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 mutable matrix, a matrix row that represents a character of S_n
- Y, a mutable matrix, 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.

Ways to use this method:

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