This is PieriMaps.info, produced by Macaulay2, version 1.25.05

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

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

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

Let V be a vector space over K. If K has characteristic 0, then given the
partition mu and the partition mu' obtained from mu by removing a single box,
there is a unique (up to nonzero scalar) GL(V)-equivariant inclusion S_mu V ->
V otimes S_mu' V, where S_mu refers to the irreducible representation of GL(V)
with highest weight mu. This can be extended uniquely to a map of P =
Sym(V)-modules P otimes S_mu V -> P otimes S_mu' V. The purpose of this package
is to write down matrix representatives for these maps. The main function for
doing so is pieri. Here is an example of the use of the package
PieriMaps, which is also designed to check whether the maps are being
constructed correctly (important, since it is notoriously difficult to get the
signs and coefficients right.) We will construct by hand the free resolution in
3 variables corresponding to the degree sequence (0,2,3,6). Let's start with
the packaged code:
+-----------------------------------------------------------------------------+
|i1 : R = QQ[a,b,c];                                                          |
+-----------------------------------------------------------------------------+
|i2 : f = pureFree({0,2,3,6}, R)                                              |
|                                                                             |
|o2 = | 12a2 0    0    6ab 0   0   6ac   0    0   2b2 0   0   2bc  0    0     |
|     | 0    12a2 0    0   6ab 0   0     6ac  0   0   2b2 0   0    2bc  0     |
|     | 0    0    12a2 0   0   6ab 0     0    6ac 0   0   2b2 0    0    2bc   |
|     | 0    0    0    0   6a2 0   -12a2 0    0   0   8ab 0   -8ab 4ac  0     |
|     | 0    0    0    0   0   6a2 0     -3a2 0   0   0   8ab 0    -2ab 4ac   |
|     | 0    0    0    0   0   0   0     0    0   0   0   2a2 0    -a2  0     |
|     ------------------------------------------------------------------------|
|     2c2   0    0   0   0   0    0   0    0   0    0   0   |                 |
|     0     2c2  0   0   0   0    0   0    0   0    0   0   |                 |
|     0     0    2c2 0   0   0    0   0    0   0    0   0   |                 |
|     -16ac 0    0   6b2 0   4bc  0   2c2  0   0    0   0   |                 |
|     0     -4ac 0   0   6b2 -b2  4bc -2bc 2c2 0    0   0   |                 |
|     2a2   0    0   0   6ab -2ab 2ac -2ac 0   12b2 6bc 2c2 |                 |
|                                                                             |
|             6      27                                                       |
|o2 : Matrix R  <-- R                                                         |
+-----------------------------------------------------------------------------+
|i3 : betti res coker f                                                       |
|                                                                             |
|            0  1  2 3                                                        |
|o3 = total: 6 27 24 3                                                        |
|         0: 6  .  . .                                                        |
|         1: . 27 24 .                                                        |
|         2: .  .  . .                                                        |
|         3: .  .  . 3                                                        |
|                                                                             |
|o3 : BettiTally                                                              |
+-----------------------------------------------------------------------------+
By the general theory from Eisenbud-Fl\o ystad-Weyman, each of these free
modules should be essentially Schur functors corresponding to the following
partitions.
+------------------------------+
|i4 : needsPackage "SchurRings"|
|                              |
|o4 = SchurRings               |
|                              |
|o4 : Package                  |
+------------------------------+
|i5 : schurRing(s,3)           |
|                              |
|o5 = schurRing (QQ, s, 3)     |
|                              |
|o5 : SchurRing                |
+------------------------------+
|i6 : dim s_{2,2}              |
|                              |
|o6 = 6                        |
+------------------------------+
|i7 : dim s_{4,2}              |
|                              |
|o7 = 27                       |
+------------------------------+
|i8 : dim s_{4,3}              |
|                              |
|o8 = 24                       |
+------------------------------+
|i9 : dim s_{4,3,3}            |
|                              |
|o9 = 3                        |
+------------------------------+
This package also provides a routine schurRank, for computing
this dimension:
+---------------------------+
|i10 : schurRank(3, {2,2})  |
|                           |
|o10 = 6                    |
+---------------------------+
|i11 : schurRank(3, {4,2})  |
|                           |
|o11 = 27                   |
+---------------------------+
|i12 : schurRank(3, {4,3})  |
|                           |
|o12 = 24                   |
+---------------------------+
|i13 : schurRank(3, {4,3,3})|
|                           |
|o13 = 3                    |
+---------------------------+
We now use pieri, to construct each of the maps of the resolution
separately.
+-----------------------------------------------------------------------------+
|i14 : f1 = pieri({4,2,0},{1,1}, R)                                           |
|                                                                             |
|o14 = | 12a2 0    0    6ab 0   0   6ac   0    0   2b2 0   0   2bc  0    0    |
|      | 0    12a2 0    0   6ab 0   0     6ac  0   0   2b2 0   0    2bc  0    |
|      | 0    0    12a2 0   0   6ab 0     0    6ac 0   0   2b2 0    0    2bc  |
|      | 0    0    0    0   6a2 0   -12a2 0    0   0   8ab 0   -8ab 4ac  0    |
|      | 0    0    0    0   0   6a2 0     -3a2 0   0   0   8ab 0    -2ab 4ac  |
|      | 0    0    0    0   0   0   0     0    0   0   0   2a2 0    -a2  0    |
|      -----------------------------------------------------------------------|
|      2c2   0    0   0   0   0    0   0    0   0    0   0   |                |
|      0     2c2  0   0   0   0    0   0    0   0    0   0   |                |
|      0     0    2c2 0   0   0    0   0    0   0    0   0   |                |
|      -16ac 0    0   6b2 0   4bc  0   2c2  0   0    0   0   |                |
|      0     -4ac 0   0   6b2 -b2  4bc -2bc 2c2 0    0   0   |                |
|      2a2   0    0   0   6ab -2ab 2ac -2ac 0   12b2 6bc 2c2 |                |
|                                                                             |
|              6      27                                                      |
|o14 : Matrix R  <-- R                                                        |
+-----------------------------------------------------------------------------+
|i15 : f2 = pieri({4,3,0},{2}, R)                                             |
|                                                                             |
|o15 = | 3b  c   0   0   0     0     0     0     0     0     0     0     0    |
|      | 0   2b  2c  0   0     0     0     0     0     0     0     0     0    |
|      | 0   0   b   3c  0     0     0     0     0     0     0     0     0    |
|      | -6a 0   0   0   3/2b  c     0     0     -3/2c 0     0     0     0    |
|      | 0   -4a 0   0   0     b     2c    0     0     -c    0     0     0    |
|      | 0   0   -2a 0   0     0     1/2b  3c    0     0     -1/2c 0     0    |
|      | 0   -2a 0   0   0     -1/2b 0     0     3b    1/2c  0     0     0    |
|      | 0   0   -4a 0   0     0     -b    0     0     2b    c     0     0    |
|      | 0   0   0   -6a 0     0     0     -3/2b 0     0     b     3/2c  0    |
|      | 0   0   0   0   -9/2a 0     0     0     0     0     0     0     c    |
|      | 0   0   0   0   0     -3a   0     0     0     0     0     0     0    |
|      | 0   0   0   0   0     0     -3/2a 0     0     0     0     0     0    |
|      | 0   0   0   0   0     -3/2a 0     0     -9/2a 0     0     0     -b   |
|      | 0   0   0   0   0     0     -3a   0     0     -3a   0     0     0    |
|      | 0   0   0   0   0     0     0     -9/2a 0     0     -3/2a 0     0    |
|      | 0   0   0   0   0     0     0     0     0     -3/2a 0     0     0    |
|      | 0   0   0   0   0     0     0     0     0     0     -3a   0     0    |
|      | 0   0   0   0   0     0     0     0     0     0     0     -9/2a 0    |
|      | 0   0   0   0   0     0     0     0     0     0     0     0     -4/3a|
|      | 0   0   0   0   0     0     0     0     0     0     0     0     0    |
|      | 0   0   0   0   0     0     0     0     0     0     0     0     0    |
|      | 0   0   0   0   0     0     0     0     0     0     0     0     0    |
|      | 0   0   0   0   0     0     0     0     0     0     0     0     0    |
|      | 0   0   0   0   0     0     0     0     0     0     0     0     0    |
|      | 0   0   0   0   0     0     0     0     0     0     0     0     0    |
|      | 0   0   0   0   0     0     0     0     0     0     0     0     0    |
|      | 0   0   0   0   0     0     0     0     0     0     0     0     0    |
|      -----------------------------------------------------------------------|
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      0   0   0     0     0     0     0     0     0     0   0     |          |
|      2c  0   -c    0     0     0     0     0     0     0   0     |          |
|      0   3c  0     -1/2c 0     0     0     0     0     0   0     |          |
|      0   0   1/2c  0     0     0     0     0     0     0   0     |          |
|      -2b 0   b     c     0     0     0     0     0     0   0     |          |
|      0   -3b 0     1/2b  3/2c  0     0     0     0     0   0     |          |
|      0   0   -1/2b 0     0     0     0     0     0     0   0     |          |
|      0   0   0     -b    0     0     0     0     0     0   0     |          |
|      0   0   0     0     -3/2b 0     0     0     0     0   0     |          |
|      0   0   0     0     0     2c    0     0     0     0   0     |          |
|      -a  0   1/3a  0     0     -1/2b 3c    -1/2c 0     0   0     |          |
|      -2a 0   0     0     0     -3b   0     c     0     0   0     |          |
|      0   -3a 0     -a    0     0     -9/2b 0     3/2c  0   0     |          |
|      0   0   0     -2a   0     0     0     -2b   0     0   0     |          |
|      0   0   0     0     -3a   0     0     0     -3b   0   0     |          |
|      0   0   0     0     0     -1/4a 0     0     0     3c  0     |          |
|      0   0   0     0     0     0     -3/2a 1/6a  0     -6b 3/2c  |          |
|      0   0   0     0     0     0     0     0     -3/2a 0   -9/2b |          |
|                                                                             |
|              27      24                                                     |
|o15 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
|i16 : f3 = pieri({4,3,3},{3,3,3}, R)                                         |
|                                                                             |
|o16 = | 6c3       0         0         |                                      |
|      | -18bc2    0         0         |                                      |
|      | 18b2c     0         0         |                                      |
|      | -6b3      0         0         |                                      |
|      | 0         6c3       0         |                                      |
|      | 72/5ac2   -72/5bc2  18/5c3    |                                      |
|      | -144/5abc 54/5b2c   -36/5bc2  |                                      |
|      | 72/5ab2   -12/5b3   18/5b2c   |                                      |
|      | -72/5ac2  -18/5bc2  12/5c3    |                                      |
|      | 144/5abc  36/5b2c   -54/5bc2  |                                      |
|      | -72/5ab2  -18/5b3   72/5b2c   |                                      |
|      | 0         0         -6b3      |                                      |
|      | 0         27ac2     0         |                                      |
|      | -108/5a2c -162/5abc 108/5ac2  |                                      |
|      | -36/5a2b  36/5ab2   -54/5abc  |                                      |
|      | -432/5a2c -108/5abc 162/5ac2  |                                      |
|      | 216/5a2b  54/5ab2   -216/5abc |                                      |
|      | 0         0         18ab2     |                                      |
|      | 0         18a2c     0         |                                      |
|      | -24/5a3   -36/5a2b  54/5a2c   |                                      |
|      | -216/5a3  -54/5a2b  216/5a2c  |                                      |
|      | 0         0         -18a2b    |                                      |
|      | 0         3/2a3     0         |                                      |
|      | 0         0         6a3       |                                      |
|                                                                             |
|              24      3                                                      |
|o16 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
Fix the degrees (i.e. make sure that the target of f2 is the source of f1,
etc). Otherwise the test of exactness below would fail.
+-----------------------------------------------------------------------------+
|i17 : f1                                                                     |
|                                                                             |
|o17 = | 12a2 0    0    6ab 0   0   6ac   0    0   2b2 0   0   2bc  0    0    |
|      | 0    12a2 0    0   6ab 0   0     6ac  0   0   2b2 0   0    2bc  0    |
|      | 0    0    12a2 0   0   6ab 0     0    6ac 0   0   2b2 0    0    2bc  |
|      | 0    0    0    0   6a2 0   -12a2 0    0   0   8ab 0   -8ab 4ac  0    |
|      | 0    0    0    0   0   6a2 0     -3a2 0   0   0   8ab 0    -2ab 4ac  |
|      | 0    0    0    0   0   0   0     0    0   0   0   2a2 0    -a2  0    |
|      -----------------------------------------------------------------------|
|      2c2   0    0   0   0   0    0   0    0   0    0   0   |                |
|      0     2c2  0   0   0   0    0   0    0   0    0   0   |                |
|      0     0    2c2 0   0   0    0   0    0   0    0   0   |                |
|      -16ac 0    0   6b2 0   4bc  0   2c2  0   0    0   0   |                |
|      0     -4ac 0   0   6b2 -b2  4bc -2bc 2c2 0    0   0   |                |
|      2a2   0    0   0   6ab -2ab 2ac -2ac 0   12b2 6bc 2c2 |                |
|                                                                             |
|              6      27                                                      |
|o17 : Matrix R  <-- R                                                        |
+-----------------------------------------------------------------------------+
|i18 : f2 = map(source f1,,f2)                                                |
|                                                                             |
|o18 = {1} | 3b  c   0   0   0     0     0     0     0     0     0     0      |
|      {1} | 0   2b  2c  0   0     0     0     0     0     0     0     0      |
|      {1} | 0   0   b   3c  0     0     0     0     0     0     0     0      |
|      {1} | -6a 0   0   0   3/2b  c     0     0     -3/2c 0     0     0      |
|      {1} | 0   -4a 0   0   0     b     2c    0     0     -c    0     0      |
|      {1} | 0   0   -2a 0   0     0     1/2b  3c    0     0     -1/2c 0      |
|      {1} | 0   -2a 0   0   0     -1/2b 0     0     3b    1/2c  0     0      |
|      {1} | 0   0   -4a 0   0     0     -b    0     0     2b    c     0      |
|      {1} | 0   0   0   -6a 0     0     0     -3/2b 0     0     b     3/2c   |
|      {1} | 0   0   0   0   -9/2a 0     0     0     0     0     0     0      |
|      {1} | 0   0   0   0   0     -3a   0     0     0     0     0     0      |
|      {1} | 0   0   0   0   0     0     -3/2a 0     0     0     0     0      |
|      {1} | 0   0   0   0   0     -3/2a 0     0     -9/2a 0     0     0      |
|      {1} | 0   0   0   0   0     0     -3a   0     0     -3a   0     0      |
