T

Breuer, Wolfgang; Schweizer, Thilo; Breuer, Claudia

doi:10.1007/978-3-531-93386-3_20

Cited by 1 publication

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“…We prove here that in fact, any irreducible H-module V is of precisely of type (1), (2), or (3). We first need an elementary lemma, most of which is well-known.…”

Section: Indicators For Involutive Hopf Algebras and Drinfel'd Doublesmentioning

confidence: 98%

“…This completes the proof for the classical Weyl groups. If W is exceptional, then Breuer [3] has used GAP to check that the result holds. Proof.…”

Section: Abelian Subgroups Of Real Reflection Groupsmentioning

confidence: 99%

“…Thus, arguing as above, we deduce that there is a nonzero N -fixed point f in U and τ (f ) = ν( V )f . We have now reduced the computation of ν( V ) to a computation in N and we proceed to prove (3).…”

Section: It Follows Easily From the Definitions Thatmentioning

confidence: 99%

“…The remaining groups are H 4 , F 4 , E 6 , E 7 and E 8 . In these cases, we use computer calculations to verify the result (see [3] for details).…”

Section: Abelian Subgroups Of Real Reflection Groupsmentioning

confidence: 99%

“…For the smaller reflection groups, we use computer calculations -some of which were quite simple and some more involved. We refer the reader to [3] and [7] for more details.…”

Section: Introductionmentioning

confidence: 99%

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

Guralnick

Montgomery

2009

Trans. Amer. Math. Soc.

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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 Q.

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“…We prove here that in fact, any irreducible H-module V is of precisely of type (1), (2), or (3). We first need an elementary lemma, most of which is well-known.…”

Section: Indicators For Involutive Hopf Algebras and Drinfel'd Doublesmentioning

confidence: 98%

“…This completes the proof for the classical Weyl groups. If W is exceptional, then Breuer [3] has used GAP to check that the result holds. Proof.…”

Section: Abelian Subgroups Of Real Reflection Groupsmentioning

confidence: 99%

Section: It Follows Easily From the Definitions Thatmentioning

confidence: 99%

“…The remaining groups are H 4 , F 4 , E 6 , E 7 and E 8 . In these cases, we use computer calculations to verify the result (see [3] for details).…”

Section: Abelian Subgroups Of Real Reflection Groupsmentioning

confidence: 99%

“…For the smaller reflection groups, we use computer calculations -some of which were quite simple and some more involved. We refer the reader to [3] and [7] for more details.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations