Rowmotion and Echelonmotion

TL;DR

This paper introduces echelonmotion, a permutation-based bijection related to rowmotion, extending its properties from distributive to semidistributive lattices and exploring its behavior on Eulerian and echelon-independent posets.

Contribution

It generalizes the equivalence of echelonmotion and rowmotion to semidistributive lattices and characterizes echelon-independent posets as exactly the semidistributive ones.

Findings

Echelonmotion agrees with rowmotion on semidistributive lattices.

Echelonmotion is an involution on Eulerian posets.

Echelon-independent posets are precisely the semidistributive lattices.

Abstract

Given a linear extension $\sigma$ of a finite poset $R$, we consider the permutation matrix indexing the Schubert cell containing the Cartan matrix of $R$ with respect to $\sigma$. This yields a bijection $\mathrm{Ech}_\sigma\colon R\to R$ that we call echelonmotion; it is the inverse of the Coxeter permutation studied by Kl\'asz, Marczinzik, and Thomas. Those authors proved that echelonmotion agrees with rowmotion when $R$ is a distributive lattice. We generalize this result to semidistributive lattices. In addition, we prove that every trim lattice has a linear extension with respect to which echelonmotion agrees with rowmotion. We also show that echelonmotion on an Eulerian poset (with respect to any linear extension) is an involution. Finally, we initiate the study of echelon-independent posets, which are posets for which echelonmotion is independent of the chosen linear extension. We prove that a lattice is echelon-independent if and only if it is semidistributive. Moreover, we show that echelon-independent connected posets are bounded and have semidistributive MacNeille completions.