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

ExteriorExtensions -- Builds a \(\mathbb{Z}_m\) -graded algebra that equivariantly extends a Lie algebra
********************************************************************************************************

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

ExteriorExtensions Builds a \(\mathbb{Z}_m\) -graded algebra that equivariantly
extends a Lie algebra \( \mathfrak{sl}_n\) via the non-zero graded piece of the
exterior algebra by defining the bracket products. Constructs matrix
representations of adjoint operators. Computes ranks of blocks coming from the
grading. Computing matrix representations of adjoint operators and computing
ranks of their blocks. The user can also call for the Killing matrix of the
algebra.

Author
======

  * Luke Oeding (see http://webhome.auburn.edu/~lao0004/ ) <oeding@auburn.edu>

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

Version 1.0 of this package was accepted for publication in volume 15 (see
https://msp.org/jsag/2025/15-1/ ) of Journal of Software for Algebra and
Geometry (see https://msp.org/jsag/ ) on 2025-07-16, in the article
ExteriorExtensions: a package for Macaulay2 (see
https://msp.org/jsag/2025/15-1/p06.xhtml ) (DOI: 10.2140/jsag.2025.15.69 (see
https://doi.org/10.2140/jsag.2025.15.69 )). That version can be obtained from
the journal (see https://msp.org/jsag/2025/15-1/jsag-v15-n1-x06-
ExteriorExtensions.m2 ).

Version
=======

This documentation describes version 1.0 of ExteriorExtensions, released April
23, 2025.

Citation
========

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

+-----------------------------------------------------------------------------+
|@misc{ExteriorExtensionsSource,                                              |
|  title = {{ExteriorExtensions: Builds an algebra extending an exterior algeb|
|  author = {Luke Oeding},                                                    |
|  howpublished = {A \emph{Macaulay2} package available at                    |
|    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}} |
|}                                                                            |
|                                                                             |
|@article{ExteriorExtensionsArticle,                                          |
|  title = {{ExteriorExtensions: a package for \emph{Macaulay2}}},            |
|  author = {Luke Oeding},                                                    |
|  journal = {Journal of Software for Algebra and Geometry},                  |
|  volume = {15},                                                             |
|  year = {2025},                                                             |
|}                                                                            |
|-----------------------------------------------------------------------------|
|ra. Version~1.0}},                                                           |
+-----------------------------------------------------------------------------+

Exports
=======

  * Types
      * ExteriorExtension -- A new algebra
        constructed from the direct sum of a Lie algebra and a module
  * Functions and commands
      * exteriorExtension -- Make an extension of an
        exterior algebra
      * getDiagonal -- Extract the diagonal of a matrix
      * KillingMatrix -- Constructs the Killing matrix of
        the ExteriorExtension
      * makeTraceless -- Project matrix to the traceless
        matrices.
      * structureTensor -- Constructs the
        structureTensor of the ExteriorExtension
  * Methods
      * "exteriorExtension(ZZ,ZZ)" -- see exteriorExtension,
        -- Make an extension of an exterior algebra
      * "exteriorExtension(ZZ,ZZ,Ring)" -- see exteriorExtension,
        -- Make an extension of an exterior algebra
      * "exteriorExtension(ZZ,ZZ,Symbol)" -- see exteriorExtension,
        -- Make an extension of an exterior algebra
      * "exteriorExtension(ZZ,ZZ,Symbol,Ring)" -- see exteriorExtension,
        -- Make an extension of an exterior algebra
      * KillingMatrix(ExteriorExtension) (missing documentation)
      * structureTensor(ExteriorExtension) (missing documentation)
  * Symbols
      * ad -- Constructs the standard matrix representation of the
        adjoint operator
      * appendage -- a placeholder for the object (exterior
        algebra) added on to the Lie algebra by exteriorExtension
      * bases -- gives a list of the bases of the different
        graded pieces of the ExteriorExtension
      * blockPowerRanks -- Computes a list of ranks of
        blocks for each power of a matrix representation of an adjoint operator
        for ExteriorExtensions
      * bracket -- Compute products in an ExteriorExtension
      * findGrade -- Find which graded piece of the algebra
        an element is in
      * getBlock -- Extract a block of the matrix
        representation of an adjoint operator for ExteriorExtensions
      * HodgeStar -- The Hodge star in an exterior algebra
      * LieAlg2Matrix -- Converts from the LieAlgebra to
        Matrix form (with the standard basis)
      * LieAlgebra -- a placeholder for the base Lie algebra
      * matrix2LieAlg -- Converts from Matrix form to the
        LieAlgebra (with the standard basis)
      * powerRanks -- Compute the ranks of powers of a
        matrix
      * powerTraces -- Compute the traces of powers of a
        matrix
      * prettyBlockPowerRanks -- same as
        blockPowerRanks, but prettier

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

The object ExteriorExtensions, is a package (Macaulay2Doc),
defined in ExteriorExtensions.m2.

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

The source of this document is in ExteriorExtensions.m2:288:0.

exteriorExtension -- Make an extension of an exterior algebra
*************************************************************

  * Usage:
        extensionAlgebra = exteriorExtension(pow, nvars)
        extensionAlgebra = exteriorExtension(pow, nvars, e)
        extensionAlgebra = exteriorExtension(pow, nvars, KK)
        extensionAlgebra = exteriorExtension(pow, nvars, e, KK)
  * Inputs:
      * pow, an integer (Macaulay2Doc),
      * nvars, an integer (Macaulay2Doc),
      * e, a symbol (Macaulay2Doc),
      * KK, a ring (Macaulay2Doc),
  * Outputs:
      * extensionAlgebra, an instance of the type ExteriorExtension,
  * Consequences:
      * Sets the value for several symbols and functions attached to the
        ExteriorExtension: "appendage", "findGrade", "HodgeStar", "bases",
        "bracket", "ad", "powerRanks", "powerTraces", "getBlock",
        "blockPowerRanks", "prettyBlockPowerRanks",

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

This module sets the functions and parameters that make the required algebra.
It constructs a natural graded algebra \(\mathfrak{a}\), with \(\mathfrak{a}_0
=  \mathfrak{sl}_n\), and  \(\mathfrak{a}_1= \bigwedge^k \mathbb{C}^n\), and
\(\mathfrak{a}_i\) for \(i>1\) defined as needed to make the algebra closed.
Provides some functionalitity for computations with the algebra. Sometimes we
build a known algebra. Let's see how this might work.

+---------------------------------------+
|i1 : ea24 = exteriorExtension(2, 4, QQ)|
|                                       |
|o1 = ExteriorExtension{...14...}       |
|                                       |
|o1 : ExteriorExtension                 |
+---------------------------------------+

We have set the algebra \(\mathfrak{sl}_2(\mathbb C) \oplus
\bigwedge^{2}\mathbb{C}\). The functions associated to this algebra are stored
by the symbol we assign the output. Let's see if the bilinear bracket satisfies
the Lie algebra axioms at least at random points in each graded piece of the
algebra: i.e. we check skew-commutativity and the Jacobi identity.

+-----------------------------------------------------------------------+
|i2 : A = makeTraceless random(QQ^4, QQ^4)                              |
|                                                                       |
|o2 = | 403/140 7/9      7        5/4      |                            |
|     | 9/4     -129/140 3/7      2/9      |                            |
|     | 3/4     7/10     -107/140 3/10     |                            |
|     | 7/4     7/3      6        -167/140 |                            |
|                                                                       |
|              4       4                                                |
|o2 : Matrix QQ  <-- QQ                                                 |
+-----------------------------------------------------------------------+
|i3 : B = makeTraceless random(QQ^4, QQ^4)                              |
|                                                                       |
|o3 = | 41/12 7/8  5/3    5/7    |                                      |
|     | 10/9  -3/4 7/2    5/9    |                                      |
|     | 10    5    -71/60 5/3    |                                      |
|     | 3/2   2/5  6/5    -89/60 |                                      |
|                                                                       |
|              4       4                                                |
|o3 : Matrix QQ  <-- QQ                                                 |
+-----------------------------------------------------------------------+
|i4 : ea24.bracket(A, B) + ea24.bracket(B, A)                           |
|                                                                       |
|o4 = 0                                                                 |
|                                                                       |
|              4       4                                                |
|o4 : Matrix QQ  <-- QQ                                                 |
+-----------------------------------------------------------------------+
|i5 : ea24.ad(ea24.bracket(A, B)) - ea24.bracket(ea24.ad(A), ea24.ad(B))|
|                                                                       |
|o5 = 0                                                                 |
|                                                                       |
|              21       21                                              |
|o5 : Matrix QQ   <-- QQ                                                |
+-----------------------------------------------------------------------+

We shouldn't be surprised that we get 0 for both since the grade 0 piece is a
Lie algebra by assumption.

+-----------------------------------------------------------------------+
|i6 : A = random(2, ea24.appendage)                                     |
|                                                                       |
|     4       3       4        9       5       5                        |
|o6 = -e e  + -e e  + -e e  + --e e  + -e e  + -e e                     |
|     3 0 1   7 0 2   7 1 2   10 0 3   9 1 3   9 2 3                    |
|                                                                       |
|o6 : QQ[e ..e ]                                                        |
|         0   3                                                         |
+-----------------------------------------------------------------------+
|i7 : B = random(2, ea24.appendage)                                     |
|                                                                       |
|     6               2       5       8       9                         |
|o7 = -e e  + 6e e  + -e e  + -e e  + -e e  + -e e                      |
|     7 0 1     0 2   9 1 2   4 0 3   5 1 3   4 2 3                     |
|                                                                       |
|o7 : QQ[e ..e ]                                                        |
|         0   3                                                         |
+-----------------------------------------------------------------------+
|i8 : ea24.bracket(A, B) + ea24.bracket(B, A)                           |
|                                                                       |
|o8 = 0                                                                 |
|                                                                       |
|                        4                 4                            |
|o8 : Matrix (QQ[e ..e ])  <-- (QQ[e ..e ])                             |
|                 0   3             0   3                               |
+-----------------------------------------------------------------------+
|i9 : ea24.ad(ea24.bracket(A, B)) - ea24.bracket(ea24.ad(A), ea24.ad(B))|
|                                                                       |
|o9 = 0                                                                 |
|                                                                       |
|              21       21                                              |
|o9 : Matrix QQ   <-- QQ                                                |
+-----------------------------------------------------------------------+

For both points in grade 1 we see the axioms hold at least for random elements.

+------------------------------------------------------------------------+
|i10 : A = makeTraceless random(QQ^4, QQ^4)                              |
|                                                                        |
|o10 = | -11/30 4      1/7   3/8   |                                     |
|      | 9/8    -23/90 7/5   5/2   |                                     |
|      | 1/8    9/10   -7/18 1/6   |                                     |
|      | 10/3   5/4    5/7   91/90 |                                     |
|                                                                        |
|               4       4                                                |
|o10 : Matrix QQ  <-- QQ                                                 |
+------------------------------------------------------------------------+
|i11 : B = random(2, ea24.appendage)                                     |
|                                                                        |
|               10       3       8       9       3                       |
|o11 = 10e e  + --e e  + -e e  + -e e  + -e e  + -e e                    |
|         0 1    9 0 2   4 1 2   3 0 3   5 1 3   5 2 3                   |
|                                                                        |
|o11 : QQ[e ..e ]                                                        |
|          0   3                                                         |
+------------------------------------------------------------------------+
|i12 : ea24.bracket(A, B) + ea24.bracket(B, A)                           |
|                                                                        |
|o12 = 0                                                                 |
|                                                                        |
|o12 : QQ[e ..e ]                                                        |
|          0   3                                                         |
+------------------------------------------------------------------------+
|i13 : ea24.ad(ea24.bracket(A, B)) - ea24.bracket(ea24.ad(A), ea24.ad(B))|
|                                                                        |
|o13 = 0                                                                 |
|                                                                        |
|               21       21                                              |
|o13 : Matrix QQ   <-- QQ                                                |
+------------------------------------------------------------------------+

For a point in grade 0 and a point in grade 1 we see the axioms hold. Caveat:
It's important to make the matrix traceless, otherwise the Jacobi identity
doesn't work.

+-----------------------------------------------------------------------------+
|i14 : printWidth =200;                                                       |
+-----------------------------------------------------------------------------+
|i15 : K = KillingMatrix ea24                                                 |
|                                                                             |
|o15 = | 20  -10 0   0  0  0  0  0  0  0  0  0  0  0  0  0   0  0   0   0  0  |
|      | -10 20  -10 0  0  0  0  0  0  0  0  0  0  0  0  0   0  0   0   0  0  |
|      | 0   -10 20  0  0  0  0  0  0  0  0  0  0  0  0  0   0  0   0   0  0  |
|      | 0   0   0   0  0  0  0  0  0  10 0  0  0  0  0  0   0  0   0   0  0  |
|      | 0   0   0   0  0  0  0  0  0  0  10 0  0  0  0  0   0  0   0   0  0  |
|      | 0   0   0   0  0  0  0  0  0  0  0  10 0  0  0  0   0  0   0   0  0  |
|      | 0   0   0   0  0  0  0  0  0  0  0  0  10 0  0  0   0  0   0   0  0  |
|      | 0   0   0   0  0  0  0  0  0  0  0  0  0  10 0  0   0  0   0   0  0  |
|      | 0   0   0   0  0  0  0  0  0  0  0  0  0  0  10 0   0  0   0   0  0  |
|      | 0   0   0   10 0  0  0  0  0  0  0  0  0  0  0  0   0  0   0   0  0  |
|      | 0   0   0   0  10 0  0  0  0  0  0  0  0  0  0  0   0  0   0   0  0  |
|      | 0   0   0   0  0  10 0  0  0  0  0  0  0  0  0  0   0  0   0   0  0  |
|      | 0   0   0   0  0  0  10 0  0  0  0  0  0  0  0  0   0  0   0   0  0  |
|      | 0   0   0   0  0  0  0  10 0  0  0  0  0  0  0  0   0  0   0   0  0  |
|      | 0   0   0   0  0  0  0  0  10 0  0  0  0  0  0  0   0  0   0   0  0  |
|      | 0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0   0  0   0   0  -10|
|      | 0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0   0  0   0   10 0  |
|      | 0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0   0  0   -10 0  0  |
|      | 0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0   0  -10 0   0  0  |
|      | 0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0   10 0   0   0  0  |
|      | 0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  -10 0  0   0   0  0  |
|                                                                             |
|               21       21                                                   |
|o15 : Matrix QQ   <-- QQ                                                     |
|-----------------------------------------------------------------------------|
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+-----------------------------------------------------------------------------+
|i16 : rank K                                                                 |
|                                                                             |
|o16 = 21                                                                     |
+-----------------------------------------------------------------------------+

The Killing matrix is \(21 \times 21\) and non-degenerate. The only possibility
is \(\mathfrak{sp}_6\). One checks that the Killing matrix has a full set of
real eigenvectors, so the isomorphism of \(\mathfrak{a}\) and
\(\mathfrak{sp}_6\) holds over the real numbers.

