## Synopsis

• Function: character
• Usage:
character(A,d)
• Inputs:
• Outputs:
• an instance of the type Character, the character of the components of a module in given degrees

## Description

Use this function to compute the characters of the finite group action on the graded components of a module. The second argument is the multidegree (as a list) or the degree (as an integer) of the desired component.

To illustrate, we compute the Betti characters of a symmetric group on the graded components of a quotient ring. The characters are determined by five permutations with cycle types, in order: 4, 31, 22, 211, 1111.

 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 : Q = R/I o4 = Q o4 : QuotientRing i5 : A = action(Q,G) o5 = QuotientRing with 5 actors o5 : ActionOnGradedModule i6 : character(A,0) o6 = Character over R (0, {0}) => | 1 1 1 1 1 | o6 : Character i7 : character(A,1) o7 = Character over R (0, {1}) => | 0 1 0 2 4 | o7 : Character

## Synopsis

• Usage:
character(A,lo,hi)
• Inputs:
• Outputs:
• an instance of the type Character, the character of the components of a module in the given range of degrees

For $\mathbb{Z}$-graded modules, one may compute characters in a range of degrees by providing the lowest and highest degrees in the range as the second and third argument.

 i8 : character(A,0,4) o8 = Character over R (0, {0}) => | 1 1 1 1 1 | (0, {1}) => | 0 1 0 2 4 | (0, {2}) => | 0 1 0 2 4 | (0, {3}) => | 0 1 0 2 4 | (0, {4}) => | 0 1 0 2 4 | o8 : Character