Paths to Understanding Birational Rowmotion on Products of Two Chains

TL;DR

This paper provides a new formula for birational rowmotion on products of two chains, simplifying proofs of its periodicity and homomesy properties, and connecting it to lattice paths and $Y$-systems.

Contribution

It introduces a lattice path formula for iterated birational rowmotion on product of chains, enabling simpler proofs of periodicity and homomesy.

Findings

Period of birational rowmotion on product of chains is r+s+2.

Established a lattice path formula for iterated birational rowmotion.

Proved birational analogue of homomesy along files for such posets.

Abstract

Birational rowmotion is an action on the space of assignments of rational functions to the elements of a finite partially-ordered set (poset). It is lifted from the well-studied rowmotion map on order ideals (equivariantly on antichains) of a poset $P$, which when iterated on special posets, has unexpectedly nice properties in terms of periodicity, cyclic sieving, and homomesy (statistics whose averages over each orbit are constant) [AST11, BW74, CF95, Pan09, PR13, RuSh12,RuWa15+,SW12, ThWi17, Yil17. In this context, rowmotion appears to be related to Auslander-Reiten translation on certain quivers, and birational rowmotion to $Y$-systems of type $A_m \times A_n$ described in Zamolodchikov periodicity. We give a formula in terms of families of non-intersecting lattice paths for iterated actions of the birational rowmotion map on a product of two chains. This allows us to give a much simpler direct proof of the key fact that the period of this map on a product of chains of lengths $r$ and $s$ is $r+s+2$ (first proved by D.~Grinberg and the second author), as well as the first proof of the birational analogue of homomesy along files for such posets.