$n$-complete algebras and McKay quivers

Abstract: Let $\Gamma^{n}$ be the cone of an $(n-1)$-complete algebra over an algebraically closed field $k$. In this paper, we prove that if the bound quiver $(Q_{n},\rho_{n})$ of $\Gamma^{n}$ is a truncation from the bound McKay quiver $(Q_{G},\rho_{G})$ of a finite subgroup $G$ of $GL(n,k)$, the bound quiver $(Q_{n+1}, \rho_{n+1})$ of $\Gamma^{n+1}$, the cone of $\Gamma^{n}$, is a truncation from the bound McKay quiver $(Q_{\widetilde{G}},\rho_{\widetilde{G}})$ of $\widetilde{G}$, where $\widetilde{G}\cong G\times \mathbb{Z}_{m}$ for some $m\in \mathbb{N}$.