 3     8 6 4     8 5 5     8 4 6    8 3 7    7 8 3     7 7 4  |
|+ T T T  + T T T  + 3T T T  + 4T T T  + 3T T T  + T T T  + T T T  + 4T T T  +|
|   0 1 2    0 1 2     0 1 2     0 1 2     0 1 2    0 1 2    0 1 2     0 1 2  |
|                                                                             |
|                                                                             |
|   9 6 6    9 5 7    8 8 5     8 7 6     8 6 7    8 5 8    7 9 5     7 8 6   |
|+ T T T  + T T T  + T T T  + 2T T T  + 2T T T  + T T T  + T T T  + 2T T T  + |
|   0 1 2    0 1 2    0 1 2     0 1 2     0 1 2    0 1 2    0 1 2     0 1 2   |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|  2   3    2 4      4         3 2       2 3        4    4 2    3 3    2 4    |
|2T T T  + T T  + T T T  + 2T T T  + 2T T T  + T T T  + T T  + T T  + T T     |
|  0 1 2    0 2    0 1 2     0 1 2     0 1 2    0 1 2    1 2    1 2    1 2    |
|                                                                             |
|                                                                             |
| 2     4 2 3     4   4    4 5     3 5       3 4 2     3 3 3     3 2 4     3  |
|T  + 7T T T  + 4T T T  + T T  + 3T T T  + 7T T T  + 9T T T  + 7T T T  + 3T T |
| 2     0 1 2     0 1 2    0 2     0 1 2     0 1 2     0 1 2     0 1 2     0 1|
|                                                                             |
|                                                                             |
|5     5 6       5 5 2      5 4 3      5 3 4     5 2 5     5   6    4 7       |
|  + 2T T T  + 7T T T  + 12T T T  + 12T T T  + 7T T T  + 2T T T  + T T T  + 5T|
|2     0 1 2     0 1 2      0 1 2      0 1 2     0 1 2     0 1 2    0 1 2     |
|                                                                             |
|                                                                             |
|6     6 7 2     6 6 3      6 5 4      6 4 5     6 3 6     6 2 7    5 8 2     |
|  + 2T T T  + 7T T T  + 12T T T  + 12T T T  + 7T T T  + 2T T T  + T T T  + 5T|
|2     0 1 2     0 1 2      0 1 2      0 1 2     0 1 2     0 1 2    0 1 2     |
|                                                                             |
|                                                                             |
|   7 6 5     7 5 6     7 4 7    7 3 8     6 8 4     6 7 5     6 6 6     6 5 7|
| 7T T T  + 7T T T  + 4T T T  + T T T  + 3T T T  + 7T T T  + 9T T T  + 7T T T |
|   0 1 2     0 1 2     0 1 2    0 1 2     0 1 2     0 1 2     0 1 2     0 1 2|
|                                                                             |
|                                                                             |
|  7 7 7     7 6 8    7 5 9    6 9 6     6 8 7     6 7 8    6 6 9    5 9 7    |
|3T T T  + 2T T T  + T T T  + T T T  + 2T T T  + 2T T T  + T T T  + T T T  + T|
|  0 1 2     0 1 2    0 1 2    0 1 2     0 1 2     0 1 2    0 1 2    0 1 2    |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
| 5    2 6       2 5 2     2 4 3     2 3 4     2 2 5    2   6      6 2       5|
|T  + T T T  + 4T T T  + 7T T T  + 7T T T  + 4T T T  + T T T  + T T T  + 3T T |
| 2    0 1 2     0 1 2     0 1 2     0 1 2     0 1 2    0 1 2    0 1 2     0 1|
|                                                                             |
|                                                                             |
|4 6 2      4 5 3      4 4 4      4 3 5     4 2 6    4   7     3 7 2     3 6 3|
| T T  + 12T T T  + 15T T T  + 12T T T  + 5T T T  + T T T  + 2T T T  + 7T T T |
|0 1 2      0 1 2      0 1 2      0 1 2     0 1 2    0 1 2     0 1 2     0 1 2|
|                                                                             |
