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 = | a2 ab b2 0 ac bc 0 0 c2 0 0 0 0 -bc 0 0 -c2 0 0 0
| 0 0 0 a2 2ab b2 0 2ac 4bc 0 c2 0 ab 0 0 -2ac -2bc 0 -c2 0
| 0 0 0 0 0 0 a2 ab b2 ac bc c2 0 0 0 0 0 0 0 0
| 0 0 0 0 0 0 0 0 0 0 0 0 a2 4ab b2 0 4ac 4bc 0 c2
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a2 2ab 0 2ac bc
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -ab 0 0
-----------------------------------------------------------------------
0 0 0 0 0 0 0 |
0 -2bc 0 0 0 0 0 |
0 0 0 0 0 0 0 |
0 -4ac -4bc 0 -2c2 0 0 |
c2 2ab b2 0 0 0 0 |
0 a2 2ab b2 ac bc 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 = | -b -c 0 0 0 0 0 0 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 0 0 0 0
| 0 0 -b -c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
| 2a 0 0 0 -b -3c -2c 0 0 0 0 0 0 0 0 0 0 0 0
| 0 4a 0 0 0 -2b -2b -4c 0 6c 0 0 0 0 0 0 0 0 0
| 0 0 2a 0 0 0 0 -b -2c b c 0 0 0 0 0 0 0 0
| 0 2a 0 0 0 3b b 0 0 -c 0 0 0 0 0 0 0 0 0
| 0 0 4a 0 0 0 0 2b 0 -6b -2c 0 0 0 0 0 0 0 0
| 0 0 0 2a 0 0 0 0 b 0 -2b -c 0 0 0 0 0 0 0
| 0 0 0 0 3a 0 0 0 0 0 0 0 -3c 0 0 0 0 0 0
| 0 0 0 0 0 6a 6a 0 0 0 0 0 0 -4c -6c 0 0 0 0
| 0 0 0 0 0 0 0 3a 0 -3a 0 0 0 0 0 -c c 0 0
| 0 0 0 0 0 0 3a 0 0 0 0 0 3b 3c 3c 0 0 0 0
| 0 0 0 0 0 0 0 6a 0 0 0 0 0 4b 6b 0 -2c 0 0
| 0 0 0 0 0 0 0 0 3a 0 3a 0 0 0 0 b -b 0 -c
| 0 0 0 0 0 0 0 0 0 3a 0 0 0 -3b -3b 0 0 0 0
| 0 0 0 0 0 0 0 0 0 0 6a 0 0 0 0 0 2b 0 0
| 0 0 0 0 0 0 0 0 0 0 0 3a 0 0 0 0 0 0 b
| 0 0 0 0 0 0 0 0 0 0 0 0 4a 0 0 0 0 0 0
| 0 0 0 0 0 0 0 0 0 0 0 0 0 a 2a 0 0 c 0
| 0 0 0 0 0 0 0 0 0 0 0 0 0 -2a 0 0 0 -6c 0
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 2a 3b 0
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4a 12b 0
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2a
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-----------------------------------------------------------------------
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
-8c 0 0 0 0 |
2b -3c 0 0 0 |
12b 6c 0 0 0 |
0 0 0 -c 0 |
0 -12b 0 0 0 |
0 0 0 2b 0 |
a 0 -c 0 0 |
0 a 2b 0 -c |
0 0 0 a 3b |
27 24
o28 : Matrix R <--- R
|
i29 : f3 = pieri({4,3,3},{3,3,3},R)
o29 = | -5c3 0 0 |
| 15bc2 0 0 |
| -15b2c 0 0 |
| 5b3 0 0 |
| 0 20c3 0 |
| -12ac2 -48bc2 3c3 |
| 24abc 36b2c -6bc2 |
| -12ab2 -8b3 3b2c |
| 12ac2 -12bc2 2c3 |
| -24abc 24b2c -9bc2 |
| 12ab2 -12b3 12b2c |
