# decomposeCharacter -- decompose a character into irreducible characters

## Synopsis

• Usage:
decomposeCharacter(c,T)
c/T
• Inputs:
• c, an instance of the type Character, of a finite group
• T, an instance of the type CharacterTable, of the same finite group
• Outputs:

## Description

Use the decomposeCharacter method to decompose a character into a linear combination of irreducible characters in a character table. The shortcut c/T is equivalent to the command decomposeCharacter(c,T).

As an example, we construct the character table of the symmetric group on 3 elements, then use it to decompose the character of the action of the same symmetric group permuting the variables of a standard graded polynomial ring.

 i1 : s = {2,3,1} o1 = {2, 3, 1} o1 : List i2 : M = matrix{{1,1,1},{-1,0,2},{1,-1,1}} o2 = | 1 1 1 | | -1 0 2 | | 1 -1 1 | 3 3 o2 : Matrix ZZ <-- ZZ i3 : R = QQ[x_1..x_3] o3 = R o3 : PolynomialRing i4 : P = {1,2,3} o4 = {1, 2, 3} o4 : List i5 : T = characterTable(s,M,R,P) o5 = Character table over R | 2 3 1 ----+----------- X0 | 1 1 1 X1 | -1 0 2 X2 | 1 -1 1 o5 : CharacterTable i6 : acts = {matrix{{x_2,x_3,x_1}},matrix{{x_2,x_1,x_3}},matrix{{x_1,x_2,x_3}}} o6 = {| x_2 x_3 x_1 |, | x_2 x_1 x_3 |, | x_1 x_2 x_3 |} o6 : List i7 : A = action(R,acts) o7 = PolynomialRing with 3 actors o7 : ActionOnGradedModule i8 : c = character(A,0,10) o8 = Character over R (0, {0}) => | 1 1 1 | (0, {1}) => | 0 1 3 | (0, {2}) => | 0 2 6 | (0, {3}) => | 1 2 10 | (0, {4}) => | 0 3 15 | (0, {5}) => | 0 3 21 | (0, {6}) => | 1 4 28 | (0, {7}) => | 0 4 36 | (0, {8}) => | 0 5 45 | (0, {9}) => | 1 5 55 | (0, {10}) => | 0 6 66 | o8 : Character i9 : decomposeCharacter(c,T) o9 = Decomposition table | X0 X1 X2 -----------+------------ (0, {0}) | 1 0 0 (0, {1}) | 1 1 0 (0, {2}) | 2 2 0 (0, {3}) | 3 3 1 (0, {4}) | 4 5 1 (0, {5}) | 5 7 2 (0, {6}) | 7 9 3 (0, {7}) | 8 12 4 (0, {8}) | 10 15 5 (0, {9}) | 12 18 7 (0, {10}) | 14 22 8 o9 : CharacterDecomposition

The results are shown in a table whose rows are indexed by pairs of homological and internal degrees, and whose columns are labeled by the irreducible characters. By default, irreducible characters in a character table are labeled as X0, X1, ..., etc, and the same labeling is inherited by the character decomposition. The user may pass custom labels in a list using the option Labels when constructing the character table.