|                                                                             |
|5 7 3      5 6 4      5 5 5      5 4 6     5 3 7    5 2 8     4 8 3     4 7 4|
| T T  + 12T T T  + 15T T T  + 12T T T  + 5T T T  + T T T  + 2T T T  + 7T T T |
|0 1 2      0 1 2      0 1 2      0 1 2     0 1 2    0 1 2     0 1 2     0 1 2|
|                                                                             |
|                                                                             |
|     6 4 8    5 9 4     5 8 5     5 7 6     5 6 7     5 5 8    5 4 9    4 9 5|
| + 3T T T  + T T T  + 4T T T  + 7T T T  + 7T T T  + 4T T T  + T T T  + T T T |
|     0 1 2    0 1 2     0 1 2     0 1 2     0 1 2     0 1 2    0 1 2    0 1 2|
|                                                                             |
|                                                                             |
|5 8 8    5 7 9                                                               |
| T T  + T T T                                                                |
|0 1 2    0 1 2                                                               |
|-----------------------------------------------------------------------------|
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
| 3       4 4       3 5      2 6    5 4    4 5                                |
|T  + 4T T T  + 3T T T  + T T T  + T T  + T T                                 |
| 2     0 1 2     0 1 2    0 1 2    1 2    1 2                                |
|                                                                             |
|                                                                             |
|      3 5 4      3 4 5     3 3 6     3 2 7     2 7 3     2 6 4     2 5 5     |
| + 12T T T  + 12T T T  + 7T T T  + 2T T T  + 2T T T  + 5T T T  + 7T T T  +   |
|      0 1 2      0 1 2     0 1 2     0 1 2     0 1 2     0 1 2     0 1 2     |
|                                                                             |
|                                                                             |
|      4 6 5      4 5 6     4 4 7     4 3 8     3 8 4     3 7 5     3 6 6     |
| + 12T T T  + 12T T T  + 7T T T  + 2T T T  + 2T T T  + 5T T T  + 7T T T  +   |
|      0 1 2      0 1 2     0 1 2     0 1 2     0 1 2     0 1 2     0 1 2     |
|                                                                             |
|                                                                             |
|     4 8 6     4 7 7     4 6 8    4 5 9    3 8 7    3 7 8                    |
| + 3T T T  + 4T T T  + 3T T T  + T T T  + T T T  + T T T                     |
|     0 1 2     0 1 2     0 1 2    0 1 2    0 1 2    0 1 2                    |
|-----------------------------------------------------------------------------|
|                                                       }                     |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|  2 4 6     2 3 7      7 4       6 5       5 6      4 7                      |
|5T T T  + 2T T T  + T T T  + 2T T T  + 2T T T  + T T T                       |
|  0 1 2     0 1 2    0 1 2     0 1 2     0 1 2    0 1 2                      |
|                                                                             |
|                                                                             |
|  3 5 7     3 4 8    2 8 5     2 7 6     2 6 7    2 5 8                      |
|5T T T  + 2T T T  + T T T  + 2T T T  + 2T T T  + T T T                       |
|  0 1 2     0 1 2    0 1 2     0 1 2     0 1 2    0 1 2                      |
+-----------------------------------------------------------------------------+

