This is SchurComplexes.info, produced by Macaulay2, version 1.24.11

SchurComplexes -- Schur functors of chain complexes
***************************************************

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

This package computes the Schur complex $S_{\lambda}F_{\bullet}$ associated to
a bounded chain complex $F_{\bullet}$ of finitely-generated free modules,  and
a partition $\lambda$. Our conventions are the transpose of the convention in
"Cohomology of Vector Bundles and Syzygies", by Jerzy Weyman, Chapter 2.4.

Authors
=======

  * Michael K. Brown (see https://www.math.wisc.edu/~mkbrown5/ )
    <mkbrown5@wisc.edu>
  * Amy Huang (see https://www.math.wisc.edu/~hhuang235/ )
    <hhuang235@math.wisc.edu>
  * Robert Laudone (see https://www.math.wisc.edu/~laudone/ )
    <robert.laudone@gmail.com>
  * Michael Perlman (see https://www3.nd.edu/~mperlman/ ) <mperlman@nd.edu>
  * Claudiu Raicu (see https://www3.nd.edu/~craicu/ ) <craicu@nd.edu>
  * Steven V Sam (see http://math.ucsd.edu/~ssam/ ) <ssam@sd.edu>
  * Joao Pedro Santos (see http://math.nd.edu/people/graduate-students/graduate
    -directory-with-photos/ ) <jsantos3@nd.edu>

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

Version 1.1 of this package was accepted for publication in volume 9 (see
https://msp.org/jsag/2019/9-2/ ) of Journal of Software for Algebra and
Geometry (see https://msp.org/jsag/ ) on 21 August 2019, in the article
Computing Schur complexes (see https://msp.org/jsag/2019/9-2/p02.xhtml ) (DOI:
10.2140/jsag.2019.9.111 (see https://doi.org/10.2140/jsag.2019.9.111 )). That
version can be obtained from the journal (see
https://msp.org/jsag/2019/9-2/jsag-v9-n2-x02-SchurComplexes.m2 ) or from the
Macaulay2 source code repository (see
https://github.com/Macaulay2/M2/blob/61384b8d76f8bfef42010911dae401d24bcc6ebe/
M2/Macaulay2/packages/SchurComplexes.m2 ).

Version
=======

This documentation describes version 1.1 of SchurComplexes.

Source code
===========

The source code from which this documentation is derived is in the file
SchurComplexes.m2 (see common/share/Macaulay2/SchurComplexes.m2 ).

Exports
=======

  * Functions and commands
      * schurComplex -- Schur functors of chain complexes
      * straightenTableau -- Straightening law for
        Z/2Z-graded tableau

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

The object SchurComplexes, is a package (Macaulay2Doc).

schurComplex -- Schur functors of chain complexes
*************************************************

Synopsis
========

  * Usage:
        G=schurComplex(lambda,F)
  * Inputs:
      * lambda, a list (Macaulay2Doc),
      * F, a chain complex (Macaulay2Doc),
  * Outputs:
      * G, a chain complex (Macaulay2Doc),

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

This function computes the Schur complex associated to a partition $\lambda$
and a bounded complex $F_{\bullet}$ of finitely-generated free modules over a
commutative ring.

The user inputs the partition $\lambda$ as a list and the chain complex
$F_{\bullet}$.

In the following example, the complex F is the free resolution of the ideal
$(x,y,z)\subset \mathbb{Z}[x,y,z]$, and lambda is the partition $(1,1)$ in the
form of a List. In this case, the Schur complex G is the second exterior power
of F.

