actors(ActionOnGradedModule,List) -- group elements acting on components of a module

Synopsis

Function: actors
Usage:

actors(A,d)
Inputs:
- A, an instance of the type ActionOnGradedModule, a finite group action on a graded module
- d, a list, a (multi)degree
Outputs:
- a list, of group elements acting in the given (multi)degree

Description

This function returns matrices describing elements of a finite group acting on the graded component of (multi)degree d of a module.

To illustrate, we compute the action of a symmetric group on the components of a monomial ideal. 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 : 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}} }

o3 = {| 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 |}

o3 : List

i4 : A = action(I,G)

o4 = Ideal with 5 actors

o4 : ActionOnGradedModule

i5 : actors(A,1)

o5 = {0, 0, 0, 0, 0}

o5 : List

i6 : actors(A,2)

o6 = {{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 |

o6 : List

i7 : actors(A,3)

o7 = {{3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |, {3} | 0 0 0 0 0 1 0 0 0 0 0 0
      {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |  {3} | 0 0 0 0 1 0 0 0 0 0 0 0
      {3} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |  {3} | 0 0 1 0 0 0 0 0 0 0 0 0
      {3} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |  {3} | 0 0 0 0 0 0 1 0 0 0 0 0
      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |  {3} | 0 0 0 0 0 0 0 0 1 0 0 0
      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |  {3} | 0 0 0 0 0 0 0 1 0 0 0 0
      {3} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |  {3} | 0 0 0 0 0 0 0 0 0 1 0 0
      {3} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  {3} | 1 0 0 0 0 0 0 0 0 0 0 0
      {3} | 0 1 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 |  {3} | 0 0 0 1 0 0 0 0 0 0 0 0
      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |  {3} | 0 0 0 0 0 0 0 0 0 0 0 0
      {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |  {3} | 0 0 0 0 0 0 0 0 0 0 0 0
      {3} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |  {3} | 0 0 0 0 0 0 0 0 0 0 1 0
      {3} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |  {3} | 0 0 0 0 0 0 0 0 0 0 0 1
      {3} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |  {3} | 0 0 0 0 0 0 0 0 0 0 0 0
      {3} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |  {3} | 0 0 0 0 0 0 0 0 0 0 0 0
     ------------------------------------------------------------------------
     0 0 0 0 |, {3} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |, {3} | 0 1 0 0 0 0 0
     0 0 0 0 |  {3} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |  {3} | 1 0 0 0 0 0 0
     0 0 0 0 |  {3} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |  {3} | 0 0 1 0 0 0 0
     0 0 0 0 |  {3} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |  {3} | 0 0 0 1 0 0 0
     0 0 0 0 |  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |  {3} | 0 0 0 0 0 0 0
     0 0 0 0 |  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |  {3} | 0 0 0 0 0 0 0
     0 0 0 0 |  {3} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |  {3} | 0 0 0 0 0 0 0
     0 0 0 0 |  {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |  {3} | 0 0 0 0 1 0 0
     0 0 0 0 |  {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |  {3} | 0 0 0 0 0 1 0
     0 0 0 0 |  {3} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |  {3} | 0 0 0 0 0 0 1
     0 0 1 0 |  {3} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |  {3} | 0 0 0 0 0 0 0
     0 0 0 1 |  {3} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |  {3} | 0 0 0 0 0 0 0
     0 0 0 0 |  {3} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |  {3} | 0 0 0 0 0 0 0
     0 0 0 0 |  {3} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |  {3} | 0 0 0 0 0 0 0
     1 0 0 0 |  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |  {3} | 0 0 0 0 0 0 0
     0 1 0 0 |  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |  {3} | 0 0 0 0 0 0 0
     ------------------------------------------------------------------------
     0 0 0 0 0 0 0 0 0 |, {3} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |}
     0 0 0 0 0 0 0 0 0 |  {3} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |  {3} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |  {3} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
     1 0 0 0 0 0 0 0 0 |  {3} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
     0 1 0 0 0 0 0 0 0 |  {3} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
     0 0 1 0 0 0 0 0 0 |  {3} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |  {3} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |  {3} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
     0 0 0 0 0 0 0 0 0 |  {3} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
     0 0 0 0 0 1 0 0 0 |  {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
     0 0 0 0 0 0 1 0 0 |  {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
     0 0 0 1 0 0 0 0 0 |  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
     0 0 0 0 1 0 0 0 0 |  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
     0 0 0 0 0 0 0 1 0 |  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
     0 0 0 0 0 0 0 0 1 |  {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

o7 : List

The degree argument can be an integer (in the case of single graded modules) or a list of integers (in the case of a multigraded module).

Ways to use this method: