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\(G(\ell ,k,d)\)-modules via groupoids

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Abstract

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 \(G(\ell ,k,d)\).

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Acknowledgments

An essential part of the research was done during the visit of both authors to the Max Planck Institute for Mathematics in Bonn. We gratefully acknowledge hospitality and support by the MPIM. For the first author, the research was partially supported by the Swedish Research Council, Knut and Alice Wallenbergs Stiftelse, and the Royal Swedish Academy of Sciences. We thank Daniel Tubbenhauer and Stuart Margolis for their comments.

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Authors and Affiliations

  1. Department of Mathematics, Uppsala University, Box 480, 751 06, Uppsala, Sweden

    Volodymyr Mazorchuk

  2. Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Catharina Stroppel

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  1. Volodymyr Mazorchuk
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  2. Catharina Stroppel
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Correspondence to Volodymyr Mazorchuk.

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Mazorchuk, V., Stroppel, C. \(G(\ell ,k,d)\)-modules via groupoids. J Algebr Comb 43, 11–32 (2016). https://doi.org/10.1007/s10801-015-0623-0

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