Haglund-Haiman-Loehr type formulas for Hall-Littlewood polynomials of type B and C. (English) Zbl 1229.05274

Lenart, Cristian

Algebra Number Theory 4, No. 7, 887-917 (2010).

Summary: In previous work [C. Lenart, Discrete Math. 311, No. 4, 258–275 (2011; Zbl 1208.17012)] 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 [C. Schwer, Int. Math. Res. Not. 2006, No. 21, Article ID 75395, 31 p. (2006; Zbl 1121.05121)] (rephrased and rederived by A. Ram [Pure Appl. Math. Q. 2, No. 4, 963–1013 (2006; Zbl 1127.20005)]) 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.

MSC:

05E05	Symmetric functions and generalizations
33D52	Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)

Keywords:

Hall-Littlewood polynomials; MacDonald polynomials; Alcove walks; Schwer’s formula; the Haglund-Haiman-Loehr formula

Citations:

Zbl 0805.17019; Zbl 0858.17023; Zbl 1211.17021; Zbl 1137.14037; Zbl 1208.17012; Zbl 1121.05121; Zbl 1127.20005