+-----------------------------------------------------------------------------+
|i17 : toList eigenvalues K                                                   |
|                                                                             |
|o17 = {34.1421, 20, 5.85786, 10, -10, 10, -10, 10, -10, 10, -10, 10, -10, 10,|
|                                                                             |
|o17 : List                                                                   |
|-----------------------------------------------------------------------------|
|-10, 10, -10, 10, -10, 10, -10}                                              |
+-----------------------------------------------------------------------------+
+----------------------------------------+
|i18 : ea36 = exteriorExtension(3, 6, QQ)|
|                                        |
|o18 = ExteriorExtension{...14...}       |
|                                        |
|o18 : ExteriorExtension                 |
+----------------------------------------+

Let's try a slightly larger example.

+-----------------------------------------------------------------------------+
|i19 : printWidth = 200                                                       |
|                                                                             |
|o19 = 200                                                                    |
+-----------------------------------------------------------------------------+
|i20 : A = ea36.ad first ea36.bases#1                                         |
|                                                                             |
|o20 = | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 -1 0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  -1  |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 1 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 1 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 1 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  -1 0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  -1 0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|      | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0   |
|                                                                             |
|               55       55                                                   |
|o20 : Matrix QQ   <-- QQ                                                     |
|-----------------------------------------------------------------------------|
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  1/2 |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  1   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  3/2 |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  1   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  1/2 |                   |
|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 -1 0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  1 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 -1 0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 -1 0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  1 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 -1 0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 -1 0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  1 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  -1 0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|-1 0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
|0  0 0 0 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0  0 0  0   |                   |
+-----------------------------------------------------------------------------+

Produces the \(55 \times 55\) adjoint matrix associated to the first basis
vector of \(\mathfra{a}_1\), and computes the ranks of the blocks for each
power.

+----------------------------------+
|i21 : ea36.prettyBlockPowerRanks A|
|                                  |
|      +---+---+---+---+-----+     |
|o21 = |g00|g10|g01|g11|total|     |
|      +---+---+---+---+-----+     |
|      |0  |10 |10 |0  |20   |     |
|      +---+---+---+---+-----+     |
|      |0  |0  |0  |1  |1    |     |
|      +---+---+---+---+-----+     |
|      |0  |0  |0  |0  |0    |     |
|      +---+---+---+---+-----+     |
|      |0  |0  |0  |0  |0    |     |
|      +---+---+---+---+-----+     |
|      |0  |0  |0  |0  |0    |     |
|      +---+---+---+---+-----+     |
+----------------------------------+

This computes the ranks of the blocks for each power of the matrix. The blocks
are determined by the dimensions of the graded pieces of the algebra.

Ways to use exteriorExtension:
==============================

  * "exteriorExtension(ZZ,ZZ)"
  * "exteriorExtension(ZZ,ZZ,Ring)"
  * "exteriorExtension(ZZ,ZZ,Symbol)"
  * "exteriorExtension(ZZ,ZZ,Symbol,Ring)"

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

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

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

The source of this document is in ExteriorExtensions.m2:415:0.

ExteriorExtension -- A new algebra constructed from the direct sum of a Lie algebra and a module
************************************************************************************************

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

A ExteriorExtension (treated as a mutable hash table) is built by the command
exteriorExtension. The menu lists the ways to interact with an
ExteriorExtension. The operation takes the Lie algebra
\(\mathfrak{sl}_n(\mathbb{F}\) of traceless \(n \times n\) matrices over a
field \(\mathbb{F}\) and appends the non-zero graded piece of an exterior
algebra \(\bigoplus_{i=1}^{n-1}\bigwedge^i \mathbb{F}^n \).

+----------------------------------+
|i1 : ea = exteriorExtension(4, 8);|
+----------------------------------+

Methods that use an object of class ExteriorExtension:
======================================================

  * KillingMatrix(ExteriorExtension) (missing documentation)
  * structureTensor(ExteriorExtension) (missing documentation)

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

The object ExteriorExtension, is a type (Macaulay2Doc),
with ancestor classes MutableHashTable (Macaulay2Doc),
< HashTable (Macaulay2Doc) <
Thing (Macaulay2Doc).

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

The source of this document is in ExteriorExtensions.m2:321:0.

LieAlgebra -- a placeholder for the base Lie algebra
****************************************************

  * Usage:
        extensionAlgebra.LieAlgebra

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

Exporting this symbol gives the ring a name that the user can see when, for
instance, matrices are constructed between modules over this ring

+-----------------------------------------------------------------------------+
|i1 : extensionAlg = exteriorExtension(3, 6, e, QQ);                          |
+-----------------------------------------------------------------------------+
|i2 : gens extensionAlg.LieAlgebra                                            |
|                                                                             |
|o2 = {h , h , h , h , h , E      , E      , E      , E      , E      , E     |
|       1   2   3   4   5   {0, 1}   {0, 2}   {0, 3}   {0, 4}   {0, 5}   {1,  |
|     ------------------------------------------------------------------------|
|       , E      , E      , E      , E      , E      , E      , E      , E    |
|     2}   {1, 3}   {1, 4}   {1, 5}   {2, 3}   {2, 4}   {2, 5}   {3, 4}   {3, |
|     ------------------------------------------------------------------------|
|       , E      , E      , E      , E      , E      , E      , E      , E    |
|     5}   {4, 5}   {1, 0}   {2, 0}   {3, 0}   {4, 0}   {5, 0}   {2, 1}   {3, |
|     ------------------------------------------------------------------------|
|       , E      , E      , E      , E      , E      , E      , E      , E    |
|     1}   {4, 1}   {5, 1}   {3, 2}   {4, 2}   {5, 2}   {4, 3}   {5, 3}   {5, |
|     ------------------------------------------------------------------------|
|       }                                                                     |
|     4}                                                                      |
|                                                                             |
|o2 : List                                                                    |
+-----------------------------------------------------------------------------+

This is the underlying matrix Lie algebra.

+-----------------------------------------------------------------------------+
|i3 : extensionAlg.bases#0                                                    |
|                                                                             |
|o3 = {| 1 0  0 0 0 0 |, | 0 0 0  0 0 0 |, | 0 0 0 0  0 0 |, | 0 0 0 0 0  0 |,|
|      | 0 -1 0 0 0 0 |  | 0 1 0  0 0 0 |  | 0 0 0 0  0 0 |  | 0 0 0 0 0  0 | |
|      | 0 0  0 0 0 0 |  | 0 0 -1 0 0 0 |  | 0 0 1 0  0 0 |  | 0 0 0 0 0  0 | |
|      | 0 0  0 0 0 0 |  | 0 0 0  0 0 0 |  | 0 0 0 -1 0 0 |  | 0 0 0 1 0  0 | |
|      | 0 0  0 0 0 0 |  | 0 0 0  0 0 0 |  | 0 0 0 0  0 0 |  | 0 0 0 0 -1 0 | |
|      | 0 0  0 0 0 0 |  | 0 0 0  0 0 0 |  | 0 0 0 0  0 0 |  | 0 0 0 0 0  0 | |
|     ------------------------------------------------------------------------|
|     | 0 0 0 0 0 0  |, | 0 1 0 0 0 0 |, | 0 0 1 0 0 0 |, | 0 0 0 1 0 0 |, | 0|
|     | 0 0 0 0 0 0  |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0|
|     | 0 0 0 0 0 0  |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0|
|     | 0 0 0 0 0 0  |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0|
|     | 0 0 0 0 1 0  |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0|
|     | 0 0 0 0 0 -1 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0|
|     ------------------------------------------------------------------------|
|     0 0 0 1 0 |, | 0 0 0 0 0 1 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 |
|     0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 1 0 0 0 |  | 0 0 0 1 0 0 |  | 0 0 0 |
|     0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 |
|     0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 |
|     0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 |
|     0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 |
|     ------------------------------------------------------------------------|
|     0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 |
|     0 1 0 |  | 0 0 0 0 0 1 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 |
|     0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 1 0 0 |  | 0 0 0 0 1 0 |  | 0 0 0 0 0 |
|     0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 |
|     0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 |
|     0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 |
|     ------------------------------------------------------------------------|
|     0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |,|
|     0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 1 0 0 0 0 0 | |
|     1 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 | |
|     0 |  | 0 0 0 0 1 0 |  | 0 0 0 0 0 1 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 | |
|     0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 1 |  | 0 0 0 0 0 0 | |
|     0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 | |
|     ------------------------------------------------------------------------|
|     | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 |
|     | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 |
|     | 1 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 |
|     | 0 0 0 0 0 0 |  | 1 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 |
|     | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 1 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 |
|     | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 1 0 0 0 0 0 |  | 0 |
|     ------------------------------------------------------------------------|
|     0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 |
|     0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 |
|     1 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 |
|     0 0 0 0 0 |  | 0 1 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 1 |
|     0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 1 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 |
|     0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 1 0 0 0 0 |  | 0 0 0 |
|     ------------------------------------------------------------------------|
|     0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 0 |, | 0 0 0 0 0 |
|     0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 |
|     0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 |
|     0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 0 0 |
|     0 0 0 |  | 0 0 1 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 1 0 0 |  | 0 0 0 0 0 |
|     0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 1 0 0 0 |  | 0 0 0 0 0 0 |  | 0 0 0 1 0 |
|     ------------------------------------------------------------------------|
|     0 |, | 0 0 0 0 0 0 |}                                                   |
|     0 |  | 0 0 0 0 0 0 |                                                    |
|     0 |  | 0 0 0 0 0 0 |                                                    |
|     0 |  | 0 0 0 0 0 0 |                                                    |
|     0 |  | 0 0 0 0 0 0 |                                                    |
|     0 |  | 0 0 0 0 1 0 |                                                    |
|                                                                             |
|o3 : List                                                                    |
+-----------------------------------------------------------------------------+

This is another way to find the basis of the Lie algebra as it is the 0-graded
piece of the Extension Algebra

See also
========

  * bracket -- Compute products in an ExteriorExtension
  * bases -- gives a list of the bases of the different graded
    pieces of the ExteriorExtension

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

The object LieAlgebra, is a symbol (Macaulay2Doc).

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

The source of this document is in ExteriorExtensions.m2:524:0.

appendage -- a placeholder for the object (exterior algebra) added on to the Lie algebra by exteriorExtension
*************************************************************************************************************

  * Usage:
        extensionAlgebra.appendage

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

Exporting this symbol gives the ring a name that the user can see when, for
instance, matrices are constructed between modules over this ring

+---------------------------------------------------+
|i1 : extensionAlg = exteriorExtension(3, 6, e, QQ);|
+---------------------------------------------------+
|i2 : gens extensionAlg.appendage                   |
|                                                   |
|o2 = {e , e , e , e , e , e }                      |
|       0   1   2   3   4   5                       |
|                                                   |
|o2 : List                                          |
+---------------------------------------------------+
|i3 : e_0*e_1                                       |
|                                                   |
|o3 = e e                                           |
|      0 1                                          |
|                                                   |
|o3 : QQ[e ..e ]                                    |
|         0   5                                     |
+---------------------------------------------------+

This is the exterior algebra.

See also
========

  * bracket -- Compute products in an ExteriorExtension
  * HodgeStar -- The Hodge star in an exterior algebra

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

The object appendage, is a symbol (Macaulay2Doc).

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

The source of this document is in ExteriorExtensions.m2:491:0.

findGrade -- Find which graded piece of the algebra an element is in
********************************************************************

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

Let's build a 7-graded algebra and find the grade for several elements

+---------------------------------------------+
|i1 : extensionAlg = exteriorExtension(3, 7); |
+---------------------------------------------+
|i2 : E = extensionAlg.appendage;             |
+---------------------------------------------+
|i3 : extensionAlg.findGrade (E_0*E_1)        |
|                                             |
|o3 = 3                                       |
+---------------------------------------------+
|i4 : extensionAlg.findGrade (E_0*E_1*E_2*E_3)|
|                                             |
|o4 = 6                                       |
+---------------------------------------------+

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

The object findGrade, is a symbol (Macaulay2Doc).

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

The source of this document is in ExteriorExtensions.m2:778:0.

HodgeStar -- The Hodge star in an exterior algebra
**************************************************

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

The Hodge star works like this:

+-----------------------------------------+
|i1 : ea = exteriorExtension(3, 6, e, QQ);|
+-----------------------------------------+
|i2 : ea.HodgeStar(first ea.bases#1)      |
|                                         |
|o2 = e e e                               |
|      3 4 5                              |
|                                         |
|o2 : QQ[e ..e ]                          |
|         0   5                           |
+-----------------------------------------+

This is the exterior algebra. HodgeStar of a module is the Hodge dual.

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

The object HodgeStar, is a symbol (Macaulay2Doc).

