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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2307.12921 |
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| _version_ | 1866913957257674752 |
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| author | Schlosser, Michael J. |
| author_facet | Schlosser, Michael J. |
| contents | We introduce an algebra of elliptic commuting variables involving a base $q$, nome $p$, and $2r$ noncommuting variables. This algebra, which for $r=1$ reduces to an algebra considered earlier by the author, is an elliptic extension of the well-known algebra of $r$ $q$-commuting variables. We present a multinomial theorem valid as an identity in this algebra, hereby extending the author's previously obtained elliptic binomial theorem to higher rank. Two essential ingredients are a consistency relation satisfied by the elliptic weights and the Weierstrass type $\mathsf A$ elliptic partial fraction decomposition. From the elliptic multinomial theorem we obtain, by convolution, an identity equivalent to Rosengren's type $\mathsf A$ extension of the Frenkel-Turaev ${}_{10}V_9$ summation. Interpreted in terms of a weighted counting of lattice paths in the integer lattice $\mathbb Z^r$, this derivation of Rosengren's $\mathsf A_r$ Frenkel-Turaev summation constitutes the first combinatorial proof of that fundamental identity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_12921 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An Algebra of Elliptic Commuting Variables and an Elliptic Extension of the Multinomial Theorem Schlosser, Michael J. Quantum Algebra Combinatorics Primary 05A10, Secondary 11B65, 33D67, 33D80, 33E90 We introduce an algebra of elliptic commuting variables involving a base $q$, nome $p$, and $2r$ noncommuting variables. This algebra, which for $r=1$ reduces to an algebra considered earlier by the author, is an elliptic extension of the well-known algebra of $r$ $q$-commuting variables. We present a multinomial theorem valid as an identity in this algebra, hereby extending the author's previously obtained elliptic binomial theorem to higher rank. Two essential ingredients are a consistency relation satisfied by the elliptic weights and the Weierstrass type $\mathsf A$ elliptic partial fraction decomposition. From the elliptic multinomial theorem we obtain, by convolution, an identity equivalent to Rosengren's type $\mathsf A$ extension of the Frenkel-Turaev ${}_{10}V_9$ summation. Interpreted in terms of a weighted counting of lattice paths in the integer lattice $\mathbb Z^r$, this derivation of Rosengren's $\mathsf A_r$ Frenkel-Turaev summation constitutes the first combinatorial proof of that fundamental identity. |
| title | An Algebra of Elliptic Commuting Variables and an Elliptic Extension of the Multinomial Theorem |
| topic | Quantum Algebra Combinatorics Primary 05A10, Secondary 11B65, 33D67, 33D80, 33E90 |
| url | https://arxiv.org/abs/2307.12921 |