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:
- a matrix, the matrix representation of the given permutation in the Specht module index by the given partition
- a hash table, 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 a method function.