Higher representation infinite algebras from McKay quivers of metacyclic groups

TL;DR

This paper constructs new examples of higher representation infinite algebras using McKay quivers of metacyclic groups embedded in special linear groups, linking to classical tame hereditary algebras for s=2.

Contribution

It introduces a method to generate higher representation infinite algebras from metacyclic groups via McKay quivers and describes their structure, including a classical case for s=2.

Findings

Examples of (s-1)- and s-representation infinite algebras are constructed.

McKay quivers with superpotentials are described for these groups.

For s=2, the examples relate to classical tame hereditary algebras of type D.

Abstract

For each prime number $s$ we introduce examples of $(s-1)$- and $s$-representation infinite algebras in the sense of Herschend, Iyama and Oppermann, which arise from skew group algebras of some metacyclic groups embedded in $\mathrm{SL}(s,\mathbb{C})$ and $\mathrm{SL}(s+1,\mathbb{C})$. For this purpose, we give a description of the McKay quiver with a superpotential of such groups. Moreover we show that for $s=2$ our examples correspond to the classical tame hereditary algebras of type $\tilde{D}$.