Quantum $\mathfrak{gl}$-weight system and its average values
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911018665377792 |
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| author | Zaitsev, Mikhail |
| author_facet | Zaitsev, Mikhail |
| contents | We present a proof of a recent conjecture due to M. Kazarian, E. Krasilnikov, S. Lando, and M. Shapiro, which describes the average value of the universal $\mathfrak{gl}$-weight system on permutations. The proof uses a quantum analogue of the $\mathfrak{gl}$-weight system on Hecke algebras of type $A$, which leads to a one-parameter deformation of the average value of the universal ${\mathfrak{gl}}$-weight system. We show that the average value of the quantum weight system is a linear combination of one-part Schur functions, with coefficients being $q$-analogues of Bernoulli polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_17706 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum $\mathfrak{gl}$-weight system and its average values Zaitsev, Mikhail Combinatorics Quantum Algebra 05E05, 20C08, 17B37 We present a proof of a recent conjecture due to M. Kazarian, E. Krasilnikov, S. Lando, and M. Shapiro, which describes the average value of the universal $\mathfrak{gl}$-weight system on permutations. The proof uses a quantum analogue of the $\mathfrak{gl}$-weight system on Hecke algebras of type $A$, which leads to a one-parameter deformation of the average value of the universal ${\mathfrak{gl}}$-weight system. We show that the average value of the quantum weight system is a linear combination of one-part Schur functions, with coefficients being $q$-analogues of Bernoulli polynomials. |
| title | Quantum $\mathfrak{gl}$-weight system and its average values |
| topic | Combinatorics Quantum Algebra 05E05, 20C08, 17B37 |
| url | https://arxiv.org/abs/2506.17706 |