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

Abstract

Let $G\le {\mathrm{SL}}_3(\mathbb{C})$ 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 ${\mathrm{GL}}_2(\mathbb{C})\hookrightarrow {\mathrm{SL}}_3(\mathbb{C})$, 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 ${\mathrm{SL}}_2(\mathbb{C})$ or ${\mathrm{PSL}}_2(\mathbb{C})$. 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 ${\mathrm{SL}}_3(\mathbb{C})$ for which $R\ast G$ is a $3$-preprojective algebra.

Improved exposition on the interaction of gradings, cuts, and loops. Revised minor details in proofs pertaining to loops. 52 pages. Comments and improvements welcome!