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

The source of this document is in ExteriorExtensions.m2:681:0.

bases -- gives a list of the bases of the different graded pieces of the ExteriorExtension
******************************************************************************************

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

Let's build a 7-graded algebra and see the bases of each piece

+-----------------------------------------------------------------------------+
|i1 : extensionAlg = exteriorExtension(3, 7);                                 |
+-----------------------------------------------------------------------------+
|i2 : extensionAlg.bases#0                                                    |
|                                                                             |
|o2 = {| 1 0  0 0 0 0 0 |, | 0 0 0  0 0 0 0 |, | 0 0 0 0  0 0 0 |, | 0 0 0 0  |
|      | 0 -1 0 0 0 0 0 |  | 0 1 0  0 0 0 0 |  | 0 0 0 0  0 0 0 |  | 0 0 0 0  |
|      | 0 0  0 0 0 0 0 |  | 0 0 -1 0 0 0 0 |  | 0 0 1 0  0 0 0 |  | 0 0 0 0  |
|      | 0 0  0 0 0 0 0 |  | 0 0 0  0 0 0 0 |  | 0 0 0 -1 0 0 0 |  | 0 0 0 1  |
|      | 0 0  0 0 0 0 0 |  | 0 0 0  0 0 0 0 |  | 0 0 0 0  0 0 0 |  | 0 0 0 0  |
|      | 0 0  0 0 0 0 0 |  | 0 0 0  0 0 0 0 |  | 0 0 0 0  0 0 0 |  | 0 0 0 0  |
|      | 0 0  0 0 0 0 0 |  | 0 0 0  0 0 0 0 |  | 0 0 0 0  0 0 0 |  | 0 0 0 0  |
|     ------------------------------------------------------------------------|
|     0  0 0 |, | 0 0 0 0 0 0  0 |, | 0 0 0 0 0 0 0  |, | 0 1 0 0 0 0 0 |, | 0|
|     0  0 0 |  | 0 0 0 0 0 0  0 |  | 0 0 0 0 0 0 0  |  | 0 0 0 0 0 0 0 |  | 0|
|     0  0 0 |  | 0 0 0 0 0 0  0 |  | 0 0 0 0 0 0 0  |  | 0 0 0 0 0 0 0 |  | 0|
|     0  0 0 |  | 0 0 0 0 0 0  0 |  | 0 0 0 0 0 0 0  |  | 0 0 0 0 0 0 0 |  | 0|
|     -1 0 0 |  | 0 0 0 0 1 0  0 |  | 0 0 0 0 0 0 0  |  | 0 0 0 0 0 0 0 |  | 0|
|     0  0 0 |  | 0 0 0 0 0 -1 0 |  | 0 0 0 0 0 1 0  |  | 0 0 0 0 0 0 0 |  | 0|
|     0  0 0 |  | 0 0 0 0 0 0  0 |  | 0 0 0 0 0 0 -1 |  | 0 0 0 0 0 0 0 |  | 0|
|     ------------------------------------------------------------------------|
|     0 1 0 0 0 0 |, | 0 0 0 1 0 0 0 |, | 0 0 0 0 1 0 0 |, | 0 0 0 0 0 1 0 |, |
|     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  |
|     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  |
|     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  |
|     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  |
|     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  |
|     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  |
|     ------------------------------------------------------------------------|
|     | 0 0 0 0 0 0 1 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0|
|     | 0 0 0 0 0 0 0 |  | 0 0 1 0 0 0 0 |  | 0 0 0 1 0 0 0 |  | 0 0 0 0 1 0 0|
|     | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0|
|     | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0|
|     | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0|
|     | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0|
|     | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0|
|     ------------------------------------------------------------------------|
|     |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 |
|     |  | 0 0 0 0 0 1 0 |  | 0 0 0 0 0 0 1 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 |
|     |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 1 0 0 0 |  | 0 0 0 0 1 |
|     |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 |
|     |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 |
|     |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 |
|     |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 |
|     ------------------------------------------------------------------------|
|     0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 |
|     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 |
|     0 0 |  | 0 0 0 0 0 1 0 |  | 0 0 0 0 0 0 1 |  | 0 0 0 0 0 0 0 |  | 0 0 0 |
|     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 1 0 0 |  | 0 0 0 |
|     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 |
|     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 |
|     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 |
|     ------------------------------------------------------------------------|
|     0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 |
|     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 |
|     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 |
|     0 0 1 0 |  | 0 0 0 0 0 0 1 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 |
|     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 1 0 |  | 0 0 0 0 0 0 1 |  | 0 |
|     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 |
|     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 |
|     ------------------------------------------------------------------------|
|     0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, |
|     0 0 0 0 0 0 |  | 1 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  |
|     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 1 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  |
|     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 1 0 0 0 0 0 0 |  |
|     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  |
|     0 0 0 0 0 1 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  |
|     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  |
|     ------------------------------------------------------------------------|
|     | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0|
|     | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0|
|     | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 1 0 0 0 0 0|
|     | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0|
|     | 1 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0|
|     | 0 0 0 0 0 0 0 |  | 1 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0|
|     | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 1 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0|
|     ------------------------------------------------------------------------|
|     |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 |
|     |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 |
|     |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 |
|     |  | 0 1 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 |
|     |  | 0 0 0 0 0 0 0 |  | 0 1 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 |
|     |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 1 0 0 0 0 0 |  | 0 0 0 0 0 |
|     |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 1 0 0 0 |
|     ------------------------------------------------------------------------|
|     0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 |
|     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 |
|     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 |
|     0 0 |  | 0 0 1 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 |
|     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 1 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 |
|     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 1 0 0 0 0 |  | 0 0 0 |
|     0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 1 |
|     ------------------------------------------------------------------------|
|     0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 |
|     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 |
|     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 |
|     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 |
|     0 0 0 0 |  | 0 0 0 1 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 |
|     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 1 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 |
|     0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 1 0 0 0 |  | 0 |
|     ------------------------------------------------------------------------|
|     0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |, | 0 0 0 0 0 0 0 |}                    |
|     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |                     |
|     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |                     |
|     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |                     |
|     0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |                     |
|     0 0 0 1 0 0 |  | 0 0 0 0 0 0 0 |  | 0 0 0 0 0 0 0 |                     |
|     0 0 0 0 0 0 |  | 0 0 0 0 1 0 0 |  | 0 0 0 0 0 1 0 |                     |
|                                                                             |
|o2 : List                                                                    |
+-----------------------------------------------------------------------------+
|i3 : extensionAlg.bases#1                                                    |
|                                                                             |
|o3 = {e e e , e e e , e e e , e e e , e e e , e e e , e e e , e e e , e e e ,|
|       0 1 2   0 1 3   0 1 4   0 1 5   0 1 6   0 2 3   0 2 4   0 2 5   0 2 6 |
|     ------------------------------------------------------------------------|
|     e e e , e e e , e e e , e e e , e e e , e e e , e e e , e e e , e e e , |
|      0 3 4   0 3 5   0 3 6   0 4 5   0 4 6   0 5 6   1 2 3   1 2 4   1 2 5  |
|     ------------------------------------------------------------------------|
|     e e e , e e e , e e e , e e e , e e e , e e e , e e e , e e e , e e e , |
|      1 2 6   1 3 4   1 3 5   1 3 6   1 4 5   1 4 6   1 5 6   2 3 4   2 3 5  |
|     ------------------------------------------------------------------------|
|     e e e , e e e , e e e , e e e , e e e , e e e , e e e , e e e }         |
|      2 3 6   2 4 5   2 4 6   2 5 6   3 4 5   3 4 6   3 5 6   4 5 6          |
|                                                                             |
|o3 : List                                                                    |
+-----------------------------------------------------------------------------+
|i4 : extensionAlg.bases#3                                                    |
|                                                                             |
|o4 = {e e , e e , e e , e e , e e , e e , e e , e e , e e , e e , e e , e e ,|
|       0 1   0 2   0 3   0 4   0 5   0 6   1 2   1 3   1 4   1 5   1 6   2 3 |
|     ------------------------------------------------------------------------|
|     e e , e e , e e , e e , e e , e e , e e , e e , e e }                   |
|      2 4   2 5   2 6   3 4   3 5   3 6   4 5   4 6   5 6                    |
|                                                                             |
|o4 : List                                                                    |
+-----------------------------------------------------------------------------+

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

The object bases, is a symbol (Macaulay2Doc).

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

The source of this document is in ExteriorExtensions.m2:797:0.

bracket -- Compute products in an ExteriorExtension
***************************************************

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

The bracket of two elements in the Exterior Extension. This is a method
function that accepts pairs from any part of the ExteriorExtension

+-----------------------------------------+
|i1 : ea = exteriorExtension(2, 4, e, QQ);|
+-----------------------------------------+
|i2 : A = first ea.bases#0                |
|                                         |
|o2 = | 1 0  0 0 |                        |
|     | 0 -1 0 0 |                        |
|     | 0 0  0 0 |                        |
|     | 0 0  0 0 |                        |
|                                         |
|              4       4                  |
|o2 : Matrix QQ  <-- QQ                   |
+-----------------------------------------+
|i3 : B = last ea.bases#1                 |
|                                         |
|o3 = e e                                 |
|      2 3                                |
|                                         |
|o3 : QQ[e ..e ]                          |
|         0   3                           |
+-----------------------------------------+
|i4 : C = ea.bracket(A, B)                |
|                                         |
|o4 = 0                                   |
|                                         |
|o4 : QQ[e ..e ]                          |
|         0   3                           |
+-----------------------------------------+

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

The object bracket, is a symbol (Macaulay2Doc).

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

The source of this document is in ExteriorExtensions.m2:545:0.

ad -- Constructs the standard matrix representation of the adjoint operator
***************************************************************************

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

The matrices constructed by this can get very large.

+-----------------------------------------+
|i1 : ea = exteriorExtension(3, 7, e, QQ);|
+-----------------------------------------+
|i2 : A = ea.ad(first ea.bases#1);        |
|                                         |
|              174       174              |
|o2 : Matrix QQ    <-- QQ                 |
+-----------------------------------------+
|i3 : rank A                              |
|                                         |
|o3 = 46                                  |
+-----------------------------------------+

This rank is an invariant

See also
========

  * powerRanks -- Compute the ranks of powers of a matrix
  * powerTraces -- Compute the traces of powers of a matrix
  * blockPowerRanks -- Computes a list of ranks of
    blocks for each power of a matrix representation of an adjoint operator for
    ExteriorExtensions
  * prettyBlockPowerRanks -- same as
    blockPowerRanks, but prettier
  * getBlock -- Extract a block of the matrix representation
    of an adjoint operator for ExteriorExtensions

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

The object ad, is a symbol (Macaulay2Doc).

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

The source of this document is in ExteriorExtensions.m2:571:0.

powerRanks -- Compute the ranks of powers of a matrix
*****************************************************

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

This is what you do:

+--------------------------------------+
|i1 : ea = exteriorExtension(3, 6, QQ);|
+--------------------------------------+
|i2 : A = ea.ad(first ea.bases#1);     |
|                                      |
|              55       55             |
|o2 : Matrix QQ   <-- QQ               |
+--------------------------------------+
|i3 : ea.powerRanks A                  |
|                                      |
|o3 = {20, 1, 0}                       |
|                                      |
|o3 : List                             |
+--------------------------------------+

The list outputs the rank of each power. These are invariants of the element x
in ad(x).

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

The object powerRanks, is a symbol (Macaulay2Doc).

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

The source of this document is in ExteriorExtensions.m2:591:0.

powerTraces -- Compute the traces of powers of a matrix
*******************************************************

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

This is what you do:

+--------------------------------------+
|i1 : ea = exteriorExtension(3, 6, QQ);|
+--------------------------------------+
|i2 : A = ea.ad(first ea.bases#1);     |
|                                      |
|              55       55             |
|o2 : Matrix QQ   <-- QQ               |
+--------------------------------------+
|i3 : ea.powerTraces A                 |
|                                      |
|     +-----+-----+----+               |
|o3 = |power|trace|rank|               |
|     +-----+-----+----+               |
|     |1    |0    |20  |               |
|     +-----+-----+----+               |
|     |2    |0    |1   |               |
|     +-----+-----+----+               |
+--------------------------------------+

The list outputs the trace of each power until the power ranks stabilize, or
until the power is equal to the dimension of the algebra. These are invariants
of the element x in ad(x).

