# innerProduct(ZZ,MutableMatrix,MutableMatrix) -- calculates the inner product for the characters of S_n

## Synopsis

• Function: innerProduct
• Usage:
innerProduct(n,X,Y)
• Inputs:
• n, an integer, the degree of the symmetric group
• X, , a matrix row that represents a character of S_n
• Y, , a matrix row that represents a character of S_n
• Outputs:
• an integer, the inner product of the two characters X and Y

## Description

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.