Frobenius-Schur indicators for subgroups and the Drinfel’d double of Weyl groups
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- by Robert Guralnick and Susan Montgomery PDF
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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 $\mathbb {Q}$.References
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Additional Information
- Robert Guralnick
- Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
- MR Author ID: 78455
- Email: guralnic@usc.edu
- Susan Montgomery
- Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
- Email: smontgom@usc.edu
- 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.
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2491893