|      {1} | 0   0   0   0   0     0     0     -9/2a 0     0     -3/2a 0      |
|      {1} | 0   0   0   0   0     0     0     0     0     -3/2a 0     0      |
|      {1} | 0   0   0   0   0     0     0     0     0     0     -3a   0      |
|      {1} | 0   0   0   0   0     0     0     0     0     0     0     -9/2a  |
|      {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      |
|      {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      |
|      {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      |
|      {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      |
|      {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     |    |
|      c     0   0   0     0     0     0     0     0     0     0   0     |    |
|      0     2c  0   -c    0     0     0     0     0     0     0   0     |    |
|      0     0   3c  0     -1/2c 0     0     0     0     0     0   0     |    |
|      -b    0   0   1/2c  0     0     0     0     0     0     0   0     |    |
|      0     -2b 0   b     c     0     0     0     0     0     0   0     |    |
|      0     0   -3b 0     1/2b  3/2c  0     0     0     0     0   0     |    |
|      0     0   0   -1/2b 0     0     0     0     0     0     0   0     |    |
|      0     0   0   0     -b    0     0     0     0     0     0   0     |    |
|      0     0   0   0     0     -3/2b 0     0     0     0     0   0     |    |
|      -4/3a 0   0   0     0     0     2c    0     0     0     0   0     |    |
|      0     -a  0   1/3a  0     0     -1/2b 3c    -1/2c 0     0   0     |    |
|      0     -2a 0   0     0     0     -3b   0     c     0     0   0     |    |
|      0     0   -3a 0     -a    0     0     -9/2b 0     3/2c  0   0     |    |
|      0     0   0   0     -2a   0     0     0     -2b   0     0   0     |    |
|      0     0   0   0     0     -3a   0     0     0     -3b   0   0     |    |
|      0     0   0   0     0     0     -1/4a 0     0     0     3c  0     |    |
|      0     0   0   0     0     0     0     -3/2a 1/6a  0     -6b 3/2c  |    |
|      0     0   0   0     0     0     0     0     0     -3/2a 0   -9/2b |    |
|                                                                             |
|              27      24                                                     |
|o18 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
|i19 : f3 = map(source f2,,f3)                                                |
|                                                                             |
|o19 = {2} | 6c3       0         0         |                                  |
|      {2} | -18bc2    0         0         |                                  |
|      {2} | 18b2c     0         0         |                                  |
|      {2} | -6b3      0         0         |                                  |
|      {2} | 0         6c3       0         |                                  |
|      {2} | 72/5ac2   -72/5bc2  18/5c3    |                                  |
|      {2} | -144/5abc 54/5b2c   -36/5bc2  |                                  |
|      {2} | 72/5ab2   -12/5b3   18/5b2c   |                                  |
|      {2} | -72/5ac2  -18/5bc2  12/5c3    |                                  |
|      {2} | 144/5abc  36/5b2c   -54/5bc2  |                                  |
|      {2} | -72/5ab2  -18/5b3   72/5b2c   |                                  |
|      {2} | 0         0         -6b3      |                                  |
|      {2} | 0         27ac2     0         |                                  |
|      {2} | -108/5a2c -162/5abc 108/5ac2  |                                  |
|      {2} | -36/5a2b  36/5ab2   -54/5abc  |                                  |
|      {2} | -432/5a2c -108/5abc 162/5ac2  |                                  |
|      {2} | 216/5a2b  54/5ab2   -216/5abc |                                  |
|      {2} | 0         0         18ab2     |                                  |
|      {2} | 0         18a2c     0         |                                  |
|      {2} | -24/5a3   -36/5a2b  54/5a2c   |                                  |
|      {2} | -216/5a3  -54/5a2b  216/5a2c  |                                  |
|      {2} | 0         0         -18a2b    |                                  |
|      {2} | 0         3/2a3     0         |                                  |
|      {2} | 0         0         6a3       |                                  |
|                                                                             |
|              24      3                                                      |
|o19 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
|i20 : f1 * f2                                                                |
|                                                                             |
|o20 = 0                                                                      |
|                                                                             |
|              6      24                                                      |
|o20 : Matrix R  <-- R                                                        |
+-----------------------------------------------------------------------------+
|i21 : f2 * f3                                                                |
|                                                                             |
|o21 = 0                                                                      |
|                                                                             |
|              27      3                                                      |
|o21 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
Check that the complex is exact.
+------------------------+
|i22 : ker f1 == image f2|
|                        |
|o22 = true              |
+------------------------+
|i23 : ker f2 == image f3|
|                        |
|o23 = true              |
+------------------------+
Looks great! Now let's try it modulo some prime numbers and see if we get
exactness.
+-----------------------------------------------------------------------------+
|i24 : p = 32003                                                              |
|                                                                             |
|o24 = 32003                                                                  |
+-----------------------------------------------------------------------------+
|i25 : R = ZZ/p[a,b,c];                                                       |
+-----------------------------------------------------------------------------+
|i26 : f1 = pieri({4,2,0},{1,1},R)                                            |
|                                                                             |
|o26 = | 7012a2+6250ab-5379b2+12089ac-8371bc-3669c2                           |
|      | 3630a2-5863ab-8195b2+3219ac-14676bc+10443c2                          |
|      | -12738a2-14389ab-3669b2+14830ac+10443bc+4994c2                       |
|      | 1962a2+704ab+42b2                                                    |
|      | 3a2                                                                  |
|      | 0                                                                    |
|      -----------------------------------------------------------------------|
|      9770a2-3164ab+14689b2+12630ac+661bc+10959c2                            |
|      -6841a2-2950ab-13606b2-14090ac-15537bc+8572c2                          |
|      15976a2-1642ab-2725b2-14788ac+7543bc-1774c2                            |
|      3793a2+6938ab+14609b2+9266ac-12554bc+1946c2                            |
|      5403a2+4637ab-2058ac+1946bc+86c2                                       |
|      a2-12863ab                                                             |
|      -----------------------------------------------------------------------|
|      1642a2+15650ab+932b2+4677ac-13229bc+10676c2                            |
|      5880a2+12908ab+12962b2-12390ac+2856bc+4469c2                           |
|      -285a2+6636ab-9146b2-10575ac-3456bc-2216c2                             |
|      3554a2+8755ab-5997b2-15690ac+6585bc-1155c2                             |
|      12831a2-7437ab+b2-15850ac-1155bc+5350c2                                |
|      102a2-9646ab                                                           |
|      -----------------------------------------------------------------------|
|      -1128a2-6948ab+12866b2+4136ac+9742bc-3952c2                            |
|      -15652a2+9995ab-13630b2+4919ac+3717bc+6340c2                           |
|      3004a2-6111ab-10090b2-12960ac-556bc+4552c2                             |
|      1723a2+2521ab+11439b2+7047ac-12864bc-3101c2                            |
|      7431a2-12074ab+b2-13792ac-3101bc+5264c2                                |
|      101a2+3217ab                                                           |
|      -----------------------------------------------------------------------|
|      7230a2-7752ab-6045b2-4786ac+6938bc-8543c2                              |
|      12526a2-7587ab+12804b2+13972ac+8090bc-1035c2                           |
|      -15894a2+3623ab+12687b2+11930ac-5873bc-3927c2                          |
|      1985a2-8539ab+14392b2-11485ac+12244bc-6993c2                           |
|      -3363a2+10655ab+b2-9676ac-6993bc+5092c2                                |
|      99a2-3060ab                                                            |
|      -----------------------------------------------------------------------|
|      10499a2+15367ab-14668b2+2246ac+2607bc+4192c2                           |
|      -9328a2+14587ab+6412b2+3856ac-14242bc+1359c2                           |
|      10229a2-12217ab-11216b2+131ac-1421bc+7570c2                            |
|      10095a2+15220ab-14490b2+1986ac+5349bc-10885c2                          |
|      -14145a2+1381ab+b2-5560ac-10885bc+4920c2                               |
|      97a2-9337ab                                                            |
|      -----------------------------------------------------------------------|
|      1047a2-9148ab-14489b2-12525ac-4085bc+2166c2                            |
|      7697a2+1003ab-1637b2+4780ac+391bc+13521c2                              |
|      -15962a2+9478ab+14126b2+15628ac+12799bc+7040c2                         |
|      -5950a2+9792ab-11201b2+15457ac-1546bc-14777c2                          |
|      7088a2-7893ab+b2-1444ac-14777bc+4748c2                                 |
|      95a2-15614ab                                                           |
|      -----------------------------------------------------------------------|
|      -11780a2-7783ab+15644b2-12502ac+9471bc+15795c2                         |
|      -6257a2-7148ab+11266b2-12116ac+13638bc+3415c2                          |
|      13379a2-9728ab-8883b2-6095ac+4751bc-5518c2                             |
|      -14147a2+7180ab-7744b2-3075ac-8441bc+13334c2                           |
|      -3670a2+14836ab+b2+2672ac+13334bc+4576c2                               |
|      93a2+10112ab                                                           |
|      -----------------------------------------------------------------------|
|      3034a2-13972ab+4905b2-4278ac-5652bc-10955c2                            |
|      -3240a2-9891ab-2428b2+461ac-6617bc-1500c2                              |
|      -1071a2+1855ab-8264b2-12598ac+1894bc-1475c2                            |
|      -1335a2+12896ab-3702b2+10400ac-15336bc+9442c2                          |
|      -14377a2+5564ab+b2+6788ac+9442bc+4404c2                                |
|      91a2+3835ab                                                            |
|      -----------------------------------------------------------------------|
|      -831a2-13029ab+6536b2+734ac-2467bc-6439c2                              |
|      10974a2-15649ab-3280b2+1186ac+2373bc+9234c2                            |
|      -10599a2+8862ab-8287b2-601ac+14687bc+1563c2                            |
|      14886a2-3252ab+10396b2-7748ac+9856bc+5550c2                            |
|      8269a2-3518ab+b2+10906ac+5550bc+4232c2                                 |
|      89a2-2463ab                                                            |
|      -----------------------------------------------------------------------|
|      -12775a2-2850ab-13594b2+4415ac-8872bc+82c2                             |
|      -9498a2+4038ab-12853b2+11380ac-7464bc-2812c2                           |
|      -6720a2-158ab-3727b2-13068ac+4744bc+997c2                              |
|      -4792a2-15924ab-5194b2-15584ac+6009bc+1659c2                           |
|      -5848a2-7444ab+b2+15112ac+1659bc+4060c2                                |
|      87a2-9502ab                                                            |
|      -----------------------------------------------------------------------|
|      -15504a2+13987ab+14845b2-3188ac+14858bc-13007c2                        |
|      10549a2+8441ab-15477b2+8136ac-11717bc-5829c2                           |
|      -13699a2-12105ab-8768b2+14289ac+4845bc+5278c2                          |
|      14817a2+6672ab+10707b2-15387ac-5252bc-2188c2                           |
|      15830a2+8648ab+b2-10655ac-2188bc+3889c2                                |
|      85a2+1314ab                                                            |
|      -----------------------------------------------------------------------|
|      4524a2+12312ab-14403b2+7223ac-12012bc+3211c2                           |
|      -6731a2-15509ab-7762b2-6386ac+9930bc+1264c2                            |
|      -12893a2+13347ab+3656b2-6061ac-15138bc+8000c2                          |
|      2048a2-15003ab+58b2-5828ac+9823bc+10672c2                              |
|      -15463a2+4694ab+42b2-801ac+10674bc-2188c2                              |
|      1902a2+5587ab+6b2+ac                                                   |
|      -----------------------------------------------------------------------|
|      -874a2+8323ab+7624b2-3641ac+7499bc+6453c2                              |
|      -2574a2-1514ab-11310b2+9477ac+8166bc-8357c2                            |
|      11944a2+3335ab+2767b2+4436ac+704bc-15849c2                             |
|      5768a2-11230ab-11596b2-3289ac-13091bc+1170c2                           |
|      14598a2+2902ab+1380b2-13881ac+1458bc-9464c2                            |
|      5137a2-3348ab+816b2+144ac+27bc+c2                                      |
|      -----------------------------------------------------------------------|
|      8798a2+6400ab-5840b2-1845ac-5373bc+3153c2                              |
|      5490a2-3378ab-15485b2-10675ac-10730bc+8660c2                           |
|      7941a2+2999ab-5200b2+11546ac-13300bc+4886c2                            |
|      312a2-8445ab-3198b2-14681ac-14607bc-15411c2                            |
|      -7665a2-10070ab+1339b2-11917ac-15125bc-3555c2                          |
|      3318a2-3010ab+810b2+143ac+27bc+c2                                      |
|      -----------------------------------------------------------------------|
|      2831a2-3560ab+571b2-13064ac+1640bc-3083c2                              |
|      -13668a2-12329ab-13022b2-4900ac+3246bc-15294c2                         |
|      -8960a2-6361ab+2810b2-2056ac+11836bc+7296c2                            |
|      13799a2+5358ab+2103b2-847ac-14917bc+2457c2                             |
|      -3911a2+7994ab+1219b2-9746ac+2737bc-7160c2                             |
|      -1896a2+12949ab+792b2+140ac+27bc+c2                                    |
|      -----------------------------------------------------------------------|
|      -10208a2-10208ab+4359b2+7863ac+12227bc-1636c2                          |
|      -6975a2-5478ab+525b2+4489ac-3792bc+2838c2                              |
|      -9436a2-12004ab-6659b2-1756ac+13227bc-12080c2                          |
|      10799a2-14805ab-5853b2+5504ac+8783bc+567c2                             |
|      1753a2+9159ab+1108b2-11225ac+841bc-12195c2                             |
|      -6399a2+6913ab+774b2+137ac+27bc+c2                                     |
|      -----------------------------------------------------------------------|
|      15978a2+15978ab-3205b2-15047ac+11129bc-1579c2                          |
|      13751a2-10331ab+10561b2+12886ac+11403bc+12979c2                        |
|      12020a2+6580ab+13037b2-5038ac-8432bc+4988c2                            |
|      -12240a2-2272ab+2215b2+3435ac+3661bc-12579c2                           |
|      137a2+8742ab+1006b2-10819ac-12311bc+9131c2                             |