+-----------------------------------------------------------------------------+
|i4 : ea = exteriorExtension(3, 9, QQ);                                       |
+-----------------------------------------------------------------------------+
|i5 : E = ea.appendage                                                        |
|                                                                             |
|o5 = E                                                                       |
|                                                                             |
|o5 : PolynomialRing, 9 skew commutative variable(s)                          |
+-----------------------------------------------------------------------------+
|i6 : A = ea.ad(E_0*E_1*E_2 + E_3*E_4*E_5 + E_6*E_7*E_8);                     |
|                                                                             |
|              248       248                                                  |
|o6 : Matrix QQ    <-- QQ                                                     |
+-----------------------------------------------------------------------------+
|i7 : ea.powerTraces A                                                        |
|                                                                             |
|     +-----+-------------------------------------------------------------+---|
|o7 = |power|trace                                                        |ran|
|     +-----+-------------------------------------------------------------+---|
|     |1    |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |2    |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |3    |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |4    |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |5    |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |6    |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |7    |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |8    |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |9    |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |10   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |11   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |12   |4536                                                         |168|
|     +-----+-------------------------------------------------------------+---|
|     |13   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |14   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |15   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |16   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |17   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |18   |-117936                                                      |168|
|     +-----+-------------------------------------------------------------+---|
|     |19   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |20   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |21   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |22   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |23   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |24   |3188808                                                      |168|
|     +-----+-------------------------------------------------------------+---|
|     |25   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |26   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |27   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |28   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |29   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |30   |-86093280                                                    |168|
|     +-----+-------------------------------------------------------------+---|
|     |31   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |32   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |33   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |34   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |35   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |36   |2324523096                                                   |168|
|     +-----+-------------------------------------------------------------+---|
|     |37   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |38   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |39   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |40   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |41   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |42   |-62762119056                                                 |168|
|     +-----+-------------------------------------------------------------+---|
|     |43   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |44   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |45   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |46   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |47   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |48   |1694577219048                                                |168|
|     +-----+-------------------------------------------------------------+---|
|     |49   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |50   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |51   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |52   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |53   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |54   |-45753584909760                                              |168|
|     +-----+-------------------------------------------------------------+---|
|     |55   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |56   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |57   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |58   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |59   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |60   |1235346792568056                                             |168|
|     +-----+-------------------------------------------------------------+---|
|     |61   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |62   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |63   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |64   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |65   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |66   |-33354363399332976                                           |168|
|     +-----+-------------------------------------------------------------+---|
|     |67   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |68   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |69   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |70   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |71   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |72   |900567811781994888                                           |168|
|     +-----+-------------------------------------------------------------+---|
|     |73   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |74   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |75   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |76   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |77   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |78   |-24315330918113857440                                        |168|
|     +-----+-------------------------------------------------------------+---|
|     |79   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |80   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |81   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |82   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |83   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |84   |656513934789074155416                                        |168|
|     +-----+-------------------------------------------------------------+---|
|     |85   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |86   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |87   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |88   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |89   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |90   |-17725876239305002191696                                     |168|
|     +-----+-------------------------------------------------------------+---|
|     |91   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |92   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |93   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |94   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |95   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |96   |478598658461235059180328                                     |168|
|     +-----+-------------------------------------------------------------+---|
|     |97   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |98   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |99   |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |100  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |101  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |102  |-12922163778453346597864320                                  |168|
|     +-----+-------------------------------------------------------------+---|
|     |103  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |104  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |105  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |106  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |107  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |108  |348898422018240358142341176                                  |168|
|     +-----+-------------------------------------------------------------+---|
|     |109  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |110  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |111  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |112  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |113  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |114  |-9420257394492489669843207216                                |168|
|     +-----+-------------------------------------------------------------+---|
|     |115  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |116  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |117  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |118  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |119  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |120  |254346949651297221085766599368                               |168|
|     +-----+-------------------------------------------------------------+---|
|     |121  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |122  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |123  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |124  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |125  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |126  |-6867367640585024969315698178400                             |168|
|     +-----+-------------------------------------------------------------+---|
|     |127  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |128  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |129  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |130  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |131  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |132  |185418926295795674171523850821336                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |133  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |134  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |135  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |136  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |137  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |138  |-5006311009986483202631143972171536                          |168|
|     +-----+-------------------------------------------------------------+---|
|     |139  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |140  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |141  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |142  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |143  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |144  |135170397269635046471040887248636008                         |168|
|     +-----+-------------------------------------------------------------+---|
|     |145  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |146  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |147  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |148  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |149  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |150  |-3649600726280146254718103955713167680                       |168|
|     +-----+-------------------------------------------------------------+---|
|     |151  |0                                                            |168|
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|     +-----+-------------------------------------------------------------+---|
|     |153  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |154  |0                                                            |168|
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|     +-----+-------------------------------------------------------------+---|
|     |156  |98539219609563948877388806804255531896                       |168|
|     +-----+-------------------------------------------------------------+---|
|     |157  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |158  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |159  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |160  |0                                                            |168|
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|     +-----+-------------------------------------------------------------+---|
|     |162  |-2660558929458226619689497783714899356656                    |168|
|     +-----+-------------------------------------------------------------+---|
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|     +-----+-------------------------------------------------------------+---|
|     |166  |0                                                            |168|
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|     +-----+-------------------------------------------------------------+---|
|     |168  |71835091095372118731616440160302282634248                    |168|
|     +-----+-------------------------------------------------------------+---|
|     |169  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |170  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |171  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
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|     |183  |0                                                            |168|
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|     +-----+-------------------------------------------------------------+---|
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|     |186  |-1413930098030209412994406391675229829086714576              |168|
|     +-----+-------------------------------------------------------------+---|
|     |187  |0                                                            |168|
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|     +-----+-------------------------------------------------------------+---|
|     |189  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
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|     +-----+-------------------------------------------------------------+---|
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|     +-----+-------------------------------------------------------------+---|
|     |196  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |197  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |198  |-1030755041464022662072922259531242545404215043840           |168|
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|     +-----+-------------------------------------------------------------+---|
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|     +-----+-------------------------------------------------------------+---|
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|     +-----+-------------------------------------------------------------+---|
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|     |203  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |204  |27830386119528611875968901007343548725913806188216           |168|
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|     |205  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |206  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |207  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |208  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |209  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |210  |-751420425227272520651160327198275815599672767077296         |168|
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|     +-----+-------------------------------------------------------------+---|
|     |213  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |214  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |215  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |216  |20288351481136358057581328834353447021191164711091528        |168|
|     +-----+-------------------------------------------------------------+---|
|     |217  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |218  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |219  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |220  |0                                                            |168|
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|     |221  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |222  |-547785489990681667554695878527543069572161447199466720      |168|
|     +-----+-------------------------------------------------------------+---|
|     |223  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |224  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |225  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |226  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |227  |0                                                            |168|
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|     |228  |14790208229748405023976788720243662878448359074385605976     |168|
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|     |229  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |230  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |231  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |232  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |233  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |234  |-399335622203206935647373295446578897718105695008411356816   |168|
|     +-----+-------------------------------------------------------------+---|
|     |235  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |236  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |237  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |238  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |239  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |240  |10782061799486587262479078977057630238388853765227106638568  |168|
|     +-----+-------------------------------------------------------------+---|
|     |241  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |242  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |243  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |244  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |245  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |246  |-291115668586137856086935132380556016436499051661131879236800|168|
|     +-----+-------------------------------------------------------------+---|
|     |247  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|     |248  |0                                                            |168|
|     +-----+-------------------------------------------------------------+---|
|-----------------------------------------------------------------------------|
|-+                                                                           |
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+-----------------------------------------------------------------------------+

The list could be quite long, as this example shows.

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

The object powerTraces, is a symbol (Macaulay2Doc).

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

The source of this document is in ExteriorExtensions.m2:619:0.

getBlock -- Extract a block of the matrix representation of an adjoint operator for ExteriorExtensions
******************************************************************************************************

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

The blocks get extracted based on their grade

+-----------------------------------------------------------------------------+
|i1 : ea = exteriorExtension(3, 6, e, QQ);                                    |
+-----------------------------------------------------------------------------+
|i2 : A = ea.ad(first ea.bases#1);                                            |
|                                                                             |
|              55       55                                                    |
|o2 : Matrix QQ   <-- QQ                                                      |
+-----------------------------------------------------------------------------+
|i3 : printWidth = 300;                                                       |
+-----------------------------------------------------------------------------+
|i4 : ea.getBlock(1, 1, A)                                                    |
|                                                                             |
|o4 = 0                                                                       |
|                                                                             |
|              20       20                                                    |
|o4 : Matrix QQ   <-- QQ                                                      |
+-----------------------------------------------------------------------------+
|i5 : ea.getBlock(1, 0, A)                                                    |
|                                                                             |
|o5 = | 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0  0 |
|     | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 -1 0  0 |
|     | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  -1 0 |
|     | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0  -1|
|     | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 1 0 0 0  0  0 |
|     | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 1 0 0  0  0 |
|     | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 1 0  0  0 |
|     | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0  0 |
|     | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0  0 |
|     | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0  0 |
|     | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0  0  0 0 0 0 0  0  0 |
|     | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  -1 0  0 0 0 0 0  0  0 |
|     | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  -1 0 0 0 0 0  0  0 |
|     | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0  0 |
|     | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0  0 |
|     | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0  0 |
|     | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0  0 |
|     | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0  0 |
|     | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0  0 |
|     | 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0  0  0 0 0 0 0  0  0 |
|                                                                             |
|              20       35                                                    |
|o5 : Matrix QQ   <-- QQ                                                      |
|-----------------------------------------------------------------------------|
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
|0 0 0 |                                                                      |
+-----------------------------------------------------------------------------+

This extracts blocks of the operator. The sizes are determined by the graded
pieces of the algebra.

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

The object getBlock, is a symbol (Macaulay2Doc).

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

The source of this document is in ExteriorExtensions.m2:708:0.

blockPowerRanks -- Computes a list of ranks of blocks for each power of a matrix representation of an adjoint operator for ExteriorExtensions
*********************************************************************************************************************************************

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

+-----------------------------------------------------------------------------+
|i1 : ea = exteriorExtension(3, 6, e, QQ);                                    |
+-----------------------------------------------------------------------------+
|i2 : A = ea.ad(first ea.bases#1);                                            |
|                                                                             |
|              55       55                                                    |
|o2 : Matrix QQ   <-- QQ                                                      |
+-----------------------------------------------------------------------------+
|i3 : ea.blockPowerRanks(A)                                                   |
|                                                                             |
|o3 = {{0, 10, 10, 0, 20}, {0, 0, 0, 1, 1}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, |
|     ------------------------------------------------------------------------|
|     {0, 0, 0, 0, 0}}                                                        |
|                                                                             |
|o3 : List                                                                    |
+-----------------------------------------------------------------------------+

This computes a list of ranks of blocks for each power of a matrix
representation of an adjoint operator for ExteriorExtensions

See also
========

  * prettyBlockPowerRanks -- same as
    blockPowerRanks, but prettier

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

The object blockPowerRanks, is a symbol (Macaulay2Doc).

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

The source of this document is in ExteriorExtensions.m2:733:0.

prettyBlockPowerRanks -- same as blockPowerRanks, but prettier
**************************************************************

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

+-----------------------------------------+
|i1 : ea = exteriorExtension(3, 6, e, QQ);|
+-----------------------------------------+
|i2 : A = ea.ad(first ea.bases#1);        |
|                                         |
|              55       55                |
|o2 : Matrix QQ   <-- QQ                  |
+-----------------------------------------+
|i3 : ea.prettyBlockPowerRanks(A)         |
|                                         |
|     +---+---+---+---+-----+             |
|o3 = |g00|g10|g01|g11|total|             |
|     +---+---+---+---+-----+             |
|     |0  |10 |10 |0  |20   |             |
|     +---+---+---+---+-----+             |
|     |0  |0  |0  |1  |1    |             |
|     +---+---+---+---+-----+             |
|     |0  |0  |0  |0  |0    |             |
|     +---+---+---+---+-----+             |
|     |0  |0  |0  |0  |0    |             |
|     +---+---+---+---+-----+             |
|     |0  |0  |0  |0  |0    |             |
|     +---+---+---+---+-----+             |
+-----------------------------------------+

This computes a list of ranks of blocks for each power of a matrix
representation of an adjoint operator for ExteriorExtensions and displays it as
a pretty table.

See also
========

  * blockPowerRanks -- Computes a list of ranks of
    blocks for each power of a matrix representation of an adjoint operator for
    ExteriorExtensions

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

The object prettyBlockPowerRanks, is a
symbol (Macaulay2Doc).

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

The source of this document is in ExteriorExtensions.m2:759:0.

structureTensor -- Constructs the structureTensor of the ExteriorExtension
**************************************************************************

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

The structure tensor of an algebra is the representative of the bilinear
product as a hypermatrix. The matrices constructed by this can get very large,
and take a long time to compute. The output can be very large, so we return a
hashTable with only the non-zero values.

