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 (Contemp Math 138:189–204, 1992) as multivariate analogue of Askey–Wilson polynomials. They are now understood as the Macdonald polynomials associated with the affine root system of type $$(C^\vee _n,C_n)$$ 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 $$(C^\vee _n,C_n)$$ -analogue of Yip’s alcove-walk formulas (Math Z 272:1259–1290, 2012) which were given in the case of reduced affine root systems.