Geodesic
$G(\ell,k,d)$-modules via groupoids
Journal of Algebraic Combinatorics, Tome 43 (2016) no. 1, pp. 11-32.

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In this note, we describe a seemingly new approach to the complex representation theory of the wreath product $G\wr S_d$, 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 $G\wr S_d$. This directly implies a classification of simple modules. As an application, we get a Gelfand model for $G\wr S_d$ 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 $G(\ell,k,d)$.
Classification : 05E10
Keywords: Schur-Weyl duality, wreath product, simple module, groupoid 
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     author = {Mazorchuk, Volodymyr and Stroppel, Catharina},
     title = {$G(\ell,k,d)$-modules via groupoids},
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Mazorchuk, Volodymyr; Stroppel, Catharina. $G(\ell,k,d)$-modules via groupoids. Journal of Algebraic Combinatorics, Tome 43 (2016) no. 1, pp. 11-32. http://geodesic.mathdoc.fr/item/JAC_2016__43_1_a10/