Unitary representation theory of the lie superalgebra gl(m|n)

Abstract

This thesis is based on our published articles in which we investigate the representation theory of the Lie superalgebra gl(m|n). Specifically, we construct explicit formulae for the eigenvalues of certain invariants of the Lie superalgebra gl(m|n) using characteristic identity and shift operator methods that were developed by Gould, Jarvis and Green. From these invariants we may then give matrix element formulae for all gl(m|n) generators, including the non-elementary generators, on finite dimensional type 1 unitary irreducible representations. We compare our formulae with previous results, all of which only present matrix element formulae for elementary generators and only for a restricted class of type 1 unitary representations. Finally, we give matrix elements for dual (type 2 unitary) representations and investigate the associated type 2 unitary branching rules. A method of obtaining the phases of the matrix elements under the Baird and Biedenharn phase convention is also given for both type 1 and type 2 unitary cases.