STANDARD TABLEAUX AND KRONECKER PROJECTIONS OF SPECHT MODULES

Year 2011, Volume: 10 Issue: 10, 123 - 150, 01.12.2011

Jaydeep Chipalkatti Tagreed Mohammed

Abstract

Given a partition λ of a positive integer d, let Vλ denote the
corresponding irreducible rational representation of the symmetric group Sd.
When λ is a hook partition or a two-rowed partition, we explicitly describe the
equivariant morphism Vλ ⊗ Vλ −→ V(d) in terms of the standard tableau basis
of Vλ. We give similar descriptions for the morphism Vλ ⊗ Vλ′ −→ V(1d), as
well as for the projection morphisms onto the irreducible factors of the tensor
product V(d−1,1) ⊗ V(d−1,1). Our results can be interpreted as giving formulae
for certain Clebsch-Gordan coefficients for the symmetric group.

Keywords

Clebsch-Gordan coefficients, Kronecker product, Specht module