The Coxeter transformation on Cominuscule Posets

TL;DR

This paper proves that the Coxeter transformation on the Grothendieck group of the derived category of incidence algebras of cominuscule posets has finite order, specifically related to the Coxeter number of the associated root system.

Contribution

It establishes the finite order of the Coxeter transformation for posets derived from cominuscule root systems, extending understanding of their algebraic and combinatorial properties.

Findings

The Coxeter transformation satisfies u^{h+1}=\u00b1 id.

Finite order of the Coxeter transformation is confirmed for all relevant cominuscule posets.

The order relates directly to the Coxeter number of the associated root system.

Abstract

Let $\mathsf{J(C)}$ be the poset of order ideals of a cominuscule poset $\mathsf{C}$ where $\mathsf{C}$ comes from two of the three infinite families of cominuscule posets or the exceptional cases. We show that the Auslander-Reiten translation $\tau$ on the Grothendieck group of the bounded derived category for the incidence algebra of the poset $\mathsf{J(C)}$, which is called the \emph{Coxeter transformation} in this context, has finite order. Specifically, we show that $\tau^{h+1}=\pm id$ where $h$ is the Coxeter number for the relevant root system.