+-------------------------------------------------------------+
|i1 : R=ZZ[x,y,z];                                            |
+-------------------------------------------------------------+
|i2 : I=ideal(x,y,z);                                         |
|                                                             |
|o2 : Ideal of R                                              |
+-------------------------------------------------------------+
|i3 : F=res I;                                                |
+-------------------------------------------------------------+
|i4 : lambda={1,1};                                           |
+-------------------------------------------------------------+
|i5 : G=schurComplex(lambda,F)                                |
|                                                             |
|      3      9      10      6      3      1                  |
|o5 = R  <-- R  <-- R   <-- R  <-- R  <-- R                   |
|                                                             |
|     1      2      3       4      5      6                   |
|                                                             |
|o5 : ChainComplex                                            |
+-------------------------------------------------------------+
|i6 : G.dd                                                    |
|                                                             |
|          3                                9                 |
|o6 = 1 : R  <---------------------------- R  : 2             |
|               | z y x 0 0 0 0  y  x  |                      |
|               | 0 z 0 y x 0 x  -z 0  |                      |
|               | 0 0 z 0 y x -y 0  -z |                      |
|                                                             |
|          9                                            10    |
|     2 : R  <---------------------------------------- R   : 3|
|               | 0  0  2y 2x 0  0   0  0   0  0   |          |
|               | 0  x  -z 0  0  y   x  0   0  0   |          |
|               | 0  -y 0  -z 0  0   0  0   y  x   |          |
|               | 0  0  0  0  2x -2z 0  0   0  0   |          |
|               | 0  0  0  0  -y 0   -z x   -z 0   |          |
|               | 0  0  0  0  0  0   0  -2y 0  -2z |          |
|               | z  -z 0  0  -y 0   0  -x  0  0   |          |
|               | x  0  -z 0  0  -y  0  0   -x 0   |          |
|               | -y 0  0  -z 0  0   -y 0   0  -x  |          |
|                                                             |
|          10                             6                   |
|     3 : R   <------------------------- R  : 4               |
|                | z  y  x  0  0  0  |                        |
|                | z  0  0  y  x  0  |                        |
|                | x  0  0  0  0  x  |                        |
|                | -y 0  0  0  0  -y |                        |
|                | 0  z  0  -z 0  0  |                        |
|                | 0  x  0  -x 0  0  |                        |
|                | 0  -y 0  0  -x z  |                        |
|                | 0  0  z  0  -z 0  |                        |
|                | 0  0  x  y  0  -z |                        |
|                | 0  0  -y 0  y  0  |                        |
|                                                             |
|          6                    3                             |
|     4 : R  <---------------- R  : 5                         |
|               | 0  y  x  |                                  |
|               | x  -z 0  |                                  |
|               | -y 0  -z |                                  |
|               | x  -z 0  |                                  |
|               | -y 0  -z |                                  |
|               | 0  -y -x |                                  |
|                                                             |
|          3              1                                   |
|     5 : R  <---------- R  : 6                               |
|               | z  |                                        |
|               | x  |                                        |
|               | -y |                                        |
|                                                             |
|o6 : ChainComplexMap                                         |
+-------------------------------------------------------------+

As a second example, we consider the ring of polynomial functions
$R=\mathbb{Q}[x_{i,j}]$ on the space of 2 x 4 generic matrices. We set the
complex F to be the map $R^4\to R^2$ given by the generic matrix $(x_{i,j})$.
We compute the third symmetric power G of F, in which case lambda is the
partition $(3)$. By Weyman "Cohomology of Vector Bundles and Syzygies",
Exercise 6.34(d), the Schur complex G is exact except in degree zero. We verify
this by computing the Hilbert series of each homology module of G.

