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.