+--------------------------------------+
|i1 : printWidth = 450;                |
+--------------------------------------+
|i2 : ea = exteriorExtension(3, 6, QQ);|
+--------------------------------------+
|i3 : B = structureTensor ea           |
|                                      |
|o3 = HashTable{(0, 5, 5) => 2    }    |
|               (0, 6, 6) => 1         |
|               (0, 7, 7) => 1         |
|               (0, 8, 8) => 1         |
|               (0, 9, 9) => 1         |
|               (0, 10, 10) => -1      |
|               (0, 11, 11) => -1      |
|               (0, 12, 12) => -1      |
|               (0, 13, 13) => -1      |
|               (0, 20, 20) => -2      |
|               (0, 21, 21) => -1      |
|               (0, 22, 22) => -1      |
|               (0, 23, 23) => -1      |
|               (0, 24, 24) => -1      |
|               (0, 25, 25) => 1       |
|               (0, 26, 26) => 1       |
|               (0, 27, 27) => 1       |
|               (0, 28, 28) => 1       |
|               (0, 39, 39) => 1       |
|               (0, 40, 40) => 1       |
|               (0, 41, 41) => 1       |
|               (0, 42, 42) => 1       |
|               (0, 43, 43) => 1       |
|               (0, 44, 44) => 1       |
|               (0, 45, 45) => -1      |
|               (0, 46, 46) => -1      |
|               (0, 47, 47) => -1      |
|               (0, 48, 48) => -1      |
|               (0, 49, 49) => -1      |
|               (0, 50, 50) => -1      |
|               (1, 5, 5) => -1        |
|               (1, 6, 6) => 1         |
|               (1, 10, 10) => 2       |
|               (1, 11, 11) => 1       |
|               (1, 12, 12) => 1       |
|               (1, 13, 13) => 1       |
|               (1, 14, 14) => -1      |
|               (1, 15, 15) => -1      |
|               (1, 16, 16) => -1      |
|               (1, 20, 20) => 1       |
|               (1, 21, 21) => -1      |
|               (1, 25, 25) => -2      |
|               (1, 26, 26) => -1      |
|               (1, 27, 27) => -1      |
|               (1, 28, 28) => -1      |
|               (1, 29, 29) => 1       |
|               (1, 30, 30) => 1       |
|               (1, 31, 31) => 1       |
|               (1, 36, 36) => 1       |
|               (1, 37, 37) => 1       |
|               (1, 38, 38) => 1       |
|               (1, 39, 39) => -1      |
|               (1, 40, 40) => -1      |
|               (1, 41, 41) => -1      |
|               (1, 48, 48) => 1       |
|               (1, 49, 49) => 1       |
|               (1, 50, 50) => 1       |
|               (1, 51, 51) => -1      |
|               (1, 52, 52) => -1      |
|               (1, 53, 53) => -1      |
|               (2, 6, 6) => -1        |
|               (2, 7, 7) => 1         |
|               (2, 10, 10) => -1      |
|               (2, 11, 11) => 1       |
|               (2, 14, 14) => 2       |
|               (2, 15, 15) => 1       |
|               (2, 16, 16) => 1       |
|               (2, 17, 17) => -1      |
|               (2, 18, 18) => -1      |
|               (2, 21, 21) => 1       |
|               (2, 22, 22) => -1      |
|               (2, 25, 25) => 1       |
|               (2, 26, 26) => -1      |
|               (2, 29, 29) => -2      |
|               (2, 30, 30) => -1      |
|               (2, 31, 31) => -1      |
|               (2, 32, 32) => 1       |
|               (2, 33, 33) => 1       |
|               (2, 35, 35) => 1       |
|               (2, 36, 36) => -1      |
|               (2, 40, 40) => 1       |
|               (2, 41, 41) => 1       |
|               (2, 42, 42) => -1      |
|               (2, 43, 43) => -1      |
|               (2, 46, 46) => 1       |
|               (2, 47, 47) => 1       |
|               (2, 48, 48) => -1      |
|               (2, 49, 49) => -1      |
|               (2, 53, 53) => 1       |
|               (2, 54, 54) => -1      |
|               (3, 7, 7) => -1        |
|               (3, 8, 8) => 1         |
|               (3, 11, 11) => -1      |
|               (3, 12, 12) => 1       |
|               (3, 14, 14) => -1      |
|               (3, 15, 15) => 1       |
|               (3, 17, 17) => 2       |
|               (3, 18, 18) => 1       |
|               (3, 19, 19) => -1      |
|               (3, 22, 22) => 1       |
|               (3, 23, 23) => -1      |
|               (3, 26, 26) => 1       |
|               (3, 27, 27) => -1      |
|               (3, 29, 29) => 1       |
|               (3, 30, 30) => -1      |
|               (3, 32, 32) => -2      |
|               (3, 33, 33) => -1      |
|               (3, 34, 34) => 1       |
|               (3, 36, 36) => 1       |
|               (3, 37, 37) => -1      |
|               (3, 39, 39) => 1       |
|               (3, 40, 40) => -1      |
|               (3, 43, 43) => 1       |
|               (3, 44, 44) => -1      |
|               (3, 45, 45) => 1       |
|               (3, 46, 46) => -1      |
|               (3, 49, 49) => 1       |
|               (3, 50, 50) => -1      |
|               (3, 52, 52) => 1       |
|               (3, 53, 53) => -1      |
|               (4, 8, 8) => -1        |
|               (4, 9, 9) => 1         |
|               (4, 12, 12) => -1      |
|               (4, 13, 13) => 1       |
|               (4, 15, 15) => -1      |
|               (4, 16, 16) => 1       |
|               (4, 17, 17) => -1      |
|               (4, 18, 18) => 1       |
|               (4, 19, 19) => 2       |
|               (4, 23, 23) => 1       |
|               (4, 24, 24) => -1      |
|               (4, 27, 27) => 1       |
|               (4, 28, 28) => -1      |
|               (4, 30, 30) => 1       |
|               (4, 31, 31) => -1      |
|               (4, 32, 32) => 1       |
|               (4, 33, 33) => -1      |
|               (4, 34, 34) => -2      |
|               (4, 37, 37) => 1       |
|               (4, 38, 38) => -1      |
|               (4, 40, 40) => 1       |
|               (4, 41, 41) => -1      |
|               (4, 42, 42) => 1       |
|               (4, 43, 43) => -1      |
|               (4, 46, 46) => 1       |
|               (4, 47, 47) => -1      |
|               (4, 48, 48) => 1       |
|               (4, 49, 49) => -1      |
|               (4, 51, 51) => 1       |
|               (4, 52, 52) => -1      |
|               (5, 0, 20) => 1        |
|               (5, 5, 0) => -2        |
|               (5, 5, 1) => 1         |
|               (5, 6, 10) => 1        |
|               (5, 7, 11) => 1        |
|               (5, 8, 12) => 1        |
|               (5, 9, 13) => 1        |
|               (5, 25, 21) => -1      |
|               (5, 26, 22) => -1      |
|               (5, 27, 23) => -1      |
|               (5, 28, 24) => -1      |
|               (5, 39, 45) => 1       |
|               (5, 40, 46) => 1       |
|               (5, 41, 47) => 1       |
|               (5, 42, 48) => 1       |
|               (5, 43, 49) => 1       |
|               (5, 44, 50) => 1       |
|               (6, 0, 21) => 1        |
|               (6, 1, 21) => 1        |
|               (6, 5, 25) => 1        |
|               (6, 6, 0) => -1        |
|               (6, 6, 1) => -1        |
|               (6, 6, 2) => 1         |
|               (6, 7, 14) => 1        |
|               (6, 8, 15) => 1        |
|               (6, 9, 16) => 1        |
|               (6, 10, 20) => -1      |
|               (6, 29, 22) => -1      |
|               (6, 30, 23) => -1      |
|               (6, 31, 24) => -1      |
|               (6, 36, 45) => -1      |
|               (6, 37, 46) => -1      |
|               (6, 38, 47) => -1      |
|               (6, 42, 51) => 1       |
|               (6, 43, 52) => 1       |
|               (6, 44, 53) => 1       |
|               (7, 0, 22) => 1        |
|               (7, 1, 22) => 1        |
|               (7, 2, 22) => 1        |
|               (7, 5, 26) => 1        |
|               (7, 6, 29) => 1        |
|               (7, 7, 0) => -1        |
|               (7, 7, 2) => -1        |
|               (7, 7, 3) => 1         |
|               (7, 8, 17) => 1        |
|               (7, 9, 18) => 1        |
|               (7, 11, 20) => -1      |
|               (7, 14, 21) => -1      |
|               (7, 32, 23) => -1      |
|               (7, 33, 24) => -1      |
|               (7, 35, 45) => 1       |
|               (7, 37, 48) => -1      |
|               (7, 38, 49) => -1      |
|               (7, 40, 51) => -1      |
|               (7, 41, 52) => -1      |
|               (7, 44, 54) => 1       |
|               (8, 0, 23) => 1        |
|               (8, 1, 23) => 1        |
|               (8, 2, 23) => 1        |
|               (8, 3, 23) => 1        |
|               (8, 5, 27) => 1        |
|               (8, 6, 30) => 1        |
|               (8, 7, 32) => 1        |
|               (8, 8, 0) => -1        |
|               (8, 8, 3) => -1        |
|               (8, 8, 4) => 1         |
|               (8, 9, 19) => 1        |
|               (8, 12, 20) => -1      |
|               (8, 15, 21) => -1      |
|               (8, 17, 22) => -1      |
|               (8, 34, 24) => -1      |
|               (8, 35, 46) => 1       |
|               (8, 36, 48) => 1       |
|               (8, 38, 50) => -1      |
|               (8, 39, 51) => 1       |
|               (8, 41, 53) => -1      |
|               (8, 43, 54) => -1      |
|               (9, 0, 24) => 1        |
|               (9, 1, 24) => 1        |
|               (9, 2, 24) => 1        |
|               (9, 3, 24) => 1        |
|               (9, 4, 24) => 1        |
|               (9, 5, 28) => 1        |
|               (9, 6, 31) => 1        |
|               (9, 7, 33) => 1        |
|               (9, 8, 34) => 1        |
|               (9, 9, 0) => -1        |
|               (9, 9, 4) => -1        |
|               (9, 13, 20) => -1      |
|               (9, 16, 21) => -1      |
|               (9, 18, 22) => -1      |
|               (9, 19, 23) => -1      |
|               (9, 35, 47) => 1       |
|               (9, 36, 49) => 1       |
|               (9, 37, 50) => 1       |
|               (9, 39, 52) => 1       |
|               (9, 40, 53) => 1       |
|               (9, 42, 54) => 1       |
|               (10, 1, 25) => 1       |
|               (10, 6, 5) => -1       |
|               (10, 10, 0) => 1       |
|               (10, 10, 1) => -2      |
|               (10, 10, 2) => 1       |
|               (10, 11, 14) => 1      |
|               (10, 12, 15) => 1      |
|               (10, 13, 16) => 1      |
|               (10, 20, 21) => 1      |
|               (10, 29, 26) => -1     |
|               (10, 30, 27) => -1     |
|               (10, 31, 28) => -1     |
|               (10, 36, 39) => 1      |
|               (10, 37, 40) => 1      |
|               (10, 38, 41) => 1      |
|               (10, 48, 51) => 1      |
|               (10, 49, 52) => 1      |
|               (10, 50, 53) => 1      |
|               (11, 1, 26) => 1       |
|               (11, 2, 26) => 1       |
|               (11, 7, 5) => -1       |
|               (11, 10, 29) => 1      |
|               (11, 11, 0) => 1       |
|               (11, 11, 1) => -1      |
|               (11, 11, 2) => -1      |
|               (11, 11, 3) => 1       |
|               (11, 12, 17) => 1      |
|               (11, 13, 18) => 1      |
|               (11, 14, 25) => -1     |
|               (11, 20, 22) => 1      |
|               (11, 32, 27) => -1     |
|               (11, 33, 28) => -1     |
|               (11, 35, 39) => -1     |
|               (11, 37, 42) => 1      |
|               (11, 38, 43) => 1      |
|               (11, 46, 51) => -1     |
|               (11, 47, 52) => -1     |
|               (11, 50, 54) => 1      |
|               (12, 1, 27) => 1       |
|               (12, 2, 27) => 1       |
|               (12, 3, 27) => 1       |
|               (12, 8, 5) => -1       |
|               (12, 10, 30) => 1      |
|               (12, 11, 32) => 1      |
|               (12, 12, 0) => 1       |
|               (12, 12, 1) => -1      |
|               (12, 12, 3) => -1      |
|               (12, 12, 4) => 1       |
|               (12, 13, 19) => 1      |
|               (12, 15, 25) => -1     |
|               (12, 17, 26) => -1     |
|               (12, 20, 23) => 1      |
|               (12, 34, 28) => -1     |
|               (12, 35, 40) => -1     |
|               (12, 36, 42) => -1     |
|               (12, 38, 44) => 1      |
|               (12, 45, 51) => 1      |
|               (12, 47, 53) => -1     |
|               (12, 49, 54) => -1     |
|               (13, 1, 28) => 1       |
|               (13, 2, 28) => 1       |
|               (13, 3, 28) => 1       |
|               (13, 4, 28) => 1       |
|               (13, 9, 5) => -1       |
|               (13, 10, 31) => 1      |
|               (13, 11, 33) => 1      |
|               (13, 12, 34) => 1      |
|               (13, 13, 0) => 1       |
|               (13, 13, 1) => -1      |
|               (13, 13, 4) => -1      |
|               (13, 16, 25) => -1     |
|               (13, 18, 26) => -1     |
|               (13, 19, 27) => -1     |
|               (13, 20, 24) => 1      |
|               (13, 35, 41) => -1     |
|               (13, 36, 43) => -1     |
|               (13, 37, 44) => -1     |
|               (13, 45, 52) => 1      |
|               (13, 46, 53) => 1      |
|               (13, 48, 54) => 1      |
|               (14, 2, 29) => 1       |
|               (14, 7, 6) => -1       |
|               (14, 11, 10) => -1     |
|               (14, 14, 1) => 1       |
|               (14, 14, 2) => -2      |
|               (14, 14, 3) => 1       |
|               (14, 15, 17) => 1      |
|               (14, 16, 18) => 1      |
|               (14, 21, 22) => 1      |
|               (14, 25, 26) => 1      |
|               (14, 32, 30) => -1     |
|               (14, 33, 31) => -1     |
|               (14, 35, 36) => 1      |
|               (14, 40, 42) => 1      |
|               (14, 41, 43) => 1      |
|               (14, 46, 48) => 1      |
|               (14, 47, 49) => 1      |
|               (14, 53, 54) => 1      |
|               (15, 2, 30) => 1       |
|               (15, 3, 30) => 1       |
|               (15, 8, 6) => -1       |
|               (15, 12, 10) => -1     |
|               (15, 14, 32) => 1      |
|               (15, 15, 1) => 1       |
|               (15, 15, 2) => -1      |
|               (15, 15, 3) => -1      |
|               (15, 15, 4) => 1       |
|               (15, 16, 19) => 1      |
|               (15, 17, 29) => -1     |
|               (15, 21, 23) => 1      |
|               (15, 25, 27) => 1      |
|               (15, 34, 31) => -1     |
|               (15, 35, 37) => 1      |
|               (15, 39, 42) => -1     |
|               (15, 41, 44) => 1      |
|               (15, 45, 48) => -1     |
|               (15, 47, 50) => 1      |
|               (15, 52, 54) => -1     |
|               (16, 2, 31) => 1       |
|               (16, 3, 31) => 1       |
|               (16, 4, 31) => 1       |
|               (16, 9, 6) => -1       |
|               (16, 13, 10) => -1     |
|               (16, 14, 33) => 1      |
|               (16, 15, 34) => 1      |
|               (16, 16, 1) => 1       |
|               (16, 16, 2) => -1      |
|               (16, 16, 4) => -1      |
|               (16, 18, 29) => -1     |
|               (16, 19, 30) => -1     |
|               (16, 21, 24) => 1      |
|               (16, 25, 28) => 1      |
|               (16, 35, 38) => 1      |
|               (16, 39, 43) => -1     |
|               (16, 40, 44) => -1     |
|               (16, 45, 49) => -1     |
|               (16, 46, 50) => -1     |
|               (16, 51, 54) => 1      |
|               (17, 3, 32) => 1       |
|               (17, 8, 7) => -1       |
|               (17, 12, 11) => -1     |
|               (17, 15, 14) => -1     |
|               (17, 17, 2) => 1       |
|               (17, 17, 3) => -2      |
|               (17, 17, 4) => 1       |
|               (17, 18, 19) => 1      |
|               (17, 22, 23) => 1      |
|               (17, 26, 27) => 1      |
|               (17, 29, 30) => 1      |
|               (17, 34, 33) => -1     |
|               (17, 36, 37) => 1      |
|               (17, 39, 40) => 1      |
|               (17, 43, 44) => 1      |
|               (17, 45, 46) => 1      |
|               (17, 49, 50) => 1      |
|               (17, 52, 53) => 1      |
|               (18, 3, 33) => 1       |
|               (18, 4, 33) => 1       |
|               (18, 9, 7) => -1       |
|               (18, 13, 11) => -1     |
|               (18, 16, 14) => -1     |
|               (18, 17, 34) => 1      |
|               (18, 18, 2) => 1       |
|               (18, 18, 3) => -1      |
|               (18, 18, 4) => -1      |
|               (18, 19, 32) => -1     |
|               (18, 22, 24) => 1      |
|               (18, 26, 28) => 1      |
|               (18, 29, 31) => 1      |
|               (18, 36, 38) => 1      |
|               (18, 39, 41) => 1      |
|               (18, 42, 44) => -1     |
|               (18, 45, 47) => 1      |
|               (18, 48, 50) => -1     |
|               (18, 51, 53) => -1     |
|               (19, 4, 34) => 1       |
|               (19, 9, 8) => -1       |
|               (19, 13, 12) => -1     |
|               (19, 16, 15) => -1     |
|               (19, 18, 17) => -1     |
|               (19, 19, 3) => 1       |
|               (19, 19, 4) => -2      |
|               (19, 23, 24) => 1      |
|               (19, 27, 28) => 1      |
|               (19, 30, 31) => 1      |
|               (19, 32, 33) => 1      |
|               (19, 37, 38) => 1      |
|               (19, 40, 41) => 1      |
|               (19, 42, 43) => 1      |
|               (19, 46, 47) => 1      |
|               (19, 48, 49) => 1      |
|               (19, 51, 52) => 1      |
|               (20, 0, 5) => -1       |
|               (20, 10, 6) => 1       |
|               (20, 11, 7) => 1       |
|               (20, 12, 8) => 1       |
|               (20, 13, 9) => 1       |
