schurComplex -- Schur functors of chain complexes

Usage:

G=schurComplex(lambda,F)
Inputs:
- lambda, a list,
- F, a complex,
Outputs:
- G, a complex,

Description

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

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

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

i1 : R=ZZ[x,y,z];

i2 : I=ideal(x,y,z);

o2 : Ideal of R

i3 : F=freeResolution 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 : Complex

i6 : G.dd

          3                                    9
o6 = 1 : R  <-------------------------------- R  : 2
               {1} | z y x 0 0 0 0  y  x  |
               {1} | 0 z 0 y x 0 x  -z 0  |
               {1} | 0 0 z 0 y x -y 0  -z |

          9                                                10
     2 : R  <-------------------------------------------- R   : 3
               {2} | 0  0  2y 2x 0  0   0  0   0  0   |
               {2} | 0  x  -z 0  0  y   x  0   0  0   |
               {2} | 0  -y 0  -z 0  0   0  0   y  x   |
               {2} | 0  0  0  0  2x -2z 0  0   0  0   |
               {2} | 0  0  0  0  -y 0   -z x   -z 0   |
               {2} | 0  0  0  0  0  0   0  -2y 0  -2z |
               {2} | -z -z 0  0  -y 0   0  -x  0  0   |
               {2} | -x 0  -z 0  0  -y  0  0   -x 0   |
               {2} | y  0  0  -z 0  0   -y 0   0  -x  |

          10                                 6
     3 : R   <----------------------------- R  : 4
                {3} | z  y  x  0  0  0  |
                {3} | -z 0  0  y  x  0  |
                {3} | -x 0  0  0  0  x  |
                {3} | y  0  0  0  0  -y |
                {3} | 0  -z 0  -z 0  0  |
                {3} | 0  -x 0  -x 0  0  |
                {3} | 0  y  0  0  -x z  |
                {3} | 0  0  -z 0  -z 0  |
                {3} | 0  0  -x y  0  -z |
                {3} | 0  0  y  0  y  0  |

          6                        3
     4 : R  <-------------------- R  : 5
               {4} | 0  y  x  |
               {4} | x  -z 0  |
               {4} | -y 0  -z |
               {4} | -x z  0  |
               {4} | y  0  z  |
               {4} | 0  y  x  |

          3                  1
     5 : R  <-------------- R  : 6
               {5} | -z |
               {5} | -x |
               {5} | y  |

o6 : ComplexMap

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

i7 : R=QQ[x11,x21,x12,x22,x13,x23,x14,x24];

i8 : M=genericMatrix(R,x11,2,4);

             2      4
o8 : Matrix R  <-- R

i9 : F = complex {M};

i10 : lambda={3};

i11 : G=schurComplex(lambda,F)

       4      12      12      4
o11 = R  <-- R   <-- R   <-- R
                              
      0      1       2       3

o11 : Complex

i12 : G.dd

           4                                                           12
o12 = 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
                 {1} | -x13 0    -x12 0    -x11 0    0    0    0    0    0    0    |
                 {1} | -x23 -x13 -x22 -x12 -x21 -x11 0    0    0    0    0    0    |
                 {1} | 0    -x23 0    -x22 0    -x21 0    0    0    0    0    0    |
                 {1} | x14  0    0    0    0    0    -x12 0    -x11 0    0    0    |
                 {1} | x24  x14  0    0    0    0    -x22 -x12 -x21 -x11 0    0    |
                 {1} | 0    x24  0    0    0    0    0    -x22 0    -x21 0    0    |
                 {1} | 0    0    x14  0    0    0    x13  0    0    0    -x11 0    |
                 {1} | 0    0    x24  x14  0    0    x23  x13  0    0    -x21 -x11 |
                 {1} | 0    0    0    x24  0    0    0    x23  0    0    0    -x21 |
                 {1} | 0    0    0    0    x14  0    0    0    x13  0    x12  0    |
                 {1} | 0    0    0    0    x24  x14  0    0    x23  x13  x22  x12  |
                 {1} | 0    0    0    0    0    x24  0    0    0    x23  0    x22  |

           12                                   4
      2 : R   <------------------------------- R  : 3
                 {2} | x12  x11  0    0    |
                 {2} | x22  x21  0    0    |
                 {2} | -x13 0    x11  0    |
                 {2} | -x23 0    x21  0    |
                 {2} | 0    -x13 -x12 0    |
                 {2} | 0    -x23 -x22 0    |
                 {2} | x14  0    0    x11  |
                 {2} | x24  0    0    x21  |
                 {2} | 0    x14  0    -x12 |
                 {2} | 0    x24  0    -x22 |
                 {2} | 0    0    x14  x13  |
                 {2} | 0    0    x24  x23  |

o12 : ComplexMap

i13 : apply(1+length G,i->reduceHilbert hilbertSeries HH_i(G))

           4     0  0  0
o13 = {--------, -, -, -}
              5  1  1  1
       (1 - T)

o13 : List

We compute a third example.

