Involution models of finite Coxeter groups

Vinroot, C. Ryan

doi:10.1515/jgt.2008.018

Cited by 11 publications

(14 citation statements)

References 13 publications

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“…Vinroot, elaborating upon the work of Baddeley [5], describes in [20] all finite Coxeter groups with involution models in the classical sense; in particular, the only irreducible finite Coxeter groups which fail to have involution models are those of type D 2n (n > 1), E 6 , E 7 , E 8 , F 4 , and H 4 . If G is a finite…”

Section: Lemma 52 (See Marin Michelmentioning

confidence: 99%

“…Hence, by Lemmas 5.1 and 5.2, a finite Coxeter group has a generalized involution model if and only if it has an involution model, and we are left with the following corollary of Theorem 1 in [20].…”

Section: Lemma 52 (See Marin Michelmentioning

confidence: 99%

“…Not long after, Baddeley classified in his doctoral thesis which irreducible Weyl groups have involution models: in particular, only those of types D 2n (n > 1), F 4 , E 6 , E 7 , and E 8 do not have involution models [5]. Vinroot recently extended this classification to finite Coxeter groups, in particular showing that the Coxeter groups of type I 2 (n) and H 3 each have an involution model while the group of type H 4 does not [20].…”

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confidence: 99%

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Automorphisms and generalized involution models of finite complex reflection groups

Marberg

2011

Journal of Algebra

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We prove that a finite complex reflection group has a generalized involution model, as defined by Bump and Ginzburg, if and only if each of its irreducible factors is either G(r, p, n) with gcd(p, n) = 1; G(r, p, 2) with r/p odd; or G 23 , the Coxeter group of type H 3 . We additionally provide explicit formulas for all automorphisms of G (r, p, n), and construct new Gelfand models for the groups G(r, p, n) with gcd(p, n) = 1.

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Section: Lemma 52 (See Marin Michelmentioning

confidence: 99%

Section: Lemma 52 (See Marin Michelmentioning

confidence: 99%

mentioning

confidence: 99%

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Automorphisms and generalized involution models of finite complex reflection groups

Marberg

2011

Journal of Algebra

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“…As a corollary to the preceding two theorems we note the following result, which is interesting to compare with [43,Theorem 1]. Here, a Gelfand model is a representation of a finite group which is the multiplicity-free sum of all of the group's irreducible representations.…”

Section: Introductionmentioning

confidence: 85%

How to compute the Frobenius–Schur indicator of a unipotent character of a finite Coxeter system

Marberg¹

2013

Advances in Mathematics

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For each finite, irreducible Coxeter system (W, S), Lusztig has associated a set of "unipotent characters" Uch(W ). There is also a notion of a "Fourier transform" on the space of functions Uch(W ) → R, due to Lusztig for Weyl groups and to Broué, Lusztig, and Malle in the remaining cases. This paper concerns a certain W -representation ̺ W in the vector space generated by the involutions of W . Our main result is to show that the irreducible multiplicities of ̺ W are given by the Fourier transform of a unique function ǫ : Uch(W ) → {−1, 0, 1}, which for various reasons serves naturally as a heuristic definition of the Frobenius-Schur indicator on Uch(W ). The formula we obtain for ǫ extends prior work of Casselman, Kottwitz, Lusztig, and Vogan addressing the case in which W is a Weyl group. We include in addition a succinct description of the irreducible decomposition of ̺ W derived by Kottwitz when (W, S) is classical, and prove that ̺ W defines a Gelfand model if and only if (W, S) has type A n , H 3 , or I 2 (m) with m odd. We show finally that a conjecture of Kottwitz connecting the decomposition of ̺ W to the left cells of W holds in all non-crystallographic types, and observe that a weaker form of Kottwitz's conjecture holds in general. In giving these results, we carefully survey the construction and notable properties of the set Uch(W ) and its attached Fourier transform.

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“…It follows that End W (M, M) is commutative. [18]), one would expect that the representation defined in this section is not a Gelfand model for W (D 2n ). We note that our module M is not mutiplicity-free in this case.…”

Section: A Gelfand Modelmentioning

confidence: 95%

Gelfand models for classical Weyl groups

Araujo

Bratten

2014

Journal of Algebra

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In a recent preprint Kodiyalam and Verma give a particularly simple Gelfand model for the symmetric group that is built naturally on the space of involutions. In this manuscript we give a natural extension of Kodiyalam and Verma's model to a Gelfand model for Weyl groups of type Bn and D2n+1. Then we define an explicit isomorphism between this Gelfand model and the polynomial model using a technique we call telescopic decomposition.

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Involution models of finite Coxeter groups

Cited by 11 publications

References 13 publications

Automorphisms and generalized involution models of finite complex reflection groups

Automorphisms and generalized involution models of finite complex reflection groups

How to compute the Frobenius–Schur indicator of a unipotent character of a finite Coxeter system

Gelfand models for classical Weyl groups

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