# Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials

Abstract : 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.
Keywords :
Document type :
Conference papers
Domain :

Cited literature [17 references]

https://hal.inria.fr/hal-01185387
Contributor : Coordination Episciences Iam <>
Submitted on : Thursday, August 20, 2015 - 11:07:05 AM
Last modification on : Tuesday, August 13, 2019 - 11:00:06 AM
Long-term archiving on: : Wednesday, April 26, 2017 - 10:00:45 AM

### File

dmAK0146.pdf
Publisher files allowed on an open archive

### Identifiers

• HAL Id : hal-01185387, version 1

### Citation

Cristian Lenart. Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. pp.549-560. ⟨hal-01185387⟩

Record views