Equivariant Projection Morphisms of Specht Modules

Abstract

This thesis is devoted to a problem in the representation theory of the symmetric group over C (the field of the complex numbers). Let d be a positive integer, and let S_d denote the symmetric group on d letters. Given a partition k of d, the Specht module V_k is a finite dimensional vector space over C which admits a natural basis indexed by all standard tableaux of shape k with entries in {1, 2, ..., d}. It affords an irreducible representation of the symmetric group S_d, and conversely every irreducible representation of S_d is isomorphic to V_k for some partition k. Given two Specht modules V_k, V_t their tensor product representation is in general reducible, and hence it splits into a direct sum of irreducibles. This raises the problem of describing the S_d equivariant projection morphisms (alternately called S_d-homomorphisms) in terms of the standard tableaux basis. In this work we give explicit formulae describing this morphism in the following cases: k=(d-1, 1), (d-2, 1,1), (2, 1,... ,1). Finally, we present a conjecture formula for the q-morphism in the case k=(d-r, 1, ..., 1).