McKay Quivers and Lusztig Algebras of Some Finite Groups

Abstract

Abstract We are interested in the McKay quiver Γ( 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 $\mathsf {G} \subseteq \text {GL}(V)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi><mml:mo>⊆</mml:mo><mml:mtext>GL</mml:mtext><mml:mo>(</mml:mo><mml:mi>V</mml:mi><mml:mo>)</mml:mo></mml:math> 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 &gt; 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 $\widetilde {A}(\mathsf {G})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mo>~</mml:mo></mml:mover><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:math> of a finite group $\mathsf {G} \subseteq \text {GL}(V)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi><mml:mo>⊆</mml:mo><mml:mtext>GL</mml:mtext><mml:mo>(</mml:mo><mml:mi>V</mml:mi><mml:mo>)</mml:mo></mml:math> , 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 .

Locations

• Algebras and Representation Theory - View - PDF