A degeneration of the $q$-Garnier system of fourth order arises from confluences in quivers

TL;DR

This paper explores how a fourth-order $q$-Garnier system degenerates through confluences in quivers, linking cluster algebra structures to system reductions.

Contribution

It introduces a degeneration framework for the $q$-Garnier system based on quiver confluences, connecting cluster algebra and integrable systems.

Findings

Degeneration structure of the $q$-Garnier system elucidated.

Connection established between quiver confluences and system reductions.

Framework for analyzing degenerations in discrete integrable systems.

Abstract

The $q$-Garnier system was first proposed by Sakai and its other directions of discrete time evolutions were given by Nagao and Yamada. Recently, it was shown that all of those directions of discrete time evolutions are derived from a birational representation of an extended affine Weyl group which arises from the cluster algebraic construction established by Masuda, Okubo and Tsuda. In this article, we investigate a degeneration structure of the $q$-Garnier system of fourth order by using confluences in quivers.