|               (20, 20, 0) => 2       |
|               (20, 20, 1) => -1      |
|               (20, 21, 25) => -1     |
|               (20, 22, 26) => -1     |
|               (20, 23, 27) => -1     |
|               (20, 24, 28) => -1     |
|               (20, 45, 39) => 1      |
|               (20, 46, 40) => 1      |
|               (20, 47, 41) => 1      |
|               (20, 48, 42) => 1      |
|               (20, 49, 43) => 1      |
|               (20, 50, 44) => 1      |
|               (21, 0, 6) => -1       |
|               (21, 1, 6) => -1       |
|               (21, 14, 7) => 1       |
|               (21, 15, 8) => 1       |
|               (21, 16, 9) => 1       |
|               (21, 20, 10) => -1     |
|               (21, 21, 0) => 1       |
|               (21, 21, 1) => 1       |
|               (21, 21, 2) => -1      |
|               (21, 22, 29) => -1     |
|               (21, 23, 30) => -1     |
|               (21, 24, 31) => -1     |
|               (21, 25, 5) => 1       |
|               (21, 45, 36) => -1     |
|               (21, 46, 37) => -1     |
|               (21, 47, 38) => -1     |
|               (21, 51, 42) => 1      |
|               (21, 52, 43) => 1      |
|               (21, 53, 44) => 1      |
|               (22, 0, 7) => -1       |
|               (22, 1, 7) => -1       |
|               (22, 2, 7) => -1       |
|               (22, 17, 8) => 1       |
|               (22, 18, 9) => 1       |
|               (22, 20, 11) => -1     |
|               (22, 21, 14) => -1     |
|               (22, 22, 0) => 1       |
|               (22, 22, 2) => 1       |
|               (22, 22, 3) => -1      |
|               (22, 23, 32) => -1     |
|               (22, 24, 33) => -1     |
|               (22, 26, 5) => 1       |
|               (22, 29, 6) => 1       |
|               (22, 45, 35) => 1      |
|               (22, 48, 37) => -1     |
|               (22, 49, 38) => -1     |
|               (22, 51, 40) => -1     |
|               (22, 52, 41) => -1     |
|               (22, 54, 44) => 1      |
|               (23, 0, 8) => -1       |
|               (23, 1, 8) => -1       |
|               (23, 2, 8) => -1       |
|               (23, 3, 8) => -1       |
|               (23, 19, 9) => 1       |
|               (23, 20, 12) => -1     |
|               (23, 21, 15) => -1     |
|               (23, 22, 17) => -1     |
|               (23, 23, 0) => 1       |
|               (23, 23, 3) => 1       |
|               (23, 23, 4) => -1      |
|               (23, 24, 34) => -1     |
|               (23, 27, 5) => 1       |
|               (23, 30, 6) => 1       |
|               (23, 32, 7) => 1       |
|               (23, 46, 35) => 1      |
|               (23, 48, 36) => 1      |
|               (23, 50, 38) => -1     |
|               (23, 51, 39) => 1      |
|               (23, 53, 41) => -1     |
|               (23, 54, 43) => -1     |
|               (24, 0, 9) => -1       |
|               (24, 1, 9) => -1       |
|               (24, 2, 9) => -1       |
|               (24, 3, 9) => -1       |
|               (24, 4, 9) => -1       |
|               (24, 20, 13) => -1     |
|               (24, 21, 16) => -1     |
|               (24, 22, 18) => -1     |
|               (24, 23, 19) => -1     |
|               (24, 24, 0) => 1       |
|               (24, 24, 4) => 1       |
|               (24, 28, 5) => 1       |
|               (24, 31, 6) => 1       |
|               (24, 33, 7) => 1       |
|               (24, 34, 8) => 1       |
|               (24, 47, 35) => 1      |
|               (24, 49, 36) => 1      |
|               (24, 50, 37) => 1      |
|               (24, 52, 39) => 1      |
|               (24, 53, 40) => 1      |
|               (24, 54, 42) => 1      |
|               (25, 1, 10) => -1      |
|               (25, 5, 6) => -1       |
|               (25, 14, 11) => 1      |
|               (25, 15, 12) => 1      |
|               (25, 16, 13) => 1      |
|               (25, 21, 20) => 1      |
|               (25, 25, 0) => -1      |
|               (25, 25, 1) => 2       |
|               (25, 25, 2) => -1      |
|               (25, 26, 29) => -1     |
|               (25, 27, 30) => -1     |
|               (25, 28, 31) => -1     |
|               (25, 39, 36) => 1      |
|               (25, 40, 37) => 1      |
|               (25, 41, 38) => 1      |
|               (25, 51, 48) => 1      |
|               (25, 52, 49) => 1      |
|               (25, 53, 50) => 1      |
|               (26, 1, 11) => -1      |
|               (26, 2, 11) => -1      |
|               (26, 5, 7) => -1       |
|               (26, 17, 12) => 1      |
|               (26, 18, 13) => 1      |
|               (26, 22, 20) => 1      |
|               (26, 25, 14) => -1     |
|               (26, 26, 0) => -1      |
|               (26, 26, 1) => 1       |
|               (26, 26, 2) => 1       |
|               (26, 26, 3) => -1      |
|               (26, 27, 32) => -1     |
|               (26, 28, 33) => -1     |
|               (26, 29, 10) => 1      |
|               (26, 39, 35) => -1     |
|               (26, 42, 37) => 1      |
|               (26, 43, 38) => 1      |
|               (26, 51, 46) => -1     |
|               (26, 52, 47) => -1     |
|               (26, 54, 50) => 1      |
|               (27, 1, 12) => -1      |
|               (27, 2, 12) => -1      |
|               (27, 3, 12) => -1      |
|               (27, 5, 8) => -1       |
|               (27, 19, 13) => 1      |
|               (27, 23, 20) => 1      |
|               (27, 25, 15) => -1     |
|               (27, 26, 17) => -1     |
|               (27, 27, 0) => -1      |
|               (27, 27, 1) => 1       |
|               (27, 27, 3) => 1       |
|               (27, 27, 4) => -1      |
|               (27, 28, 34) => -1     |
|               (27, 30, 10) => 1      |
|               (27, 32, 11) => 1      |
|               (27, 40, 35) => -1     |
|               (27, 42, 36) => -1     |
|               (27, 44, 38) => 1      |
|               (27, 51, 45) => 1      |
|               (27, 53, 47) => -1     |
|               (27, 54, 49) => -1     |
|               (28, 1, 13) => -1      |
|               (28, 2, 13) => -1      |
|               (28, 3, 13) => -1      |
|               (28, 4, 13) => -1      |
|               (28, 5, 9) => -1       |
|               (28, 24, 20) => 1      |
|               (28, 25, 16) => -1     |
|               (28, 26, 18) => -1     |
|               (28, 27, 19) => -1     |
|               (28, 28, 0) => -1      |
|               (28, 28, 1) => 1       |
|               (28, 28, 4) => 1       |
|               (28, 31, 10) => 1      |
|               (28, 33, 11) => 1      |
|               (28, 34, 12) => 1      |
|               (28, 41, 35) => -1     |
|               (28, 43, 36) => -1     |
|               (28, 44, 37) => -1     |
|               (28, 52, 45) => 1      |
|               (28, 53, 46) => 1      |
|               (28, 54, 48) => 1      |
|               (29, 2, 14) => -1      |
|               (29, 6, 7) => -1       |
|               (29, 10, 11) => -1     |
|               (29, 17, 15) => 1      |
|               (29, 18, 16) => 1      |
|               (29, 22, 21) => 1      |
|               (29, 26, 25) => 1      |
|               (29, 29, 1) => -1      |
|               (29, 29, 2) => 2       |
|               (29, 29, 3) => -1      |
|               (29, 30, 32) => -1     |
|               (29, 31, 33) => -1     |
|               (29, 36, 35) => 1      |
|               (29, 42, 40) => 1      |
|               (29, 43, 41) => 1      |
|               (29, 48, 46) => 1      |
|               (29, 49, 47) => 1      |
|               (29, 54, 53) => 1      |
|               (30, 2, 15) => -1      |
|               (30, 3, 15) => -1      |
|               (30, 6, 8) => -1       |
|               (30, 10, 12) => -1     |
|               (30, 19, 16) => 1      |
|               (30, 23, 21) => 1      |
|               (30, 27, 25) => 1      |
|               (30, 29, 17) => -1     |
|               (30, 30, 1) => -1      |
|               (30, 30, 2) => 1       |
|               (30, 30, 3) => 1       |
|               (30, 30, 4) => -1      |
|               (30, 31, 34) => -1     |
|               (30, 32, 14) => 1      |
|               (30, 37, 35) => 1      |
|               (30, 42, 39) => -1     |
|               (30, 44, 41) => 1      |
|               (30, 48, 45) => -1     |
|               (30, 50, 47) => 1      |
|               (30, 54, 52) => -1     |
|               (31, 2, 16) => -1      |
|               (31, 3, 16) => -1      |
|               (31, 4, 16) => -1      |
|               (31, 6, 9) => -1       |
|               (31, 10, 13) => -1     |
|               (31, 24, 21) => 1      |
|               (31, 28, 25) => 1      |
|               (31, 29, 18) => -1     |
|               (31, 30, 19) => -1     |
|               (31, 31, 1) => -1      |
|               (31, 31, 2) => 1       |
|               (31, 31, 4) => 1       |
|               (31, 33, 14) => 1      |
|               (31, 34, 15) => 1      |
|               (31, 38, 35) => 1      |
|               (31, 43, 39) => -1     |
|               (31, 44, 40) => -1     |
|               (31, 49, 45) => -1     |
|               (31, 50, 46) => -1     |
|               (31, 54, 51) => 1      |
|               (32, 3, 17) => -1      |
|               (32, 7, 8) => -1       |
|               (32, 11, 12) => -1     |
|               (32, 14, 15) => -1     |
|               (32, 19, 18) => 1      |
|               (32, 23, 22) => 1      |
|               (32, 27, 26) => 1      |
|               (32, 30, 29) => 1      |
|               (32, 32, 2) => -1      |
|               (32, 32, 3) => 2       |
|               (32, 32, 4) => -1      |
|               (32, 33, 34) => -1     |
|               (32, 37, 36) => 1      |
|               (32, 40, 39) => 1      |
|               (32, 44, 43) => 1      |
|               (32, 46, 45) => 1      |
|               (32, 50, 49) => 1      |
|               (32, 53, 52) => 1      |
|               (33, 3, 18) => -1      |
|               (33, 4, 18) => -1      |
|               (33, 7, 9) => -1       |
|               (33, 11, 13) => -1     |
|               (33, 14, 16) => -1     |
|               (33, 24, 22) => 1      |
|               (33, 28, 26) => 1      |
|               (33, 31, 29) => 1      |
|               (33, 32, 19) => -1     |
|               (33, 33, 2) => -1      |
|               (33, 33, 3) => 1       |
|               (33, 33, 4) => 1       |
|               (33, 34, 17) => 1      |
|               (33, 38, 36) => 1      |
|               (33, 41, 39) => 1      |
|               (33, 44, 42) => -1     |
|               (33, 47, 45) => 1      |
|               (33, 50, 48) => -1     |
|               (33, 53, 51) => -1     |
|               (34, 4, 19) => -1      |
|               (34, 8, 9) => -1       |
|               (34, 12, 13) => -1     |
|               (34, 15, 16) => -1     |
|               (34, 17, 18) => -1     |
|               (34, 24, 23) => 1      |
|               (34, 28, 27) => 1      |
|               (34, 31, 30) => 1      |
|               (34, 33, 32) => 1      |
|               (34, 34, 3) => -1      |
|               (34, 34, 4) => 2       |
|               (34, 38, 37) => 1      |
|               (34, 41, 40) => 1      |
|               (34, 43, 42) => 1      |
|               (34, 47, 46) => 1      |
|               (34, 49, 48) => 1      |
|               (34, 52, 51) => 1      |
|                              1       |
|               (35, 0, 54) => -       |
|                              2       |
|               (35, 1, 54) => 1       |
|                              3       |
|               (35, 2, 54) => -       |
|                              2       |
|               (35, 3, 54) => 1       |
|                              1       |
|               (35, 4, 54) => -       |
|                              2       |
|               (35, 7, 44) => -1      |
|               (35, 8, 43) => 1       |
|               (35, 9, 42) => -1      |
|               (35, 11, 50) => -1     |
|               (35, 12, 49) => 1      |
|               (35, 13, 48) => -1     |
|               (35, 14, 53) => -1     |
|               (35, 15, 52) => 1      |
|               (35, 16, 51) => -1     |
|               (35, 35, 2) => -1      |
|               (35, 36, 29) => -1     |
|               (35, 37, 30) => -1     |
|               (35, 38, 31) => -1     |
|               (35, 39, 26) => 1      |
|               (35, 40, 27) => 1      |
|               (35, 41, 28) => 1      |
|               (35, 45, 22) => -1     |
|               (35, 46, 23) => -1     |
|               (35, 47, 24) => -1     |
|                                1     |
|               (36, 0, 53) => - -     |
|                                2     |
|               (36, 1, 53) => -1      |
|                                1     |
|               (36, 2, 53) => - -     |
|                                2     |
|               (36, 3, 53) => -1      |
|                                1     |
|               (36, 4, 53) => - -     |
|                                2     |
|               (36, 6, 44) => 1       |
|               (36, 8, 41) => -1      |
|               (36, 9, 40) => 1       |
|               (36, 10, 50) => 1      |
|               (36, 12, 47) => -1     |
|               (36, 13, 46) => 1      |
|               (36, 17, 52) => 1      |
|               (36, 18, 51) => -1     |
|               (36, 29, 54) => 1      |
|               (36, 35, 14) => -1     |
|               (36, 36, 1) => -1      |
|               (36, 36, 2) => 1       |
|               (36, 36, 3) => -1      |
|               (36, 37, 32) => -1     |
|               (36, 38, 33) => -1     |
|               (36, 39, 25) => -1     |
|               (36, 42, 27) => 1      |
|               (36, 43, 28) => 1      |
|               (36, 45, 21) => 1      |
|               (36, 48, 23) => -1     |
|               (36, 49, 24) => -1     |
|                              1       |
|               (37, 0, 52) => -       |
|                              2       |
|               (37, 1, 52) => 1       |
|                              1       |
|               (37, 2, 52) => -       |
|                              2       |
|                              1       |
|               (37, 4, 52) => -       |
|                              2       |
|               (37, 6, 43) => -1      |
|               (37, 7, 41) => 1       |
|               (37, 9, 39) => -1      |
|               (37, 10, 49) => -1     |
|               (37, 11, 47) => 1      |
|               (37, 13, 45) => -1     |
|               (37, 19, 51) => -1     |
|               (37, 30, 54) => 1      |
|               (37, 32, 53) => -1     |
|               (37, 35, 15) => -1     |
|               (37, 36, 17) => -1     |
|               (37, 37, 1) => -1      |
|               (37, 37, 3) => 1       |
|               (37, 37, 4) => -1      |
|               (37, 38, 34) => -1     |
|               (37, 40, 25) => -1     |
|               (37, 42, 26) => -1     |
|               (37, 44, 28) => 1      |
|               (37, 46, 21) => 1      |
|               (37, 48, 22) => 1      |
|               (37, 50, 24) => -1     |
|                                1     |
|               (38, 0, 51) => - -     |
|                                2     |
|               (38, 1, 51) => -1      |
|                                1     |
|               (38, 2, 51) => - -     |
|                                2     |
|                              1       |
|               (38, 4, 51) => -       |
|                              2       |
|               (38, 6, 42) => 1       |
|               (38, 7, 40) => -1      |
|               (38, 8, 39) => 1       |
|               (38, 10, 48) => 1      |
|               (38, 11, 46) => -1     |
|               (38, 12, 45) => 1      |
|               (38, 31, 54) => 1      |
|               (38, 33, 53) => -1     |
|               (38, 34, 52) => 1      |
|               (38, 35, 16) => -1     |
|               (38, 36, 18) => -1     |
|               (38, 37, 19) => -1     |
|               (38, 38, 1) => -1      |
|               (38, 38, 4) => 1       |
|               (38, 41, 25) => -1     |
|               (38, 43, 26) => -1     |
|               (38, 44, 27) => -1     |
|               (38, 47, 21) => 1      |
|               (38, 49, 22) => 1      |
|               (38, 50, 23) => 1      |
|                              1       |
|               (39, 0, 50) => -       |
|                              2       |
|                              1       |
|               (39, 2, 50) => -       |
|                              2       |
|               (39, 3, 50) => 1       |
|                              1       |
|               (39, 4, 50) => -       |
|                              2       |
|               (39, 5, 44) => -1      |
|               (39, 8, 38) => 1       |
|               (39, 9, 37) => -1      |
|               (39, 15, 47) => -1     |
|               (39, 16, 46) => 1      |
|               (39, 17, 49) => -1     |
|               (39, 18, 48) => 1      |
|               (39, 25, 53) => -1     |
|               (39, 26, 54) => -1     |
|               (39, 35, 11) => 1      |
|               (39, 36, 10) => -1     |
|               (39, 39, 0) => -1      |
|               (39, 39, 1) => 1       |
|               (39, 39, 3) => -1      |
|               (39, 40, 32) => -1     |
|               (39, 41, 33) => -1     |
|               (39, 42, 30) => 1      |
|               (39, 43, 31) => 1      |
|               (39, 45, 20) => -1     |
|               (39, 51, 23) => -1     |
|               (39, 52, 24) => -1     |
|                                1     |
|               (40, 0, 49) => - -     |
|                                2     |
|                                1     |
|               (40, 2, 49) => - -     |
|                                2     |
|                                1     |
|               (40, 4, 49) => - -     |
|                                2     |
|               (40, 5, 43) => 1       |
|               (40, 7, 38) => -1      |
|               (40, 9, 36) => 1       |
|               (40, 14, 47) => 1      |
