$n$-complete algebras and McKay quivers

Let $Γ^{n}$ 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 $(Q_{n},ρ_{n})$ of $Γ^{n}$ is a truncation from the bound McKay quiver $(Q_{G},ρ_{G})$ of a finite subgroup $G$ of $GL(n,k)$, the bound quiver $(Q_{n+1}, ρ_{n+1})$ of $Γ^{n+1}$, the cone of $Γ^{n}$, is a truncation from the bound McKay quiver $(Q_{\widetilde{G}},ρ_{\widetilde{G}})$ of $\widetilde{G}$, where $\widetilde{G}\cong G\times \mathbb{Z}_{m}$ for some $m\in \mathbb{N}$.

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