Representations of Hecke Algebras at Roots of Unity

Geck, Meinolf; Jacon, Nicolas

doi:10.1007/978-0-85729-716-7

Cited by 127 publications

(225 citation statements)

References 28 publications

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“…There exist others well-known deformations of complex reflection groups of type G(d, 1, n) which have been intensively studied during the last past years : the Ariki-Koike algebras. These algebras turn out to have a deep representation theory (in both semisimple and modular cases) which is now quite well-understood (see for example [6] for an overview).…”

Section: Introductionmentioning

confidence: 99%

An isomorphism theorem for Yokonuma–Hecke algebras and applications to link invariants

Jacon

d'Andecy

2015

Math. Z.

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We develop several applications of the fact that the Yokonuma-Hecke algebra of the general linear group GL is isomorphic to a direct sum of matrix algebras associated to Iwahori-Hecke algebras of type A. This includes a description of the semisimple and modular representation theory of the Yokonuma-Hecke algebras of GL and a complete classification of all the Markov traces for them. Finally, from these Markov traces, we construct 3-variables polynomials which are invariants for framed and classical knots and links, and investigate their properties with the help of the isomorphism.

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Section: Introductionmentioning

confidence: 99%

An isomorphism theorem for Yokonuma–Hecke algebras and applications to link invariants

Jacon

d'Andecy

2015

Math. Z.

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“…If C is a conjugacy class of reflections, the above result is mentioned in [GeJa,Exp. 3.3.8]; in this case, it is due to L. Iancu (unpublished).…”

Section: Hecke Algebras Involutions Cellsmentioning

confidence: 99%

“…Then, as already mentioned in Example 3.8, all the two-sided cells in W are smooth, and so we now recover a known result of Schützenberger [Sch] in this case. An elementary proof that Lusztig's P Conjectures for (W, S, ϕ) hold is given in [Ge3] (see also [GeJa,§2.8…”

Section: Bonnafé and M Geckmentioning

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 refer to [13], [12], and [5]. As mentioned in the introduction, Geck [11] used generic triviality of decomposition maps in the context of James's conjecture. §3.…”

Section: Proposition Suppose Thatmentioning

confidence: 99%

“…This technique has been extended by Geck and Rouquier [14] to a decent class of algebras over integral domains. It became an important tool for studying algebras involving parameters, so for example Hecke algebras (see [13], [12], and [7]) and, more recently, rational Cherednik algebras (see [2], [15], and [28]). We list several further examples in §2A.…”

Section: Introductionmentioning

confidence: 99%

Decomposition Matrices are Generically Trivial

Thiel¹

2015

Int Math Res Notices

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ABSTRACT. We establish a genericity property in the representation theory of a flat family of finite-dimensional algebras in the sense of Cline-Parshal-Scott. More precisely, we show that the decomposition matrices as introduced by Geck and Rouquier of an algebra which is free of finite dimension over a noetherian integral domain and which splits over the fraction field of this ring are generically trivial, i.e., they are trivial in an open neighborhood of the generic point of the spectrum of the base ring. This generalizes a classical result by Brauer in modular representation theory of finite groups. We furthermore show that this is true precisely on an open set in case all fibers of the algebra split. In this way we get a stratification of the base scheme such that decomposition maps are trivial on each stratum. Moreover, this defines a new discriminant of such algebras which generalizes Schur elements of simple modules for symmetric split semisimple algebras. We provide some extensions to the theory of decomposition maps allowing us to work without the usual normality assumption on the base ring.

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Representations of Hecke Algebras at Roots of Unity

Cited by 127 publications

References 28 publications

An isomorphism theorem for Yokonuma–Hecke algebras and applications to link invariants

An isomorphism theorem for Yokonuma–Hecke algebras and applications to link invariants

Conjugacy classes of involutions and Kazhdan–Lusztig cells

Decomposition Matrices are Generically Trivial

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