A Littlewood-Richardson rule for Koornwinder polynomials

Abstract

Koornwinder polynomials are $q$-orthogonal polynomials equipped with extra five parameters and the $B{C}_n$-type Weyl group symmetry, which were introduced by Koornwinder (1992) as multivariate analogue of Askey-Wilson polynomials. They are now understood as the Macdonald polynomials associated to the affine root system of type $(C_n^{\vee },C_n)$ via the Macdonald-Cherednik theory of double affine Hecke algebras. In this paper we give explicit formulas of Littlewood-Richardson coefficients for Koornwinder polynomials, i.e., the structure constants of the product as invariant polynomials. Our formulas are natural $(C_n^{\vee },C_n)$-analogue of Yip's alcove-walk formulas (2012) which were given in the case of reduced affine root systems.

32 pages