# matrixRepresentation -- the matrix representation of a permutation in the Specht Module

## Synopsis

• Usage:
matrixRepresentation(perm,standard)
matrixRepresentation(perm,parti)
matrixRepresentation(standard)
matrixRepresentation(parti)
• Inputs:
• perm, a list, a permutation
• standard, an instance of the type TableauList, a list of standard tableaux of a given partition
• parti, an instance of the type Partition, a partition
• Outputs:
• , the matrix representation of the given permutation in the Specht module index by the given partition
• , if no permutation is given then it calculates the representation for all the permutations in S_n

## Description

The matrix representation for a permutation is calculated by studying the action of the permutation on the basis of standard polytabloids.

The permuted polytabloids are then written as a linear combination of standard polytabloids using the straightening algorithm.

 i1 : p = new Partition from {2,1} o1 = Partition{2, 1} o1 : Partition i2 : l = {0,2,1} o2 = {0, 2, 1} o2 : List i3 : matrixRepresentation (l,p) o3 = | 0 1 | | 1 0 | 2 2 o3 : Matrix QQ <--- QQ i4 : stan = standardTableaux p o4 = {| 0 1 |, | 0 2 |} | 2 | | 1 | o4 : TableauList i5 : matrixRepresentation (l,stan) o5 = | 0 1 | | 1 0 | 2 2 o5 : Matrix QQ <--- QQ i6 : matrixRepresentation stan o6 = HashTable{{0, 1, 2} => | 1 0 | } | 0 1 | {0, 2, 1} => | 0 1 | | 1 0 | {1, 0, 2} => | 1 0 | | -1 -1 | {1, 2, 0} => | 0 1 | | -1 -1 | {2, 0, 1} => | -1 -1 | | 1 0 | {2, 1, 0} => | -1 -1 | | 0 1 | o6 : HashTable

## Ways to use matrixRepresentation :

• "matrixRepresentation(List,Partition)"
• "matrixRepresentation(List,TableauList)"
• "matrixRepresentation(Partition)"
• "matrixRepresentation(TableauList)"

## For the programmer

The object matrixRepresentation is .