Smash products of group weighted bound quivers and Brauer graphs

Abstract

Let $\Bbbk$ be a field, $G$ a group, and $(Q, I)$ a bound quiver. A map $W\colon Q_1 \to G$ is called a $G$-weight on $Q$, which defines a $G$-graded $\Bbbk$-category $\Bbbk(Q, W)$, and $W$ is called homogeneous if $I$ is a homogeneous ideal of the $G$-graded $\Bbbk$-category $\Bbbk(Q, W)$. Then we have a $G$-graded $\Bbbk$-category $\Bbbk(Q, I, W):= \Bbbk(Q, W)/I$. We can then form a smash product $\Bbbk(Q, I, W)\# G$ of $\Bbbk(Q, I, W)$ and $G$, which canonically defines a Galois covering $\Bbbk(Q, I, W)\# G \to \Bbbk(Q, I)$ with group $G$ (we will see that all such Galois coverings to $\Bbbk(Q, I)$ have this form for some $W$). First we give a quiver presentation $\Bbbk(Q_{G, W}, I_{G, W}) \cong \Bbbk(Q, I, W)\# G$ of the smash product $\Bbbk(Q, I, W)\# G$. Next if $(Q, I, W)$ is defined by a Brauer graph with an admissible weight then the smash product $\Bbbk(Q, I, W)\# G$ is again defined by a Brauer graph, which will be computed explicitly. The computation is simplified by introducing a concept of Brauer permutations as an intermediate one between Brauer graphs and Brauer bound quivers. This extends and simplifies the result by Green--Schroll--Snashall on the computation of coverings of Brauer graphs, which dealt with the case that $G$ is a finite abelian group, while in our case $G$ is an arbitrary group. In particular, it enables us to delete all cycles in Brauer graphs to transform it to an infinite Brauer tree.