Probabilistic operators for non-attacking tableaux and a compact formula for the symmetric Macdonald polynomials
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arXiv
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| Format: | Preprint |
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2024
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| author | Mandelshtam, Olya |
| author_facet | Mandelshtam, Olya |
| contents | We prove a new tableaux formula for the symmetric Macdonald polynomials $P_λ(X;q,t)$ that has considerably fewer terms and simpler weights than previously existing formulas. Our formula is a sum over certain sorted non-attacking tableaux, weighted by the queue inversion statistic quinv. The quinv statistic originates from a formula for the modified Macdonald polynomials $\widetilde{H}_λ(X;q,t)$ due to Ayyer, Martin, and the author (2022), and is naturally related to the dynamics of the asymmetric simple exclusion process (ASEP) on a circle.
We prove our results by introducing probabilistic operators that act on non-attacking tableaux to generate a set of tableaux whose weighted sum equals $P_λ(X;q,t)$. These operators are a modification of the inversion flip operators of Loehr and Niese (2012), which yield an involution on tableaux that preserves the major index statistic, but fails to preserve the non-attacking condition. Our tableaux are in bijection with the multiline queues introduced by Martin (2020), allowing us to derive an alternative multiline queue formula for $P_λ(X;q,t)$. Finally, our formula recovers an alternative formula for the Jack polynomials $J_λ(X;α)$ due to Knop and Sahi (1996) using the same queue inversion statistic. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_17223 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Probabilistic operators for non-attacking tableaux and a compact formula for the symmetric Macdonald polynomials Mandelshtam, Olya Combinatorics We prove a new tableaux formula for the symmetric Macdonald polynomials $P_λ(X;q,t)$ that has considerably fewer terms and simpler weights than previously existing formulas. Our formula is a sum over certain sorted non-attacking tableaux, weighted by the queue inversion statistic quinv. The quinv statistic originates from a formula for the modified Macdonald polynomials $\widetilde{H}_λ(X;q,t)$ due to Ayyer, Martin, and the author (2022), and is naturally related to the dynamics of the asymmetric simple exclusion process (ASEP) on a circle. We prove our results by introducing probabilistic operators that act on non-attacking tableaux to generate a set of tableaux whose weighted sum equals $P_λ(X;q,t)$. These operators are a modification of the inversion flip operators of Loehr and Niese (2012), which yield an involution on tableaux that preserves the major index statistic, but fails to preserve the non-attacking condition. Our tableaux are in bijection with the multiline queues introduced by Martin (2020), allowing us to derive an alternative multiline queue formula for $P_λ(X;q,t)$. Finally, our formula recovers an alternative formula for the Jack polynomials $J_λ(X;α)$ due to Knop and Sahi (1996) using the same queue inversion statistic. |
| title | Probabilistic operators for non-attacking tableaux and a compact formula for the symmetric Macdonald polynomials |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2401.17223 |