This function computes the Schur complex associated to a partition $\lambda$ and a bounded complex $F_{\bullet}$ of finitely-generated free modules over a commutative ring.
The user inputs the partition $\lambda$ as a list and the chain complex $F_{\bullet}$.
In the following example, the complex F is the free resolution of the ideal $(x,y,z)\subset \mathbb{Z}[x,y,z]$, and lambda is the partition $(1,1)$ in the form of a List. In this case, the Schur complex G is the second exterior power of F.
i1 : R=ZZ[x,y,z]; |
i2 : I=ideal(x,y,z); o2 : Ideal of R |
i3 : F=res I; |
i4 : lambda={1,1};
|
i5 : G=schurComplex(lambda,F)
3 9 10 6 3 1
o5 = R <-- R <-- R <-- R <-- R <-- R
1 2 3 4 5 6
o5 : ChainComplex
|
i6 : G.dd
3 9
o6 = 1 : R <---------------------------- R : 2
| z y x 0 0 0 0 y x |
| 0 z 0 y x 0 x -z 0 |
| 0 0 z 0 y x -y 0 -z |
9 10
2 : R <---------------------------------------- R : 3
| 0 0 2y 2x 0 0 0 0 0 0 |
| 0 x -z 0 0 y x 0 0 0 |
| 0 -y 0 -z 0 0 0 0 y x |
| 0 0 0 0 2x -2z 0 0 0 0 |
| 0 0 0 0 -y 0 -z x -z 0 |
| 0 0 0 0 0 0 0 -2y 0 -2z |
| z -z 0 0 -y 0 0 -x 0 0 |
| x 0 -z 0 0 -y 0 0 -x 0 |
| -y 0 0 -z 0 0 -y 0 0 -x |
10 6
3 : R <------------------------- R : 4
| z y x 0 0 0 |
| z 0 0 y x 0 |
| x 0 0 0 0 x |
| -y 0 0 0 0 -y |
| 0 z 0 -z 0 0 |
| 0 x 0 -x 0 0 |
| 0 -y 0 0 -x z |
| 0 0 z 0 -z 0 |
| 0 0 x y 0 -z |
| 0 0 -y 0 y 0 |
6 3
4 : R <---------------- R : 5
| 0 y x |
| x -z 0 |
| -y 0 -z |
| x -z 0 |
| -y 0 -z |
| 0 -y -x |
3 1
5 : R <---------- R : 6
| z |
| x |
| -y |
o6 : ChainComplexMap
|
As a second example, we consider the ring of polynomial functions $R=\mathbb{Q}[x_{i,j}]$ on the space of 2 x 4 generic matrices. We set the complex F to be the map $R^4\to R^2$ given by the generic matrix $(x_{i,j})$. We compute the third symmetric power G of F, in which case lambda is the partition $(3)$. By Weyman "Cohomology of Vector Bundles and Syzygies", Exercise 6.34(d), the Schur complex G is exact except in degree zero. We verify this by computng the Hilbert series of each homology module of G.
i7 : R=QQ[x11,x21,x12,x22,x13,x23,x14,x24]; |
i8 : M=genericMatrix(R,x11,2,4);
2 4
o8 : Matrix R <--- R
|
i9 : F = new ChainComplex; F.ring = R; F#0=target M; F#1=source M; F.dd#1=M;
2 4
o13 : Matrix R <--- R
|
i14 : lambda={3};
|
i15 : G=schurComplex(lambda,F)
4 12 12 4
o15 = R <-- R <-- R <-- R
0 1 2 3
o15 : ChainComplex
|
i16 : G.dd
4 12
o16 = 0 : R <------------------------------------------------------- R : 1
