Birational Rowmotion and the Octahedron Recurrence

TL;DR

This paper simplifies the understanding of birational rowmotion on product of two chains using the octahedron recurrence, revealing predictable weight shifts and introducing generalized Stanley-Thomas words to characterize the dynamics.

Contribution

It provides a simplified proof of birational rowmotion formula, introduces generalized Stanley-Thomas words, and explores their relation to RSK and Greene's theorem.

Findings

Weights of chains in rectangles shift predictably under rowmotion

Generalized Stanley-Thomas words determine birational rowmotion uniquely

Established a birational analogue of Greene's theorem

Abstract

We use the octahedron recurrence to give a simplified statement and proof of a formula for iterated birational rowmotion on a product of two chains, first described by Musiker and Roby. Using this, we show that weights of certain chains in rectangles shift in a predictable way under the action of rowmotion. We then define generalized Stanley-Thomas words whose cyclic rotation uniquely determines birational rowmotion on the product of two chains. We also discuss the relationship between rowmotion and birational RSK and give a birational analogue of Greene's theorem in this setting.