| 0 0 -5b3 |
| 0 90ac2 0 |
| 18a2c -108abc 18ac2 |
| 6a2b 24ab2 -9abc |
| 72a2c -72abc 27ac2 |
| -36a2b 36ab2 -36abc |
| 0 0 15ab2 |
| 0 60a2c 0 |
| 4a3 -24a2b 9a2c |
| 36a3 -36a2b 36a2c |
| 0 0 -15a2b |
| 0 5a3 0 |
| 0 0 5a3 |
24 3
o29 : Matrix R <--- R
|
i30 : f2 = map(source f1,,f2)
o30 = | -b -c 0 0 0 0 0 0 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 0 0 0 0
| 0 0 -b -c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
| 2a 0 0 0 -b -3c -2c 0 0 0 0 0 0 0 0 0 0 0 0
| 0 4a 0 0 0 -2b -2b -4c 0 6c 0 0 0 0 0 0 0 0 0
| 0 0 2a 0 0 0 0 -b -2c b c 0 0 0 0 0 0 0 0
| 0 2a 0 0 0 3b b 0 0 -c 0 0 0 0 0 0 0 0 0
| 0 0 4a 0 0 0 0 2b 0 -6b -2c 0 0 0 0 0 0 0 0
| 0 0 0 2a 0 0 0 0 b 0 -2b -c 0 0 0 0 0 0 0
| 0 0 0 0 3a 0 0 0 0 0 0 0 -3c 0 0 0 0 0 0
| 0 0 0 0 0 6a 6a 0 0 0 0 0 0 -4c -6c 0 0 0 0
| 0 0 0 0 0 0 0 3a 0 -3a 0 0 0 0 0 -c c 0 0
| 0 0 0 0 0 0 3a 0 0 0 0 0 3b 3c 3c 0 0 0 0
| 0 0 0 0 0 0 0 6a 0 0 0 0 0 4b 6b 0 -2c 0 0
| 0 0 0 0 0 0 0 0 3a 0 3a 0 0 0 0 b -b 0 -c
| 0 0 0 0 0 0 0 0 0 3a 0 0 0 -3b -3b 0 0 0 0
| 0 0 0 0 0 0 0 0 0 0 6a 0 0 0 0 0 2b 0 0
| 0 0 0 0 0 0 0 0 0 0 0 3a 0 0 0 0 0 0 b
| 0 0 0 0 0 0 0 0 0 0 0 0 4a 0 0 0 0 0 0
| 0 0 0 0 0 0 0 0 0 0 0 0 0 a 2a 0 0 c 0
| 0 0 0 0 0 0 0 0 0 0 0 0 0 -2a 0 0 0 -6c 0
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a 2a 3b 0
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4a 12b 0
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2a
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-----------------------------------------------------------------------
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
0 0 0 0 0 |
-8c 0 0 0 0 |
2b -3c 0 0 0 |
12b 6c 0 0 0 |
0 0 0 -c 0 |
0 -12b 0 0 0 |
0 0 0 2b 0 |
a 0 -c 0 0 |
0 a 2b 0 -c |
0 0 0 a 3b |
27 24
o30 : Matrix R <--- R
|
i31 : f3 = map(source f2,,f3)
o31 = {1} | -5c3 0 0 |
{1} | 15bc2 0 0 |
{1} | -15b2c 0 0 |
{1} | 5b3 0 0 |
{1} | 0 20c3 0 |
{1} | -12ac2 -48bc2 3c3 |
{1} | 24abc 36b2c -6bc2 |
{1} | -12ab2 -8b3 3b2c |
{1} | 12ac2 -12bc2 2c3 |
{1} | -24abc 24b2c -9bc2 |
{1} | 12ab2 -12b3 12b2c |
{1} | 0 0 -5b3 |
{1} | 0 90ac2 0 |
{1} | 18a2c -108abc 18ac2 |
{1} | 6a2b 24ab2 -9abc |
{1} | 72a2c -72abc 27ac2 |
{1} | -36a2b 36ab2 -36abc |
{1} | 0 0 15ab2 |
{1} | 0 60a2c 0 |
{1} | 4a3 -24a2b 9a2c |
{1} | 36a3 -36a2b 36a2c |
{1} | 0 0 -15a2b |
{1} | 0 5a3 0 |
{1} | 0 0 5a3 |
24 3
