character(ActionOnComplex) -- compute all Betti characters of minimal free resolution

Synopsis

Function: character
Usage:

character(A)
Inputs:
- A, an instance of the type ActionOnComplex, a finite group action on a minimal free resolution
Outputs:
- an instance of the type Character, Betti characters of the resolution

Description

Use this function to compute all nonzero Betti characters of a finite group action on a minimal free resolution. This function calls character(ActionOnComplex,ZZ) on all nonzero homological degrees and then assembles the outputs in a hash table indexed by homological degree.

To illustrate, we compute the Betti characters of a symmetric group on the resolution of a monomial ideal. The ideal is the symbolic square of the ideal generated by all squarefree monomials of degree three in four variables. The symmetric group acts by permuting the four variables of the ring. The characters are determined by five permutations with cycle types, in order: 4, 31, 22, 211, 1111.

i1 : R = QQ[x_1..x_4]

o1 = R

o1 : PolynomialRing

i2 : J = intersect(apply(subsets(gens R,3),x->(ideal x)^2))

                                             2 2   2 2   2 2   2 2   2 2 
o2 = ideal (x x x , x x x , x x x , x x x , x x , x x , x x , x x , x x ,
             2 3 4   1 3 4   1 2 4   1 2 3   3 4   2 4   1 4   2 3   1 3 
     ------------------------------------------------------------------------
      2 2
     x x )
      1 2

o2 : Ideal of R

i3 : RJ = res J

      1      10      15      6
o3 = R  <-- R   <-- R   <-- R  <-- 0
                                    
     0      1       2       3      4

o3 : ChainComplex

i4 : G = { matrix{{x_2,x_3,x_4,x_1}},
           matrix{{x_2,x_3,x_1,x_4}},
           matrix{{x_2,x_1,x_4,x_3}},
           matrix{{x_2,x_1,x_3,x_4}},
           matrix{{x_1,x_2,x_3,x_4}} }

o4 = {| x_2 x_3 x_4 x_1 |, | x_2 x_3 x_1 x_4 |, | x_2 x_1 x_4 x_3 |, | x_2
     ------------------------------------------------------------------------
     x_1 x_3 x_4 |, | x_1 x_2 x_3 x_4 |}

o4 : List

i5 : A = action(RJ,G)

o5 = ChainComplex with 5 actors

o5 : ActionOnComplex

i6 : character(A)

o6 = Character over R
      
     (0, {0}) => | 1 1 1 1 1 |
     (1, {3}) => | 0 1 0 2 4 |
     (1, {4}) => | 0 0 2 2 6 |
     (2, {4}) => | -1 0 -1 1 3 |
     (2, {5}) => | 0 0 0 2 12 |
     (3, {6}) => | 0 0 -2 0 6 |

o6 : Character

See character(ActionOnComplex,ZZ) for more details on this example.

Ways to use this method: