Highest weight vectors of irreducible representations of the quantum superalgebra U_q(gl(m,n))

TL;DR

This paper explores the interplay between the Iwahori-Hecke algebra and the quantum superalgebra U_q(gl(m,n)) on tensor spaces, constructing highest weight vectors for irreducible components using algebraic actions and symmetrizers.

Contribution

It demonstrates the commuting actions of the Hecke algebra and U_q(gl(m,n)) and constructs highest weight vectors for all irreducible summands in tensor spaces.

Findings

Actions of Hecke algebra and quantum superalgebra commute

Construction of highest weight vectors for irreducible modules

Application of Gyoja's q-analogue of Young symmetrizer

Abstract

The Iwahori-Hecke algebra of type A acts on tensor product space of the natural representation of the quantum superalgebra U_q(gl(m,n)). We show this action of the Hecke algebra and the action of U_q(gl(m,n)) on the same space determine commuting actions of each other. Together with this result and Gyoja's q-analogue of the Young symmetrizer, we construct a highest weight vector of each irreducible summmand of the tensor product space.