Simplical complexes associated to quivers arising from finite groups

Abstract

In this paper, we will introduce a simplicial complex $\mathrm{T}_{Q}(\mathcal{H})$ defined by a quiver $Q$ and a family $\mathcal{H}$ of paths in $Q$. We call $\mathrm{T}_{Q}(\mathcal{H})$ a path complex of $\mathcal{H}$ in $Q$. Let $G$ be a finite group, and denote by $\mathrm{Sgp}(G)$ and $\mathrm{Coset}(G)$ respectively the totality of subgroups of $G$, and that of left cosets $gL \in G/L$ of subgroups $L$ of $G$. We will particularly focus on quivers $Q_{G}$ and $Q_{\mathit{CG}}$ obtained naturally from posets $\mathrm{Sgp}(G)$ and $\mathrm{Coset}(G)$ ordered by the inclusion-relation. Then various properties of path complexes associated to $Q_{G}$ and $Q_{\mathit{CG}}$ will be studied.