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- W3084846892 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 $A$." @default.
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- W3084846892 date "2020-09-14" @default.
- W3084846892 modified "2023-10-18" @default.
- W3084846892 title "McKay quivers and Lusztig algebras of some finite groups" @default.
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