# Labels -- custom labels for irreducible characters

## Description

This optional input is used with the method characterTable provided by the package BettiCharacters.

By default, irreducible characters in a character table are labeled as X0, X1, ..., etc. The user may pass custom labels in a list using this option.

The next example sets up the character table of the dihedral group $D_4$, generated by an order 4 rotation $r$ and an order 2 reflection $s$ with the relation $srs=r^3$. The representatives of the conjugacy classes are, in order: the identity, $r^2$, $r$, $s$, and $rs$. Besides the trivial representation, $D_4$ has three irreducible one-dimensional representations, corresponding to the three normal subgroups of index two: $\langle r\rangle$, $\langle r^,,s\rangle$, and $\langle r^2,rs\rangle$. The characters of these representations send the elements of the corresponding subgroup to 1, and the other elements to -1. We denote those characters rho1,rho2,rho3. Finally, there is a unique irreducible representation of dimension 2.

 i1 : R = QQ[x,y] o1 = R o1 : PolynomialRing i2 : D8 = { matrix{{x,y}}, matrix{{-x,-y}}, matrix{{-y,x}}, matrix{{x,-y}}, matrix{{y,x}} } o2 = {| x y |, | -x -y |, | -y x |, | x -y |, | y x |} o2 : List i3 : M = matrix {{1,1,1,1,1}, {1,1,1,-1,-1}, {1,1,-1,1,-1}, {1,1,-1,-1,1}, {2,-2,0,0,0}}; 5 5 o3 : Matrix ZZ <--- ZZ i4 : T = characterTable({1,1,2,2,2},M,R,{1,2,3,4,5}, Labels=>{"triv","rho1","rho2","rho3","dim2"}) o4 = Character table over R | 1 1 2 2 2 ------+------------------- triv | 1 1 1 1 1 rho1 | 1 1 1 -1 -1 rho2 | 1 1 -1 1 -1 rho3 | 1 1 -1 -1 1 dim2 | 2 -2 0 0 0 o4 : CharacterTable

The same labels are automatically used when decomposing characters against a labeled character table.

 i5 : A = action(R,D8) o5 = PolynomialRing with 5 actors o5 : ActionOnGradedModule i6 : c = character(A,0,8) o6 = Character over R (0, {0}) => | 1 1 1 1 1 | (0, {1}) => | 2 -2 0 0 0 | (0, {2}) => | 3 3 -1 1 1 | (0, {3}) => | 4 -4 0 0 0 | (0, {4}) => | 5 5 1 1 1 | (0, {5}) => | 6 -6 0 0 0 | (0, {6}) => | 7 7 -1 1 1 | (0, {7}) => | 8 -8 0 0 0 | (0, {8}) => | 9 9 1 1 1 | o6 : Character i7 : decomposeCharacter(c,T) o7 = Decomposition table | triv rho1 rho2 rho3 dim2 ----------+------------------------------ (0, {0}) | 1 0 0 0 0 (0, {1}) | 0 0 0 0 1 (0, {2}) | 1 0 1 1 0 (0, {3}) | 0 0 0 0 2 (0, {4}) | 2 1 1 1 0 (0, {5}) | 0 0 0 0 3 (0, {6}) | 2 1 2 2 0 (0, {7}) | 0 0 0 0 4 (0, {8}) | 3 2 2 2 0 o7 : CharacterDecomposition

The labels are stored in the character table under the key labels.