Haglund–Haiman–Loehr type formulas for Hall–Littlewood polynomials of type B and C

Lenart, Cristian

doi:10.2140/ant.2010.4.887

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Haglund–Haiman–Loehr type formulas for Hall–Littlewood polynomials of type $B$ and $C$

Cristian Lenart

Vol. 4 (2010), No. 7, 887–917

DOI: 10.2140/ant.2010.4.887

Abstract

In previous work we showed that two apparently unrelated formulas for the Hall–Littlewood polynomials of type $A$ are, in fact, closely related. The first is the tableau formula obtained by specializing $q = 0$ in the Haglund–Haiman–Loehr formula for Macdonald polynomials. The second is the type $A$ instance of Schwer’s formula (rephrased and rederived by Ram) for Hall–Littlewood polynomials of arbitrary finite type; Schwer’s formula is in terms of so-called alcove walks, which originate in the work of Gaussent and Littelmann and of the author with Postnikov on discrete counterparts to the Littelmann path model. We showed that the tableau formula follows by “compressing” Ram’s version of Schwer’s formula. In this paper, we derive new tableau formulas for the Hall–Littlewood polynomials of type $B$ and $C$ by compressing the corresponding instances of Schwer’s formula.

Keywords

Hall–Littlewood polynomials, Macdonald polynomials, alcove walks, Schwer's formula, the Haglund–Haiman–Loehr formula

Mathematical Subject Classification 2000

Primary: 05E05

Secondary: 33D52

Milestones

Received: 16 July 2009

Revised: 11 July 2010

Accepted: 13 October 2010

Published: 29 January 2011

Authors

Cristian Lenart
Department of Mathematics and Statistics State University of New York at Albany 1400 Washington Avenue Albany, NY 12222 United States
http://www.albany.edu/~lenart

Vol. 4, No. 7, 2010