|      -10245a2-7910ab+756b2+134ac+27bc+c2                                    |
|      -----------------------------------------------------------------------|
|      8a2+8ab-6399b2-9590ac+2115bc-5865c2                                    |
|      -15989a2+3346ab+2178b2+12477ac-5945bc-3518c2                           |
|      -14213a2-12550ab+5406b2-7599ac+5172bc+13573c2                          |
|      -9477a2+252ab-4635b2+3027ac+11793bc+15836c2                            |
|      -2787a2-4434ab+913b2-14623ac-15905bc-11076c2                           |
|      -13488a2-10036ab+738b2+131ac+27bc+c2                                   |
|      -----------------------------------------------------------------------|
|      4045a2+4045ab+809b2-2113ac+1696bc+1963c2                               |
|      -1304a2+12277ab-3177b2+973ac+7104bc-15817c2                            |
|      -4410a2-7310ab-14593b2+12296ac-7956bc-10499c2                          |
|      -15500a2+12511ab-5323b2-1496ac+4892bc-10046c2                          |
|      8205a2-1470ab+829b2+15722ac-9790bc-12374c2                             |
|      15821a2-394ab+720b2+128ac+27bc+c2                                      |
|      -----------------------------------------------------------------------|
|      15219a2+15219ab+15845b2-11552ac-1366bc-2180c2                          |
|      4293a2-1686ab+6688b2-10945ac-10702bc+1327c2                            |
|      4900a2+4550ab+10451b2+5419ac+1789bc-6152c2                             |
|      -10585a2+213ab-7314b2-3964ac-4016bc-15700c2                            |
|      -5979a2-11500ab+754b2+924ac-15450bc+1997c2                             |
|      13622a2+1758ab+702b2+125ac+27bc+c2                                     |
|      -----------------------------------------------------------------------|
|      -2422a2-2422ab-6885b2+7401ac+4968bc-5617c2                             |
|      516a2-8522ab-3268b2+14073ac+8136bc-9874c2                              |
|      -6986a2-6472ab-10454b2+11582ac-7971bc-5982c2                           |
|      8679a2-941ab+3008b2-2798ac+11988bc-13214c2                             |
|      2493a2+14054ab+688b2+3806ac-12970bc-2882c2                             |
|      11864a2-2309ab+684b2+122ac+27bc+c2                                     |
|      -----------------------------------------------------------------------|
|      6487a2+6487ab+7698b2-6463ac+10096bc+983c2                              |
|      1235a2+15775ab-1860b2-10278ac-6994bc+15653c2                           |
|      -9492a2+14649ab+5162b2+2860ac-12427bc+11060c2                          |
|      -3302a2-15821ab-4033b2+10151ac-13273bc-4590c2                          |
|      -12215a2-4361ab+582b2+7846ac-4366bc-11113c2                            |
|      9642a2+11901ab+626b2+112ac+26bc+c2                                     |
|      -----------------------------------------------------------------------|
|      -12758a2-12758ab+3849b2-15384ac+5255bc-6838c2                          |
|      -11535a2+14459ab+12641b2-534ac-8196bc+6473c2                           |
|      12756a2-4746ab+9071b2-3909ac-380bc-2164c2                              |
|      13224a2-2459ab+12661b2+6309ac+15482bc-12673c2                          |
|      -4479a2+12477ab+401b2-13706ac-12495bc-1648c2                           |
|      6331a2+12532ab+494b2+89ac+23bc+c2                                      |
|      -----------------------------------------------------------------------|
|      3849ac-7137bc+4496c2                                                   |
|      3849a2-4713ab+13571b2-5152ac-1772bc-5389c2                             |
|      -930a2-14501ab-3597b2-15426ac+13417bc-2977c2                           |
|      -12411a2-13177ab+6020b2-7509ac+14376bc+4715c2                          |
|      -11925a2-12616ab+135b2+5609ac+4813bc-9068c2                            |
|      1785a2-3459ab+266b2+49ac+17bc+c2                                       |
|      -----------------------------------------------------------------------|
|      -7344bc-4176c2                                0               |        |
|      4717ab-9757ac+2927bc+11966c2                  0               |        |
|      13571a2+13571ab-10087b2-10087ac-8418bc+4717c2 0               |        |
|      4717a2-2627ab+7344b2+7259ac+8259bc+15083c2    -10c2           |        |
|      2448a2-11272ab+25b2-8765ac+15125bc+3025c2     0               |        |
|      275a2+5961ab+110b2+21ac+11bc+c2               25b2+5ac+5bc+c2 |        |
|                                                                             |
|              6      27                                                      |
|o26 : Matrix R  <-- R                                                        |
+-----------------------------------------------------------------------------+
|i27 : betti res coker f1                                                     |
|                                                                             |
|             0  1  2 3                                                       |
|o27 = total: 6 27 24 3                                                       |
|          0: 6  .  . .                                                       |
|          1: . 27 24 .                                                       |
|          2: .  .  . .                                                       |
|          3: .  .  . 3                                                       |
|                                                                             |
|o27 : BettiTally                                                             |
+-----------------------------------------------------------------------------+
|i28 : f2 = pieri({4,3,0},{2},R)                                              |
|                                                                             |
|o28 = | 6986b+4510c           -1451b-7220c         13064b+9527c              |
|      | 9020b-10741c          -14440b+3016c        -12949b-15317c            |
|      | 10631b+2953c          1508b+10808c         8343b+684c                |
|      | -13972a+14166b+3105c  2902a-9397b-5451c    5875a-5254b+4348c         |
|      | 13963a+923b+2263c     -3123a-12640b-10200c -6105a-4110b+15487c       |
|      | 10741a+10088b-320c    -3016a+9732b+14476c  15317a-12189b+420c        |
|      | -9020a+11176b+3043c   14440a-8058b-7439c   12949a-14861b+15883c      |
|      | -10521a-8004b-15157c  -6032a+14786b+14710c -1369a-8099b+10532c       |
|      | -5906a-3207b+8650c    10387a-12207b+8371c  -1368a+7689b+1079c        |
|      | -10495a-11636c        -3812a-11660c        15762a-3966c              |
|      | -2769a+2034c          5917a+2034c          12330a+1147c              |
|      | 1739a+15880c          2807a+15880c         4564a+13355c              |
|      | -10840a+11636b-13565c 8524a+11660b-13565c  -464a+3966b+11577c        |
|      | -14780a-2034b+11327c  -13758a-2034b+11327c 9839a-1147b-4667c         |
|      | 10581a-15880b-270c    -6807a-15880b-270c   7676a-13355b-10233c       |
|      | -9129a+13565b         -9686a+13565b        -15646a-11577b            |
|      | 13468a-11327b         -12127a-11327b       407a+4667b                |
|      | 6053a+270b            6890a+270b           -3237a+10233b             |
|      | 4847a-24c             4879a-24c            5288a-1392c               |
|      | -7163a+6b+2034c       -7163a+6b+2034c      11569a+348b-2296c         |
|      | -6907a+36b-12204c     -6907a+36b-12204c    7702a+2088b+13764c        |
|      | 15134a+6102b          15134a+6102b         9647a-6879b               |
|      | 9349a-7595b           9349a-7595b          9334a+4475b               |
|      | 540a                  540a                 -11537a                   |
|      | 3a                    3a                   174a                      |
|      | 0                     0                    a                         |
|      | 0                     0                    0                         |
|      -----------------------------------------------------------------------|
|      -15183b-7919c       4969b-6623c          15163b-9429c                  |
|      -15838b-4094c       -13246b+10515c       13145b-2015c                  |
|      -2047b-12069c       -10744b-5499c        14994b+30c                    |
|      -1637a+7830b-12120c -9938a+8975b+14045c  1677a-13275b-6277c            |
|      -327a-10408b+12432c -5511a+13349b+6604c  5713a+7996b-3832c             |
|      4094a+11292b-3503c  -10515a+11413b+7642c 2015a+15497b+8124c            |
|      15838a+8628b-15635c 13246a+7935b-12479c  -13145a-7455b+907c            |
|      8188a+10885b-5888c  10973a-8736b+10767c  4030a+4637b+4339c             |
|      -7865a+13337b+7255c 10998a+12255b+10478c -60a+7193b-12886c             |
|      8513a+14347c        5078a+11453c         7822a-12082c                  |
|      -779a-299c          -8044a-4319c         8015a-8243c                   |
|      -1873a+13436c       -2236a-2363c         -14488a+13760c                |
|      10476a-14347b-648c  -1934a-11453b+1094c  9193a+12082b-15937c           |
|      -5945a+299b+9117c   6396a+4319b+11574c   -2415a+8243b+247c             |
|      2501a-13436b-10053c 4315a+2363b-9603c    -13948a-13760b-9333c          |
|      14902a+648b         5434a-1094b          -2721a+15937b                 |
|      -14339a-9117b       -298a-11574b         -13017a-247b                  |
|      10238a+10053b       569a+9603b           6655a+9333b                   |
|      2206a-1376c         -4603a-1336c         -5226a-1312c                  |
|      -15636a+344b-3652c  -3539a+334b-7042c    4100a+328b-9076c              |
|      1595a+2064b-10103c  2131a+2004b+10237c   8114a+1968b-9562c             |
|      -11110a-10947b      -14998a+10886b       1871a+4784b                   |
|      13769a-11797b       8855a+11529b         -494a-12879b                  |
|      -11897a             -12797a              -13337a                       |
|      172a                167a                 164a                          |
|      a                   a                    a                             |
|      0                   0                    0                             |
|      -----------------------------------------------------------------------|
|      -15488b-2190c         -6016b-15512c        -10796b-3322c               |
|      -4380b+3603c          979b+8643c           -6644b+13540c               |
|      -14200b+3911c         -11680b+13264c       6770b+594c                  |
|      -1027a+13161b+1107c   12032a+12462b+8085c  -10411a-8151b+73c           |
|      8760a-6030b+5120c     -1958a+13732b+1140c  13288a+11081b+15412c        |
|      -3603a+3224b-8085c    -8643a+188b+1322c    -13540a-146b+14592c         |
|      4380a-11259b+3888c    -979a+4428b-194c     6644a+474b-7998c            |
|      -7206a+9104b+401c     14717a-1152b-7492c   4923a+303b+6218c            |
|      -7822a+12344b+6148c   5475a-6280b-1448c    -1188a+13369b-8693c         |
|      -7480a-6710c          -5383a-4866c         -7550a-6982c                |
|      -13913a-13787c        -9193a+11052c        -1240a+2268c                |
|      -9672a-2120c          -564a+14003c         438a-1877c                  |
|      -1547a+6710b+241c     -5536a+4866b-14378c  -1641a+6982b+4212c          |
|      -10669a+13787b-11080c 36a-11052b+9596c     -15142a-2268b-1731c         |
|      -12777a+2120b-9063c   14874a-14003b-8793c  12126a+1877b-8523c          |
|      -11664a-241b          582a+14378b          -8009a-4212b                |
|      -1203a+11080b         -9527a-9596b         13349a+1731b                |
|      13559a+9063b          4344a+8793b          -5924a+8523b                |
|      -1721a-1288c          6488a-1264c          -12026a-1240c               |
|      12147a+322b-11110c    -11401a+316b-13144c  -2538a+310b-15178c          |
|      13305a+1932b+2642c    -14299a+1896b+14846c -10692a+1860b-4953c         |
|      -13263a-1318b         3606a-7420b          -11528a-13522b              |
|      -9843a-5284b          12811a+2311b         3462a+9906b                 |
|      -13877a               -14417a              -14957a                     |
|      161a                  158a                 155a                        |
|      a                     a                    a                           |
|      0                     0                    0                           |
|      -----------------------------------------------------------------------|
|      -5877b-6258c         -8813b-14176c        9247b+1060c                  |
|      -12516b+12885c       3651b-8309c          2120b+10717c                 |
|      -9559b+5151c         11847b-9902c         -10643b+15903c               |
|      11754a-5269b-13526c  -14377a+10506b-6944c 13509a+1070b-4108c           |
|      -6971a-4888b+6606c   -7302a-12365b+14458c -4240a+14466b-7935c          |
|      -12885a-6457b+8767c  8309a+3644b+324c     -10717a+5242b+6182c          |
|      12516a-12283b+15786c -3651a+11342b+59c    -2120a-2088b-1550c           |
|      6233a+12052b+3081c   -15385a-7052b+12629c 10569a+15319b+7046c          |
|      -10302a+5928b+11394c -12199a+1309b-9825c  197a-13808b+9463c            |
|      15807a-13490c        485a+7187c           -3210a-5592c                 |
|      14664a-8136c         5092a+11835c         -11395a-9899c                |
|      -12632a+14246c       -10932a-1634c        -15726a+14484c               |
|      13421a+13490b-7995c  -13194a-7187b+13013c -13415a+5592b+9233c          |
|      8032a+8136b-13058c   9785a-11835b+7618c   9851a+9899b-3711c            |
|      -12082a-14246b-8253c -4899a+1634b-7983c   -9125a-14484b-7713c          |
|      -15355a+7995b        -177a-13013b         4650a-9233b                  |
|      -9243a+13058b        -5884a-7618b         10865a+3711b                 |
|      -2179a+8253b         -2528a+7983b         3614a+7713b                  |
|      7319a-1216c          1085a-1192c          7456a-1168c                  |
|      6733a+304b+14791c    -15589a+298b+12757c  -3462a+292b+10723c           |
|      -7877a+1824b+7251c   -5858a+1788b-12548c  -8567a+1752b-344c            |
|      5341a+12379b         -9793a+6277b         7084a+175b                   |
|      -5887a-14502b        -15236a-6907b        7422a+688b                   |
|      -15497a              15966a               15426a                       |
|      152a                 149a                 146a                         |
|      a                    a                    a                            |
|      0                    0                    0                            |
|      -----------------------------------------------------------------------|
|      8346b+14184c         -9595b+9348c         -13631b+10629c               |
|      -3635b+3555c         -13307b+15643c       -10745b+3953c                |
|      -14224b+9547c        -8180b+3540c         -14025b-10748c               |
|      15311a+7734b-3965c   -12813a+9990b-9574c  -4741a+2846b+13893c          |
|      7270a-8118b+15295c   -5389a-15886b+2496c  -10513a+4749b+11970c         |
|      -3555a+11902b-6621c  -15643a+1686b+83c    -3953a+11113b-5917c          |
|      3635a-4247b+155c     13307a-4681b+2124c   10745a-4661b-15762c          |
|      -7110a+8819b-14645c  717a+5124b-523c      -7906a+10735b+15973c         |
|      12909a-15674b-15693c -7080a+7567b+10712c  -10507a-9064b+15358c         |
|      8801a-643c           2033a-10155c         -8538a+8235c                 |
|      -7649a-5644c         15655a+15019c        -14247a-9613c                |
|      -3703a-1484c         -5058a+13376c        -1336a-1110c                 |
|      -7367a+643b-6563c    10762a+10155b+14319c 4307a-8235b+12905c           |
|      -8336a+5644b-15074c  9143a-15019b+14327c  -4109a+9613b+15512c          |
|      2879a+1484b-7443c    9053a-13376b-6634c   12940a+1110b-2619c           |
|      -465a+6563b          -6372a-14319b        15283a-12905b                |
|      11932a+15074b        1569a-14327b         -15916a-15512b               |
|      15076a+7443b         -133a+6634b          -14071a+2619b                |
|      11525a-1144c         13540a-1072c         -10980a-7952c                |
|      -5658a+286b+8690c    3719a+268b+2616c     -4829a+1988b+8796c           |
|      -13233a+1716b+11854c -11654a+1608b-15708c 15806a+11928b+11062c         |
|      -7908a-5924b         13138a+7857b         9845a-5489b-c                |
