Matrix Bailey Lemma and the Star-Triangle Relation

TL;DR

This paper demonstrates how the integral Bailey lemma can be reduced to a matrix form using residue calculus, revealing its interpretation as a star-triangle relation and a Coxeter relation for permutation groups.

Contribution

It explicitly connects matrix and integral Bailey lemmas, showing the matrix version as a star-triangle relation and Coxeter relation, unifying different mathematical frameworks.

Findings

Integral Bailey lemma reduces to matrix form via residue calculus.

Matrix Bailey lemma interpreted as a star-triangle relation.

Establishes connection to Coxeter relations for permutation groups.

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.