AR-Components of domestic finite group schemes: McKay-Quivers and Ramification

TL;DR

This paper characterizes the Euclidean components of the Auslander-Reiten quiver for domestic finite group schemes using McKay-quivers and explores the relationship between ramification indices and the structure of these quivers.

Contribution

It provides a direct description of Euclidean components via McKay-quivers and links ramification indices to the ranks of tubes in Auslander-Reiten quivers for subgroup schemes.

Findings

Euclidean components described via McKay-quivers

Connection established between ramification indices and quiver structure

Insight into the representation theory of finite group schemes

Abstract

For a domestic finite group scheme, we give a direct description of the Euclidean components in its Auslander-Reiten quiver via the McKay-quiver of a finite linearly reductive subgroup scheme of $SL(2)$. Moreover, for a normal subgroup scheme $\mathcal{N}$ of a finite group scheme $\mathcal{G}$, we show that there is a connection between the ramification indices of the restriction morphism $\mathbb{P}(\mathcal{V}_{\mathcal{N}})\rightarrow\mathbb{P}(\mathcal{V}_{\mathcal{G}})$ between their projectivized cohomological support varieties and the ranks of the tubes in their Auslander-Reiten quivers.