Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials of Types A and C. Extended Abstract

Details Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials of Types A and C. Extended Abstract Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials of Types A and C. Extended Abstract

2008-11-25

arxiv.org/abs/0811.4152v1

Mathematics

Combinatorics

Scientific paper

A breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of fillings of Young diagrams. Recently, Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of the corresponding affine Weyl group. In this paper, we show that a Haglund-Haiman-Loehr type formula follows naturally from the more general Ram-Yip formula, via compression. Then we extend this approach to the Hall-Littlewood polynomials of type C, which are specializations of the corresponding Macdonald polynomials at q=0. We note that no analog of the Haglund-Haiman-Loehr formula exists beyond type A, so our work is a first step towards finding such a formula.

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