Rowmotion in slow motion

TL;DR

This paper generalizes the concept of rowmotion using trim lattices, introduces a new flag simplicial complex, and connects these ideas to representation theory, revealing structural properties of torsion classes in certain algebras.

Contribution

It extends rowmotion to trim lattices, introduces a related flag simplicial complex, and links these structures to representation theory and semidistributive lattices.

Findings

Extends rowmotion to trim lattices.

Introduces a flag simplicial complex related to semidistributive lattices.

Shows torsion classes form a trim lattice in certain algebras.

Abstract

Rowmotion is a simple cyclic action on the distributive lattice of order ideals of a poset: it sends the order ideal x to the order ideal generated by the minimal elements not in x. It can also be computed in "slow motion" as a sequence of local moves. We use the setting of trim lattices to generalize both definitions of rowmotion, proving many structural results along the way. We introduce a flag simplicial complex (similar to the canonical join complex of a semidistributive lattice), and relate our results to recent work of Barnard by proving that extremal semidistributive lattices are trim. As a corollary, we prove that if A is a representation finite algebra and mod A has no cycles, then the torsion classes of A ordered by inclusion form a trim lattice.