+-----------------------------------------------------------------------------+
|i7 : R=QQ[x11,x21,x12,x22,x13,x23,x14,x24];                                  |
+-----------------------------------------------------------------------------+
|i8 : M=genericMatrix(R,x11,2,4);                                             |
|                                                                             |
|             2      4                                                        |
|o8 : Matrix R  <-- R                                                         |
+-----------------------------------------------------------------------------+
|i9 : F = new ChainComplex; F.ring = R; F#0=target M; F#1=source M; F.dd#1=M; |
|                                                                             |
|              2      4                                                       |
|o13 : Matrix R  <-- R                                                        |
+-----------------------------------------------------------------------------+
|i14 : lambda={3};                                                            |
+-----------------------------------------------------------------------------+
|i15 : G=schurComplex(lambda,F)                                               |
|                                                                             |
|       4      12      12      4                                              |
|o15 = R  <-- R   <-- R   <-- R                                               |
|                                                                             |
|      0      1       2       3                                               |
|                                                                             |
|o15 : ChainComplex                                                           |
+-----------------------------------------------------------------------------+
|i16 : G.dd                                                                   |
|                                                                             |
|           4                                                           12    |
|o16 = 0 : R  <------------------------------------------------------- R   : 1|
|                | x14 0   0   x13 0   0   x12 0   0   x11 0   0   |          |
|                | x24 x14 0   x23 x13 0   x22 x12 0   x21 x11 0   |          |
|                | 0   x24 x14 0   x23 x13 0   x22 x12 0   x21 x11 |          |
|                | 0   0   x24 0   0   x23 0   0   x22 0   0   x21 |          |
|                                                                             |
|           12                                                                |
|      1 : R   <--------------------------------------------------------------|
|                 | -x13 0    -x12 0    -x11 0    0    0    0    0    0    0  |
|                 | -x23 -x13 -x22 -x12 -x21 -x11 0    0    0    0    0    0  |
|                 | 0    -x23 0    -x22 0    -x21 0    0    0    0    0    0  |
|                 | x14  0    0    0    0    0    -x12 0    -x11 0    0    0  |
|                 | x24  x14  0    0    0    0    -x22 -x12 -x21 -x11 0    0  |
|                 | 0    x24  0    0    0    0    0    -x22 0    -x21 0    0  |
|                 | 0    0    x14  0    0    0    x13  0    0    0    -x11 0  |
|                 | 0    0    x24  x14  0    0    x23  x13  0    0    -x21 -x1|
|                 | 0    0    0    x24  0    0    0    x23  0    0    0    -x2|
|                 | 0    0    0    0    x14  0    0    0    x13  0    x12  0  |
|                 | 0    0    0    0    x24  x14  0    0    x23  x13  x22  x12|
|                 | 0    0    0    0    0    x24  0    0    0    x23  0    x22|
|                                                                             |
|           12                               4                                |
|      2 : R   <--------------------------- R  : 3                            |
|                 | x12  x11  0    0    |                                     |
|                 | x22  x21  0    0    |                                     |
|                 | -x13 0    x11  0    |                                     |
|                 | -x23 0    x21  0    |                                     |
|                 | 0    -x13 -x12 0    |                                     |
|                 | 0    -x23 -x22 0    |                                     |
|                 | x14  0    0    x11  |                                     |
|                 | x24  0    0    x21  |                                     |
|                 | 0    x14  0    -x12 |                                     |
|                 | 0    x24  0    -x22 |                                     |
|                 | 0    0    x14  x13  |                                     |
|                 | 0    0    x24  x23  |                                     |
|                                                                             |
|o16 : ChainComplexMap                                                        |
|-----------------------------------------------------------------------------|
|       12                                                                    |
|----- R   : 2                                                                |
|  |                                                                          |
|  |                                                                          |
|  |                                                                          |
|  |                                                                          |
|  |                                                                          |
|  |                                                                          |
|  |                                                                          |
|1 |                                                                          |
|1 |                                                                          |
|  |                                                                          |
|  |                                                                          |
|  |                                                                          |
+-----------------------------------------------------------------------------+
|i17 : apply((length G)+1,i->reduceHilbert hilbertSeries HH_(i)(G))           |
|                                                                             |
|           4     0  0  0                                                     |
|o17 = {--------, -, -, -}                                                    |
|              5  1  1  1                                                     |
|       (1 - T)                                                               |
|                                                                             |
|o17 : List                                                                   |
+-----------------------------------------------------------------------------+

We compute a third example.