Ways to use multiBetti:
=======================

  * "multiBetti(ZZ,ZZ,ZZ)"

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

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

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

The source of this document is in SchurVeronese/doc.m2:211:0.

numDistinctRepsBetti -- a hash table containing the number of distinct Schur functors of a Veronese embedding
*************************************************************************************************************

  * Usage:
        numDistinctRepsBetti(d,n,b)
  * Inputs:
      * d, an integer (Macaulay2Doc),
      * n, an integer (Macaulay2Doc),
      * b, an integer (Macaulay2Doc),
  * Outputs:
      * a hash table (Macaulay2Doc),

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

This function returns a hash table $H$ whose keys are pairs $(p,q)$ such that
the corresponding value $H#(p,q)$ is the number of distinct Schur functors
appearing in the Schur functor decomposition of $K_{p,q}(\mathbb{P}^n, d;b)$.
Note that the function numRepsBetti, is similar, though it
counts Schur functors appearing with multiplicity.

+-------------------------------------+
|i1 : totalBettiTally(3,2,0)          |
|                                     |
|            0  1   2   3   4   5  6 7|
|o1 = total: 1 27 105 189 189 105 27 1|
|         0: 1  .   .   .   .   .  . .|
|         1: . 27 105 189 189 105 27 .|
|         2: .  .   .   .   .   .  . 1|
|                                     |
|o1 : BettiTally                      |
+-------------------------------------+
|i2 : numDistinctRepsBetti(3,2,0)     |
|                                     |
|o2 = HashTable{(0, 0) => 1}          |
|               (0, 1) => 0           |
|               (0, 2) => 0           |
|               (1, 0) => 0           |
|               (1, 1) => 1           |
|               (1, 2) => 0           |
|               (2, 0) => 0           |
|               (2, 1) => 4           |
|               (2, 2) => 0           |
|               (3, 0) => 0           |
|               (3, 1) => 7           |
|               (3, 2) => 0           |
|               (4, 0) => 0           |
|               (4, 1) => 7           |
|               (4, 2) => 0           |
|               (5, 0) => 0           |
|               (5, 1) => 4           |
|               (5, 2) => 0           |
|               (6, 0) => 0           |
|               (6, 1) => 1           |
|               (6, 2) => 0           |
|               (7, 0) => 0           |
|               (7, 1) => 0           |
|               (7, 2) => 1           |
|                                     |
|o2 : HashTable                       |
+-------------------------------------+
|i3 : makeBettiTally oo               |
|                                     |
|            0 1 2 3 4 5 6 7          |
|o3 = total: 1 1 4 7 7 4 1 1          |
|         0: 1 . . . . . . .          |
|         1: . 1 4 7 7 4 1 .          |
|         2: . . . . . . . 1          |
|                                     |
|o3 : BettiTally                      |
+-------------------------------------+

This example shows that the $\beta_{3,4}(3,2,0)$ entry, which is $189$,
consists of $7$ distinct Schur functors.  The specific Schur functors that
appear can be computed using schurBetti.  Here is a more
complicated example:

+-----------------------------------------------------------------------------+
|i4 : totalBettiTally(5,2,1)                                                  |
|                                                                             |
|            0  1   2    3     4     5      6      7      8      9     10     |
|o4 = total: 3 35 525 5865 29988 97920 231540 417690 590070 661232 590070     |
|         0: 3 35 120    .     .     .      .      .      .      .      .     |
|         1: .  . 405 5865 29988 97920 231540 417690 590070 661232 590070     |
|         2: .  .   .    .     .     .      .      .      .      .      .     |
|                                                                             |
|o4 : BettiTally                                                              |
|-----------------------------------------------------------------------------|
|    11     12    13    14   15  16 17 18                                     |
|417690 231540 97920 29988 5865 525 35  3                                     |
|     .      .     .     .    .   .  .  .                                     |
|417690 231540 97920 29988 5865 405  .  .                                     |
|     .      .     .     .    . 120 35  3                                     |
+-----------------------------------------------------------------------------+
|i5 : makeBettiTally numDistinctRepsBetti(5,2,1)                              |
|                                                                             |
|            0 1 2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18            |
|o5 = total: 1 1 5 26 43 53 63 68 72 73 72 68 63 53 43 26  5  1  1            |
|         0: 1 1 3  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .            |
|         1: . . 2 26 43 53 63 68 72 73 72 68 63 53 43 26  2  .  .            |
|         2: . . .  .  .  .  .  .  .  .  .  .  .  .  .  .  3  1  1            |
|                                                                             |
|o5 : BettiTally                                                              |
+-----------------------------------------------------------------------------+

Ways to use numDistinctRepsBetti:
=================================

  * "numDistinctRepsBetti(ZZ,ZZ,ZZ)"

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

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

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

The source of this document is in SchurVeronese/doc.m2:307:0.

numRepsBetti -- a hash table containing the number of Schur functors of a Veronese embedding
********************************************************************************************

  * Usage:
        numRepsBetti(d,n,b)
  * Inputs:
      * d, an integer (Macaulay2Doc),
      * n, an integer (Macaulay2Doc),
      * b, an integer (Macaulay2Doc),
  * Outputs:
      * a hash table (Macaulay2Doc),

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

This function returns a hash table $H$ whose keys are pairs $(p,q)$ such that
the corresponding value $H#(p,q)$ is the number of Schur functors appearing in
the Schur functor decomposition of $K_{p,q}(\mathbb{P}^n, d;b)$ counted with
multiplicity. Note that the function numDistinctRepsBetti,
is similar, though that ignores the multiplicities of the
Schur functors.