| x14 0 0 x13 0 0 x12 0 0 x11 0 0 |
| x24 x14 0 x23 x13 0 x22 x12 0 x21 x11 0 |
| 0 x24 x14 0 x23 x13 0 x22 x12 0 x21 x11 |
| 0 0 x24 0 0 x23 0 0 x22 0 0 x21 |
12 12
1 : R <------------------------------------------------------------------- R : 2
| -x13 0 -x12 0 -x11 0 0 0 0 0 0 0 |
| -x23 -x13 -x22 -x12 -x21 -x11 0 0 0 0 0 0 |
| 0 -x23 0 -x22 0 -x21 0 0 0 0 0 0 |
| x14 0 0 0 0 0 -x12 0 -x11 0 0 0 |
| x24 x14 0 0 0 0 -x22 -x12 -x21 -x11 0 0 |
| 0 x24 0 0 0 0 0 -x22 0 -x21 0 0 |
| 0 0 x14 0 0 0 x13 0 0 0 -x11 0 |
| 0 0 x24 x14 0 0 x23 x13 0 0 -x21 -x11 |
| 0 0 0 x24 0 0 0 x23 0 0 0 -x21 |
| 0 0 0 0 x14 0 0 0 x13 0 x12 0 |
| 0 0 0 0 x24 x14 0 0 x23 x13 x22 x12 |
| 0 0 0 0 0 x24 0 0 0 x23 0 x22 |
12 4
2 : R <--------------------------- R : 3
| x12 x11 0 0 |
| x22 x21 0 0 |
| -x13 0 x11 0 |
| -x23 0 x21 0 |
| 0 -x13 -x12 0 |
| 0 -x23 -x22 0 |
| x14 0 0 x11 |
| x24 0 0 x21 |
| 0 x14 0 -x12 |
| 0 x24 0 -x22 |
| 0 0 x14 x13 |
| 0 0 x24 x23 |
o16 : ChainComplexMap
|
i17 : apply((length G)+1,i->reduceHilbert hilbertSeries HH_(i)(G))
4 0 0 0
o17 = {--------, -, -, -}
5 1 1 1
(1 - T)
o17 : List
|
We compute a third example.
i18 : R=ZZ/7[x,y,z,w]; |
i19 : I=ideal(x*z-y^2,x*w-y*z, y*w-z^2); o19 : Ideal of R |
i20 : F=res I; |
i21 : lambda={2,1};
|
i22 : G=schurComplex(lambda,F)
3 11 20 22 12 2
o22 = R <-- R <-- R <-- R <-- R <-- R
1 2 3 4 5 6
o22 : ChainComplex
|
i23 : G.dd
3 11
o23 = 1 : R <------------------------------------------------------------------- R : 2
| yz-xw y2-xz z2-yw yz-xw y2-xz 0 0 0 0 -x y |
| 0 0 0 z2-yw 0 y2-xz yz-xw y2-xz 0 y -z |
| 0 0 0 0 z2-yw 0 0 yz-xw y2-xz -z w |
11 20
2 : R <--------------------------------------------------------------------------------------------------------------------------- R : 3
| z2-yw yz-xw y2-xz 0 0 0 0 0 y -z 0 0 0 0 x -y 0 0 0 0 |
| 0 0 0 z2-yw yz-xw y2-xz 0 0 -z w 0 0 0 0 0 0 0 0 x -y |
| -yz+xw 0 0 -y2+xz 0 0 -2x 2y -x y 0 0 0 0 0 0 0 0 0 0 |
| 0 -yz+xw 0 0 -y2+xz 0 y -z 0 0 0 0 -x y -x y 0 0 0 0 |
| 0 0 -yz+xw 0 0 -y2+xz -z w 0 0 0 0 0 0 0 0 -x y -x y |
| 0 0 -z2+yw 0 z2-yw 0 0 0 0 0 yz-xw y2-xz 0 0 -z w 0 0 -y z |
| 0 z2-yw 0 0 0 0 0 0 0 0 -y2+xz 0 2y -2z y -z 0 0 0 0 |
| 0 0 z2-yw 0 0 0 0 0 0 0 0 -y2+xz -z w 0 0 y -z y -z |
| 0 0 0 0 0 z2-yw 0 0 0 0 0 yz-xw 0 0 0 0 -2z 2w -z w |
| 0 0 0 0 0 0 -z2+yw 0 0 0 0 0 -yz+xw 0 0 0 -y2+xz 0 0 0 |
| 0 0 0 0 0 0 0 -z2+yw 0 0 0 0 0 -yz+xw 0 0 0 -y2+xz 0 0 |
20 22
