Bisymmetric functions, Macdonald polynomials and sl3 basic hypergeometric series
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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