Kostka-Foulkes polynomials and Macdonald spherical functions

Details Kostka-Foulkes polynomials and Macdonald spherical functions Kostka-Foulkes polynomials and Macdonald spherical functions

2004-01-22

arxiv.org/abs/math/0401298v1

Surveys in Combinatorics 2003 , C. Wensley ed., London Math. Soc. Lect. Notes 307, Cambridge University Press, 2003, 325--370.

Mathematics

Representation Theory

Scientific paper

Generalized Hall-Littlewood polynomials (Macdonald spherical functions) and generalized Kostka-Foulkes polynomials ($q$-weight multiplicities) arise in many places in combinatorics, representation theory, geometry, and mathematical physics. This paper attempts to organize the different definitions of these objects and prove the fundamental combinatorial results from ``scratch'', in a presentation which, hopefully, will be accessible and useful for both the nonexpert and researchers currently working in this very active field. The combinatorics of the affine Hecke algebra plays a central role. The final section of this paper can be read independently of the rest of the paper. It presents, with proof, Lascoux and Sch\"utzenberger's positive formula for the Kostka-Foulkes poynomials in the type A case.

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