Symmetric functions and infinite dimensional algebras

Abstract

Infinite-dimensional algebras and symmetric functions arise in many diverse areas of mathematics and physics. In this thesisseveral problems in these two areas are studied. We investigate the concept of replicated and q-replicated arguments in Schur and Hall-Littlewood symmetric functions. A description of "dual" compound symmetric functions is obtained with the help of functions of a replicated argument, while Schur and Hall-Littlewood functions of a q-replicated argument are both shown to be related to Macdonald's symmetric functions. Various tensor product decompositions and winding subalgebra branching rules for the N = 1 and N = 2 superconformal algebras are examined by using the triple and quintuple product identities and various generalizations thereof, concentrating on the particular cases when these decompositions are finite or multiplicity free. The boson-fermion correspondence is utilized to develop an algorithm for the calculation of outer products of Schur and Q-functions with power sum symmetric functions, and general (outer) multiplication of S-functions. A procedure is also developed for the evaluation of (outer) plethysms of Schur functions and power sums. A few examples are given which demonstrate the usefulness of this method for calculating plethysms between Schur functions. By examining the vertex operator realization of Hall-Littlewood functions we are also able to generate an algorithm for expressing Hall-Littlewood functions in terms of Schur functions. The operation of outer plethysm is defined for Hall-Littlewood functions and the algorithm developed for S-functions is extended to this case as well.