$2$-representation infinite algebras from non-abelian subgroups of $\operatorname{SL}_3$. Part II: Central extensions and exceptionals

TL;DR

This paper classifies when skew-group algebras from finite subgroups of SL_3(C) are 3-preprojective algebras, focusing on non-abelian groups, especially from GL_2(C) and exceptional types, expanding the understanding of 2-representation infinite algebras.

Contribution

It provides a detailed classification of subgroups of SL_3(C) that yield 3-preprojective algebras, including new examples and algorithmic approaches for computing McKay quivers and cuts.

Findings

A 3-preprojective cut exists iff G is not a subgroup of SL_2(C) or PSL_2(C) for type (B) groups.

All groups of types (E)-(L) admit a 3-preprojective cut except types (H) and (I).

Explicit computations of McKay quivers and cuts for exceptional subgroups, enabling algorithmic analysis.

Abstract

Let $G \leq \operatorname{SL}_3(\mathbb{C})$ be a non-trivial finite group, acting on $R = \mathbb{C}[x_1, x_2, x_3]$. We continue our investigation from arXiv:2505.10683 [math.RT] into when the resulting skew-group algebra $R \ast G$ is a $3$-preprojective algebra of a $2$-representation infinite algebra, defined by a so-called cut. We consider the subgroups arising from $\operatorname{GL}_2(\mathbb{C}) \hookrightarrow \operatorname{SL}_3(\mathbb{C})$, called type (B), as well as the exceptional subgroups, called types (E) -- (L). For groups of type (B), we show that a $3$-preprojective cut exists on $R \ast G$ if and only if $G$ is not isomorphic to a subgroup of $\operatorname{SL}_2(\mathbb{C})$ or $\operatorname{PSL}_2(\mathbb{C})$. For groups $G$ of the remaining types (E) -- (L), every $R \ast G$ admits a $3$-preprojective cut, except for type (H) and (I). To prove our results for type (B), we explore how the notion of isoclinism interacts with the shape of McKay quivers. We compute the McKay quivers in detail, using a knitting-style heuristic. For the exceptional subgroups, we compute the McKay quivers directly, as well as cuts, and we discuss how this task can be done algorithmically. This provides many new examples of $2$-representation infinite algebras, and together with arXiv:2401.10720 [math.RT], arXiv:2505.10683 [math.RT] completes the classification of finite subgroups of $\operatorname{SL}_3(\mathbb{C})$ for which $R \ast G$ is a $3$-preprojective algebra.