# actors -- group elements of an action

## Description

This function is provided by the package BettiCharacters.

When called (without additional arguments) on an object of type Action, this function returns the list of group elements originally provided by the user to act on a module or in a given homological degree of a resolution. Note that these group elements are assumed to trivial, unless otherwise indicated when constructing the action.

The user may specify additional arguments to obtain elements of the group acting in other degrees. See the specific use cases for more details.

 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 : M = module I o3 = image | x_1x_2 x_1x_3 x_2x_3 x_1x_4 x_2x_4 x_3x_4 | 1 o3 : R-module, submodule of R i4 : RM = res M 6 8 3 o4 = R <-- R <-- R <-- 0 0 1 2 3 o4 : ChainComplex i5 : 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}} } o5 = {| 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 |} o5 : List i6 : G' = { (id_(R^6))_{2,4,5,0,1,3}, (id_(R^6))_{2,0,1,4,5,3}, (id_(R^6))_{0,4,3,2,1,5}, (id_(R^6))_{0,2,1,4,3,5}, id_(R^6) } o6 = {| 0 0 0 1 0 0 |, | 0 1 0 0 0 0 |, | 1 0 0 0 0 0 |, | 1 0 0 0 0 0 |, | 1 | 0 0 0 0 1 0 | | 0 0 1 0 0 0 | | 0 0 0 0 1 0 | | 0 0 1 0 0 0 | | 0 | 1 0 0 0 0 0 | | 1 0 0 0 0 0 | | 0 0 0 1 0 0 | | 0 1 0 0 0 0 | | 0 | 0 0 0 0 0 1 | | 0 0 0 0 0 1 | | 0 0 1 0 0 0 | | 0 0 0 0 1 0 | | 0 | 0 1 0 0 0 0 | | 0 0 0 1 0 0 | | 0 1 0 0 0 0 | | 0 0 0 1 0 0 | | 0 | 0 0 1 0 0 0 | | 0 0 0 0 1 0 | | 0 0 0 0 0 1 | | 0 0 0 0 0 1 | | 0 ------------------------------------------------------------------------ 0 0 0 0 0 |} 1 0 0 0 0 | 0 1 0 0 0 | 0 0 1 0 0 | 0 0 0 1 0 | 0 0 0 0 1 | o6 : List i7 : A = action(RM,G,G',0) o7 = ChainComplex with 5 actors o7 : ActionOnComplex i8 : actors(A) o8 = {{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 | o8 : List i9 : B = action(M,G) o9 = Module with 5 actors o9 : ActionOnGradedModule i10 : actors(B) o10 = {| 1 |, | 1 |, | 1 |, | 1 |, | 1 |} o10 : List