# Wendy Baratta (2008-2010)

>## Special function aspects of Macdonald Polynomial theory

### Abstract:

The nonsymmetric Macdonald polynomials generalise the symmetric Macdonald polynomials and the nonsymmetric Jack polynomials. Consequently results obtained in nonsymmetric Macdonald polynomial theory specialise to analogous results in the theory of these other polynomials. The converse, however, does not always apply. Thus some properties of symmetric Macdonald, and nonsymmetric Jack, polynomials have no known nonsymmetric Macdonald polynomial analogues in the existing literature. Such examples are contained in the theory of Jack polynomials with prescribed symmetry and the study of Pieri-type formulas for the symmetric Macdonald polynomials. The study of Jack polynomials with prescribed symmetry considers a special class of Jack polynomials that are symmetric with respect to some variables and antisymmetric with respect to others. The Pieri-type formulas provide the branching coefficients for the expansion of the product of elementary symmetric function and a symmetric Macdonald polynomial. In this thesis we generalise both studies to the theory of nonsymmetric Macdonald polynomials.

In relation to the Macdonald polynomials with prescribed symmetry, we use our new results to prove a special case of a constant term conjecture posed in the late 1990s. A highlight of our study of Pieri-type formulas is the development of an alternative viewpoint to the existing literature on nonsymmetric interpolation Macdonald polynomials. This allows for greater extensions, and more simplified derivation, than otherwise would be possible. As a consequence of obtaining the Pieri-type formulas we are able to deduce explicit formulas for the related generalised binomial coefficients.

A feature of the various polynomials studied within this thesis is that they allow for explicit computation. Having this knowledge provides an opportunity to experimentally seek new properties, and to check our working in some of the analytic work. In the last chapter a computer software program that was developed for these purposes is detailed.

#### Published papers:

Pieri type formulas for nonsymmetric Macdonald polynomials, International Mathematics Research Notices **2009**, 2829-2854 (2009)

Some properties of Macdonald polynomials with prescribed symmetry, Kyushu Journal of Mathematics **64**, 323-343 (2010)

Further Pieri type formulas for nonsymmetric Macdonald polynomials, Journal of Algebraic Combinatorics **36**, 45-66 (2012)