$R$-systems

TL;DR

This paper introduces $R$-systems, a generalization of birational rowmotion on directed graphs, exploring their integrability, singularity confinement, and connections to Laurent phenomena, Somos, and Gale-Robinson sequences.

Contribution

It generalizes birational rowmotion to arbitrary strongly connected directed graphs and investigates its integrability and Laurent properties, revealing new sequences and symmetries.

Findings

Many $R$-systems exhibit singularity confinement.

$R$-systems often demonstrate the Laurent property.

Some $R$-systems reduce to known sequences like Somos and Gale-Robinson.

Abstract

Birational toggling on Gelfand-Tsetlin patterns appeared first in the study of geometric crystals and geometric Robinson-Schensted-Knuth correspondence. Based on these birational toggle relations, Einstein and Propp introduced a discrete dynamical system called birational rowmotion associated with a partially ordered set. We generalize birational rowmotion to the class of arbitrary strongly connected directed graphs, calling the resulting discrete dynamical system the $R$-system. We study its integrability from the points of view of singularity confinement and algebraic entropy. We show that in many cases, singularity confinement in an $R$-system reduces to the Laurent phenomenon either in a cluster algebra, or in a Laurent phenomenon algebra, or beyond both of those generalities, giving rise to many new sequences with the Laurent property possessing rich groups of symmetries. Some special cases of $R$-systems reduce to Somos and Gale-Robinson sequences.