Paths to Understanding Birational Rowmotion on Products of Two Chains

Gregg Musiker,Tom Roby

doi:10.5802/alco.43

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.

Highlights

The rowmotion map ρ, defined on the set J(P ) of order ideals of a poset P, has been thoroughly studied by a number of combinatorialists and representation theorists
When iterated on special posets, root posets andminuscule posets associated with representations of finite-dimensional
In this paper we give a formula in terms of families of non-intersecting lattice paths for iterated actions of the birational rowmotion map ρB on a product of two chains

Summary

Introduction

The rowmotion map ρ, defined on the set J(P ) of order ideals (equivalently on antichains) of a poset P , has been thoroughly studied by a number of combinatorialists and representation theorists. In this paper we give a formula in terms of families of non-intersecting lattice paths for iterated actions of the birational rowmotion map ρB on a product of two chains. This allows us to give a direct and significantly simpler proof that ρB is periodic, with the same period as ordinary (combinatorial) rowmotion (Corollary 2.12). Our methods yield the first proof of a birational homomesy result along files of our poset, namely that the product over all iterates of birational rowmotion over all elements of a given file is equal to 1 (Theorem 2.16).

Definitions and main result

Proof of Main Theorems

H F where

Findings

Connections to other works and future directions