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Frobenius-Schur indicators for subgroups and the Drinfel'd double of Weyl groups


Authors: Robert Guralnick and Susan Montgomery
Journal: Trans. Amer. Math. Soc. 361 (2009), 3611-3632
MSC (2000): Primary 16W30, 20C15, 20G42
DOI: https://doi.org/10.1090/S0002-9947-09-04659-5
Published electronically: February 4, 2009
MathSciNet review: 2491893
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Abstract: If $ G$ is any finite group and $ k$ is a field, there is a natural construction of a Hopf algebra over $ k$ associated to $ G$, the Drinfel'd double $ D(G)$. We prove that if $ G$ is any finite real reflection group, with Drinfel'd double $ D(G)$ over an algebraically closed field $ k$ of characteristic not $ 2$, then every simple $ D(G)$-module has Frobenius-Schur indicator +1. This generalizes the classical results for modules over the group itself. We also prove some new results about Weyl groups. In particular, we prove that any abelian subgroup is inverted by some involution. Also, if $ E$ is any elementary abelian $ 2$-subgroup of the Weyl group $ W$, then all representations of $ C_W(E)$ are defined over $ \mathbb{Q}$.


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Additional Information

Robert Guralnick
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
Email: guralnic@usc.edu

Susan Montgomery
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
Email: smontgom@usc.edu

DOI: https://doi.org/10.1090/S0002-9947-09-04659-5
Keywords: Drinfel'd double, Schur indicator, Weyl groups, reflection groups, rationality of representations
Received by editor(s): March 26, 2007
Published electronically: February 4, 2009
Additional Notes: The authors were supported by NSF grants DMS 0140578 and DMS 0401399.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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