+-------------------------------------+
|i1 : totalBettiTally(3,2,0)          |
|                                     |
|            0  1   2   3   4   5  6 7|
|o1 = total: 1 27 105 189 189 105 27 1|
|         0: 1  .   .   .   .   .  . .|
|         1: . 27 105 189 189 105 27 .|
|         2: .  .   .   .   .   .  . 1|
|                                     |
|o1 : BettiTally                      |
+-------------------------------------+
|i2 : numRepsBetti(3,2,0)             |
|                                     |
|o2 = HashTable{(0, 0) => 1}          |
|               (0, 1) => 0           |
|               (0, 2) => 0           |
|               (1, 0) => 0           |
|               (1, 1) => 1           |
|               (1, 2) => 0           |
|               (2, 0) => 0           |
|               (2, 1) => 4           |
|               (2, 2) => 0           |
|               (3, 0) => 0           |
|               (3, 1) => 7           |
|               (3, 2) => 0           |
|               (4, 0) => 0           |
|               (4, 1) => 7           |
|               (4, 2) => 0           |
|               (5, 0) => 0           |
|               (5, 1) => 4           |
|               (5, 2) => 0           |
|               (6, 0) => 0           |
|               (6, 1) => 1           |
|               (6, 2) => 0           |
|               (7, 0) => 0           |
|               (7, 1) => 0           |
|               (7, 2) => 1           |
|                                     |
|o2 : HashTable                       |
+-------------------------------------+
|i3 : makeBettiTally oo               |
|                                     |
|            0 1 2 3 4 5 6 7          |
|o3 = total: 1 1 4 7 7 4 1 1          |
|         0: 1 . . . . . . .          |
|         1: . 1 4 7 7 4 1 .          |
|         2: . . . . . . . 1          |
|                                     |
|o3 : BettiTally                      |
+-------------------------------------+

This example shows that the $\beta_{3,4}(3,2,0)$ entry, which is $189$,
consists of $7$ Schur functors.  (In this case, the Schur functors all appear
with multiplicity 1.) The specific Schur functors that appear can be computed
using schurBetti.  Here is an example where some Schur
functors appear with higher multiplicity.

+-----------------------------------------------------------------------------+
|i4 : totalBettiTally(5,2,1)                                                  |
|                                                                             |
|            0  1   2    3     4     5      6      7      8      9     10     |
|o4 = total: 3 35 525 5865 29988 97920 231540 417690 590070 661232 590070     |
|         0: 3 35 120    .     .     .      .      .      .      .      .     |
|         1: .  . 405 5865 29988 97920 231540 417690 590070 661232 590070     |
|         2: .  .   .    .     .     .      .      .      .      .      .     |
|                                                                             |
|o4 : BettiTally                                                              |
|-----------------------------------------------------------------------------|
|    11     12    13    14   15  16 17 18                                     |
|417690 231540 97920 29988 5865 525 35  3                                     |
|     .      .     .     .    .   .  .  .                                     |
|417690 231540 97920 29988 5865 405  .  .                                     |
|     .      .     .     .    . 120 35  3                                     |
+-----------------------------------------------------------------------------+
|i5 : makeBettiTally numRepsBetti(5,2,1)                                      |
|                                                                             |
|            0 1 2  3   4   5    6    7    8    9   10   11   12  13  14 15 16|
|o5 = total: 1 1 5 38 165 477 1027 1733 2355 2608 2355 1733 1027 477 165 38  5|
|         0: 1 1 3  .   .   .    .    .    .    .    .    .    .   .   .  .  .|
|         1: . . 2 38 165 477 1027 1733 2355 2608 2355 1733 1027 477 165 38  2|
|         2: . . .  .   .   .    .    .    .    .    .    .    .   .   .  .  3|
|                                                                             |
|o5 : BettiTally                                                              |
|-----------------------------------------------------------------------------|
|17 18                                                                        |
| 1  1                                                                        |
| .  .                                                                        |
| .  .                                                                        |
| 1  1                                                                        |
+-----------------------------------------------------------------------------+

Ways to use numRepsBetti:
=========================

  * "numRepsBetti(ZZ,ZZ,ZZ)"

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

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

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

The source of this document is in SchurVeronese/doc.m2:344:0.

schurBetti -- a hash table for Schur module decomposition of Veronese Betti tables
**********************************************************************************

