The Coxeter transformation on Cominuscule Posets

Abstract

Let $\mathsf{J}(\mathsf{C})$ 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 $\mathsf{J}(\mathsf{C})$, 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.

arXiv comment24 pages, 11 figures

References in corpus (2)

• On Orbits of Order Ideals of Minuscule Posets
• On the periodicity of Coxeter transformations and the non-negativity of their Euler forms