$qt$RSK${}^*$: A probabilistic dual RSK correspondence for Macdonald polynomials
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916174797733888 |
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| author | Frieden, Gabriel Schreier-Aigner, Florian |
| author_facet | Frieden, Gabriel Schreier-Aigner, Florian |
| contents | We introduce a probabilistic generalization of the dual Robinson--Schensted--Knuth correspondence, called $qt$RSK${}^*$, depending on two parameters $q$ and $t$. This correspondence extends the $q$RS$t$ correspondence, recently introduced by the authors, and allows the first tableaux-theoretic proof of the dual Cauchy identity for Macdonald polynomials. By specializing $q$ and $t$, one recovers the row and column insertion version of the classical dual RSK correspondence as well as of $q$- and $t$-deformations thereof which are connected to $q$-Whittaker and Hall--Littlewood polynomials. When restricting to Jack polynomials and $\{0,1\}$-matrices corresponding to words, we prove that the insertion tableaux obtained by $qt$RSK${}^*$ are invariant under swapping letters in the input word. Our approach is based on Fomin's growth diagrams and the notion of probabilistic bijections. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_16243 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $qt$RSK${}^*$: A probabilistic dual RSK correspondence for Macdonald polynomials Frieden, Gabriel Schreier-Aigner, Florian Combinatorics Probability 05E05, 60C05, 33C52 We introduce a probabilistic generalization of the dual Robinson--Schensted--Knuth correspondence, called $qt$RSK${}^*$, depending on two parameters $q$ and $t$. This correspondence extends the $q$RS$t$ correspondence, recently introduced by the authors, and allows the first tableaux-theoretic proof of the dual Cauchy identity for Macdonald polynomials. By specializing $q$ and $t$, one recovers the row and column insertion version of the classical dual RSK correspondence as well as of $q$- and $t$-deformations thereof which are connected to $q$-Whittaker and Hall--Littlewood polynomials. When restricting to Jack polynomials and $\{0,1\}$-matrices corresponding to words, we prove that the insertion tableaux obtained by $qt$RSK${}^*$ are invariant under swapping letters in the input word. Our approach is based on Fomin's growth diagrams and the notion of probabilistic bijections. |
| title | $qt$RSK${}^*$: A probabilistic dual RSK correspondence for Macdonald polynomials |
| topic | Combinatorics Probability 05E05, 60C05, 33C52 |
| url | https://arxiv.org/abs/2403.16243 |