  * Usage:
        schurBetti(d,n,b)
  * Inputs:
      * d, an integer (Macaulay2Doc),
      * n, an integer (Macaulay2Doc),
      * b, an integer (Macaulay2Doc),
  * Outputs:
      * a hash table (Macaulay2Doc),

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

This function returns a hash table with the Schur functor decompositions of the
syzygies of $\mathcal{O}(b)$ under the embedding by $\mathcal{O}(d)$. The keys
of the hash table $H$ are pairs $(p,q)$ where $H#(p,q)$ gives the Schur functor
decomposition of $K_{p,q}(\mathbb{P}^n, d;b)$. $\mathcal{O}(b)$ the Schur
functor decomposition as a list of tuples $(\{a_1,a_2,a_3\},b)$ where
$\{a_1,a_2,a_3\}$ specifies the weight of the Schur functor and $m$ the
multiplicity with which that particular Schur functor appears in the
decomposition of $K_{p,q}(\mathbb{P}^n, d;b)$.

Some tables are incomplete and we mark unknown entries with ({0,0,0},infinity).

+-----------------------------------------------------------------------------+
|i1 : schurBetti(3,2,0)                                                       |
|                                                                             |
|o1 = HashTable{(0, 0) => {({0, 0, 0}, 1)}                                    |
|               (0, 1) => {}                                                  |
|               (0, 2) => {}                                                  |
|               (1, 0) => {}                                                  |
|               (1, 1) => {({4, 2, 0}, 1)}                                    |
|               (1, 2) => {}                                                  |
|               (2, 0) => {}                                                  |
|               (2, 1) => {({6, 2, 1}, 1), ({5, 4, 0}, 1), ({5, 3, 1}, 1),    |
|               (2, 2) => {}                                                  |
|               (3, 0) => {}                                                  |
|               (3, 1) => {({7, 4, 1}, 1), ({7, 3, 2}, 1), ({6, 5, 1}, 1),    |
|               (3, 2) => {}                                                  |
|               (4, 0) => {}                                                  |
|               (4, 1) => {({8, 5, 2}, 1), ({8, 4, 3}, 1), ({7, 6, 2}, 1),    |
|               (4, 2) => {}                                                  |
|               (5, 0) => {}                                                  |
|               (5, 1) => {({9, 5, 4}, 1), ({8, 7, 3}, 1), ({8, 6, 4}, 1),    |
|               (5, 2) => {}                                                  |
|               (6, 0) => {}                                                  |
|               (6, 1) => {({9, 7, 5}, 1)}                                    |
|               (6, 2) => {}                                                  |
|               (7, 0) => {}                                                  |
|               (7, 1) => {}                                                  |
|               (7, 2) => {({9, 9, 9}, 1)}                                    |
|                                                                             |
|o1 : HashTable                                                               |
|-----------------------------------------------------------------------------|
|                                                               }             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|                                                                             |
|({4, 3, 2}, 1)}                                                              |
|                                                                             |
|                                                                             |
|({6, 4, 2}, 1), ({6, 3, 3}, 1), ({5, 5, 2}, 1), ({5, 4, 3}, 1)}              |
|                                                                             |
|                                                                             |
|({7, 5, 3}, 1), ({7, 4, 4}, 1), ({6, 6, 3}, 1), ({6, 5, 4}, 1)}              |
|                                                                             |
|                                                                             |
|({7, 6, 5}, 1)}                                                              |
+-----------------------------------------------------------------------------+

Ways to use schurBetti:
=======================

  * "schurBetti(ZZ,ZZ,ZZ)"

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

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

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

The source of this document is in SchurVeronese/doc.m2:180:0.

totalBetti -- a hash table containing the graded Betti numbers of a Veronese embedding
**************************************************************************************

  * Usage:
        totalBetti(d,n,b)
  * Inputs:
      * d, an integer (Macaulay2Doc),
      * n, an integer (Macaulay2Doc),
      * b, an integer (Macaulay2Doc),
  * Outputs:
      * a hash table (Macaulay2Doc),

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

This is a hash table for the total numbers of $\mathcal{O}(b)$ under the
embedding by $\mathcal{O}(d)$. The keys of the hash table $H$ are pairs $(p,q)$
where $H#(p,q)$ gives the rank of $K_{p,q}(\mathbb{P}^n, d;b)$. This equals the
Betti number $\beta_{p,p+q}(d,\mathbb{P}^n,b)$.Some tables are incomplete and
we mark unknown entries with infinity.

