actors(ActionOnComplex,ZZ) -- group elements of action on resolution

Synopsis

Function: actors
Usage:

actors(A,i)
Inputs:
- A, an instance of the type ActionOnComplex, a finite group action on a minimal free resolution
- i, an integer, a homological degree
Outputs:
- a list, of group elements acting in homological degree i

Description

This function returns matrices describing elements of a finite group acting on a minimal free resolution in a given homological degree. If the homological degree is the one where the user originally defined the action, then the user provided elements are returned. Otherwise, suitable elements are computed as indicated in F. Galetto - Finite group characters on free resolutions.

To illustrate, we compute the action of a symmetric group on the resolution of a monomial ideal. The ideal is generated by all squarefree monomials of degree two in four variables. The symmetric group acts by permuting the four variables of the ring. We only consider five permutations with cycle types, in order: 4, 31, 22, 211, 1111 (since these are enough to determine the characters of the action).

i1 : R = QQ[x_1..x_4]

o1 = R

o1 : PolynomialRing

i2 : I = ideal apply(subsets(gens R,2),product)

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

o2 : Ideal of R

i3 : RI = res I

      1      6      8      3
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(RI,G)

o5 = ChainComplex with 5 actors

o5 : ActionOnComplex

i6 : actors(A,0)

o6 = {| 1 |, | 1 |, | 1 |, | 1 |, | 1 |}

o6 : List

i7 : actors(A,1)

o7 = {{2} | 0 0 0 1 0 0 |, {2} | 0 1 0 0 0 0 |, {2} | 1 0 0 0 0 0 |, {2} | 1
      {2} | 0 0 0 0 1 0 |  {2} | 0 0 1 0 0 0 |  {2} | 0 0 0 0 1 0 |  {2} | 0
      {2} | 1 0 0 0 0 0 |  {2} | 1 0 0 0 0 0 |  {2} | 0 0 0 1 0 0 |  {2} | 0
      {2} | 0 0 0 0 0 1 |  {2} | 0 0 0 0 0 1 |  {2} | 0 0 1 0 0 0 |  {2} | 0
      {2} | 0 1 0 0 0 0 |  {2} | 0 0 0 1 0 0 |  {2} | 0 1 0 0 0 0 |  {2} | 0
      {2} | 0 0 1 0 0 0 |  {2} | 0 0 0 0 1 0 |  {2} | 0 0 0 0 0 1 |  {2} | 0
     ------------------------------------------------------------------------
     0 0 0 0 0 |, {2} | 1 0 0 0 0 0 |}
     0 1 0 0 0 |  {2} | 0 1 0 0 0 0 |
     1 0 0 0 0 |  {2} | 0 0 1 0 0 0 |
     0 0 0 1 0 |  {2} | 0 0 0 1 0 0 |
     0 0 1 0 0 |  {2} | 0 0 0 0 1 0 |
     0 0 0 0 1 |  {2} | 0 0 0 0 0 1 |

o7 : List

i8 : actors(A,2)

o8 = {{3} | 0 0 -1 1 0  0 0  0 |, {3} | -1 1 0 0 0 0  0 0  |, {3} | 0 0  1 0 
      {3} | 0 0 -1 0 0  0 0  0 |  {3} | -1 0 0 0 0 0  0 0  |  {3} | 0 0  1 -1
      {3} | 0 0 0  0 -1 1 0  0 |  {3} | 0  0 0 0 1 0  0 0  |  {3} | 1 0  0 0 
      {3} | 0 0 0  0 -1 0 0  0 |  {3} | 0  0 0 0 1 -1 0 0  |  {3} | 1 -1 0 0 
      {3} | 0 0 0  0 0  0 -1 1 |  {3} | 0  0 0 0 0 0  1 0  |  {3} | 0 0  0 0 
      {3} | 0 0 0  0 0  0 -1 0 |  {3} | 0  0 0 0 0 0  1 -1 |  {3} | 0 0  0 0 
      {3} | 1 0 0  0 0  0 0  0 |  {3} | 0  0 1 0 0 0  0 0  |  {3} | 0 0  0 0 
      {3} | 0 1 0  0 0  0 0  0 |  {3} | 0  0 0 1 0 0  0 0  |  {3} | 0 0  0 0 
     ------------------------------------------------------------------------
     0  0 0  0 |, {3} | 1 0  0 0  0 0 0 0 |, {3} | 1 0 0 0 0 0 0 0 |}
     0  0 0  0 |  {3} | 1 -1 0 0  0 0 0 0 |  {3} | 0 1 0 0 0 0 0 0 |
     0  0 0  0 |  {3} | 0 0  1 0  0 0 0 0 |  {3} | 0 0 1 0 0 0 0 0 |
     0  0 0  0 |  {3} | 0 0  1 -1 0 0 0 0 |  {3} | 0 0 0 1 0 0 0 0 |
     0  0 -1 1 |  {3} | 0 0  0 0  0 0 1 0 |  {3} | 0 0 0 0 1 0 0 0 |
     0  0 0  1 |  {3} | 0 0  0 0  0 0 0 1 |  {3} | 0 0 0 0 0 1 0 0 |
     -1 1 0  0 |  {3} | 0 0  0 0  1 0 0 0 |  {3} | 0 0 0 0 0 0 1 0 |
     0  1 0  0 |  {3} | 0 0  0 0  0 1 0 0 |  {3} | 0 0 0 0 0 0 0 1 |

o8 : List

i9 : actors(A,3)

o9 = {{4} | 0 -1 1 |, {4} | 0  1  0 |, {4} | 0  -1 1  |, {4} | 0 1 0  |, {4}
      {4} | 1 1  0 |  {4} | -1 -1 0 |  {4} | -1 0  -1 |  {4} | 1 0 0  |  {4}
      {4} | 0 1  0 |  {4} | 0  -1 1 |  {4} | 0  0  -1 |  {4} | 0 0 -1 |  {4}
     ------------------------------------------------------------------------
     | 1 0 0 |}
     | 0 1 0 |
     | 0 0 1 |

o9 : List

Caveat

When applied to a minimal free resolution $F_\bullet$, this function returns matrices that induce the action of group elements on the representations $F_i/\mathfrak{m}F_i$, where $\mathfrak{m}$ is the maximal ideal generated by the variables of the polynomial ring. While these matrices often represent the action of the same group elements on the modules $F_i$ of the resolution, this is in general not a guarantee.

Ways to use this method: