G(`, k, d)-MODULES VIA GROUPOIDS
نویسنده
چکیده
In this note we describe a seemingly new approach to the complex representation theory of the wreath product G o Sd where G is a finite abelian group. The approach is motivated by an appropriate version of Schur-Weyl duality. We construct a combinatorially defined groupoid in which all endomorphism algebras are direct products of symmetric groups and prove that the groupoid algebra is isomorphic to the group algebra of G o Sd. This directly implies a classification of simple modules. As an application, we get a Gelfand model for G oSd from the classical involutive Gelfand model for the symmetric group. We describe the Schur-Weyl duality which motivates our approach and relate it to various Schur-Weyl dualities in the literature. Finally, we discuss an extension of these methods to all complex reflection groups of type G(`, k, d).
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