|               (40, 16, 45) => -1     |
|               (40, 19, 48) => 1      |
|               (40, 25, 52) => 1      |
|               (40, 27, 54) => -1     |
|               (40, 32, 50) => 1      |
|               (40, 35, 12) => 1      |
|               (40, 37, 10) => -1     |
|               (40, 39, 17) => -1     |
|               (40, 40, 0) => -1      |
|               (40, 40, 1) => 1       |
|               (40, 40, 2) => -1      |
|               (40, 40, 3) => 1       |
|               (40, 40, 4) => -1      |
|               (40, 41, 34) => -1     |
|               (40, 42, 29) => -1     |
|               (40, 44, 31) => 1      |
|               (40, 46, 20) => -1     |
|               (40, 51, 22) => 1      |
|               (40, 53, 24) => -1     |
|                              1       |
|               (41, 0, 48) => -       |
|                              2       |
|                              1       |
|               (41, 2, 48) => -       |
|                              2       |
|                                1     |
|               (41, 4, 48) => - -     |
|                                2     |
|               (41, 5, 42) => -1      |
|               (41, 7, 37) => 1       |
|               (41, 8, 36) => -1      |
|               (41, 14, 46) => -1     |
|               (41, 15, 45) => 1      |
|               (41, 25, 51) => -1     |
|               (41, 28, 54) => -1     |
|               (41, 33, 50) => 1      |
|               (41, 34, 49) => -1     |
|               (41, 35, 13) => 1      |
|               (41, 38, 10) => -1     |
|               (41, 39, 18) => -1     |
|               (41, 40, 19) => -1     |
|               (41, 41, 0) => -1      |
|               (41, 41, 1) => 1       |
|               (41, 41, 2) => -1      |
|               (41, 41, 4) => 1       |
|               (41, 43, 29) => -1     |
|               (41, 44, 30) => -1     |
|               (41, 47, 20) => -1     |
|               (41, 52, 22) => 1      |
|               (41, 53, 23) => 1      |
|                              1       |
|               (42, 0, 47) => -       |
|                              2       |
|                                1     |
|               (42, 2, 47) => - -     |
|                                2     |
|                              1       |
|               (42, 4, 47) => -       |
|                              2       |
|               (42, 5, 41) => -1      |
|               (42, 6, 38) => 1       |
|               (42, 9, 35) => -1      |
|               (42, 18, 45) => -1     |
|               (42, 19, 46) => -1     |
|               (42, 26, 52) => 1      |
|               (42, 27, 53) => 1      |
|               (42, 29, 49) => -1     |
|               (42, 30, 50) => -1     |
|               (42, 36, 12) => 1      |
|               (42, 37, 11) => -1     |
|               (42, 39, 15) => 1      |
|               (42, 40, 14) => -1     |
|               (42, 42, 0) => -1      |
|               (42, 42, 2) => 1       |
|               (42, 42, 4) => -1      |
|               (42, 43, 34) => -1     |
|               (42, 44, 33) => 1      |
|               (42, 48, 20) => -1     |
|               (42, 51, 21) => -1     |
|               (42, 54, 24) => -1     |
|                                1     |
|               (43, 0, 46) => - -     |
|                                2     |
|                              1       |
|               (43, 2, 46) => -       |
|                              2       |
|                              1       |
|               (43, 4, 46) => -       |
|                              2       |
|               (43, 5, 40) => 1       |
|               (43, 6, 37) => -1      |
|               (43, 8, 35) => 1       |
|               (43, 17, 45) => 1      |
|               (43, 26, 51) => -1     |
|               (43, 28, 53) => 1      |
|               (43, 29, 48) => 1      |
|               (43, 31, 50) => -1     |
|               (43, 34, 47) => 1      |
|               (43, 36, 13) => 1      |
|               (43, 38, 11) => -1     |
|               (43, 39, 16) => 1      |
|               (43, 41, 14) => -1     |
|               (43, 42, 19) => -1     |
|               (43, 43, 0) => -1      |
|               (43, 43, 2) => 1       |
|               (43, 43, 3) => -1      |
|               (43, 43, 4) => 1       |
|               (43, 44, 32) => -1     |
|               (43, 49, 20) => -1     |
|               (43, 52, 21) => -1     |
|               (43, 54, 23) => 1      |
|                              1       |
|               (44, 0, 45) => -       |
|                              2       |
|                                1     |
|               (44, 2, 45) => - -     |
|                                2     |
|               (44, 3, 45) => -1      |
|                                1     |
|               (44, 4, 45) => - -     |
|                                2     |
|               (44, 5, 39) => -1      |
|               (44, 6, 36) => 1       |
|               (44, 7, 35) => -1      |
|               (44, 27, 51) => -1     |
|               (44, 28, 52) => -1     |
|               (44, 30, 48) => 1      |
|               (44, 31, 49) => 1      |
|               (44, 32, 46) => -1     |
|               (44, 33, 47) => -1     |
|               (44, 37, 13) => 1      |
|               (44, 38, 12) => -1     |
|               (44, 40, 16) => 1      |
|               (44, 41, 15) => -1     |
|               (44, 42, 18) => 1      |
|               (44, 43, 17) => -1     |
|               (44, 44, 0) => -1      |
|               (44, 44, 3) => 1       |
|               (44, 50, 20) => -1     |
|               (44, 53, 21) => -1     |
|               (44, 54, 22) => -1     |
|                              1       |
|               (45, 0, 44) => -       |
|                              2       |
|                                1     |
|               (45, 2, 44) => - -     |
|                                2     |
|               (45, 3, 44) => -1      |
|                                1     |
|               (45, 4, 44) => - -     |
|                                2     |
|               (45, 12, 38) => 1      |
|               (45, 13, 37) => -1     |
|               (45, 15, 41) => 1      |
|               (45, 16, 40) => -1     |
|               (45, 17, 43) => 1      |
|               (45, 18, 42) => -1     |
|               (45, 20, 50) => 1      |
|               (45, 21, 53) => 1      |
|               (45, 22, 54) => 1      |
|               (45, 35, 7) => -1      |
|               (45, 36, 6) => 1       |
|               (45, 39, 5) => -1      |
|               (45, 45, 0) => 1       |
|               (45, 45, 3) => -1      |
|               (45, 46, 32) => -1     |
|               (45, 47, 33) => -1     |
|               (45, 48, 30) => 1      |
|               (45, 49, 31) => 1      |
|               (45, 51, 27) => -1     |
|               (45, 52, 28) => -1     |
|                                1     |
|               (46, 0, 43) => - -     |
|                                2     |
|                              1       |
|               (46, 2, 43) => -       |
|                              2       |
|                              1       |
|               (46, 4, 43) => -       |
|                              2       |
|               (46, 11, 38) => -1     |
|               (46, 13, 36) => 1      |
|               (46, 14, 41) => -1     |
|               (46, 16, 39) => 1      |
|               (46, 19, 42) => -1     |
|               (46, 20, 49) => -1     |
|               (46, 21, 52) => -1     |
|               (46, 23, 54) => 1      |
|               (46, 32, 44) => -1     |
|               (46, 35, 8) => -1      |
|               (46, 37, 6) => 1       |
|               (46, 40, 5) => -1      |
|               (46, 45, 17) => -1     |
|               (46, 46, 0) => 1       |
|               (46, 46, 2) => -1      |
|               (46, 46, 3) => 1       |
|               (46, 46, 4) => -1      |
|               (46, 47, 34) => -1     |
|               (46, 48, 29) => -1     |
|               (46, 50, 31) => 1      |
|               (46, 51, 26) => 1      |
|               (46, 53, 28) => -1     |
|                              1       |
|               (47, 0, 42) => -       |
|                              2       |
|                                1     |
|               (47, 2, 42) => - -     |
|                                2     |
|                              1       |
|               (47, 4, 42) => -       |
|                              2       |
|               (47, 11, 37) => 1      |
|               (47, 12, 36) => -1     |
|               (47, 14, 40) => 1      |
|               (47, 15, 39) => -1     |
|               (47, 20, 48) => 1      |
|               (47, 21, 51) => 1      |
|               (47, 24, 54) => 1      |
|               (47, 33, 44) => -1     |
|               (47, 34, 43) => 1      |
|               (47, 35, 9) => -1      |
|               (47, 38, 6) => 1       |
|               (47, 41, 5) => -1      |
|               (47, 45, 18) => -1     |
|               (47, 46, 19) => -1     |
|               (47, 47, 0) => 1       |
|               (47, 47, 2) => -1      |
|               (47, 47, 4) => 1       |
|               (47, 49, 29) => -1     |
|               (47, 50, 30) => -1     |
|               (47, 52, 26) => 1      |
|               (47, 53, 27) => 1      |
|                              1       |
|               (48, 0, 41) => -       |
|                              2       |
|                              1       |
|               (48, 2, 41) => -       |
|                              2       |
|                                1     |
|               (48, 4, 41) => - -     |
|                                2     |
|               (48, 10, 38) => 1      |
|               (48, 13, 35) => -1     |
|               (48, 18, 39) => 1      |
|               (48, 19, 40) => 1      |
|               (48, 20, 47) => 1      |
|               (48, 22, 52) => -1     |
|               (48, 23, 53) => -1     |
|               (48, 29, 43) => 1      |
|               (48, 30, 44) => 1      |
|               (48, 36, 8) => -1      |
|               (48, 37, 7) => 1       |
|               (48, 42, 5) => -1      |
|               (48, 45, 15) => 1      |
|               (48, 46, 14) => -1     |
|               (48, 48, 0) => 1       |
|               (48, 48, 1) => -1      |
|               (48, 48, 2) => 1       |
|               (48, 48, 4) => -1      |
|               (48, 49, 34) => -1     |
|               (48, 50, 33) => 1      |
|               (48, 51, 25) => -1     |
|               (48, 54, 28) => -1     |
|                                1     |
|               (49, 0, 40) => - -     |
|                                2     |
|                                1     |
|               (49, 2, 40) => - -     |
|                                2     |
|                                1     |
|               (49, 4, 40) => - -     |
|                                2     |
|               (49, 10, 37) => -1     |
|               (49, 12, 35) => 1      |
|               (49, 17, 39) => -1     |
|               (49, 20, 46) => -1     |
|               (49, 22, 51) => 1      |
|               (49, 24, 53) => -1     |
|               (49, 29, 42) => -1     |
|               (49, 31, 44) => 1      |
|               (49, 34, 41) => -1     |
|               (49, 36, 9) => -1      |
|               (49, 38, 7) => 1       |
|               (49, 43, 5) => -1      |
|               (49, 45, 16) => 1      |
|               (49, 47, 14) => -1     |
|               (49, 48, 19) => -1     |
|               (49, 49, 0) => 1       |
|               (49, 49, 1) => -1      |
|               (49, 49, 2) => 1       |
|               (49, 49, 3) => -1      |
|               (49, 49, 4) => 1       |
|               (49, 50, 32) => -1     |
|               (49, 52, 25) => -1     |
|               (49, 54, 27) => 1      |
|                              1       |
|               (50, 0, 39) => -       |
|                              2       |
|                              1       |
|               (50, 2, 39) => -       |
|                              2       |
|               (50, 3, 39) => 1       |
|                              1       |
|               (50, 4, 39) => -       |
|                              2       |
|               (50, 10, 36) => 1      |
|               (50, 11, 35) => -1     |
|               (50, 20, 45) => 1      |
|               (50, 23, 51) => 1      |
|               (50, 24, 52) => 1      |
|               (50, 30, 42) => -1     |
|               (50, 31, 43) => -1     |
|               (50, 32, 40) => 1      |
|               (50, 33, 41) => 1      |
|               (50, 37, 9) => -1      |
|               (50, 38, 8) => 1       |
|               (50, 44, 5) => -1      |
|               (50, 46, 16) => 1      |
|               (50, 47, 15) => -1     |
|               (50, 48, 18) => 1      |
|               (50, 49, 17) => -1     |
|               (50, 50, 0) => 1       |
|               (50, 50, 1) => -1      |
|               (50, 50, 3) => 1       |
|               (50, 53, 25) => -1     |
|               (50, 54, 26) => -1     |
|                                1     |
|               (51, 0, 38) => - -     |
|                                2     |
|               (51, 1, 38) => -1      |
|                                1     |
|               (51, 2, 38) => - -     |
|                                2     |
|                              1       |
|               (51, 4, 38) => -       |
|                              2       |
|               (51, 16, 35) => -1     |
|               (51, 18, 36) => -1     |
|               (51, 19, 37) => -1     |
|               (51, 21, 47) => 1      |
|               (51, 22, 49) => 1      |
|               (51, 23, 50) => 1      |
|               (51, 25, 41) => -1     |
|               (51, 26, 43) => -1     |
|               (51, 27, 44) => -1     |
|               (51, 39, 8) => -1      |
|               (51, 40, 7) => 1       |
|               (51, 42, 6) => -1      |
|               (51, 45, 12) => -1     |
|               (51, 46, 11) => 1      |
|               (51, 48, 10) => -1     |
|               (51, 51, 1) => 1       |
|               (51, 51, 4) => -1      |
|               (51, 52, 34) => -1     |
|               (51, 53, 33) => 1      |
|               (51, 54, 31) => -1     |
|                              1       |
|               (52, 0, 37) => -       |
|                              2       |
|               (52, 1, 37) => 1       |
|                              1       |
|               (52, 2, 37) => -       |
|                              2       |
|                              1       |
|               (52, 4, 37) => -       |
|                              2       |
|               (52, 15, 35) => 1      |
|               (52, 17, 36) => 1      |
|               (52, 21, 46) => -1     |
|               (52, 22, 48) => -1     |
|               (52, 24, 50) => 1      |
|               (52, 25, 40) => 1      |
|               (52, 26, 42) => 1      |
|               (52, 28, 44) => -1     |
|               (52, 34, 38) => 1      |
|               (52, 39, 9) => -1      |
|               (52, 41, 7) => 1       |
|               (52, 43, 6) => -1      |
|               (52, 45, 13) => -1     |
|               (52, 47, 11) => 1      |
|               (52, 49, 10) => -1     |
|               (52, 51, 19) => -1     |
|               (52, 52, 1) => 1       |
|               (52, 52, 3) => -1      |
|               (52, 52, 4) => 1       |
|               (52, 53, 32) => -1     |
|               (52, 54, 30) => 1      |
|                                1     |
|               (53, 0, 36) => - -     |
|                                2     |
|               (53, 1, 36) => -1      |
|                                1     |
|               (53, 2, 36) => - -     |
|                                2     |
|               (53, 3, 36) => -1      |
|                                1     |
|               (53, 4, 36) => - -     |
|                                2     |
|               (53, 14, 35) => -1     |
|               (53, 21, 45) => 1      |
|               (53, 23, 48) => -1     |
|               (53, 24, 49) => -1     |
|               (53, 25, 39) => -1     |
|               (53, 27, 42) => 1      |
|               (53, 28, 43) => 1      |
|               (53, 32, 37) => -1     |
|               (53, 33, 38) => -1     |
|               (53, 40, 9) => -1      |
|               (53, 41, 8) => 1       |
|               (53, 44, 6) => -1      |
|               (53, 46, 13) => -1     |
|               (53, 47, 12) => 1      |
|               (53, 50, 10) => -1     |
|               (53, 51, 18) => 1      |
|               (53, 52, 17) => -1     |
|               (53, 53, 1) => 1       |
|               (53, 53, 2) => -1      |
|               (53, 53, 3) => 1       |
|               (53, 54, 29) => -1     |
|                              1       |
|               (54, 0, 35) => -       |
|                              2       |
|               (54, 1, 35) => 1       |
|                              3       |
|               (54, 2, 35) => -       |
|                              2       |
|               (54, 3, 35) => 1       |
|                              1       |
|               (54, 4, 35) => -       |
|                              2       |
|               (54, 22, 45) => 1      |
|               (54, 23, 46) => 1      |
|               (54, 24, 47) => 1      |
|               (54, 26, 39) => -1     |
|               (54, 27, 40) => -1     |
|               (54, 28, 41) => -1     |
|               (54, 29, 36) => 1      |
|               (54, 30, 37) => 1      |
|               (54, 31, 38) => 1      |
|               (54, 42, 9) => -1      |
|               (54, 43, 8) => 1       |
|               (54, 44, 7) => -1      |
|               (54, 48, 13) => -1     |
|               (54, 49, 12) => 1      |
|               (54, 50, 11) => -1     |
|               (54, 51, 16) => -1     |
|               (54, 52, 15) => 1      |
|               (54, 53, 14) => -1     |
|               (54, 54, 2) => 1       |
|                                      |
|o3 : HashTable                        |
+--------------------------------------+

