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
|