i14 : R=ZZ/7[x,y,z,w];

i15 : I=ideal(x*z-y^2,x*w-y*z, y*w-z^2);

o15 : Ideal of R

i16 : F=freeResolution I;

i17 : lambda={2,1};

i18 : G=schurComplex(lambda,F)

       3      11      20      22      12      2
o18 = R  <-- R   <-- R   <-- R   <-- R   <-- R
                                              
      1      2       3       4       5       6

o18 : Complex

i19 : G.dd

           3                                                                           11
o19 = 1 : R  <----------------------------------------------------------------------- R   : 2
                {2} | yz-xw y2-xz z2-yw yz-xw y2-xz 0     0     0     0     -x y  |
                {2} | 0     0     0     z2-yw 0     y2-xz yz-xw y2-xz 0     y  -z |
                {2} | 0     0     0     0     z2-yw 0     0     yz-xw y2-xz -z w  |

           11                                                                                                                                   20
      2 : R   <------------------------------------------------------------------------------------------------------------------------------- R   : 3
                 {4} | z2-yw  yz-xw  y2-xz  0      0      0      0      0      y  -z 0      0      0      0      x  -y 0      0      0  0  |
                 {4} | 0      0      0      z2-yw  yz-xw  y2-xz  0      0      -z w  0      0      0      0      0  0  0      0      x  -y |
                 {4} | -yz+xw 0      0      -y2+xz 0      0      -2x    2y     -x y  0      0      0      0      0  0  0      0      0  0  |
                 {4} | 0      -yz+xw 0      0      -y2+xz 0      y      -z     0  0  0      0      -x     y      -x y  0      0      0  0  |
                 {4} | 0      0      -yz+xw 0      0      -y2+xz -z     w      0  0  0      0      0      0      0  0  -x     y      -x y  |
                 {4} | 0      0      -z2+yw 0      z2-yw  0      0      0      0  0  yz-xw  y2-xz  0      0      -z w  0      0      -y z  |
                 {4} | 0      z2-yw  0      0      0      0      0      0      0  0  -y2+xz 0      2y     -2z    y  -z 0      0      0  0  |
                 {4} | 0      0      z2-yw  0      0      0      0      0      0  0  0      -y2+xz -z     w      0  0  y      -z     y  -z |
                 {4} | 0      0      0      0      0      z2-yw  0      0      0  0  0      yz-xw  0      0      0  0  -2z    2w     -z w  |
                 {3} | 0      0      0      0      0      0      -z2+yw 0      0  0  0      0      -yz+xw 0      0  0  -y2+xz 0      0  0  |
                 {3} | 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
                 {6} | -2x    2y     0      0      -y    -x    0     z     y     0     0      0      0     0     0     0     0     0     0   0  0  0   |
                 {6} | y      -z     0      0      0     -y    0     0     z     0     0      0      -x    0     y     0     0     0     0   0  0  0   |
                 {6} | -z     w      0      0      0     0     -y    0     0     z     0      0      0     -x    0     y     0     0     0   0  0  0   |
                 {6} | 0      0      -2x    2y     z     0     -x    -w    0     y     0      0      0     0     0     0     0     0     0   0  0  0   |
                 {6} | 0      0      y      -z     0     z     0     0     -w    0     -x     y      0     -x    0     y     0     0     0   0  0  0   |
                 {6} | 0      0      -z     w      0     0     z     0     0     -w    0      0      0     0     0     0     -x    y     0   0  0  0   |
                 {5} | yz-xw  0      y2-xz  0      0     0     0     0     0     0     0      0      0     0     0     0     0     0     x   0  -y 0   |
                 {5} | 0      yz-xw  0      y2-xz  0     0     0     0     0     0     0      0      0     0     0     0     0     0     0   x  0  -y  |
                 {5} | 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   |
                 {5} | 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  |
                 {6} | 0      0      0      0      0     0     0     0     0     0     2y     -2z    z     y     -w    -z    0     0     0   0  0  0   |
                 {6} | 0      0      0      0      0     0     0     0     0     0     -z     w      0     z     0     -w    y     -z    0   0  0  0   |
                 {5} | -z2+yw 0      0      0      0     0     0     0     0     0     y2-xz  0      0     0     0     0     0     0     -y  0  z  0   |
                 {5} | 0      -z2+yw 0      0      0     0     0     0     0     0     0      y2-xz  0     0     0     0     0     0     0   -y 0  z   |
                 {5} | 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   |
                 {5} | 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 |
                 {5} | 0      0      -z2+yw 0      0     0     0     0     0     0     -yz+xw 0      0     0     0     0     0     0     z   0  -w 0   |
                 {5} | 0      0      0      -z2+yw 0     0     0     0     0     0     0      -yz+xw 0     0     0     0     0     0     0   z  0  -w  |
                 {5} | 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   |
                 {5} | 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
                 {7} | y      0      -z     0      x      0      -y     0      0      0      0      0      |
                 {7} | 0      y      0      -z     0      x      0      -y     0      0      0      0      |
                 {7} | -z     0      w      0      0      0      0      0      x      0      -y     0      |
                 {7} | 0      -z     0      w      0      0      0      0      0      x      0      -y     |
                 {7} | -3x    2y     y      0      0      0      0      0      0      0      0      0      |
                 {7} | y      -z     0      0      -2x    y      y      0      0      0      0      0      |
                 {7} | -z     w      0      0      0      0      0      0      -2x    y      y      0      |
                 {7} | 0      -x     -2x    3y     0      0      0      0      0      0      0      0      |
                 {7} | 0      0      y      -z     0      -x     -x     2y     0      0      0      0      |
                 {7} | 0      0      -z     w      0      0      0      0      0      -x     -x     2y     |
                 {7} | 0      0      0      0      -z     0      w      0      -y     0      z      0      |
                 {7} | 0      0      0      0      0      -z     0      w      0      -y     0      z      |
                 {7} | 0      0      0      0      3y     -2z    -z     0      0      0      0      0      |
                 {7} | 0      0      0      0      -z     w      0      0      2y     -z     -z     0      |
                 {7} | 0      0      0      0      0      y      2y     -3z    0      0      0      0      |
                 {7} | 0      0      0      0      0      0      -z     w      0      y      y      -2z    |
                 {7} | 0      0      0      0      0      0      0      0      -3z    2w     w      0      |
                 {7} | 0      0      0      0      0      0      0      0      0      -z     -2z    3w     |
                 {6} | -z2+yw 0      0      0      -yz+xw 0      0      0      -y2+xz 0      0      0      |
                 {6} | 0      -z2+yw 0      0      0      -yz+xw 0      0      0      -y2+xz 0      0      |
                 {6} | 0      0      -z2+yw 0      0      0      -yz+xw 0      0      0      -y2+xz 0      |
                 {6} | 0      0      0      -z2+yw 0      0      0      -yz+xw 0      0      0      -y2+xz |

