The character table for two characters $X$ and $Y$ of $G$ is calculated using the formula $<X,Y> = \sum_{g \in G} X(g)Y(g) = \sum_{C \in Cl(G)} |C|X(g_C)Y(g_C) $ where the second sum is taken over all conjugacy classes of $G$ and $g_c$ is an element in the conjugacy class.
As an example we calculate the inner product between the character of the regular representation of $S_4$ 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.