Yang–Baxter equation, parameter permutations, and the elliptic beta integral

This paper presents a construction of an infinite-dimensional solution of the Yang–Baxter equation of rank 1 which is represented as an integral operator with an elliptic hypergeometric kernel acting in the space of functions of two complex variables. This $\mathrm{R}$-operator intertwines the product of two standard $\mathrm{L}$-operators associated with the Sklyanin algebra, an elliptic deformation of the algebra $\operatorname{sl}(2)$. The solution is constructed from three basic operators $\mathrm{S}_1$, $\mathrm{S}_2$, and $\mathrm{S}_3$ generating the permutation group $\mathfrak{S}_4$ on four parameters. Validity of the key Coxeter relations (including a star-triangle relation) is based on the formula for computing an elliptic beta integral and the Bailey lemma associated with an elliptic Fourier transformation. The operators $\mathrm{S}_j$ are determined uniquely with the help of the elliptic modular double.
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