The character table for two characters X and Y of G is calculated using the formula <X,Y> = ∑g ∈G X(g)Y(g) = ∑C ∈Cl(G) |C|X(gC)Y(gC) where the second sum is taken over all conjugacy classes of G and gc is an element in the conjugacy class.
As an example we calculate the inner product between the character of the regular representation of S4 and the character indexed by partition 2,1,1.
i1 : n = 4 o1 = 4 |
i2 : X = mutableMatrix {{0,0,0,0,24}} o2 = | 0 0 0 0 24 | o2 : MutableMatrix |
i3 : Y = mutableMatrix {{1,0,-1,-1,3}} o3 = | 1 0 -1 -1 3 | o3 : MutableMatrix |
i4 : innerProduct(4,X,Y) o4 = 3 |
As expected this inner product is equal to 3.