+-----------------------------------------------------------------------------+
|i18 : R=ZZ/7[x,y,z,w];                                                       |
+-----------------------------------------------------------------------------+
|i19 : I=ideal(x*z-y^2,x*w-y*z, y*w-z^2);                                     |
|                                                                             |
|o19 : Ideal of R                                                             |
+-----------------------------------------------------------------------------+
|i20 : F=res I;                                                               |
+-----------------------------------------------------------------------------+
|i21 : lambda={2,1};                                                          |
+-----------------------------------------------------------------------------+
|i22 : G=schurComplex(lambda,F)                                               |
|                                                                             |
|       3      11      20      22      12      2                              |
|o22 = R  <-- R   <-- R   <-- R   <-- R   <-- R                               |
|                                                                             |
|      1      2       3       4       5       6                               |
|                                                                             |
|o22 : ChainComplex                                                           |
+-----------------------------------------------------------------------------+
|i23 : G.dd                                                                   |
|                                                                             |
|           3                                                                 |
|o23 = 1 : R  <---------------------------------------------------------------|
|                | yz-xw y2-xz z2-yw yz-xw y2-xz 0     0     0     0     -x y |
|                | 0     0     0     z2-yw 0     y2-xz yz-xw y2-xz 0     y  -z|
|                | 0     0     0     0     z2-yw 0     0     yz-xw y2-xz -z w |
|                                                                             |
|           11                                                                |
|      2 : R   <--------------------------------------------------------------|
|                 | z2-yw  yz-xw  y2-xz  0      0      0      0      0      y |
|                 | 0      0      0      z2-yw  yz-xw  y2-xz  0      0      -z|
|                 | -yz+xw 0      0      -y2+xz 0      0      -2x    2y     -x|
|                 | 0      -yz+xw 0      0      -y2+xz 0      y      -z     0 |
|                 | 0      0      -yz+xw 0      0      -y2+xz -z     w      0 |
|                 | 0      0      -z2+yw 0      z2-yw  0      0      0      0 |
|                 | 0      z2-yw  0      0      0      0      0      0      0 |
|                 | 0      0      z2-yw  0      0      0      0      0      0 |
|                 | 0      0      0      0      0      z2-yw  0      0      0 |
|                 | 0      0      0      0      0      0      -z2+yw 0      0 |
|                 | 0      0      0      0      0      0      0      -z2+yw 0 |
|                                                                             |
|           20                                                                |
|      3 : R   <--------------------------------------------------------------|
|                 | -2x    2y     0      0      -y    -x    0     z     y     |
|                 | y      -z     0      0      0     -y    0     0     z     |
|                 | -z     w      0      0      0     0     -y    0     0     |
|                 | 0      0      -2x    2y     z     0     -x    -w    0     |
|                 | 0      0      y      -z     0     z     0     0     -w    |
|                 | 0      0      -z     w      0     0     z     0     0     |
|                 | yz-xw  0      y2-xz  0      0     0     0     0     0     |
|                 | 0      yz-xw  0      y2-xz  0     0     0     0     0     |
|                 | 0      0      0      0      z2-yw yz-xw y2-xz 0     0     |
|                 | 0      0      0      0      0     0     0     z2-yw yz-xw |
|                 | 0      0      0      0      0     0     0     0     0     |
|                 | 0      0      0      0      0     0     0     0     0     |
|                 | -z2+yw 0      0      0      0     0     0     0     0     |
|                 | 0      -z2+yw 0      0      0     0     0     0     0     |
|                 | z2-yw  0      0      0      0     z2-yw 0     0     0     |
|                 | 0      z2-yw  0      0      0     0     0     0     z2-yw |
|                 | 0      0      -z2+yw 0      0     0     0     0     0     |
|                 | 0      0      0      -z2+yw 0     0     0     0     0     |
|                 | 0      0      z2-yw  0      0     0     z2-yw 0     0     |
|                 | 0      0      0      z2-yw  0     0     0     0     0     |
|                                                                             |
|           22                                                                |
|      4 : R   <--------------------------------------------------------------|
|                 | y      0      -z     0      x      0      -y     0      0 |
|                 | 0      y      0      -z     0      x      0      -y     0 |
|                 | -z     0      w      0      0      0      0      0      x |
|                 | 0      -z     0      w      0      0      0      0      0 |
|                 | -3x    2y     y      0      0      0      0      0      0 |
|                 | y      -z     0      0      -2x    y      y      0      0 |
|                 | -z     w      0      0      0      0      0      0      -2|
|                 | 0      -x     -2x    3y     0      0      0      0      0 |
|                 | 0      0      y      -z     0      -x     -x     2y     0 |
|                 | 0      0      -z     w      0      0      0      0      0 |
|                 | 0      0      0      0      -z     0      w      0      -y|
|                 | 0      0      0      0      0      -z     0      w      0 |
|                 | 0      0      0      0      3y     -2z    -z     0      0 |
|                 | 0      0      0      0      -z     w      0      0      2y|
|                 | 0      0      0      0      0      y      2y     -3z    0 |
|                 | 0      0      0      0      0      0      -z     w      0 |
|                 | 0      0      0      0      0      0      0      0      -3|
|                 | 0      0      0      0      0      0      0      0      0 |
|                 | -z2+yw 0      0      0      -yz+xw 0      0      0      -y|
|                 | 0      -z2+yw 0      0      0      -yz+xw 0      0      0 |
|                 | 0      0      -z2+yw 0      0      0      -yz+xw 0      0 |
|                 | 0      0      0      -z2+yw 0      0      0      -yz+xw 0 |
|                                                                             |
|           12                   2                                            |
|      5 : R   <--------------- R  : 6                                        |
|                 | y   0   |                                                 |
|                 | 2x  -y  |                                                 |
|                 | -x  2y  |                                                 |
|                 | 0   x   |                                                 |
|                 | -z  0   |                                                 |
|                 | -2y z   |                                                 |
|                 | y   -2z |                                                 |
|                 | 0   -y  |                                                 |
|                 | w   0   |                                                 |
|                 | 2z  -w  |                                                 |
|                 | -z  2w  |                                                 |
|                 | 0   z   |                                                 |
|                                                                             |
|o23 : ChainComplexMap                                                        |
|-----------------------------------------------------------------------------|
|      11                                                                     |
|---- R   : 2                                                                 |
| |                                                                           |
| |                                                                           |
| |                                                                           |
|                                                                             |
|                                                               20            |
|------------------------------------------------------------- R   : 3        |
| -z 0      0      0      0      x  -y 0      0      0  0  |                  |
| w  0      0      0      0      0  0  0      0      x  -y |                  |
| y  0      0      0      0      0  0  0      0      0  0  |                  |
