Matrix Bailey Lemma and the Star-Triangle Relation

Abstract

We compare previously found finite-dimensional matrix and integral operator realizations of the Bailey lemma employing univariate elliptic hypergeometric functions. With the help of residue calculus we explicitly show how the integral Bailey lemma can be reduced to its matrix version. As a consequence, we demonstrate that the matrix Bailey lemma can be interpreted as a star-triangle relation, or as a Coxeter relation for a permutation group.

Bailey pairs for the q-hypergeometric integral pentagon identity

“…A generalization of the Bailey pairs approach to the integral identities is firstly done by Spiridinov in [18,19]. The construction of integral Bailey pairs yields new powerful verifications of various supersymmetric dualities [20,21], generating solutions to Yang-Baxter equations [22][23][24][25], etc.…”