           12                       2
      5 : R   <------------------- R  : 6
                 {8} | y   0   |
                 {8} | 2x  -y  |
                 {8} | -x  2y  |
                 {8} | 0   x   |
                 {8} | -z  0   |
                 {8} | -2y z   |
                 {8} | y   -2z |
                 {8} | 0   -y  |
                 {8} | w   0   |
                 {8} | 2z  -w  |
                 {8} | -z  2w  |
                 {8} | 0   z   |

o19 : ComplexMap

If the input complex F has a (multi)grading, the resulting Schur complex does as well.

i20 : needsPackage "NormalToricVarieties"

o20 = NormalToricVarieties

o20 : Package

i21 : R = ring hirzebruchSurface 3;
warning: clearing value of symbol x to allow access to subscripted variables based on it
       : debug with expression   debug 9868   or with command line option   --debug 9868

i22 : F = koszulComplex {x_0^4*x_1 - x_2*x_3, x_0^2};

i23 : lambda = {2,2};

i24 : G = schurComplex(lambda, F);

i25 : G.dd

           1                                                                  4
o25 = 2 : R  <-------------------------------------------------------------- R  : 3
                {3, 1} | -x_0^2 -x_0^4x_1+x_2x_3 -x_0^2 -x_0^4x_1+x_2x_3 |

           4                                                                                                                6
      3 : R  <------------------------------------------------------------------------------------------------------------ R  : 4
                {5, 1} | -x_0^4x_1+x_2x_3 -x_0^4x_1+x_2x_3 x_0^2  x_0^4x_1-x_2x_3  0                0                  |
                {4, 2} | x_0^2            2x_0^2           0      x_0^2            x_0^4x_1-x_2x_3  0                  |
                {5, 1} | 0                -x_0^4x_1+x_2x_3 -x_0^2 -x_0^4x_1+x_2x_3 0                -2x_0^4x_1+2x_2x_3 |
                {4, 2} | 0                0                0      -x_0^2           -x_0^4x_1+x_2x_3 2x_0^2             |

           6                                                                                        4
      4 : R  <------------------------------------------------------------------------------------ R  : 5
                {6, 2} | 2x_0^2          2x_0^4x_1-2x_2x_3 0                  0                |
                {6, 2} | -x_0^2          -x_0^4x_1+x_2x_3  -x_0^2             -x_0^4x_1+x_2x_3 |
                {7, 1} | x_0^4x_1-x_2x_3 0                 -3x_0^4x_1+3x_2x_3 0                |
                {6, 2} | 0               x_0^4x_1-x_2x_3   2x_0^2             -x_0^4x_1+x_2x_3 |
                {5, 3} | 0               -x_0^2            0                  3x_0^2           |
                {6, 2} | 0               0                 x_0^2              x_0^4x_1-x_2x_3  |

           4                                     1
      5 : R  <--------------------------------- R  : 6
                {8, 2} | -3x_0^4x_1+3x_2x_3 |
                {7, 3} | 3x_0^2             |
                {8, 2} | -x_0^4x_1+x_2x_3   |
                {7, 3} | x_0^2              |

o25 : ComplexMap

i26 : assert isHomogeneous G

For the programmer

The object schurComplex is a function closure.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurComplexes.m2:597:0.

schurComplex -- Schur functors of chain complexes

Description

See also

For the programmer