Note that totalBetti differs from totalBettiTally, only
in that the output is a hash table instead of a Betti tally. One can convert
the output of totalBetti into a Betti tally via the makeBettiTally,
function.

In example below we generate a hash table showing the total graded Betti
numbers of $\mathbb{P}^{2}$ embedded by $\mathcal{O}(3)$.

+-----------------------------+
|i1 : B = totalBetti(3,2,0)   |
|                             |
|o1 = HashTable{(0, 0) => 1  }|
|               (0, 1) => 0   |
|               (0, 2) => 0   |
|               (1, 0) => 0   |
|               (1, 1) => 27  |
|               (1, 2) => 0   |
|               (2, 0) => 0   |
|               (2, 1) => 105 |
|               (2, 2) => 0   |
|               (3, 0) => 0   |
|               (3, 1) => 189 |
|               (3, 2) => 0   |
|               (4, 0) => 0   |
|               (4, 1) => 189 |
|               (4, 2) => 0   |
|               (5, 0) => 0   |
|               (5, 1) => 105 |
|               (5, 2) => 0   |
|               (6, 0) => 0   |
|               (6, 1) => 27  |
|               (6, 2) => 0   |
|               (7, 0) => 0   |
|               (7, 1) => 0   |
|               (7, 2) => 1   |
|                             |
|o1 : HashTable               |
+-----------------------------+

If we wish to view these graded Betti numbers in the usual fashion, we can use
makeBettiTally to convert the hash table above to a Betti tally.

+-------------------------------------+
|i2 : makeBettiTally B                |
|                                     |
|            0  1   2   3   4   5  6 7|
|o2 = total: 1 27 105 189 189 105 27 1|
|         0: 1  .   .   .   .   .  . .|
|         1: . 27 105 189 189 105 27 .|
|         2: .  .   .   .   .   .  . 1|
|                                     |
|o2 : BettiTally                      |
+-------------------------------------+

Ways to use totalBetti:
=======================

  * "totalBetti(ZZ,ZZ,ZZ)"

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

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

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

The source of this document is in SchurVeronese/doc.m2:110:0.

totalBettiTally -- a Betti tally containing the graded Betti numbers of a Veronese embedding
********************************************************************************************

  * Usage:
        totalBettiTally(d,n,b)
  * Inputs:
      * d, an integer (Macaulay2Doc),
      * n, an integer (Macaulay2Doc),
      * b, an integer (Macaulay2Doc),
  * Outputs:
      * a Betti tally (Macaulay2Doc),

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

This function returns a Betti tally for the total graded Betti numbers of
$\mathcal{O}(b)$ under the embedding by $\mathcal{O}(d)$. Some tables are
incomplete and we mark unknown entries with infinity.

Note that totalBettiTally differs from totalBetti, only in
that the output is a Betti tally instead of a hash table.

In example below we generate a hash table showing the total graded Betti
numbers of $\mathbb{P}^{2}$ embedded by $\mathcal{O}(3)$.

+-------------------------------------+
|i1 : totalBettiTally(3,2,0)          |
|                                     |
|            0  1   2   3   4   5  6 7|
|o1 = total: 1 27 105 189 189 105 27 1|
|         0: 1  .   .   .   .   .  . .|
|         1: . 27 105 189 189 105 27 .|
|         2: .  .   .   .   .   .  . 1|
|                                     |
|o1 : BettiTally                      |
+-------------------------------------+

We can also produce the Betti tables of the pushforwards of line bundles.  For
instance, the following example computes the Betti table of the pushforward of
$\mathcal{O}(1)$ under the 3-uple embedding.

+------------------------------------+
|i2 : totalBettiTally(3,2,1)         |
|                                    |
|            0  1  2   3   4   5  6 7|
|o2 = total: 3 15 42 105 147 105 39 6|
|         0: 3 15 21   .   .   .  . .|
|         1: .  . 21 105 147 105 39 6|
|         2: .  .  .   .   .   .  . .|
|                                    |
|o2 : BettiTally                     |
+------------------------------------+

Ways to use totalBettiTally:
============================

  * "totalBettiTally(ZZ,ZZ,ZZ)"

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

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

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

The source of this document is in SchurVeronese/doc.m2:147:0.