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Bisymmetric functions, Macdonald polynomials and $\mathfrak {sl}_3$ basic hypergeometric series

Part of: Algebraic combinatorics Basic hypergeometric functions

Published online by Cambridge University Press:  01 March 2008

S. Ole Warnaar
S. Ole Warnaar*
Affiliation:
Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia (email: warnaar@ms.unimelb.edu.au)
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Abstract

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A new type of $\mathfrak {sl}_3$ basic hypergeometric series based on Macdonald polynomials is introduced. Besides a pair of Macdonald polynomials attached to two different sets of variables, a key ingredient in the $\mathfrak {sl}_3$ basic hypergeometric series is a bisymmetric function related to Macdonald’s commuting family of q-difference operators, to the $\mathfrak {sl}_3$ Selberg integrals of Tarasov and Varchenko, and to alternating sign matrices. Our main result for $\mathfrak {sl}_3$ series is a multivariable generalization of the celebrated q-binomial theorem. In the limit this q-binomial sum yields a new $\mathfrak {sl}_3$ Selberg integral for Jack polynomials.

Keywords

basic hypergeometric seriesMacdonald polynomialssl3 Selberg integrals

MSC classification

Secondary: 05E05: Symmetric functions and generalizations 33D67: Basic hypergeometric functions associated with root systems

Information

Type
Research Article
Information
Compositio Mathematica , Volume 144 , Issue 2 , March 2008 , pp. 271 - 303
Copyright
Copyright © Foundation Compositio Mathematica 2008

References

Work supported by the Australian Research Council.