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

## Synopsis

• Function: actors
• Usage:
actors(A,d)
• Inputs:
• Outputs:
• a list, of group elements acting in the given (multi)degree

## Description

This function is provided by the package BettiCharacters.

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).