| 0  0      0      -x     y      -x y  0      0      0  0  |                  |
| 0  0      0      0      0      0  0  -x     y      -x y  |                  |
| 0  yz-xw  y2-xz  0      0      -z w  0      0      -y z  |                  |
| 0  -y2+xz 0      2y     -2z    y  -z 0      0      0  0  |                  |
| 0  0      -y2+xz -z     w      0  0  y      -z     y  -z |                  |
| 0  0      yz-xw  0      0      0  0  -2z    2w     -z w  |                  |
| 0  0      0      -yz+xw 0      0  0  -y2+xz 0      0  0  |                  |
| 0  0      0      0      -yz+xw 0  0  0      -y2+xz 0  0  |                  |
|                                                                             |
|                                                                           22|
|------------------------------------------------------------------------- R  |
|0     0      0      0     0     0     0     0     0     0   0  0  0   |      |
|0     0      0      -x    0     y     0     0     0     0   0  0  0   |      |
|z     0      0      0     -x    0     y     0     0     0   0  0  0   |      |
|y     0      0      0     0     0     0     0     0     0   0  0  0   |      |
|0     -x     y      0     -x    0     y     0     0     0   0  0  0   |      |
|-w    0      0      0     0     0     0     -x    y     0   0  0  0   |      |
|0     0      0      0     0     0     0     0     0     x   0  -y 0   |      |
|0     0      0      0     0     0     0     0     0     0   x  0  -y  |      |
|0     0      0      0     0     0     0     0     0     -2x y  y  0   |      |
|y2-xz 0      0      0     0     0     0     0     0     0   -x -x 2y  |      |
|0     2y     -2z    z     y     -w    -z    0     0     0   0  0  0   |      |
|0     -z     w      0     z     0     -w    y     -z    0   0  0  0   |      |
|0     y2-xz  0      0     0     0     0     0     0     -y  0  z  0   |      |
|0     0      y2-xz  0     0     0     0     0     0     0   -y 0  z   |      |
|0     0      0      yz-xw y2-xz 0     0     0     0     2y  -z -z 0   |      |
|0     0      0      0     0     yz-xw y2-xz 0     0     0   y  y  -2z |      |
|0     -yz+xw 0      0     0     0     0     0     0     z   0  -w 0   |      |
|0     0      -yz+xw 0     0     0     0     0     0     0   z  0  -w  |      |
|0     yz-xw  0      0     yz-xw 0     0     y2-xz 0     -2z w  w  0   |      |
|z2-yw 0      yz-xw  0     0     0     yz-xw 0     y2-xz 0   -z -z 2w  |      |
|                                                                             |
|                               12                                            |
|----------------------------- R   : 5                                        |
|     0      0      0      |                                                  |
|     0      0      0      |                                                  |
|     0      -y     0      |                                                  |
|     x      0      -y     |                                                  |
|     0      0      0      |                                                  |
|     0      0      0      |                                                  |
|x    y      y      0      |                                                  |
|     0      0      0      |                                                  |
|     0      0      0      |                                                  |
|     -x     -x     2y     |                                                  |
|     0      z      0      |                                                  |
|     -y     0      z      |                                                  |
|     0      0      0      |                                                  |
|     -z     -z     0      |                                                  |
|     0      0      0      |                                                  |
|     y      y      -2z    |                                                  |
|z    2w     w      0      |                                                  |
|     -z     -2z    3w     |                                                  |
|2+xz 0      0      0      |                                                  |
|     -y2+xz 0      0      |                                                  |
|     0      -y2+xz 0      |                                                  |
|     0      0      -y2+xz |                                                  |
|-----------------------------------------------------------------------------|
|: 4                                                                          |
+-----------------------------------------------------------------------------+

