On the interaction of the Coxeter transformation and the rowmotion bijection

TL;DR

This paper proves a conjecture relating rowmotion and Coxeter matrices in finite posets, and explores their interaction in more general algebraic settings, revealing different behaviors depending on algebraic properties.

Contribution

It establishes the identity $( ho^{-1} C)^2=id$ for finite posets and extends the analysis to higher Auslander algebras, showing new algebraic relations based on parity of $n$.

Findings

Proves the conjecture $( ho^{-1} C)^2=id$ for finite posets.

Shows $(R^{-1} C)^2=id$ for even $n$ in higher Auslander algebras.

Demonstrates $(R^{-1}C+id)^2=0$ for odd $n$ in higher Auslander algebras.

Abstract

Let $P$ be a finite poset and $L$ the associated distributive lattice of order ideals of $P$. Let $\rho$ denote the rowmotion bijection of the order ideals of $P$ viewed as a permutation matrix and $C$ the Coxeter matrix for the incidence algebra $kL$ of $L$. Then we show the identity $(\rho^{-1} C)^2=id$, as was originally conjectured by Sam Hopkins. Recently it was noted that the rowmotion bijection is a special case of the much more general grade bijection $R$ that exists for any Auslander regular algebra. This motivates to study the interaction of the grade bijection and the Coxeter matrix for general Auslander regular algebras. For the class of higher Auslander algebras coming from $n$-representation finite algebras we show that $(R^{-1} C)^2=id$ if $n$ is even and $(R^{-1}C+id)^2=0$ when $n$ is odd.