|      -1855a+8295b         3349a-587b           979a+9879b                   |
|      14886a               13268a               5238a+2b                     |
|      143a                 134a                 994a-2c                      |
|      a                    a                    14a+4b                       |
|      0                    0                    a                            |
|      -----------------------------------------------------------------------|
|      13944b+8113c         -12487b-10199c       11493b-3230c                 |
|      -15777b-3330c        11605b+15016c        -6460b+10930c                |
|      -1665b-9464c         7508b-2013c          5465b+9843c                  |
|      4115a+3655b-13627c   -7029a+11202b-9870c  9017a-13905b-10840c          |
|      -449a-96b-10941c     8793a-4578b+5692c    12920a+12644b-2954c          |
|      3330a+1715b+9216c    -15016a-1300b-13233c -10930a+10268b-5069c         |
|      15777a-4893b-7101c   -11605a+12873b-5446c 6460a+8643b-9990c            |
|      6660a+169b+6374c     1971a+12819b-4774c   10143a+3510b+14230c          |
|      -13075a-15829b-1522c 4026a+3036b-13463c   12317a+13207b+10645c         |
|      -10965a-3751c        -1603a+15007c        9712a-3181c                  |
|      288a-2428c           13734a+7603c         -5929a+4719c                 |
|      -5145a-3628c         3900a-15438c         1199a-5707c                  |
|      -8446a+3751b-10743c  -9009a-15007b-13513c 6591a+3181b-7312c            |
|      313a+2428b+14400c    8473a-7603b+6775c    -1668a-4719b+6030c           |
|      -12164a+3628b-8838c  -1412a+15438b-12058c 7589a+5707b+3309c            |
|      -10700a+10743b       -15665a+13513b       -2033a+7312b                 |
|      12881a-14400b        14322a-6775b         -10687a-6030b                |
|      4566a+8838b          8386a+12058b         68a-3309b                    |
|      15669a-7533c         1326a-581c           14909a+11467c                |
|      4795a+9884b-8050c    -4631a+8146b+11934c  -5256a+5134b+3257c           |
|      -8089a-4702b+15121c  -12580a-15130b-8618c 9905a-1199b+11729c           |
|      14031a+8735b-16c     -10726a+4564b-15c    -3338a+10320b-13c            |
|      3203a+1761b          -13550a-14767b       -12060a+8545b                |
|      -14327a+32b          -7887a+30b           -6618a+26b                   |
|      4942a-30c            4073a-28c            2567a-24c                    |
|      98a+60b-c            85a+56b-c            61a+48b-c                    |
|      16a+3b               15a+3b               13a+3b                       |
|      -----------------------------------------------------------------------|
|      -12152b+5475c         -7500b-3750c          3601b-14201c               |
|      10950b-4201c          -7500b-12651c         3601b-14201c               |
|      13901b+9024c          9676b+7975c           8901b+14976c               |
|      -7699a-11453b-530c    15000a-7001b-1228c    -7202a+6075b+12181c        |
|      10103a-1952b-5791c    15000a+9022b-1078c    -7202a-8095b-2134c         |
|      4201a+421b-6692c      12651a+13839b+2171c   14201a+11919b+9122c        |
|      -10950a-1868b-12264c  7500a+15989b-3786c    -3601a+11500b-7098c        |
|      8402a+14108b-605c     -6701a-10819b-13931c  -3601a+11776b+12273c       |
|      13955a+10893b+6492c   -15950a-3533b+171c    2051a-3357b-13273c         |
|      2356a-10975c          -11000a+10754c        13778a-13697c              |
|      5856a+9441c           4937a-6991c           -7718a+6289c               |
|      -1263a+9943c          -9514a-7889c          -3754a+14309c              |
|      7194a+10975b-15119c   -12280a-10754b+15534c -7037a+13697b+1965c        |
|      7052a-9441b-3531c     3688a+6991b+2007c     3077a-6289b+8197c          |
|      -12603a-9943b+1032c   4086a+7889b-13825c    14708a-14309b-5059c        |
|      4789a+15119b          11358a-15534b         -10709a-1965b              |
|      1815a+3531b           9790a-2007b           -4816a-8197b               |
|      12527a-1032b          -513a+13825b          7816a+5059b                |
|      -6702a-11912c         -3671a-6136c          7595a-2632c                |
|      5653a+2978b+7658c     14319a+1534b-5087c    12202a+658b+8068c          |
|      -11206a-14135b-14437c 7926a+9204b-1781c     -10219a+3948b+15442c       |
|      11355a-8660b-11c      -11123a-15036b-9c     -10603a-7682b-7c           |
|      7062a-3129b           -4014a+3562b          15609a+1119b               |
|      -2064a+22b            -4353a+18b            10118a+14b                 |
|      1489a-20c             767a-16c              329a-12c                   |
|      41a+40b-c             25a+32b-c             13a+24b-c                  |
|      11a+3b                9a+3b                 7a+3b                      |
|      -----------------------------------------------------------------------|
|      0                  0             0    |                                |
|      0                  0             0    |                                |
|      8901c              0             0    |                                |
|      8901b+4661c        0             0    |                                |
|      -4240b-8732c       0             0    |                                |
|      2077b-12513c       0             0    |                                |
|      -15682b+8520c      0             0    |                                |
|      -2077b+3408c       0             0    |                                |
|      14201a-7189b+1704c 0             0    |                                |
|      5300a+639c         0             0    |                                |
|      12720a-5184c       0             0    |                                |
|      -6231a-8527c       -748c         0    |                                |
|      1060a-639b-13516c  0             0    |                                |
|      424a+5184b-4170c   -374c         0    |                                |
|      -4900a+8527b-1337c 748b-187c     0    |                                |
|      6443a+13516b       0             0    |                                |
|      -10224a+4170b      374b          0    |                                |
|      -5112a+1337b       187b          0    |                                |
|      -852a-824c         -136c         0    |                                |
|      -14238a+206b+1696c 34b+184c      0    |                                |
|      213a+1236b-10236c  204b-1116c    0    |                                |
|      14782a+5133b-5c    1309a+561b-3c -c   |                                |
|      8340a-11531b       748a+2232b    0    |                                |
|      2674a+10b          374a+6b       2b   |                                |
|      103a-8c            17a-4c        -c   |                                |
|      5a+16b-c           a+8b-c        2b-c |                                |
|      5a+3b              3a+3b         a+3b |                                |
|                                                                             |
|              27      24                                                     |
|o28 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
|i29 : f3 = pieri({4,3,3},{3,3,3},R)                                          |
|                                                                             |
|o29 = | -20c3         -15c3         0      |                                 |
|      | 60bc2         45bc2         0      |                                 |
|      | -60b2c        -45b2c        0      |                                 |
|      | 20b3          15b3          0      |                                 |
|      | 20c3          20c3          0      |                                 |
|      | -48ac2-48bc2  -36ac2-48bc2  3c3    |                                 |
|      | 96abc+36b2c   72abc+36b2c   -6bc2  |                                 |
|      | -48ab2-8b3    -36ab2-8b3    3b2c   |                                 |
|      | 48ac2-12bc2   36ac2-12bc2   2c3    |                                 |
|      | -96abc+24b2c  -72abc+24b2c  -9bc2  |                                 |
|      | 48ab2-12b3    36ab2-12b3    12b2c  |                                 |
|      | 0             0             -5b3   |                                 |
|      | 90ac2         90ac2         0      |                                 |
|      | 72a2c-108abc  54a2c-108abc  18ac2  |                                 |
|      | 24a2b+24ab2   18a2b+24ab2   -9abc  |                                 |
|      | 288a2c-72abc  216a2c-72abc  27ac2  |                                 |
|      | -144a2b+36ab2 -108a2b+36ab2 -36abc |                                 |
|      | 0             0             15ab2  |                                 |
|      | 60a2c         60a2c         0      |                                 |
|      | 16a3-24a2b    12a3-24a2b    9a2c   |                                 |
|      | 144a3-36a2b   108a3-36a2b   36a2c  |                                 |
|      | 0             0             -15a2b |                                 |
|      | 5a3           5a3           0      |                                 |
|      | 0             0             5a3    |                                 |
|                                                                             |
|              24      3                                                      |
|o29 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
|i30 : f2 = map(source f1,,f2)                                                |
|                                                                             |
|o30 = | 6986b+4510c           -1451b-7220c         13064b+9527c              |
|      | 9020b-10741c          -14440b+3016c        -12949b-15317c            |
|      | 10631b+2953c          1508b+10808c         8343b+684c                |
|      | -13972a+14166b+3105c  2902a-9397b-5451c    5875a-5254b+4348c         |
|      | 13963a+923b+2263c     -3123a-12640b-10200c -6105a-4110b+15487c       |
|      | 10741a+10088b-320c    -3016a+9732b+14476c  15317a-12189b+420c        |
|      | -9020a+11176b+3043c   14440a-8058b-7439c   12949a-14861b+15883c      |
|      | -10521a-8004b-15157c  -6032a+14786b+14710c -1369a-8099b+10532c       |
|      | -5906a-3207b+8650c    10387a-12207b+8371c  -1368a+7689b+1079c        |
|      | -10495a-11636c        -3812a-11660c        15762a-3966c              |
|      | -2769a+2034c          5917a+2034c          12330a+1147c              |
|      | 1739a+15880c          2807a+15880c         4564a+13355c              |
|      | -10840a+11636b-13565c 8524a+11660b-13565c  -464a+3966b+11577c        |
|      | -14780a-2034b+11327c  -13758a-2034b+11327c 9839a-1147b-4667c         |
|      | 10581a-15880b-270c    -6807a-15880b-270c   7676a-13355b-10233c       |
|      | -9129a+13565b         -9686a+13565b        -15646a-11577b            |
|      | 13468a-11327b         -12127a-11327b       407a+4667b                |
|      | 6053a+270b            6890a+270b           -3237a+10233b             |
|      | 4847a-24c             4879a-24c            5288a-1392c               |
|      | -7163a+6b+2034c       -7163a+6b+2034c      11569a+348b-2296c         |
|      | -6907a+36b-12204c     -6907a+36b-12204c    7702a+2088b+13764c        |
|      | 15134a+6102b          15134a+6102b         9647a-6879b               |
|      | 9349a-7595b           9349a-7595b          9334a+4475b               |
|      | 540a                  540a                 -11537a                   |
|      | 3a                    3a                   174a                      |
|      | 0                     0                    a                         |
|      | 0                     0                    0                         |
|      -----------------------------------------------------------------------|
|      -15183b-7919c       4969b-6623c          15163b-9429c                  |
|      -15838b-4094c       -13246b+10515c       13145b-2015c                  |
|      -2047b-12069c       -10744b-5499c        14994b+30c                    |
|      -1637a+7830b-12120c -9938a+8975b+14045c  1677a-13275b-6277c            |
|      -327a-10408b+12432c -5511a+13349b+6604c  5713a+7996b-3832c             |
|      4094a+11292b-3503c  -10515a+11413b+7642c 2015a+15497b+8124c            |
|      15838a+8628b-15635c 13246a+7935b-12479c  -13145a-7455b+907c            |
|      8188a+10885b-5888c  10973a-8736b+10767c  4030a+4637b+4339c             |
|      -7865a+13337b+7255c 10998a+12255b+10478c -60a+7193b-12886c             |
|      8513a+14347c        5078a+11453c         7822a-12082c                  |
|      -779a-299c          -8044a-4319c         8015a-8243c                   |
|      -1873a+13436c       -2236a-2363c         -14488a+13760c                |
|      10476a-14347b-648c  -1934a-11453b+1094c  9193a+12082b-15937c           |
|      -5945a+299b+9117c   6396a+4319b+11574c   -2415a+8243b+247c             |
|      2501a-13436b-10053c 4315a+2363b-9603c    -13948a-13760b-9333c          |
|      14902a+648b         5434a-1094b          -2721a+15937b                 |
|      -14339a-9117b       -298a-11574b         -13017a-247b                  |
|      10238a+10053b       569a+9603b           6655a+9333b                   |
|      2206a-1376c         -4603a-1336c         -5226a-1312c                  |
|      -15636a+344b-3652c  -3539a+334b-7042c    4100a+328b-9076c              |
|      1595a+2064b-10103c  2131a+2004b+10237c   8114a+1968b-9562c             |
|      -11110a-10947b      -14998a+10886b       1871a+4784b                   |
|      13769a-11797b       8855a+11529b         -494a-12879b                  |
|      -11897a             -12797a              -13337a                       |
|      172a                167a                 164a                          |
|      a                   a                    a                             |
|      0                   0                    0                             |
|      -----------------------------------------------------------------------|
|      -15488b-2190c         -6016b-15512c        -10796b-3322c               |
|      -4380b+3603c          979b+8643c           -6644b+13540c               |
|      -14200b+3911c         -11680b+13264c       6770b+594c                  |
|      -1027a+13161b+1107c   12032a+12462b+8085c  -10411a-8151b+73c           |
|      8760a-6030b+5120c     -1958a+13732b+1140c  13288a+11081b+15412c        |
|      -3603a+3224b-8085c    -8643a+188b+1322c    -13540a-146b+14592c         |
|      4380a-11259b+3888c    -979a+4428b-194c     6644a+474b-7998c            |
|      -7206a+9104b+401c     14717a-1152b-7492c   4923a+303b+6218c            |
|      -7822a+12344b+6148c   5475a-6280b-1448c    -1188a+13369b-8693c         |
|      -7480a-6710c          -5383a-4866c         -7550a-6982c                |
|      -13913a-13787c        -9193a+11052c        -1240a+2268c                |
|      -9672a-2120c          -564a+14003c         438a-1877c                  |
|      -1547a+6710b+241c     -5536a+4866b-14378c  -1641a+6982b+4212c          |
|      -10669a+13787b-11080c 36a-11052b+9596c     -15142a-2268b-1731c         |
|      -12777a+2120b-9063c   14874a-14003b-8793c  12126a+1877b-8523c          |
|      -11664a-241b          582a+14378b          -8009a-4212b                |
|      -1203a+11080b         -9527a-9596b         13349a+1731b                |
|      13559a+9063b          4344a+8793b          -5924a+8523b                |
|      -1721a-1288c          6488a-1264c          -12026a-1240c               |
|      12147a+322b-11110c    -11401a+316b-13144c  -2538a+310b-15178c          |
|      13305a+1932b+2642c    -14299a+1896b+14846c -10692a+1860b-4953c         |
|      -13263a-1318b         3606a-7420b          -11528a-13522b              |
|      -9843a-5284b          12811a+2311b         3462a+9906b                 |
|      -13877a               -14417a              -14957a                     |
|      161a                  158a                 155a                        |
|      a                     a                    a                           |
|      0                     0                    0                           |
|      -----------------------------------------------------------------------|
|      -5877b-6258c         -8813b-14176c        9247b+1060c                  |
