An Algebra of Elliptic Commuting Variables and an Elliptic Extension of the Multinomial Theorem

Michael J. Schlosser

Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria

Аннотация: 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.

Ключевые слова: multinomial theorem, commutation relations, elliptic weights, elliptic hypergeometric series.

MSC: 05A10, 11B65, 33D67, 33D80, 33E90

Поступила: 26 декабря 2024 г.; в окончательном варианте 23 июня 2025 г.; опубликована 6 июля 2025 г.

Язык публикации: английский

DOI: 10.3842/SIGMA.2025.052

ArXiv: 2307.12921