Generalized $q$-Painlevé VI systems of type $(A_{2n+1}+A_1+A_1{)}^{(1)}$ arising from cluster algebra

Abstract

In this article we formulate a group of birational transformations which is isomorphic to an extended affine Weyl group of type $(A_{2n+1}+A_1+A_1{)}^{(1)}$ with the aid of mutations and permutations of vertices to a mutation-periodic quiver on a torus. This group provides a class of higher order generalizations of Jimbo-Sakai's $q$-Painlevé VI equation as translations on a root lattice. Then the known three systems are obtained again; the $q$-Garnier system, a similarity reduction of the lattice $q$-UC hierarchy and a similarity reduction of the $q$-Drinfeld-Sokolov hierarchy.

arXiv comment34 pages

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Cited by in corpus (3)

• Cluster realizations of Weyl groups and higher Teichmüller theory
• Cluster integrable systems and spin chains
• Two variations on $(A_3\times A_1\times A_1{)}^{(1)}$ type discrete Painlevé equations