McKay quivers and Lusztig algebras of some finite groups

TL;DR

This paper develops combinatorial and algebraic methods to construct and analyze McKay quivers and Lusztig algebras for finite groups, especially complex reflection groups, enhancing understanding of their skew group rings.

Contribution

It introduces a Young diagram-based combinatorial method for McKay quivers of $G(r,p,n)$ and defines Lusztig algebras for arbitrary finite groups, linking them to skew group rings.

Findings

Constructed McKay quivers for $G(r,p,n)$ using Young diagrams.

Established a Morita equivalence between Lusztig algebras and skew group rings.

Provided a conceptual framework for McKay quivers of finite groups.

Abstract

We are interested in the McKay quiver $\Gamma(G)$ and skew group rings $A*G$, where $G$ is a finite subgroup of $\mathrm{GL}(V)$, where $V$ is a finite dimensional vector space over a field $K$, and $A$ is a $K-G$-algebra. These skew group rings appear in Auslander's version of the McKay correspondence. In the first part of this paper we consider complex reflection groups $G \subseteq \mathrm{GL}(V)$ and find a combinatorial method, making use of Young diagrams, to construct the McKay quivers for the groups $G(r,p,n)$. We first look at the case $G(1,1,n)$, which is isomorphic to the symmetric group $S_n$, followed by $G(r,1,n)$ for $r >1$. Then, using Clifford theory, we can determine the McKay quiver for any $G(r,p,n)$ and thus for all finite irreducible complex reflection groups up to finitely many exceptions. In the second part of the paper we consider a more conceptual approach to McKay quivers of arbitrary finite groups: we define the Lusztig algebra $\widetilde A(G)$ of a finite group $G \subseteq \mathrm{GL}(V)$, which is Morita equivalent to the skew group ring $A*G$. This description gives us an embedding of the basic algebra Morita equivalent to $A*G$ into a matrix algebra over $A$.