Quadrirational Yang-Baxter maps and the elliptic Cremona system

TL;DR

This paper explores the relationship between quadrirational Yang-Baxter maps and the elliptic Cremona system, revealing a novel connection between two classes of integrable systems via birational Coxeter groups.

Contribution

It establishes a new link between two-dimensional Yang-Baxter maps and higher-dimensional elliptic systems using Coxeter group formulations.

Findings

Identifies a natural connection between integrable systems of different dimensions.

Formulates both systems in terms of birational Coxeter groups.

Provides a framework for understanding integrable systems beyond traditional reductions.

Abstract

This paper connects the quadrirational Yang-Baxter maps, which are two-dimensional integrable discrete systems of KdV type, and the elliptic Cremona system, which is a higher analogue of discrete Painlev\'e equations associated with $\tilde{E}_8$ symmetry. This is a natural connection between integrable systems in different dimensions that is outside of the usual paradigm of reductions. Our approach is based on formulation of both systems in terms of birational Coxeter groups.