Bisymmetric functions, Macdonald polynomials and sl₃ basic hypergeometric series

2005/11/30 by S. Ole Warnaar · 1 citation
Mathematics · #math.CO #math.QA #msc:05E05 #msc:33D67

paper · pdf

published as Compositio Math. 144 (2008), 271-303 · 33 pages; major revision of earlier, much longer paper; to appear in Compositio Mathematica

arxiv created 2007/07/24 · arxiv updated 2009/12/01

Abstract

A new type of sl₃ 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 sl₃ basic hypergeometric series is a bisymmetric function related to Macdonald's commuting family of q-difference operators, to the sl₃ Selberg integrals of Tarasov and Varchenko, and to alternating sign matrices. Our main result for sl₃ series is a multivariable generalization of the celebrated q-binomial theorem. In the limit this q-binomial sum yields a new sl₃ Selberg integral for Jack polynomials.

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