Young

Introduction

Young is a GAP system for finding characteristic zero representations of the symmetric groups. These representations are indexed by partitions.

Example

The group S₁₀ has two irreducible 315-dimensional representations, indexed by the partitions [6,3,1] and its dual [3,2²,1³].

gap> DualPartition([6,3,1]);
[ 3, 2, 2, 1, 1, 1 ]
gap> HookLengthFormula([6,3,1]);
315
gap>

We can find the representation [6,3,1] on standard generators of S₁₀ in two ways:

gap> R1 := YoungRepresentation([6,3,1], [(1,2), (2,3,4,5,6,7,8,9,10)]);;
gap> R2 := YoungRepresentationByStandardGeneratorsSn([6,3,1]);;
gap> R1 = R2;
true

The matrices can be found by looking at the entries of the list:

gap> a := R1[1];;
gap> b := R1[2];;
gap> Size(a);
315
gap> Order(a);
2
gap> Order(b);
9
gap> Order(a*b);
10
gap> Trace(a);
91
gap> Trace(b);
0
gap> Trace(a*b);
0

Source code

• young.g: GAP source code

Last updated 27th September 2005