o31 : Matrix R <--- R
|
i32 : f1 * f2
o32 = | -a2b -2ab2+3a2c -b3 -b2c+2ac2 0 -2abc
| 2a3 8a2b 2ab2+8a2c 8abc -a2b -4ab2-3a2c+6ac2
| 0 2a3 4a2b 2ab2 3a2c 3a2b+6abc
| 0 0 0 0 0 0
| 0 0 0 0 0 0
| 0 0 0 0 0 0
-----------------------------------------------------------------------
-2abc -6abc-b2c-4ac2 -bc2 b2c+6ac2 -6ac2-bc2
3a2b-4ab2-2a2c+6ac2 -b3-4abc 2b2c b3-6a2c -12abc-7b2c-4ac2
a2b+6abc 2ab2+3ac2 b3 -6ab2-a2c-3ac2 -2b3-2abc
3a3 24a2b 3ab2 0 3ab2+24a2c
0 0 0 3a3 12a2b
0 0 0 0 0
-----------------------------------------------------------------------
-c3 0 -4b2c -6b2c 0 0
-4bc2 3ab2-4ac2 9abc-4c3 9abc-6c3 -2abc -4abc-4b2c
-b2c -3ac2 -4bc2 -6bc2 -c3 c3
12abc 3a2b 16ab2+3a2c+ac2 24ab2+3a2c+2ac2 b3-4a2c -b3-8a2c-16abc
0 8a2c -3a2b+abc-2ac2 -3a2b+2abc 2a2b 4a2b+8ab2
-3a2b 0 0 0 a3 2a3+8a2b
-----------------------------------------------------------------------
0 0 0 0 0
-6b2c 0 8c3 0 0
0 0 0 0 0
-12abc-48b2c+c3 3b2c -2ac2+2bc2 48b2c-3c3 2c3
6ab2+12b3+bc2-6c3 0 2b2c-16ac2+12bc2 -12b3-3bc2+6c3 0
3a2b+24ab2 ab2 a2c -24ab2+abc 2b2c-ac2
-----------------------------------------------------------------------
0 0 |
2bc2 0 |
0 0 |
4ac2 0 |
-2abc 0 |
2b3-a2c+ac2 2bc2 |
6 24
o32 : Matrix R <--- R
|
i33 : f2 * f3
o33 = | -10bc3 0
| 0 0
| 10b3c 0
| -48abc2+26ac3 -72b2c2+124bc3
| -60abc2 -40b3c+240b2c2
| 12ab3-42ab2c-24ac3 8b4+12b3c+24bc3
| 24ab2c+18abc2 36b3c-168b2c2
| -24ab3+60ab2c -16b4-120b3c
| -14ab3+12abc2 24b4-12b2c2
| 0 -210ac3
| 108a2bc-144a2c2 72ab2c+144abc2
| -36a2b2+36a2bc-72a2c2 -24ab3-36ab2c+72abc2
| 90a2bc+54a2c2 180ab2c-54abc2
| -36a2b2+144a2bc 96ab3-504ab2c
| 72a2b2+72a2bc+36a2c2 -72ab3-72ab2c-60a2c2-36abc2
| -18a2b2-126a2bc -72ab3+396ab2c
| 0 0
| 0 60a2bc
| -32a3c 192a2bc+360a2c2
| 20a3b-90a3c 0
| 48a3b+180a3c -288a2b2
| -72a3b+72a3c 72a2b2-5a3c-72a2bc
| -576a3b 576a2b2
| 0 10a3b+120a3c
| 4a4 -24a3b
| 36a4 -36a3b
| 0 5a4
-----------------------------------------------------------------------
0 |
0 |
0 |
12bc3-9c4 |
-60bc3 |
-3b3c+3b2c2-4c4 |
-6b2c2+18bc3 |
6b3c+30b2c2 |
-19b3c+2bc3 |
0 |
18abc2-54ac3 |
9ab2c-9abc2-27ac3 |
-45abc2+54ac3 |
-36ab2c+144abc2 |
72ab2c+27abc2+6ac3 |
27ab2c-81abc2 |
0 |
-15ab3 |
-72a2c2 |
15ab2c-90a2c2 |
108a2bc-90ab2c+180a2c2 |
45ab3-72a2bc+27a2c2 |
180ab3-576a2bc |
0 |
9a3c+15a2bc |
-30a2b2+31a3c |
15a3b |
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.
