Bisymmetric functions, Macdonald polynomials and sl_3 basic hypergeometric series

Details Bisymmetric functions, Macdonald polynomials and sl_3 basic hypergeometric series Bisymmetric functions, Macdonald polynomials and sl_3 basic hypergeometric series

2005-11-14

arxiv.org/abs/math/0511333v2

Compositio Math. 144 (2008), 271-303

Mathematics

Combinatorics

33 pages; major revision of earlier, much longer paper; to appear in Compositio Mathematica

Scientific paper

A new type of 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 sl_3 basic hypergeometric series is a bisymmetric function related to Macdonald's commuting family of q-difference operators, to the sl_3 Selberg integrals of Tarasov and Varchenko, and to alternating sign matrices. Our main result for sl_3 series is a multivariable generalization of the celebrated q-binomial theorem. In the limit this q-binomial sum yields a new sl_3 Selberg integral for Jack polynomials.

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