Michael J. Schlosser

An algebra of elliptic commuting variables and an elliptic extension of the multinomial theorem

(16 pages)

Abstract. 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 A elliptic partial fraction decomposition. From the elliptic multinomial theorem we obtain, by convolution, an identity equivalent to Rosengren's type A extension of the Frenkel-Turaev ₁₀V₉ summation. Interpreted in terms of a weighted counting of lattice paths in the integer lattice ℤ, this derivation of Rosengren's A_r Frenkel-Turaev summation constitutes the first combinatorial proof of that fundamental identity.

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