Vol. 4, No. 7, 2010

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Haglund–Haiman–Loehr type formulas for Hall–Littlewood polynomials of type $B$ and $C$

Cristian Lenart

Vol. 4 (2010), No. 7, 887–917
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