|      -12516b+12885c       3651b-8309c          2120b+10717c                 |
|      -9559b+5151c         11847b-9902c         -10643b+15903c               |
|      11754a-5269b-13526c  -14377a+10506b-6944c 13509a+1070b-4108c           |
|      -6971a-4888b+6606c   -7302a-12365b+14458c -4240a+14466b-7935c          |
|      -12885a-6457b+8767c  8309a+3644b+324c     -10717a+5242b+6182c          |
|      12516a-12283b+15786c -3651a+11342b+59c    -2120a-2088b-1550c           |
|      6233a+12052b+3081c   -15385a-7052b+12629c 10569a+15319b+7046c          |
|      -10302a+5928b+11394c -12199a+1309b-9825c  197a-13808b+9463c            |
|      15807a-13490c        485a+7187c           -3210a-5592c                 |
|      14664a-8136c         5092a+11835c         -11395a-9899c                |
|      -12632a+14246c       -10932a-1634c        -15726a+14484c               |
|      13421a+13490b-7995c  -13194a-7187b+13013c -13415a+5592b+9233c          |
|      8032a+8136b-13058c   9785a-11835b+7618c   9851a+9899b-3711c            |
|      -12082a-14246b-8253c -4899a+1634b-7983c   -9125a-14484b-7713c          |
|      -15355a+7995b        -177a-13013b         4650a-9233b                  |
|      -9243a+13058b        -5884a-7618b         10865a+3711b                 |
|      -2179a+8253b         -2528a+7983b         3614a+7713b                  |
|      7319a-1216c          1085a-1192c          7456a-1168c                  |
|      6733a+304b+14791c    -15589a+298b+12757c  -3462a+292b+10723c           |
|      -7877a+1824b+7251c   -5858a+1788b-12548c  -8567a+1752b-344c            |
|      5341a+12379b         -9793a+6277b         7084a+175b                   |
|      -5887a-14502b        -15236a-6907b        7422a+688b                   |
|      -15497a              15966a               15426a                       |
|      152a                 149a                 146a                         |
|      a                    a                    a                            |
|      0                    0                    0                            |
|      -----------------------------------------------------------------------|
|      8346b+14184c         -9595b+9348c         -13631b+10629c               |
|      -3635b+3555c         -13307b+15643c       -10745b+3953c                |
|      -14224b+9547c        -8180b+3540c         -14025b-10748c               |
|      15311a+7734b-3965c   -12813a+9990b-9574c  -4741a+2846b+13893c          |
|      7270a-8118b+15295c   -5389a-15886b+2496c  -10513a+4749b+11970c         |
|      -3555a+11902b-6621c  -15643a+1686b+83c    -3953a+11113b-5917c          |
|      3635a-4247b+155c     13307a-4681b+2124c   10745a-4661b-15762c          |
|      -7110a+8819b-14645c  717a+5124b-523c      -7906a+10735b+15973c         |
|      12909a-15674b-15693c -7080a+7567b+10712c  -10507a-9064b+15358c         |
|      8801a-643c           2033a-10155c         -8538a+8235c                 |
|      -7649a-5644c         15655a+15019c        -14247a-9613c                |
|      -3703a-1484c         -5058a+13376c        -1336a-1110c                 |
|      -7367a+643b-6563c    10762a+10155b+14319c 4307a-8235b+12905c           |
|      -8336a+5644b-15074c  9143a-15019b+14327c  -4109a+9613b+15512c          |
|      2879a+1484b-7443c    9053a-13376b-6634c   12940a+1110b-2619c           |
|      -465a+6563b          -6372a-14319b        15283a-12905b                |
|      11932a+15074b        1569a-14327b         -15916a-15512b               |
|      15076a+7443b         -133a+6634b          -14071a+2619b                |
|      11525a-1144c         13540a-1072c         -10980a-7952c                |
|      -5658a+286b+8690c    3719a+268b+2616c     -4829a+1988b+8796c           |
|      -13233a+1716b+11854c -11654a+1608b-15708c 15806a+11928b+11062c         |
|      -7908a-5924b         13138a+7857b         9845a-5489b-c                |
|      -1855a+8295b         3349a-587b           979a+9879b                   |
|      14886a               13268a               5238a+2b                     |
|      143a                 134a                 994a-2c                      |
|      a                    a                    14a+4b                       |
|      0                    0                    a                            |
|      -----------------------------------------------------------------------|
|      13944b+8113c         -12487b-10199c       11493b-3230c                 |
|      -15777b-3330c        11605b+15016c        -6460b+10930c                |
|      -1665b-9464c         7508b-2013c          5465b+9843c                  |
|      4115a+3655b-13627c   -7029a+11202b-9870c  9017a-13905b-10840c          |
|      -449a-96b-10941c     8793a-4578b+5692c    12920a+12644b-2954c          |
|      3330a+1715b+9216c    -15016a-1300b-13233c -10930a+10268b-5069c         |
|      15777a-4893b-7101c   -11605a+12873b-5446c 6460a+8643b-9990c            |
|      6660a+169b+6374c     1971a+12819b-4774c   10143a+3510b+14230c          |
|      -13075a-15829b-1522c 4026a+3036b-13463c   12317a+13207b+10645c         |
|      -10965a-3751c        -1603a+15007c        9712a-3181c                  |
|      288a-2428c           13734a+7603c         -5929a+4719c                 |
|      -5145a-3628c         3900a-15438c         1199a-5707c                  |
|      -8446a+3751b-10743c  -9009a-15007b-13513c 6591a+3181b-7312c            |
|      313a+2428b+14400c    8473a-7603b+6775c    -1668a-4719b+6030c           |
|      -12164a+3628b-8838c  -1412a+15438b-12058c 7589a+5707b+3309c            |
|      -10700a+10743b       -15665a+13513b       -2033a+7312b                 |
|      12881a-14400b        14322a-6775b         -10687a-6030b                |
|      4566a+8838b          8386a+12058b         68a-3309b                    |
|      15669a-7533c         1326a-581c           14909a+11467c                |
|      4795a+9884b-8050c    -4631a+8146b+11934c  -5256a+5134b+3257c           |
|      -8089a-4702b+15121c  -12580a-15130b-8618c 9905a-1199b+11729c           |
|      14031a+8735b-16c     -10726a+4564b-15c    -3338a+10320b-13c            |
|      3203a+1761b          -13550a-14767b       -12060a+8545b                |
|      -14327a+32b          -7887a+30b           -6618a+26b                   |
|      4942a-30c            4073a-28c            2567a-24c                    |
|      98a+60b-c            85a+56b-c            61a+48b-c                    |
|      16a+3b               15a+3b               13a+3b                       |
|      -----------------------------------------------------------------------|
|      -12152b+5475c         -7500b-3750c          3601b-14201c               |
|      10950b-4201c          -7500b-12651c         3601b-14201c               |
|      13901b+9024c          9676b+7975c           8901b+14976c               |
|      -7699a-11453b-530c    15000a-7001b-1228c    -7202a+6075b+12181c        |
|      10103a-1952b-5791c    15000a+9022b-1078c    -7202a-8095b-2134c         |
|      4201a+421b-6692c      12651a+13839b+2171c   14201a+11919b+9122c        |
|      -10950a-1868b-12264c  7500a+15989b-3786c    -3601a+11500b-7098c        |
|      8402a+14108b-605c     -6701a-10819b-13931c  -3601a+11776b+12273c       |
|      13955a+10893b+6492c   -15950a-3533b+171c    2051a-3357b-13273c         |
|      2356a-10975c          -11000a+10754c        13778a-13697c              |
|      5856a+9441c           4937a-6991c           -7718a+6289c               |
|      -1263a+9943c          -9514a-7889c          -3754a+14309c              |
|      7194a+10975b-15119c   -12280a-10754b+15534c -7037a+13697b+1965c        |
|      7052a-9441b-3531c     3688a+6991b+2007c     3077a-6289b+8197c          |
|      -12603a-9943b+1032c   4086a+7889b-13825c    14708a-14309b-5059c        |
|      4789a+15119b          11358a-15534b         -10709a-1965b              |
|      1815a+3531b           9790a-2007b           -4816a-8197b               |
|      12527a-1032b          -513a+13825b          7816a+5059b                |
|      -6702a-11912c         -3671a-6136c          7595a-2632c                |
|      5653a+2978b+7658c     14319a+1534b-5087c    12202a+658b+8068c          |
|      -11206a-14135b-14437c 7926a+9204b-1781c     -10219a+3948b+15442c       |
|      11355a-8660b-11c      -11123a-15036b-9c     -10603a-7682b-7c           |
|      7062a-3129b           -4014a+3562b          15609a+1119b               |
|      -2064a+22b            -4353a+18b            10118a+14b                 |
|      1489a-20c             767a-16c              329a-12c                   |
|      41a+40b-c             25a+32b-c             13a+24b-c                  |
|      11a+3b                9a+3b                 7a+3b                      |
|      -----------------------------------------------------------------------|
|      0                  0             0    |                                |
|      0                  0             0    |                                |
|      8901c              0             0    |                                |
|      8901b+4661c        0             0    |                                |
|      -4240b-8732c       0             0    |                                |
|      2077b-12513c       0             0    |                                |
|      -15682b+8520c      0             0    |                                |
|      -2077b+3408c       0             0    |                                |
|      14201a-7189b+1704c 0             0    |                                |
|      5300a+639c         0             0    |                                |
|      12720a-5184c       0             0    |                                |
|      -6231a-8527c       -748c         0    |                                |
|      1060a-639b-13516c  0             0    |                                |
|      424a+5184b-4170c   -374c         0    |                                |
|      -4900a+8527b-1337c 748b-187c     0    |                                |
|      6443a+13516b       0             0    |                                |
|      -10224a+4170b      374b          0    |                                |
|      -5112a+1337b       187b          0    |                                |
|      -852a-824c         -136c         0    |                                |
|      -14238a+206b+1696c 34b+184c      0    |                                |
|      213a+1236b-10236c  204b-1116c    0    |                                |
|      14782a+5133b-5c    1309a+561b-3c -c   |                                |
|      8340a-11531b       748a+2232b    0    |                                |
|      2674a+10b          374a+6b       2b   |                                |
|      103a-8c            17a-4c        -c   |                                |
|      5a+16b-c           a+8b-c        2b-c |                                |
|      5a+3b              3a+3b         a+3b |                                |
|                                                                             |
|              27      24                                                     |
|o30 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
|i31 : f3 = map(source f2,,f3)                                                |
|                                                                             |
|o31 = {1} | -20c3         -15c3         0      |                             |
|      {1} | 60bc2         45bc2         0      |                             |
|      {1} | -60b2c        -45b2c        0      |                             |
|      {1} | 20b3          15b3          0      |                             |
|      {1} | 20c3          20c3          0      |                             |
|      {1} | -48ac2-48bc2  -36ac2-48bc2  3c3    |                             |
|      {1} | 96abc+36b2c   72abc+36b2c   -6bc2  |                             |
|      {1} | -48ab2-8b3    -36ab2-8b3    3b2c   |                             |
|      {1} | 48ac2-12bc2   36ac2-12bc2   2c3    |                             |
|      {1} | -96abc+24b2c  -72abc+24b2c  -9bc2  |                             |
|      {1} | 48ab2-12b3    36ab2-12b3    12b2c  |                             |
|      {1} | 0             0             -5b3   |                             |
|      {1} | 90ac2         90ac2         0      |                             |
|      {1} | 72a2c-108abc  54a2c-108abc  18ac2  |                             |
|      {1} | 24a2b+24ab2   18a2b+24ab2   -9abc  |                             |
|      {1} | 288a2c-72abc  216a2c-72abc  27ac2  |                             |
|      {1} | -144a2b+36ab2 -108a2b+36ab2 -36abc |                             |
|      {1} | 0             0             15ab2  |                             |
|      {1} | 60a2c         60a2c         0      |                             |
|      {1} | 16a3-24a2b    12a3-24a2b    9a2c   |                             |
|      {1} | 144a3-36a2b   108a3-36a2b   36a2c  |                             |
|      {1} | 0             0             -15a2b |                             |
|      {1} | 5a3           5a3           0      |                             |
|      {1} | 0             0             5a3    |                             |
|                                                                             |
|              24      3                                                      |
|o31 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
|i32 : f1 * f2                                                                |
|                                                                             |
|o32 = | 3253a3+5047a2b-1601ab2-12426b3+6717a2c+6959abc-4464b2c-5683ac2-3930bc|
|      | 267a3+9095a2b-93ab2-14086b3-11447a2c-14714abc-2149b2c-10471ac2+11716b|
|      | 11576a3-10838a2b-12579ab2+2734b3-10134a2c+10361abc+8410b2c-4926ac2+11|
|      | 5577a3+971a2b-1520ab2-8124b3-4824a2c-9075abc-15841b2c+11537ac2-12867b|
|      | -299a3+4752a2b+10383ab2+11115b3+4920a2c-2145abc-1583b2c+212ac2-5152bc|
|      | 6023a3-1770a2b-15405ab2+9867b3+15602a2c+5903abc-1371b2c+12587ac2+6444|
|      -----------------------------------------------------------------------|
|      2-6379c3     -10844a3+10340a2b+4202ab2+15438b3+4976a2c-2293abc-13300b2c|
|      c2+12911c3   -8459a3-14651a2b-7616ab2+7859b3+2835a2c+9117abc+3093b2c-15|
|      220bc2-826c3 6483a3-6927a2b+8397ab2+10240b3+8686a2c+4987abc+13256b2c+58|
|      c2-8831c3    15419a3+15273a2b+7039ab2-3395b3+5236a2c+15383abc-4577b2c+4|
|      2+3108c3     -1568a3+13761a2b-5619ab2-7923b3-14571a2c-7211abc+13872b2c+|
|      bc2+863c3    11655a3-12102a2b-7669ab2+10011b3+6370a2c-15951abc-1371b2c+|
|      -----------------------------------------------------------------------|
|      -11659ac2+15014bc2+4334c3                                              |
|      798ac2-9663bc2+15851c3                                                 |
|      24ac2+12168bc2+11899c3                                                 |
|      216ac2+6510bc2+3985c3                                                  |
|      3274ac2-14865bc2+14248c3                                               |
|      2941ac2+6444bc2+863c3                                                  |
|      -----------------------------------------------------------------------|
|      -12893a3+11254a2b+9293ab2                                              |
|      -13273a3-9154a2b-2421ab2-                                              |
|      -8742a3-10835a2b-12394ab2                                              |
|      1057a3+15620a2b+10655ab2-                                              |
|      -2510a3+12559a2b-8149ab2+                                              |
|      9381a3-10179a2b+11313ab2-                                              |
|      -----------------------------------------------------------------------|
|      +6315b3+6513a2c+8291abc+5745b2c+1999ac2+7067bc2-15927c3                |
|      11012b3-11023a2c-1747abc+4939b2c-1254ac2+2368bc2-15065c3               |
|      +14008b3+7277a2c+15288abc-15126b2c-14357ac2-11433bc2-14903c3           |
|      12440b3-12125a2c-8473abc-13417b2c+10782ac2-10863bc2+7756c3             |
|      2209b3-7337a2c-15633abc+13837b2c+7241ac2-11889bc2-12698c3              |
|      8922b3+4063a2c+13041abc-6571b2c+15768ac2-13383bc2-4824c3               |
|      -----------------------------------------------------------------------|
|      10432a3-6305a2b-8548ab2-683b3+8196a2c-2128abc-8250b2c+12467ac2-6360bc2+|
|      13831a3+7064a2b-11476ab2-11340b3+1496a2c-11410abc-307b2c+14972ac2+7752b|
