Kazhdan–Lusztig cells and the Frobenius–Schur indicator

Geck, Meinolf

doi:10.1016/j.jalgebra.2013.01.019

Cited by 5 publications

(7 citation statements)

References 12 publications

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“…We discuss the relation between Theorem 3.8 and the results of [10,15,28]. Let A = (A, S, g) be a pivotal algebra such that the algebra A is symmetric with trace form φ : A → k. By definition, the map…”

Section: Separable Pivotal Algebrasmentioning

confidence: 98%

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Frobenius–Schur Indicator for Categories with Duality

Shimizu

2012

Axioms

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Abstract:We introduce the Frobenius-Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius-Schur theorem including that for semisimple quasi-Hopf algebras, weak Hopf C * -algebras and association schemes. Our framework also clarifies a mechanism of how the "twisted" theory arises from the ordinary case. As a demonstration, we establish twisted versions of the Frobenius-Schur theorem for various algebraic objects. We also give several applications to the quantum SL 2 .

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Section: Separable Pivotal Algebrasmentioning

confidence: 98%

“…Doi [10] reconstructed the results of [28] with an emphasis on the use of the theory of symmetric algebras. Recently, Geck [15] proved a result similar to Doi and gave some applications to finite Coxeter groups.…”

Section: Introductionmentioning

confidence: 98%

Frobenius–Schur Indicator for Categories with Duality

Shimizu

2012

Axioms

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“…On the other hand, it easily follows from Proposition 2.6 that |C (2) | = χ 1 (1) + · · · + χ n (1); see [Ge5,Cor. 3.12], Hence, we have…”

Section: Leading Coefficientsmentioning

confidence: 99%

Conjugacy classes of involutions and Kazhdan–Lusztig cells

Bonnafé¹,

Geck

2014

Represent. Theory

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Abstract. According to an old result of Schützenberger, the involutions in a given two-sided cell of the symmetric group S n are all conjugate. In this paper, we study possible generalizations of this property to other types of Coxeter groups. We show that Schützenberger's result is a special case of a general result on "smooth" two-sided cells. Furthermore, we consider Kottwitz's conjecture concerning the intersections of conjugacy classes of involutions with the left cells in a finite Coxeter group. Our methods lead to a proof of this conjecture for classical types which, combined with further recent work, settles this conjecture in general.

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“…We note here that a weakened version of the conjecture follows immediately from Theorem 1.2 and recent work of Geck [18]. We organize this article as follows.…”

Section: Introductionmentioning

confidence: 98%

“…Alvis notes how to extend Lusztig's proof to type H 4[2, Proposition 3.4], and his argument remains valid in types H 3 and I 2 (m) (noting the explicit description of the left cells for these groups given in the following sections.) Alternatively, Geck has given a general proof of this theorem; see[18, Corollary 3.9].Corollary 5.2. If Γ is a left cell in a finite Coxeter group W , then χ Γ is multiplicity-free if and only if every w ∈ Γ ∩ Γ −1 is an involution.…”

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confidence: 99%

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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Kazhdan–Lusztig cells and the Frobenius–Schur indicator

Cited by 5 publications

References 12 publications

Frobenius–Schur Indicator for Categories with Duality

Frobenius–Schur Indicator for Categories with Duality

Conjugacy classes of involutions and Kazhdan–Lusztig cells

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

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