### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 121, 13 pages      arXiv:1810.10806      https://doi.org/10.3842/SIGMA.2018.121
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

### Matrix Bailey Lemma and the Star-Triangle Relation

Kamil Yu. Magadov a and Vyacheslav P. Spiridonov bc
a) Deceased; Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, Russia
b) Laboratory of Theoretical Physics, JINR, Dubna, Moscow Region, 141980 Russia
c) National Research University Higher School of Economics, Moscow, Russia

Received August 10, 2018, in final form October 30, 2018; Published online November 10, 2018

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.

Key words: elliptic hypergeometric functions; Bailey lemma; star-triangle relation.

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