|      7519a3+11323a2b+8321ab2-13808b3+15193a2c-443abc-14102b2c+15854ac2-15195|
|      -7764a3+15894a2b+2935ab2-11060b3+9707a2c-2953abc+15212b2c-3471ac2-3332b|
|      11581a3-708a2b+8334ab2-4803b3+7982a2c+15687abc-243b2c-5403ac2-9747bc2+6|
|      -5351a3+544a2b-10469ab2+5548b3-3586a2c+15343abc-4634b2c-1317ac2+14347bc|
|      -----------------------------------------------------------------------|
|      11673c3     14229a3-5657a2b                                            |
|      c2+13183c3  8899a3-7400a2b+                                            |
|      bc2+12224c3 2407a3-6580a2b+                                            |
|      c2-15634c3  4425a3-11834a2b                                            |
|      7c3         9232a3-15000a2b                                            |
|      2+15936c3   7646a3+11764a2b                                            |
|      -----------------------------------------------------------------------|
|      -731ab2-11568b3-7671a2c-13273abc-8756b2c+1366ac2-5118bc2-1302c3        |
|      1534ab2-4742b3+537a2c+8801abc-3226b2c+9844ac2+3689bc2-6585c3           |
|      568ab2-5261b3-5882a2c-15147abc-6891b2c+5926ac2+3281bc2+6489c3          |
|      -10836ab2+10920b3-6937a2c-2897abc-7641b2c+14895ac2-7544bc2-12339c3     |
|      -14636ab2+6196b3+16a2c-5212abc+9336b2c+3599ac2-4423bc2+15238c3         |
|      +10992ab2+9435b3-11569a2c-143abc+4812b2c+9342ac2+3768bc2+3830c3        |
|      -----------------------------------------------------------------------|
|      7671a3+4827a2b+879ab2-14812b3+14268a2c+11388abc+6628b2c-10653ac2+9294bc|
|      1163a3-14544a2b-925ab2-12106b3+11335a2c+15325abc-14257b2c-8917ac2-14630|
|      -2226a3+4183a2b+10664ab2-9775b3+6707a2c-1362abc-7235b2c-15270ac2-11936b|
|      -6935a3+8207a2b-5528ab2+1688b3+2610a2c+15332abc+14151b2c+11188ac2+2890b|
|      15556a3-11286a2b+7548ab2-3982b3-367a2c-2865abc+4109b2c+1307ac2-14871bc2|
|      14355a3+9489a2b-985ab2-1570b3-8034a2c+8065abc-4337b2c-1728ac2-2193bc2+2|
|      -----------------------------------------------------------------------|
|      2-7463c3   12906a3+12736a2b+7065ab2                                    |
|      bc2+5399c3 -4058a3-9133a2b+7088ab2-                                    |
|      c2+3947c3  -10475a3+6079a2b-6788ab2                                    |
|      c2+7298c3  -13796a3-9895a2b-9911ab2                                    |
|      -9516c3    13371a3+9285a2b-2923ab2-                                    |
|      967c3      7519a3-12292a2b-12788ab2                                    |
|      -----------------------------------------------------------------------|
|      +15342b3+11045a2c-14954abc-8387b2c-355ac2-3169bc2-9756c3               |
|      7990b3-10773a2c+1218abc+6606b2c+2722ac2+7937bc2+7435c3                 |
|      -15739b3-2385a2c+2010abc+11769b2c+4234ac2-3177bc2-5692c3               |
|      +10102b3-11040a2c+9420abc-135b2c-1329ac2-15303bc2-8515c3               |
|      1106b3-2079a2c-9751abc-12049b2c+5151ac2-14552bc2-10484c3               |
|      -11267b3-10778a2c-15745abc+4928b2c+4703ac2-7740bc2+2104c3              |
|      -----------------------------------------------------------------------|
|      5650a3+5484a2b-7668ab2+10288b3+3029a2c+670abc-11707b2c+14062ac2+4403bc2|
|      -1210a3-2881a2b+3550ab2+12803b3+945a2c+11551abc+12875b2c-1289ac2+3636bc|
|      10599a3-12691a2b-8232ab2+14005b3+3182a2c-14925abc+15614b2c-4736ac2-1463|
|      4235a3-6922a2b+6616ab2+10735b3-3296a2c+3614abc-2633b2c-3554ac2-7006bc2+|
|      -15506a3-11539a2b+4278ab2+15942b3+2475a2c-11680abc+7820b2c-14465ac2-734|
|      -8649a3-14167a2b-2321ab2+14939b3+8326a2c-15416abc+604b2c+15028ac2-12873|
|      -----------------------------------------------------------------------|
|      +13213c3    -2353a3-15525a2b+13178ab2+8936b3+5483a2c-12960abc-11958b2c+|
|      2-1472c3    5807a3+487a2b-7366ab2-8686b3-1836a2c-4182abc-9678b2c+4991ac|
|      4bc2-7958c3 -9966a3+2858a2b-4076ab2-2761b3-4878a2c-12583abc+5833b2c-255|
|      2873c3      8957a3+11679a2b-15648ab2-2245b3+3892a2c-14493abc+8661b2c-66|
|      7bc2+1602c3 -15113a3+3397a2b+8533ab2+11354b3+11895a2c+2805abc-5733b2c+5|
|      bc2+1241c3  -13746a3+3192a2b-13762ab2+15634b3-12432a2c-8571abc+14694b2c|
|      -----------------------------------------------------------------------|
|      12671ac2+15570bc2+2596c3                                               |
|      2+12477bc2-9719c3                                                      |
|      0ac2-9414bc2+14942c3                                                   |
|      22ac2+20bc2+10764c3                                                    |
|      237ac2+7282bc2+5180c3                                                  |
|      -15517ac2+14411bc2+378c3                                               |
|      -----------------------------------------------------------------------|
|      -13868a3-4297a2b+2974ab2-6824b3-6079a2c-9352abc-9913b2c+49ac2+12922bc2+|
|      15301a3+12293a2b-14636ab2-12572b3+14418a2c+9548abc+14285b2c+7799ac2+271|
|      -15968a3+5103a2b-15086ab2-14693b3-8694a2c+5856abc+10687b2c+8471ac2+1079|
|      -9344a3+15556a2b+7793ab2+6852b3-8467a2c+10046abc-2755b2c+12556ac2+13704|
|      241a3+10771a2b-4401ab2+4454b3+8570a2c-14136abc-1276b2c-4367ac2-7818bc2-|
|      -15064a3-9475a2b-15076ab2-6590b3+12675a2c-6167abc+15195b2c+11304ac2+101|
|      -----------------------------------------------------------------------|
|      10004c3      -3290a3-13425a2b-4342ab2+7276b3-6779a2c+                  |
|      4bc2+13919c3 6190a3-5185a2b+7550ab2+12249b3+11643a2c-                  |
|      7bc2+12958c3 -14630a3+63a2b-10355ab2+12513b3-1086a2c-                  |
|      bc2-5092c3   11510a3-1757a2b-13134ab2-2031b3-6080a2c-                  |
|      9127c3       -6042a3-9830a2b+6158ab2-3094b3-14827a2c+                  |
|      06bc2-485c3  -9123a3+191a2b-11126ab2-15398b3-14699a2c                  |
|      -----------------------------------------------------------------------|
|      2901abc-6364b2c+8832ac2-10473bc2-10420c3                               |
|      12201abc-10924b2c-4203ac2+15873bc2+3674c3                              |
|      15127abc+3205b2c-2029ac2+5637bc2+13024c3                               |
|      80abc-2931b2c+1075ac2+2276bc2-12480c3                                  |
|      11833abc+9897b2c+5373ac2-5717bc2+14146c3                               |
|      -9472abc+2197b2c+3717ac2+6217bc2-1348c3                                |
|      -----------------------------------------------------------------------|
|      -8012a3+4342a2b-3093ab2-1300b3-7823a2c-8326abc+5125b2c-7611ac2-3781bc2-|
|      2144a3-10955a2b+8130ab2+14568b3-10249a2c-9343abc-10688b2c+3489ac2-1212b|
|      -3559a3+14046a2b+15269ab2+7582b3+10327a2c+11878abc-6114b2c+12224ac2-614|
|      9309a3-4576a2b+298ab2-4786b3-13038a2c-6878abc-14214b2c-14461ac2-117bc2+|
|      -12075a3-5922a2b+6244ab2-4842b3-12375a2c-4435abc+3871b2c-5036ac2-2895bc|
|      1423a3-5688a2b+12657ab2-15545b3-8719a2c-3591abc+2186b2c+9826ac2+4768bc2|
|      -----------------------------------------------------------------------|
|      15124c3     -12193a3+8854a2b+2142ab2+14078b3+2728a2c-9601abc-12831b2c+2|
|      c2+6997c3   -323a3-7291a2b-13103ab2+12554b3+5771a2c+7057abc+15111b2c-56|
|      7bc2+9397c3 8094a3-3742a2b-13325ab2-2931b3-14207a2c+15902abc+3966b2c-11|
|      10903c3     -1435a3+9142a2b+10638ab2-3574b3+13837a2c+12959abc+10780b2c+|
|      2-9339c3    -9513a3+4720a2b-6329ab2+1728b3+13726a2c-6042abc+12552b2c-74|
|      -2213c3     4047a3-6300a2b-2637ab2+3796b3+997a2c+4811abc+5169b2c-10303a|
|      -----------------------------------------------------------------------|
|      514ac2-3367bc2-4049c3                                                  |
|      22ac2-13413bc2-4512c3                                                  |
|      733ac2+9698bc2+9857c3                                                  |
|      13503ac2-3238bc2-3017c3                                                |
|      40ac2-14165bc2-2035c3                                                  |
|      c2-11575bc2-3117c3                                                     |
|      -----------------------------------------------------------------------|
|      13323a3-1873a2b+5179ab2+15531b3-2685a2c-549abc+3779b2c+5846ac2+14768bc2|
|      11894a3+12256a2b-4976ab2+15861b3-4991a2c+2617abc-10062b2c+9698ac2+4312b|
|      -1179a3+13369a2b+6846ab2+1067b3+4241a2c+14428abc+8485b2c+10751ac2-2797b|
|      2277a3+387a2b-2835ab2+4249b3+7407a2c-12988abc-14540b2c-11889ac2-5823bc2|
|      -991a3+15960a2b+4943ab2+2131b3-14798a2c-11947abc-13658b2c+15863ac2-7111|
|      291a3+12162a2b-13458ab2+124b3-9892a2c+8079abc-4451b2c-15530ac2+8043bc2-|
|      -----------------------------------------------------------------------|
|      -8771c3    13479a3-2673a2b+8186ab2-7869b3-6472a2c+4045abc+9313b2c-10778|
|      c2+348c3   1191a3+14474a2b+3119ab2+3625b3-2603a2c+4966abc+11921b2c+1251|
|      c2-7585c3  -14967a3-4339a2b+9699ab2-2730b3+2157a2c-13109abc+11927b2c+40|
|      -1840c3    -6819a3+12142a2b-2172ab2+3448b3-2571a2c-2685abc+13957b2c-151|
|      bc2-6084c3 -5032a3+5168a2b+3912ab2+4394b3-4291a2c-2358abc-388b2c-3471ac|
|      6471c3     -12663a3+10566a2b+8887ab2+10773b3-975a2c-1605abc+6359b2c-102|
|      -----------------------------------------------------------------------|
|      ac2-9874bc2+14937c3                                                    |
|      9ac2-14658bc2-1402c3                                                   |
|      85ac2+12262bc2-15188c3                                                 |
|      88ac2+2238bc2-3094c3                                                   |
|      2-13714bc2-4572c3                                                      |
|      98ac2+686bc2-7207c3                                                    |
|      -----------------------------------------------------------------------|
|      13168a3+786a2b+13215ab2+5939b3+8856a2c-12310abc-9093b2c-4497ac2+15683bc|
|      805a3-14218a2b+11609ab2-403b3-9697a2c+4951abc-1903b2c+15458ac2-4805bc2-|
|      1194a3+5762a2b+12765ab2+3415b3+2489a2c+4864abc+2150b2c-9933ac2-1342bc2-|
|      3864a3+2859a2b+11409ab2-12452b3-731a2c-10818abc+14326b2c-7744ac2+14185b|
|      -3087a3+14005a2b-8485ab2-4328b3+1747a2c-2169abc-12619b2c-6276ac2+2294bc|
|      13527a3+14393a2b-10837ab2-3763b3+7557a2c+3547abc+13651b2c+11653ac2+3742|
|      -----------------------------------------------------------------------|
|      2-9242c3   2206a3+5191a2b+3792ab2+12204b3-2642a2c-11609abc-3522b2c+6597|
|      6518c3     -15326a3-3718a2b+14495ab2+15256b3-7099a2c-4043abc-13729b2c-6|
|      7123c3     7514a3+436a2b-14241ab2+14798b3+6510a2c-12122abc-2128b2c+1247|
|      c2-6792c3  13822a3-7511a2b+3419ab2-13285b3+2645a2c+2104abc-2951b2c+443a|
|      2-2364c3   10294a3-11777a2b-691ab2+12756b3+14535a2c-14905abc-10428b2c+3|
|      bc2+5053c3 -13160a3+13348a2b-11663ab2+6603b3+5647a2c+1762abc-351b2c-497|
|      -----------------------------------------------------------------------|
|      ac2-8682bc2-7612c3                                                     |
|      174ac2-11643bc2-12072c3                                                |
|      8ac2-6741bc2-9805c3                                                    |
|      c2+11334bc2-12747c3                                                    |
|      316ac2+12116bc2+12228c3                                                |
|      1ac2+3851bc2-2592c3                                                    |
|      -----------------------------------------------------------------------|
|      1490a3-7028a2b-867ab2+15524b3-15341a2c                                 |
|      -1064a3+9823a2b-5852ab2-8089b3+8359a2c                                 |
|      6548a3-6654a2b-5503ab2+15986b3+136a2c-                                 |
|      -7792a3-11541a2b+437ab2+2978b3-7578a2c                                 |
|      15813a3-9657a2b-6303ab2-6330b3+9805a2c                                 |
|      1953a3+858a2b+15816ab2+4986b3+9192a2c-                                 |
|      -----------------------------------------------------------------------|
|      +10883abc+5463b2c-810ac2+10379bc2+7684c3                               |
|      +10828abc+10341b2c-10325ac2-12190bc2-15790c3                           |
|      14927abc+4275b2c+8500ac2-5614bc2-13485c3                               |
|      +3393abc-8394b2c+12776ac2-4378bc2-11814c3                              |
|      +14185abc-5099b2c+2659ac2-714bc2+13689c3                               |
|      5886abc-7448b2c+5621ac2-4641bc2+3751c3                                 |
|      -----------------------------------------------------------------------|
|      -14618a3+5980a2b-8488ab2+9007b3+7300a2c-4212abc-679b2c-10439ac2+13602bc|
|      -10529a3-12894a2b+1116ab2-7308b3+13622a2c+15225abc+11997b2c+11361ac2+12|
|      -13229a3+9124a2b-2252ab2-9649b3-12812a2c+12094abc+4646b2c+10714ac2+7719|
|      15887a3+14099a2b+6519ab2+14350b3-6307a2c-13167abc+4278b2c+2762ac2+7172b|
|      -4201a3-15945a2b+9210ab2+2519b3-15231a2c-14362abc-2581b2c+14925ac2-6448|
|      2504a3-160a2b-487ab2-615b3+940a2c+11163abc+1273b2c+8254ac2+10632bc2+107|
|      -----------------------------------------------------------------------|
|      2+10226c3      10049a3-14286a2b                                        |
|      823bc2-12909c3 -5069a3+5393a2b-                                        |
|      bc2+11052c3    14871a3+4636a2b-                                        |
|      c2-11151c3     12876a3-4698a2b-                                        |
|      bc2-15150c3    -11577a3-5736a2b                                        |
|      81c3           10625a3-14342a2b                                        |
|      -----------------------------------------------------------------------|
|      +11005ab2-13930b3-1058a2c+9559abc-3698b2c+8326ac2-5257bc2-7889c3       |
|      8502ab2-1087b3-3595a2c+2865abc+4407b2c+4457ac2-15699bc2+15492c3        |
|      11213ab2-8339b3+10251a2c+14831abc+13739b2c+6731ac2+15350bc2+298c3      |
|      13580ab2-9883b3-2577a2c-35abc+1265b2c+13376ac2-2844bc2+9988c3          |
|      -9028ab2+1370b3+12545a2c+14902abc-2735b2c-13701ac2+11920bc2-10797c3    |
|      +2064ab2-11826b3-5008a2c+1660abc-14605b2c+4693ac2+11824bc2+7155c3      |
|      -----------------------------------------------------------------------|
|      -13610a3+4417a2b-5385ab2-1174b3-7698a2c+5290abc-8615b2c-3373ac2-178bc2-|
|      6546a3-10112a2b+4114ab2+9977b3-3918a2c-5632abc+5684b2c-13949ac2+6130bc2|
|      -10040a3-13438a2b-1613ab2-7103b3+8760a2c-5258abc-11467b2c+10352ac2+926b|
|      7298a3+13745a2b+474ab2-7454b3+13945a2c+8819abc-2848b2c+2002ac2-14025bc2|
|      13822a3+10660a2b+1494ab2+4950b3+1046a2c-3780abc+3631b2c-9126ac2-14046bc|
|      -14049a3+2458a2b-15382ab2+2871b3+15070a2c+10050abc+8655b2c+6431ac2+1235|
|      -----------------------------------------------------------------------|
|      10789c3     2535a3-473a2b+9322ab2+7574b3+1243a2c-3242abc-3058b2c+4676ac|
|      +6655c3     9239a3-14867a2b-11414ab2+3128b3+15678a2c+1410abc-330b2c-731|
|      c2-15613c3  7124a3-13944a2b-7560ab2+2728b3-6872a2c+14114abc-7155b2c+142|
|      -2394c3     -4869a3-2380a2b+15526ab2+4547b3-9399a2c+9740abc+10443b2c+14|
|      2+1470c3    -2959a3+10334a2b+14997ab2+11856b3+416a2c-777abc-12168b2c+64|
|      4bc2-8007c3 -9794a3-14999a2b+230ab2-2656b3+1294a2c+11721abc-14178b2c+52|
|      -----------------------------------------------------------------------|
|      2-12956bc2+9434c3                                                      |
|      4ac2-1741bc2-15645c3                                                   |
|      90ac2+6585bc2+14317c3                                                  |
|      865ac2-8765bc2-2036c3                                                  |
|      00ac2-11550bc2+4668c3                                                  |
|      83ac2+10044bc2-14885c3                                                 |
|      -----------------------------------------------------------------------|
|      14677a3+1508a2b-5369ab2+15800b3-5266a2c-6770abc-                       |
|      6816a3-3106a2b+14665ab2-14894b3-7378a2c+2323abc+                       |
|      12892a3+11448a2b+1861ab2-3997b3+3432a2c-8070abc+                       |
|      -2857a3+14361a2b+3505ab2+5812b3+3174a2c+4496abc+                       |
|      -4180a3-1058a2b+3822ab2-14641b3+15496a2c+3527abc                       |
|      -13752a3-7719a2b-2773ab2+13012b3+5821a2c+5315abc                       |
|      -----------------------------------------------------------------------|
