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
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x_1 x_3 x_4 |, | x_1 x_2 x_3 x_4 |}
o3 : List
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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
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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
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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
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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
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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
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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).