3 : R <--------------------------------------------------------------------------------------------------------------------------------------- R : 4
| -2x 2y 0 0 -y -x 0 z y 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| y -z 0 0 0 -y 0 0 z 0 0 0 -x 0 y 0 0 0 0 0 0 0 |
| -z w 0 0 0 0 -y 0 0 z 0 0 0 -x 0 y 0 0 0 0 0 0 |
| 0 0 -2x 2y z 0 -x -w 0 y 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 y -z 0 z 0 0 -w 0 -x y 0 -x 0 y 0 0 0 0 0 0 |
| 0 0 -z w 0 0 z 0 0 -w 0 0 0 0 0 0 -x y 0 0 0 0 |
| yz-xw 0 y2-xz 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 -y 0 |
| 0 yz-xw 0 y2-xz 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x 0 -y |
| 0 0 0 0 z2-yw yz-xw y2-xz 0 0 0 0 0 0 0 0 0 0 0 -2x y y 0 |
| 0 0 0 0 0 0 0 z2-yw yz-xw y2-xz 0 0 0 0 0 0 0 0 0 -x -x 2y |
| 0 0 0 0 0 0 0 0 0 0 2y -2z z y -w -z 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 -z w 0 z 0 -w y -z 0 0 0 0 |
| -z2+yw 0 0 0 0 0 0 0 0 0 y2-xz 0 0 0 0 0 0 0 -y 0 z 0 |
| 0 -z2+yw 0 0 0 0 0 0 0 0 0 y2-xz 0 0 0 0 0 0 0 -y 0 z |
| z2-yw 0 0 0 0 z2-yw 0 0 0 0 0 0 yz-xw y2-xz 0 0 0 0 2y -z -z 0 |
| 0 z2-yw 0 0 0 0 0 0 z2-yw 0 0 0 0 0 yz-xw y2-xz 0 0 0 y y -2z |
| 0 0 -z2+yw 0 0 0 0 0 0 0 -yz+xw 0 0 0 0 0 0 0 z 0 -w 0 |
| 0 0 0 -z2+yw 0 0 0 0 0 0 0 -yz+xw 0 0 0 0 0 0 0 z 0 -w |
| 0 0 z2-yw 0 0 0 z2-yw 0 0 0 yz-xw 0 0 yz-xw 0 0 y2-xz 0 -2z w w 0 |
| 0 0 0 z2-yw 0 0 0 0 0 z2-yw 0 yz-xw 0 0 0 yz-xw 0 y2-xz 0 -z -z 2w |
22 12
4 : R <------------------------------------------------------------------------------------------- R : 5
| y 0 -z 0 x 0 -y 0 0 0 0 0 |
| 0 y 0 -z 0 x 0 -y 0 0 0 0 |
| -z 0 w 0 0 0 0 0 x 0 -y 0 |
| 0 -z 0 w 0 0 0 0 0 x 0 -y |
| -3x 2y y 0 0 0 0 0 0 0 0 0 |
| y -z 0 0 -2x y y 0 0 0 0 0 |
| -z w 0 0 0 0 0 0 -2x y y 0 |
| 0 -x -2x 3y 0 0 0 0 0 0 0 0 |
| 0 0 y -z 0 -x -x 2y 0 0 0 0 |
| 0 0 -z w 0 0 0 0 0 -x -x 2y |
| 0 0 0 0 -z 0 w 0 -y 0 z 0 |
| 0 0 0 0 0 -z 0 w 0 -y 0 z |
| 0 0 0 0 3y -2z -z 0 0 0 0 0 |
| 0 0 0 0 -z w 0 0 2y -z -z 0 |
| 0 0 0 0 0 y 2y -3z 0 0 0 0 |
| 0 0 0 0 0 0 -z w 0 y y -2z |
| 0 0 0 0 0 0 0 0 -3z 2w w 0 |
| 0 0 0 0 0 0 0 0 0 -z -2z 3w |
| -z2+yw 0 0 0 -yz+xw 0 0 0 -y2+xz 0 0 0 |
| 0 -z2+yw 0 0 0 -yz+xw 0 0 0 -y2+xz 0 0 |
| 0 0 -z2+yw 0 0 0 -yz+xw 0 0 0 -y2+xz 0 |
| 0 0 0 -z2+yw 0 0 0 -yz+xw 0 0 0 -y2+xz |
12 2
5 : R <--------------- R : 6
| y 0 |
| 2x -y |
| -x 2y |
| 0 x |
| -z 0 |
| -2y z |
| y -2z |
| 0 -y |
| w 0 |
| 2z -w |
| -z 2w |
| 0 z |
o23 : ChainComplexMap
|
The object schurComplex is a function closure.