Representations of Path Algebras with Applications to Subgroup Lattices and Group Characters

Abstract

Let $Q$ be a quiver, and $w$ a weight function on the set of arrows of $Q$. In this paper, we will introduce an $R$-algebra ${\sf UD}(Q,w;R)$ over a ring $R$ in which the information how vertices of $Q$ are joined by its arrows with weights should be reflected well. This algebra is obtained by using ${\mathbb Z} Q$-modules where ${\mathbb Z} Q$ is the path algebra of $Q$ over ${\mathbb Z}$. We will particularly focus on quivers and weight functions defined by the subgroup lattice of a finite group $G$, and defined by irreducible characters of subgroups of $G$. The structure of the corresponding ${\mathbb Z}$-algebras ${\sf UD}(Q,w;{\mathbb Z})$ and relations with the group $G$ will be studied.