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arXiv · 2312.17469

Rhombic staircase tableaux and Koornwinder polynomials

Sylvie Corteel·Olya Mandelshtam·Lauren Williams
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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 $K_{\lambda} = K_{\lambda}(z_1,\dots,z_N; a,b,c,d; q,t)$, where $\lambda = (1,\dots,1,0,\dots,0)$. We also give combinatorial formulas for all ``open boundary ASEP polynomials'' $F_{\mu}$, where $\mu$ is a composition in $\{-1,0,1\}^N$; these polynomials are related to the nonsymmetric Koornwinder polynomials $E_{\mu}$ 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.

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Sylvie Corteel, Olya Mandelshtam, Lauren Williams. 2023-12-29. Rhombic staircase tableaux and Koornwinder polynomials. https://arxiv.org/abs/2312.17469

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