On the interaction of the Coxeter transformation and the rowmotion bijection

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 $({\rho }^{-1}C{)}^2=\mathrm{id}$, 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 $(R^{-1}C{)}^2=\mathrm{id}$ if $n$ is even and $(R^{-1}C+\mathrm{id}{)}^2=0$ when $n$ is odd.