|      6318b2c-10093ac2-4920bc2+15992c3                                       |
|      4675b2c+15268ac2-10292bc2+8404c3                                       |
|      6380b2c-102ac2-14388bc2-13468c3                                        |
|      14777b2c-9399ac2-7061bc2+3990c3                                        |
|      -12509b2c-5926ac2-3472bc2+2266c3                                       |
|      -7909b2c+11529ac2-7783bc2-1637c3                                       |
|      -----------------------------------------------------------------------|
|      6487a2b+6487ab2+7698b3+2422a2c+3657abc+2707b2c-11250ac2-12515bc2+5297c3|
|      8933a2b+6349ab2-6721b3-4365a2c-12064abc+11162b2c+836ac2-4416bc2+3297c3 |
|      -11352a2b-14353ab2-2032b3-5655a2c+11413abc+6542b2c+13931ac2+8078bc2+424|
|      3879a2b-10172ab2+8007b3-985a2c+11878abc-893b2c+3038ac2+2170bc2-6584c3  |
|      -4062a2b+2410ab2+852b3+6984a2c-3105abc+4412b2c-650ac2-4214bc2+8925c3   |
|      13212a2b+5008ab2+1283b3-13919a2c+47abc-977b2c-191ac2-48bc2-3c3         |
|      -----------------------------------------------------------------------|
|          |                                                                  |
|          |                                                                  |
|      2c3 |                                                                  |
|          |                                                                  |
|          |                                                                  |
|          |                                                                  |
|                                                                             |
|              6      24                                                      |
|o32 : Matrix R  <-- R                                                        |
+-----------------------------------------------------------------------------+
|i33 : f2 * f3                                                                |
|                                                                             |
|o33 = | 14508a3b-14796a2b2-14848ab3+10239b4+7254a3c-7562a2bc-10217ab2c-2502b3|
|      | 14508a3b+9541a2b2+129ab3+15632b4-7150a3c+15121a2bc+7611ab2c+11536b3c+|
|      | -3575a3b+13774a2b2+6885ab3+6345b4+11931a3c-4949a2bc+11480ab2c+2274b3c|
|      | 2987a4-7699a3b+2383a2b2+5177ab3-5162b4-1084a3c-13260a2bc-7417ab2c-921|
|      | 2987a4+15689a3b+15764a2b2+9374ab3-9621b4-11663a3c-9755a2bc-5526ab2c+1|
|      | 7150a4-9968a3b-1048a2b2-10926ab3-11410b4-7027a3c+11968a2bc+7672ab2c-1|
|      | -14508a4+14106a3b-11183a2b2-4259ab3+1032b4-14162a3c-11707a2bc+11066ab|
|      | 14300a4-4594a3b+12052a2b2-5495ab3-8490b4-14138a3c-3732a2bc+13017ab2c-|
|      | 8141a4-6785a3b+3472a2b2+9302ab3+13245b4-2135a3c-11369a2bc+5459ab2c+15|
|      | 15864a4-14070a3b-12959a2b2+15486ab3+1950a3c-1690a2bc+759ab2c+15603b3c|
|      | -8304a4+15940a3b+6098a2b2-3140ab3-12998a3c-14486a2bc-8077ab2c-12395b3|
|      | 11266a4-6163a3b-5292a2b2+2227ab3+8816a3c-780a2bc+14029ab2c-15714b3c+1|
|      | 6306a4+4669a3b-12276a2b2-8216ab3-15603b4+9626a3c-13344a2bc-9366ab2c-1|
|      | -9952a4-7556a3b+10336a2b2+2842ab3+12395b4+13948a3c+4029a2bc+1158ab2c+|
|      | 7124a4+3996a3b+220a2b2+10677ab3+15714b4-13801a3c-145a2bc-2138ab2c-112|
|      | 15761a4-4227a3b+2576a2b2-10434ab3-9914b4+11262a3c+14621a2bc+4624ab2c+|
|      | 7187a4-4198a3b+6927a2b2+10863ab3-14156b4-4773a3c-11433a2bc-7530ab2c-1|
|      | -2809a4+360a3b+7156a2b2-13655ab3+2917b4+8860a3c-2858a2bc-4940ab2c+636|
|      | 10848a4+280a3b+4917a2b2+11199ab3-864a3c+4816a2bc+4151ab2c-3104b3c-103|
|      | 2006a4+15518a3b-5296a2b2-14370ab3+776b4-3060a3c-10440a2bc-6243ab2c+50|
|      | -594a4+13032a3b+1372a2b2-15001ab3+4656b4+12648a3c-2937a2bc-12486ab2c+|
|      | -2096a4-2597a3b+15546a2b2+12441ab3-10883b4+2466a3c+13424a2bc-8708ab2c|
|      | 11008a4+10234a3b+11173a2b2+2780ab3-11529b4-12838a3c+9146a2bc-10123ab2|
|      | 13085a4+11905a3b+14836a2b2+6962ab3+1589a3c-11323a2bc+10434ab2c+13863a|
|      | -4273a4+15149a3b+10037a2b2+388ab3-15864a3c-12599a2bc-2052ab2c+2598a2c|
|      | 2277a4-8964a3b-6204a2b2+2112ab3-1412a3c+12720a2bc-4356ab2c-258a2c2+12|
|      | 1167a4-2109a3b-48a2b2+108ab3+5268a3c-108a2bc-216ab2c                 |
|      -----------------------------------------------------------------------|
|      c+9788a2c2-1925abc2-13846b2c2+1549ac3-13087bc3+1361c4                  |
|      11205a2c2-1439abc2+4342b2c2+10491ac3-10090bc3+9081c4                   |
|      -9132a2c2+9277abc2-7047b2c2-9782ac3-11609bc3-9025c4                    |
|      4b3c-7578a2c2+4043abc2+9576b2c2-3373ac3-2448bc3-5221c4                 |
|      0235b3c+3360a2c2+4006abc2-5362b2c2-9398ac3-12455bc3-9189c4             |
|      45b3c+7963a2c2-4997abc2-8012b2c2-6480ac3+9974bc3-775c4                 |
|      2c+7700b3c-6852a2c2+11615abc2+10578b2c2-173ac3-10678bc3+9590c4         |
|      2524b3c+12612a2c2+15584abc2-10604b2c2+2097ac3+9018bc3+6432c4           |
|      864b3c+5943a2c2-8595abc2-7993b2c2-8945ac3-1800bc3+4557c4               |
|      -15489a2c2-25abc2-7330b2c2-13658ac3-3877bc3+13738c4                    |
|      c+7932a2c2+11637abc2+7656b2c2-12923ac3+10443bc3+952c4                  |
|      882a2c2-7763abc2+8335b2c2-3546ac3-5186bc3-12827c4                      |
|      4759b3c+7172a2c2-15356abc2-9866b2c2+10551ac3+14806bc3+5153c4           |
|      6500b3c-10591a2c2-6379abc2+5347b2c2-3988ac3+15521bc3+4940c4            |
|      52b3c-10275a2c2+2543abc2-1183b2c2+11762ac3+2836bc3+5358c4              |
|      13743b3c-7647a2c2+5143abc2+3459b2c2+3233ac3-5153bc3                    |
|      5790b3c+3229a2c2+4501abc2+15530b2c2+12707ac3-4940bc3                   |
|      9b3c-15024a2c2-4037abc2+9991b2c2-13671ac3-5358bc3                      |
|      48a2c2-4986abc2+7968b2c2-477ac3+12410bc3+5763c4                        |
|      48b3c-2178a2c2-8414abc2+9674b2c2-14370ac3+10315bc3+10498c4             |
|      9814b3c+7970a2c2-9011abc2+735b2c2-3286ac3-14453bc3+778c4               |
|      -9675b3c+15989a2c2-6086abc2+10725b2c2+5417ac3-329bc3                   |
|      c-6697b3c-9065a2c2+15116abc2+11617b2c2-9880ac3-1556bc3                 |
|      2c2-12021abc2-10716ac3                                                 |
|      2-7392abc2+3280ac3                                                     |
|      abc2+20ac3                                                             |
|                                                                             |
|      -----------------------------------------------------------------------|
|      10881a3b+1494a2b2-13287ab3-9855b4-10561a3c+10362a2bc+12397ab2c+9032b3c-|
|      10881a3b+11746a2b2+68ab3-1187b4+10639a3c-12731a2bc+15771ab2c-2214b3c+13|
|      -10682a3b-15824a2b2+12588ab3-15423b4-15054a3c-1833a2bc-15183ab2c-4254b3|
|      10241a4-6954a3b+1421a2b2+4831ab3-12309b4+11714a3c-12120a2bc+10754ab2c+4|
|      10241a4+10587a3b+14602a2b2+4143ab3+10416b4+6785a3c+6650a2bc-5488ab2c+14|
|      -10639a4+12830a3b+2080a2b2-8862ab3-3864b4-14262a3c-5439a2bc-14944ab2c-5|
|      -10881a4+13990a3b+227a2b2-6402ab3-10105b4+1145a3c+6579a2bc-14329ab2c-90|
|      10725a4-10838a3b+2278a2b2-7638ab3+1091b4+3416a3c+10708a2bc+13800ab2c+14|
|      -1895a4-15856a3b-4796a2b2-10438ab3+10566b4+7705a3c+3744a2bc-4515ab2c-10|
|      11898a4+11453a3b+12634a2b2+4924ab3-11202a3c-8059a2bc-3456ab2c+7874b3c+1|
|      -6228a4-12211a3b-5307a2b2+755ab3-1916a3c+10655a2bc+7462ab2c-10900b3c-85|
|      -7552a4+14241a3b-8888a2b2+11592ab3-13268a3c-12871a2bc+14109ab2c+13115b3|
|      -11272a4+4502a3b+4817a2b2-11969ab3-7874b4+3119a3c+1191a2bc+9720ab2c+159|
|      -7464a4+8518a3b+10004a2b2+9960ab3+10900b4+3763a3c-4859a2bc+4852ab2c+771|
|      5343a4+7059a3b-13471a2b2+2546ab3-13115b4+8390a3c-6248a2bc-1475ab2c-1297|
|      3820a4-14941a3b-472a2b2-12276ab3-13154b4+274a3c-13663a2bc-399ab2c+103b3|
|      13391a4+13926a3b+4401a2b2-5184ab3-574b4-357a3c+2191a2bc+14044ab2c-9791b|
|      5894a4-10757a3b-4054a2b2+8881ab3-15345b4+2532a3c+15790a2bc-8929ab2c-151|
|      8136a4-14116a3b+5747a2b2+169ab3-5339a3c-7452a2bc-13402ab2c+3776b3c-7136|
|      -14497a4+7967a3b+6258a2b2+20ab3-944b4-13279a3c+45a2bc-15940ab2c-3475b3c|
|      15556a4-9562a3b-12102a2b2+10323ab3-5664b4+16a3c-2271a2bc+13141ab2c-4360|
|      8065a4-15098a3b-10464a2b2-364ab3+11849b4+4155a3c+9070a2bc-15320ab2c+151|
|      9191a4+6079a3b+12875a2b2+12548ab3+15453b4-15709a3c-12136a2bc+4664ab2c-3|
|      -13721a4+15999a3b+3341a2b2+2441ab3-5766a3c+11499a2bc-2606ab2c+1300a2c2+|
|      12818a4-4202a3b+10145a2b2-472ab3+10432a3c-13811a2bc+558ab2c+4866a2c2-74|
|      1709a4-6980a3b-4212a2b2+2107ab3-8446a3c+8364a2bc-4341ab2c-186a2c2+12abc|
|      879a4-1695a3b+60a2b2+108ab3+4116a3c-324a2bc-216ab2c                    |
|      -----------------------------------------------------------------------|
|      6543a2c2-12070abc2-9191b2c2-7729ac3+2131bc3-8092c4                     |
|      94a2c2+10755abc2-8813b2c2+15849ac3-10230bc3-12621c4                    |
|      c+499a2c2+9713abc2+12596b2c2+15453ac3+7438bc3+5740c4                   |
|      579b3c+10834a2c2-4125abc2-8270b2c2+10582ac3-9868bc3+10304c4            |
|      437b3c+15224a2c2-14511abc2+504b2c2-10496ac3-14855bc3+2126c4            |
|      450b3c+9622a2c2-7653abc2+12323b2c2+1612ac3+3289bc3-2375c4              |
|      28b3c+4699a2c2-14581abc2-14343b2c2-2419ac3-3228bc3-7198c4              |
|      40b3c+424a2c2-1629abc2-10408b2c2-9050ac3+4372bc3-5347c4                |
|      85b3c+253a2c2+15404abc2-721b2c2+7218ac3-15388bc3+15804c4               |
|      3780a2c2-9556abc2-2814b2c2+679ac3+11008bc3-12439c4                     |
|      37a2c2-5500abc2-7142b2c2+7115ac3+11936bc3+11122c4                      |
|      c+3233a2c2-9117abc2-15361b2c2+775ac3+12638bc3+2567c4                   |
|      68b3c+12874a2c2+8347abc2-11111b2c2+2590ac3-11566bc3+1334c4             |
|      6b3c-445a2c2-9376abc2-2145b2c2+9854ac3-4539bc3-2431c4                  |
|      b3c-15457a2c2+450abc2-12487b2c2-9059ac3-8508bc3+4008c4                 |
|      c-8197a2c2+8185abc2-7998b2c2-10409ac3-1334bc3                          |
|      3c+6867a2c2+2655abc2-6583b2c2-15962ac3+2431bc3                         |
|      b3c+7912a2c2-1658abc2+5941b2c2-15409ac3-4008bc3                        |
|      a2c2-12437abc2-12912b2c2-9109ac3+12770bc3+5643c4                       |
|      +1933a2c2-15198abc2+7147b2c2-8964ac3+11838bc3-11335c4                  |
|      b3c+3728a2c2+14488abc2+14637b2c2+2880ac3+8772bc3+3764c4                |
|      52b3c-14089a2c2-13424abc2+15204b2c2-14922ac3-1822bc3                   |
|      578b3c+7469a2c2-14503abc2-2470b2c2+4862ac3-7528bc3                     |
|      11882abc2-8016ac3                                                      |
|      37abc2+3295ac3                                                         |
|      2+20ac3                                                                |
|                                                                             |
|      -----------------------------------------------------------------------|
|      12380ab3-14232b4-1870a2bc+11713ab2c-1092b3c-935a2c2-4764abc2-6821b2c2+3|
|      -891ab3-10600b4-1870a2bc+2890ab2c-6836b3c+14965a2c2+6459abc2+1139b2c2-3|
|      -14034ab3-10791b4-14022a2bc+3582ab2c-4394b3c+2854a2c2+9008abc2-3810b2c2|
|      1740a2b2+11907ab3-5350b4+3740a3c-2547a2bc+8155ab2c-8020b3c-12095a2c2-68|
|      1376a2b2-13161ab3-8324b4+3740a3c-13146a2bc-15741ab2c-3503b3c-3633a2c2-1|
|      -3087a2b2+15587ab3+5793b4-14965a3c+5360a2bc+8533ab2c+13382b3c-3748a2c2+|
|      12100a2b2+12233ab3+10440b4+1870a3c+15107a2bc-10021ab2c-2877b3c-8133a2c2|
|      -8718a2b2-195ab3-12589b4+2073a3c-12384a2bc-5073ab2c+4702b3c-2873a2c2-15|
|      11006a3b+4563a2b2+5102ab3+5034b4-5708a3c-4561a2bc+1992ab2c+13559b3c+333|
|      -15494a3b-14335a2b2-15953ab3+12972a3c-3047a2bc+5962ab2c-4043b3c-15707a2|
|      1218a3b+7074a2b2-7031ab3-9394a3c-414a2bc+8301ab2c-14511b3c+5009a2c2-116|
|      -2544a3b-14018a2b2+14624ab3+3251a3c-471a2bc+5540ab2c-8414b3c-9857a2c2+1|
|      -15900a3b+12441a2b2-13222ab3+4043b4-11819a3c-11477a2bc+9721ab2c+10201b3|
|      -6360a3b-6771a2b2+7972ab3+14511b4+15952a3c+15054a2bc+11133ab2c+3394b3c+|
|      9494a3b-14070a2b2+3218ab3+8414b4-9792a3c-9651a2bc+13762ab2c-15839b3c+80|
|      -636a3b-9214a2b2-9579ab3+14162b4+4725a3c-8174a2bc+13335ab2c+14990b3c+12|
|      -6655a3b+1166a2b2+15240ab3+13448b4+10743a3c-13440a2bc-3708ab2c+7808b3c-|
|      12674a3b+12968a2b2-3699ab3-6562b4-11268a3c+3280a2bc+5082ab2c-5837b3c-71|
|      12780a3b-386a2b2-5277ab3-15643a3c-562a2bc+12471ab2c+5840b3c+4759a2c2+49|
|      -10451a3b+14079a2b2-1689ab3-1460b4-7911a3c-1790a2bc-3519ab2c+14915b3c-6|
|      -3195a3b+2023a2b2-7153ab3-8760b4-8523a3c-14936a2bc-11978ab2c-3139b3c+15|
|      2291a3b+947a2b2-8632ab3-875b4-1775a3c+13742a2bc+2225ab2c-10942b3c+6931a|
|      2912a3b-7935a2b2-4944ab3-3440b4+13750a3c+7280a2bc+10275ab2c-11945b3c-13|
|      -8097a3b-3411a2b2-12734ab3+5041a3c+13435a2bc-12772ab2c+12010a2c2-422abc|
|      -1545a3b+6502a2b2-730ab3-13261a3c+4561a2bc+1902ab2c+7258a2c2-1308abc2-8|
|      -65a3b+675a2b2+715ab3+688a3c-2019a2bc-2052ab2c+2619a2c2+1641abc2-22ac3 |
|      5a4-60a3b+150a2b2+45ab3+333a3c-414a2bc-108ab2c+432a2c2+81abc2          |
|      -----------------------------------------------------------------------|
|      279ac3-2650bc3-2928c4                                 |                |
|      54ac3-15427bc3-10968c4                                |                |
|      +210ac3-11303bc3+1278c4                               |                |
|      75abc2+5391b2c2-12001ac3-5041bc3+13318c4              |                |
|      4949abc2+1064b2c2-14746ac3-12021bc3-12675c4           |                |
|      7808abc2+14620b2c2+5270ac3+15806bc3-10450c4           |                |
|      +14624abc2-13791b2c2+12376ac3+5199bc3-13275c4         |                |
|      971abc2-2017b2c2-7398ac3-15618bc3-6550c4              |                |
|      6a2c2+2901abc2+6364b2c2-10849ac3+4892bc3+7962c4       |                |
|      c2-7684abc2+7640b2c2+12326ac3+1655bc3+13796c4         |                |
|      68abc2+15161b2c2+2330ac3-4069bc3+11810c4              |                |
|      409abc2-9602b2c2+4215ac3+12518bc3+5523c4              |                |
|      c-15798a2c2-1168abc2+15358b2c2-8022ac3-7293bc3-7384c4 |                |
|      6135a2c2+11489abc2-3739b2c2+1100ac3+12177bc3-2721c4   |                |
|      22a2c2-11801abc2-6681b2c2+8406ac3-4880bc3-13042c4     |                |
|      443a2c2-15526abc2-6503b2c2+7822ac3+7384bc3            |                |
|      8007a2c2-12230abc2+8016b2c2-12353ac3+2721bc3          |                |
|      24a2c2+8268abc2-643b2c2+8117ac3+13042bc3              |                |
|      36abc2+13907b2c2-6393ac3-13331bc3-6416c4              |                |
|      970a2c2+13461abc2+12975b2c2-3023ac3-849bc3+6422c4     |                |
|      699a2c2+7734abc2-6034b2c2+469ac3-7481bc3-6589c4       |                |
|      2c2-6710abc2-7494b2c2+14128ac3-12692bc3               |                |
|      252a2c2-10993abc2+2207b2c2+5442ac3+13178bc3           |                |
|      2-5919ac3                                             |                |
|      ac3                                                   |                |
|                                                            |                |
|                                                            |                |
|                                                                             |
|              27      3                                                      |
|o33 : Matrix R   <-- R                                                       |
+-----------------------------------------------------------------------------+
|i34 : ker f1 == image f2                                                     |
|                                                                             |
|o34 = false                                                                  |
+-----------------------------------------------------------------------------+
|i35 : ker f2 == image f3                                                     |
|                                                                             |
|o35 = false                                                                  |
+-----------------------------------------------------------------------------+
These do not piece together well. The reason is that pieri,
changes the bases of the free modules in a way which is not invertible (over
ZZ) when the ground field has positive characteristic.