See also
========

  * straightenTableau -- Straightening law for
    Z/2Z-graded tableau

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

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

straightenTableau -- Straightening law for Z/2Z-graded tableau
**************************************************************

Synopsis
========

  * Usage:
        S=straightenTableau(T,lambda)
  * Inputs:
      * T, a hash table (Macaulay2Doc), with keys $(i,j)$
        representing the box of the Young diagram in column $i$ and row $j$.
        The values of T are the entries of the boxes.
      * lambda, a list (Macaulay2Doc),
  * Outputs:
      * S, a hash table (Macaulay2Doc), with keys representing
        standard tableaux and values representing the coefficients.

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

This function takes a $\mathbb{Z}/2\mathbb{Z}$-graded Young tableau and
expresses it as a linear combination of standard tableau. Positive entries in
the tableaux correspond to even elements, and negative entries correspond to
odd elements.

The user inputs the Young tableau T in the form of a hash table, and a
partition of the same shape as T, in the form of a list. The key $(i,j)$ in the
hash table of T corresponds to the box of T in column $i$ and row $j$. The
values are the entries of the boxes of T.

The output is a hash table with keys representing standard tableaux and values
representing the coefficients in the linear combination.

+-----------------------------------------------------------------------------+
|i1 : T = new HashTable from {(1,1) => -3, (1,2) => -2, (1,3) => -2, (2,1) => |
|-----------------------------------------------------------------------------|
|1, (2,2) => 2, (2,3) => 3, (3,1) => -1, (3,2) => -1};                        |
+-----------------------------------------------------------------------------+
|i2 : lambda = {3,3,2};                                                       |
+-----------------------------------------------------------------------------+
|i3 : straightenTableau(T,lambda)                                             |
|                                                                             |
|o3 = HashTable{HashTable{(1, 1) => -3} => 1 }                                |
|                         (1, 2) => -2                                        |
|                         (1, 3) => -2                                        |
|                         (2, 1) => -1                                        |
|                         (2, 2) => -1                                        |
|                         (2, 3) => 3                                         |
|                         (3, 1) => 1                                         |
|                         (3, 2) => 2                                         |
|               HashTable{(1, 1) => -3} => -1                                 |
|                         (1, 2) => -2                                        |
|                         (1, 3) => -2                                        |
|                         (2, 1) => -1                                        |
|                         (2, 2) => -1                                        |
|                         (2, 3) => 2                                         |
|                         (3, 1) => 1                                         |
|                         (3, 2) => 3                                         |
|               HashTable{(1, 1) => -3} => 1                                  |
|                         (1, 2) => -2                                        |
|                         (1, 3) => -2                                        |
|                         (2, 1) => -1                                        |
|                         (2, 2) => -1                                        |
|                         (2, 3) => 1                                         |
|                         (3, 1) => 2                                         |
|                         (3, 2) => 3                                         |
|                                                                             |
|o3 : HashTable                                                               |
+-----------------------------------------------------------------------------+

We compute a second example.

+-----------------------------------------------------------------------------+
|i4 : T = new HashTable from {(1,1) => -1, (1,2) => -2, (1,3) => 3, (2,1) =>  |
|-----------------------------------------------------------------------------|
|2, (2,2) => 1, (2,3) => -3};                                                 |
+-----------------------------------------------------------------------------+
|i5 : lambda = {2,2,2};                                                       |
+-----------------------------------------------------------------------------+
|i6 : straightenTableau(T,lambda)                                             |
|                                                                             |
|o6 = HashTable{HashTable{(1, 1) => -3} => 1 }                                |
|                         (1, 2) => -2                                        |
|                         (1, 3) => 2                                         |
|                         (2, 1) => -1                                        |
|                         (2, 2) => 1                                         |
|                         (2, 3) => 3                                         |
|               HashTable{(1, 1) => -3} => -1                                 |
|                         (1, 2) => -2                                        |
|                         (1, 3) => 1                                         |
|                         (2, 1) => -1                                        |
|                         (2, 2) => 2                                         |
|                         (2, 3) => 3                                         |
|               HashTable{(1, 1) => -3} => 1                                  |
|                         (1, 2) => -1                                        |
|                         (1, 3) => 2                                         |
|                         (2, 1) => -2                                        |
|                         (2, 2) => 1                                         |
|                         (2, 3) => 3                                         |
|               HashTable{(1, 1) => -3} => -1                                 |
|                         (1, 2) => -1                                        |
|                         (1, 3) => 1                                         |
|                         (2, 1) => -2                                        |
|                         (2, 2) => 2                                         |
|                         (2, 3) => 3                                         |
|               HashTable{(1, 1) => -3} => 1                                  |
|                         (1, 2) => -2                                        |
|                         (1, 3) => -1                                        |
|                         (2, 1) => 1                                         |
|                         (2, 2) => 2                                         |
|                         (2, 3) => 3                                         |
|                                                                             |
|o6 : HashTable                                                               |
+-----------------------------------------------------------------------------+

See also
========

  * schurComplex -- Schur functors of chain complexes

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

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