Rhombic staircase tableaux and Koornwinder polynomials

Abstract

In this article we give a combinatorial formula for a certain class of Koornwinder polynomials, also known as Macdonald polynomials of type $\tilde{C}$. In particular, we give a combinatorial formula for the Koornwinder polynomials $K_{\hat{I}}»=K_{\hat{I}}»(z_1,\dots ,z_N;a,b,c,d;q,t)$, where $\hat{I}»=(1,\dots ,1,0,\dots ,0)$. We also give combinatorial formulas for all ``open boundary ASEP polynomials'' $F_{\hat{I}}¼$, where $\hat{I}¼$ is a composition in $\{-1,0,1{\}}^N$; these polynomials are related to the nonsymmetric Koornwinder polynomials $E_{\hat{I}}¼$ up to a triangular change of basis. Our formulas are in terms of rhombic staircase tableaux, certain tableaux that we introduced in previous work to give a formula for the stationary distribution of the two-species asymmetric simple exclusion process (ASEP) on a line with open boundaries.

27 pages, 5 figures, comments welcome, factorization property of Koornwinder polynomials at q=1 added, typos corrected