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McKay Quivers and Lusztig Algebras of Some Finite Groups

机译：McKay Quivers and Lusztig Algebras of Some Finite Groups

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摘要

We are interested in the McKay quiver Gamma(G) and skew group rings A *G, where G is a finite subgroup of 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 subset of 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 (A) over tilde (G) of a finite group G subset of 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.

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