See also
========

  * pieri -- computes a matrix representation for a Pieri
    inclusion of representations of a general linear group
  * pureFree -- computes a GL(V)-equivariant map whose
    resolution is pure, or the reduction mod p of such a map
  * schurRank -- computes the dimension of the irreducible
    GL(QQ^n) representation associated to a partition

Author
======

  * Steven V Sam (see http://math.mit.edu/~ssam/ ) <ssam@math.mit.edu>

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

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

Version
=======

This documentation describes version 1.0 of PieriMaps.

Citation
========

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

+----------------------------------------------------------------------------+
|@misc{PieriMapsSource,                                                      |
|  title = {{PieriMaps: A \emph{Macaulay2} package. Version~1.0}},           |
|  author = {Steven V Sam},                                                  |
|  howpublished = {A \emph{Macaulay2} package available at                   |
|    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}|
|}                                                                           |
|                                                                            |
|@article{PieriMapsArticle,                                                  |
|  title = {{Computing inclusions of Schur modules}},                        |
|  author = {Steven V Sam},                                                  |
|  journal = {The Journal of Software for Algebra and Geometry: Macaulay2},  |
|  volume = {1},                                                             |
|  year = {2009},                                                            |
|}                                                                           |
|                                                                            |
+----------------------------------------------------------------------------+

Exports
=======

  * Functions and commands
      * pieri -- computes a matrix representation for a Pieri
        inclusion of representations of a general linear group
      * pureFree -- computes a GL(V)-equivariant map whose
        resolution is pure, or the reduction mod p of such a map
      * schurRank -- computes the dimension of the
        irreducible GL(QQ^n) representation associated to a partition
      * standardTableaux -- list all standard tableaux
        of a certain shape with bounded labels
      * straighten -- computes straightening of a tableau
  * Methods
      * "pieri(List,List,PolynomialRing)" -- see pieri, --
        computes a matrix representation for a Pieri inclusion of
        representations of a general linear group
      * "pureFree(List,PolynomialRing)" -- see pureFree, --
        computes a GL(V)-equivariant map whose resolution is pure, or the
        reduction mod p of such a map
      * "schurRank(ZZ,List)" -- see schurRank, -- computes the
        dimension of the irreducible GL(QQ^n) representation associated to a
        partition
      * "standardTableaux(ZZ,List)" -- see standardTableaux,
        -- list all standard tableaux of a certain shape with
        bounded labels
      * "straighten(List)" -- see straighten, -- computes
        straightening of a tableau
      * "straighten(List,MutableHashTable)" -- see straighten,
        -- computes straightening of a tableau

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

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

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

The source of this document is in PieriMaps.m2:487:0.

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

  * Usage:
        pieri(mu, boxes, P)
  * Inputs:
      * mu, a list (Macaulay2Doc), a partition (mu_1, ..., mu_r)
        where mu_i is the number of boxes in the ith row
      * boxes, a list (Macaulay2Doc), a list of rows from which to
        remove boxes (boxes are always removed from the end of the row). This
        specifies which map of GL(V) representations we want. The row indices
        start from 1 and not 0, and this must specify a horizontal strip in mu
        (see description below).
      * P, a polynomial ring (Macaulay2Doc), a polynomial
        ring over a field K in n variables
  * Outputs:
      * a matrix (Macaulay2Doc), If K has characteristic 0, then
        given the partition mu and the partition mu' obtained from mu by
        removing a single box, there is a unique (up to nonzero scalar)
        GL(V)-equivariant inclusion S_mu V -> V otimes S_mu' V, where S_mu
        refers to the irreducible representation of GL(V) with highest weight
        mu. This can be extended uniquely to a map of P = Sym(V)-modules P
        otimes S_mu V -> P otimes S_mu' V. This method computes the matrix
        representation for the composition of maps that one obtains by
        iterating this procedure of removing boxes, i.e., the final output is a
        GL(V)-equivariant map P otimes S_mu V -> P otimes S_lambda V where
        lambda is the partition obtained from mu by deleting a box from row
        boxes_0, a box from row boxes_1, etc. If K has positive characteristic,
        then the corresponding map is calculated over QQ, lifted to a ZZ-form
        of the representation which has the property that the map has a
        torsion-free cokernel over ZZ, and then the coefficients are reduced to
        K.

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

Convention: the partition (d) represents the dth symmetric power, while the
partition (1,...,1) represents the dth exterior power. Using the notation from
the output, mu/lambda must be a horizontal strip. Precisely, this means that
lambda_i >= mu_(i+1) for all i. If this condition is not satisfied, the program
throws an error because a nonzero equivariant map of the desired form will not
exist.
+-----------------------------------------------------------------------------+
|i1 : pieri({3,1}, {1}, QQ[a,b,c]) -- removes the last box from row 1 of the  |
|                                                                             |
|o1 = | 3a 0  b  0  c  0  0  0  0     0 0   0  0  0  0  |                     |
|     | 0  3a 0  b  0  c  0  0  0     0 0   0  0  0  0  |                     |
|     | 0  0  2a 0  0  0  2b 0  c     0 0   0  0  0  0  |                     |
|     | 0  0  0  2a 0  0  0  2b 0     c 0   0  0  0  0  |                     |
|     | 0  0  0  0  2a 0  0  0  b     0 2c  0  0  0  0  |                     |
|     | 0  0  0  0  0  2a 0  0  0     b 0   2c 0  0  0  |                     |
|     | 0  0  0  0  0  0  0  a  -1/2a 0 0   0  3b c  0  |                     |
|     | 0  0  0  0  0  0  0  0  0     a -2a 0  0  2b 2c |                     |
|                                                                             |
|                      8               15                                     |
|o1 : Matrix (QQ[a..c])  <-- (QQ[a..c])                                       |
|-----------------------------------------------------------------------------|
|partition {3,1}                                                              |
+-----------------------------------------------------------------------------+
|i2 : res coker oo -- resolve this map                                        |
|                                                                             |
|               8               15               10               3           |
|o2 = (QQ[a..c])  <-- (QQ[a..c])   <-- (QQ[a..c])   <-- (QQ[a..c])  <-- 0     |
|                                                                             |
|     0               1                2                3               4     |
|                                                                             |
|o2 : ChainComplex                                                            |
+-----------------------------------------------------------------------------+
|i3 : betti oo -- check that the resolution is pure                           |
|                                                                             |
|            0  1  2 3                                                        |
|o3 = total: 8 15 10 3                                                        |
|         0: 8 15  . .                                                        |
|         1: .  . 10 .                                                        |
|         2: .  .  . 3                                                        |
|                                                                             |
|o3 : BettiTally                                                              |
+-----------------------------------------------------------------------------+

See also
========

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

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

  * "pieri(List,List,PolynomialRing)"

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

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

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

The source of this document is in PieriMaps.m2:555:0.

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

  * Usage:
        pureFree(d, P)
  * Inputs:
      * d, a list (Macaulay2Doc), a list of degrees (increasing
        numbers)
      * P, a polynomial ring (Macaulay2Doc), a polynomial
        ring over a field K in n variables
  * Outputs:
      * a matrix (Macaulay2Doc), A map whose cokernel has Betti
        diagram with degree sequence d if K has characteristic 0. If K has
        positive characteristic p, then the corresponding map is calculated
        over QQ and is lifted to a ZZ-form which is then reduced mod p.

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

The function translates the data of a degree sequence d for a desired pure free
resolution into the data of a Pieri map according to the formula of
Eisenbud-Fl\o ystad-Weyman and then applies the function pieri.
+-----------------------------------------------------------------------------+
|i1 : betti res coker pureFree({0,1,2,4}, QQ[a,b,c]) -- degree sequence       |
|                                                                             |
|            0 1 2 3                                                          |
|o1 = total: 3 8 6 1                                                          |
|         0: 3 8 6 .                                                          |
|         1: . . . 1                                                          |
|                                                                             |
|o1 : BettiTally                                                              |
|-----------------------------------------------------------------------------|
|{0,1,2,4}                                                                    |
+-----------------------------------------------------------------------------+
|i2 : betti res coker pureFree({0,1,2,4}, ZZ/2[a,b,c]) -- same map, but       |
|                                                                             |
|            0 1 2 3                                                          |
|o2 = total: 3 8 6 1                                                          |
|         0: 3 8 6 .                                                          |
|         1: . . . 1                                                          |
|                                                                             |
|o2 : BettiTally                                                              |
|-----------------------------------------------------------------------------|
|reduced mod 2                                                                |
+-----------------------------------------------------------------------------+
|i3 : betti res coker pureFree({0,1,2,4}, GF(4)[a,b,c]) -- can also use non   |
|                                                                             |
|            0 1 2 3                                                          |
|o3 = total: 3 8 6 1                                                          |
|         0: 3 8 6 .                                                          |
|         1: . . . 1                                                          |
|                                                                             |
|o3 : BettiTally                                                              |
|-----------------------------------------------------------------------------|
|prime fields                                                                 |
+-----------------------------------------------------------------------------+

See also
========

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

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

  * "pureFree(List,PolynomialRing)"

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

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

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

The source of this document is in PieriMaps.m2:579:0.

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

  * Usage:
        schurRank(n, mu)
  * Inputs:
      * n, an integer (Macaulay2Doc), the size of the matrix group
        GL(QQ^n)
      * mu, a list (Macaulay2Doc), a partition (mu_1, ..., m_r)
        where mu_i is the number of boxes in the ith row
  * Outputs:
      * an integer (Macaulay2Doc), The dimension of the irreducible
        GL(QQ^n) representation associated to mu

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

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

See also
========

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

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

  * "schurRank(ZZ,List)"

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

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

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

The source of this document is in PieriMaps.m2:597:0.

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

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

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

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

See also
========

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

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

  * "standardTableaux(ZZ,List)"

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

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

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

The source of this document is in PieriMaps.m2:504:0.

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

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

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

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

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

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

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

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

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

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