Rhombic staircase tableaux and Koornwinder polynomials
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909206554083328 |
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| author | Corteel, Sylvie Mandelshtam, Olya Williams, Lauren |
| author_facet | Corteel, Sylvie Mandelshtam, Olya Williams, Lauren |
| contents | 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 $K_λ = K_λ(z_1,\dots,z_N; a,b,c,d; q,t)$, where $λ= (1,\dots,1,0,\dots,0)$. We also give combinatorial formulas for all ``open boundary ASEP polynomials'' $F_μ$, where $μ$ is a composition in $\{-1,0,1\}^N$; these polynomials are related to the nonsymmetric Koornwinder polynomials $E_μ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_17469 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rhombic staircase tableaux and Koornwinder polynomials Corteel, Sylvie Mandelshtam, Olya Williams, Lauren Combinatorics Mathematical Physics Probability 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 $K_λ = K_λ(z_1,\dots,z_N; a,b,c,d; q,t)$, where $λ= (1,\dots,1,0,\dots,0)$. We also give combinatorial formulas for all ``open boundary ASEP polynomials'' $F_μ$, where $μ$ is a composition in $\{-1,0,1\}^N$; these polynomials are related to the nonsymmetric Koornwinder polynomials $E_μ$ 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. |
| title | Rhombic staircase tableaux and Koornwinder polynomials |
| topic | Combinatorics Mathematical Physics Probability |
| url | https://arxiv.org/abs/2312.17469 |