Bisymmetric functions, Macdonald polynomials and basic hypergeometric series

Abstract

AbstractA 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.