Macdonald polynomials and symmetric functions

TL;DR

This paper explores the structure of symmetric functions, introduces deformations of Schur functions including Macdonald polynomials, and proves a generating function identity for them, advancing understanding of their algebraic properties.

Contribution

It generalizes the basis of Schur functions using umbral calculus and proves a key generating function identity for Macdonald polynomials.

Findings

Deformation of Schur basis via umbral calculus.

Introduction of new bases with Littlewood--Richardson structure constants.

Proof of Kawanaka's generating function identity for Macdonald polynomials.

Abstract

The ring of symmetric functions $\Lambda$, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the symmetric group. One may define a coproduct on $\Lambda$ by the plethystic addition on alphabets. In this way the ring of symmetric functions becomes a Hopf algebra. The Littlewood--Richardson numbers may be viewed as the structure constants for the co-product in the Schur basis. In the first part of this thesis we show that by using a generalization of the classical umbral calculus of Gian-Carlo Rota, one may deform the basis of Schur functions to find many other bases for which the Littlewood--Richardson numbers as coproduct structure constants. The Macdonald polynomials are a somewhat mysterious qt-deformation of the Schur functions. The second part of this thesis contains a proof a generating function identity for the Macdonald polynomials which was originally conjectured by Kawanaka.