This tensor B can be used to compute any product of basis elements -- the
number in \(B(i,j,k)\) is the coefficient on \(e_k\) of the product of \(e_i\)
and \(e_j\).

Ways to use structureTensor:
============================

  * structureTensor(ExteriorExtension) (missing documentation)

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

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

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

The source of this document is in ExteriorExtensions.m2:640:0.

KillingMatrix -- Constructs the Killing matrix of the ExteriorExtension
***********************************************************************

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

The Killling matrix is constructed by taking the adjoint of every basis vector
of the algebra, then computing the trace of each bracket between them and
storing the results in a matrix. The matrices constructed by this can get very
large, and take a long time to compute.

+-----------------------------------------------------------------------------+
|i1 : printWidth = 450;                                                       |
+-----------------------------------------------------------------------------+
|i2 : ea = exteriorExtension(3, 6, QQ);                                       |
+-----------------------------------------------------------------------------+
|i3 : K = KillingMatrix ea                                                    |
|                                                                             |
|o3 = | 36  -18 0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | -18 36  -18 0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   -18 36  -18 0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   -18 36  -18 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   -18 36  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  18 0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  18|
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   18 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  18 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  18 0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  18 0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  18 0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  18 0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  18 0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  18 0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  18 0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  18 0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  18 0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  18 0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  18 0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  18 0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  18 0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|     | 0   0   0   0   0   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 |
|                                                                             |
|              55       55                                                    |
|o3 : Matrix QQ   <-- QQ                                                      |
|-----------------------------------------------------------------------------|
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|18 0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  18 0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  18 0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  18 0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  18 0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  18 0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  18 0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  18 0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  18 0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  18 0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  18 0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  18 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
|-----------------------------------------------------------------------------|
|0 |                                                                          |
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+-----------------------------------------------------------------------------+
|i4 : rank K                                                                  |
|                                                                             |
|o4 = 35                                                                      |
+-----------------------------------------------------------------------------+

This rank is an invariant of the algebra.

Ways to use KillingMatrix:
==========================

  * KillingMatrix(ExteriorExtension) (missing documentation)

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

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

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

The source of this document is in ExteriorExtensions.m2:662:0.

LieAlg2Matrix -- Converts from the LieAlgebra to Matrix form (with the standard basis)
**************************************************************************************

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

Let's build a 7-graded algebra and see the bases of each piece and see the
elements of the (symbolic) Lie algebra get converted to matrices

+--------------------------------------------+
|i1 : ea = exteriorExtension(3, 7, QQ);      |
+--------------------------------------------+
|i2 : sln = ea.LieAlgebra;                   |
+--------------------------------------------+
|i3 : (gens sln)#0                           |
|                                            |
|o3 = h                                      |
|      1                                     |
|                                            |
|o3 : sln                                    |
+--------------------------------------------+
|i4 : ea.LieAlg2Matrix (gens ea.LieAlgebra)#0|
|                                            |
|o4 = | 1 0  0 0 0 0 0 |                     |
|     | 0 -1 0 0 0 0 0 |                     |
|     | 0 0  0 0 0 0 0 |                     |
|     | 0 0  0 0 0 0 0 |                     |
|     | 0 0  0 0 0 0 0 |                     |
|     | 0 0  0 0 0 0 0 |                     |
|     | 0 0  0 0 0 0 0 |                     |
|                                            |
|               7        7                   |
|o4 : Matrix sln  <-- sln                    |
+--------------------------------------------+

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

The object LieAlg2Matrix, is a symbol (Macaulay2Doc).

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

The source of this document is in ExteriorExtensions.m2:816:0.

matrix2LieAlg -- Converts from Matrix form to the LieAlgebra (with the standard basis)
**************************************************************************************

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

Let's build a 7-graded algebra and see the bases of each piece getting
converted to the symbolic ements in the Lie algebra.

+--------------------------------------+
|i1 : ea = exteriorExtension(3, 7, QQ);|
+--------------------------------------+
|i2 : ea.bases#0#0                     |
|                                      |
|o2 = | 1 0  0 0 0 0 0 |               |
|     | 0 -1 0 0 0 0 0 |               |
|     | 0 0  0 0 0 0 0 |               |
|     | 0 0  0 0 0 0 0 |               |
|     | 0 0  0 0 0 0 0 |               |
|     | 0 0  0 0 0 0 0 |               |
|     | 0 0  0 0 0 0 0 |               |
|                                      |
|              7       7               |
|o2 : Matrix QQ  <-- QQ                |
+--------------------------------------+
|i3 : sln = ea.LieAlgebra              |
|                                      |
|o3 = sln                              |
|                                      |
|o3 : PolynomialRing                   |
+--------------------------------------+
|i4 : ea.matrix2LieAlg (ea.bases#0#0)  |
|                                      |
|o4 = h                                |
|      1                               |
|                                      |
|o4 : sln                              |
+--------------------------------------+
|i5 : ea.bases#0#8                     |
|                                      |
|o5 = | 0 0 0 1 0 0 0 |                |
|     | 0 0 0 0 0 0 0 |                |
|     | 0 0 0 0 0 0 0 |                |
|     | 0 0 0 0 0 0 0 |                |
|     | 0 0 0 0 0 0 0 |                |
|     | 0 0 0 0 0 0 0 |                |
|     | 0 0 0 0 0 0 0 |                |
|                                      |
|              7       7               |
|o5 : Matrix ZZ  <-- ZZ                |
+--------------------------------------+
|i6 : ea.matrix2LieAlg (ea.bases#0#8)  |
|                                      |
|o6 = E                                |
|      {0, 3}                          |
|                                      |
|o6 : sln                              |
+--------------------------------------+

If you don't name the Lie algebra something it has an annoying printout.

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

The object matrix2LieAlg, is a symbol (Macaulay2Doc).

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

The source of this document is in ExteriorExtensions.m2:839:0.

getDiagonal -- Extract the diagonal of a matrix
***********************************************

  * Usage:
        v = getDiagonal(A)
  * Inputs:
      * A, a matrix (Macaulay2Doc),
  * Outputs:
      * v, a list (Macaulay2Doc),

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

Extract the diagonal of a matrix

+---------------------------+
|i1 : A = random(QQ^4, QQ^4)|
|                           |
|o1 = | 9/2 7/9  7   5/4  | |
|     | 9/4 7/10 3/7 2/9  | |
|     | 3/4 7/10 6/7 3/10 | |
|     | 7/4 7/3  6   3/7  | |
|                           |
|              4       4    |
|o1 : Matrix QQ  <-- QQ     |
+---------------------------+
|i2 : v = getDiagonal A     |
|                           |
|      9   7  6  3          |
|o2 = {-, --, -, -}         |
|      2  10  7  7          |
|                           |
|o2 : List                  |
+---------------------------+

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

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

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

The source of this document is in ExteriorExtensions.m2:440:0.

makeTraceless -- Project matrix to the traceless matrices.
**********************************************************

  * Usage:
        Ap = makeTraceless(A)
  * Inputs:
      * A, a matrix (Macaulay2Doc),
  * Outputs:
      * Ap, a matrix (Macaulay2Doc),

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

Project matrix to the traceless matrices.

+-------------------------------------------+
|i1 : A = random(QQ^4, QQ^4)                |
|                                           |
|o1 = | 9/2 7/9  7   5/4  |                 |
|     | 9/4 7/10 3/7 2/9  |                 |
|     | 3/4 7/10 6/7 3/10 |                 |
|     | 7/4 7/3  6   3/7  |                 |
|                                           |
|              4       4                    |
|o1 : Matrix QQ  <-- QQ                     |
+-------------------------------------------+
|i2 : Ap = makeTraceless A                  |
|                                           |
|o2 = | 403/140 7/9      7        5/4      ||
|     | 9/4     -129/140 3/7      2/9      ||
|     | 3/4     7/10     -107/140 3/10     ||
|     | 7/4     7/3      6        -167/140 ||
|                                           |
|              4       4                    |
|o2 : Matrix QQ  <-- QQ                     |
+-------------------------------------------+

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

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

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

The source of this document is in ExteriorExtensions.m2:465:0.