The anti-spherical category

Libedinsky, Nicolás; Williamson, Geordie

doi:10.1016/j.aim.2022.108509

Cited by 10 publications

(3 citation statements)

References 23 publications

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“…In particular, is obtained from by killing morphisms factoring through objects , whenever begins with some element . By furthermore killing the morphism , we obtain the ( standard ) antispherical quotients and discussed in sources such as [RW18, § 1.3] and [LW22]. The snake arrow denotes passage to Karoubi envelopes of additive hulls.…”

Section: The Loop Antispherical Modulementioning

confidence: 99%

Hecke category actions via Smith–Treumann theory

Ciappara

2023

Compositio Math.

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Let $\textbf {G}$ be a simply connected semisimple algebraic group over a field of characteristic greater than the Coxeter number. We construct a monoidal action of the diagrammatic Hecke category on the principal block $\operatorname {Rep}_0(\textbf {G})$ of $\operatorname {Rep}(\textbf {G})$ by wall-crossing functors. This action was conjectured to exist by Riche and Williamson. Our method uses constructible sheaves and relies on Smith–Treumann theory.

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Section: The Loop Antispherical Modulementioning

confidence: 99%

Hecke category actions via Smith–Treumann theory

Ciappara

2023

Compositio Math.

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“…An important example is software developed by the second author and Thorge Jensen for efficient computation of the 𝑝-canonical basis of Hecke algebras (see [19,23] for details, and [27] for an application of this technology). A second instance is recent work of Libedinsky and the second author [26] which uses localization to prove the positivity of anti-spherical Kazhdan-Lusztig polynomials.…”

Section: Introductionmentioning

confidence: 99%

Localized calculus for the Hecke category

Elias,

Williamson

2023

Annales Mathématiques Blaise Pascal

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We construct a functor from the Hecke category to a groupoid built from the underlying Coxeter group. This fixes a gap in an earlier work of the authors. This functor provides an abstract realization of the localization of the Hecke category at the field of fractions. Knowing explicit formulas for the localization is a key technical tool in software for computations with Soergel bimodules. Calcul localisé pour la catégorie de Hecke RésuméNous construisons un foncteur de la catégorie de Hecke vers un groupoïde construit à partir du groupe de Coxeter sous-jacent. Cette construction corrige une lacune dans un travail antérieur des auteurs. Ce foncteur fournit une réalisation abstraite de la localisation de la catégorie de Hecke en le corps des fractions. Connaître des formules explicites pour la localisation est un outil technique clé pour le calcul algorithmique avec les bimodules de Soergel.

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“…Given w ∈ Λ h (∞,s), the standard module Δ(w) is the graded left S h -module with basis {c S | S ∈ Path + alc (w)}. The action of S h on Δ(w) is given by ac S = U∈Path + alc (w) r SU (a)c U , where the scalars r SU (a) are the scalars appearing in (3) of Theorem 6.16.Theorem 6.18[25, Section 6.4] and[41, Theorem 5.3]. For k a field, the algebra S h is quasi-hereditary with simple modules L(w) = Δ(w)/rad(Δ(w))…”

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confidence: 99%

Unitary Representations of Cyclotomic Hecke Algebras at Roots of Unity: Combinatorial Classification and BGG Resolutions

Bowman

Norton²,

Simental

2022

J. Inst. Math. Jussieu

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We relate the classes of unitary and calibrated representations of cyclotomic Hecke algebras, and, in particular, we show that for the most important deformation parameters these two classes coincide. We classify these representations in terms of both multipartition combinatorics and as the points in the fundamental alcove under the action of an affine Weyl group. Finally, we cohomologically construct these modules via BGG resolutions.

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The anti-spherical category

Cited by 10 publications

References 23 publications

Hecke category actions via Smith–Treumann theory

Hecke category actions via Smith–Treumann theory

Localized calculus for the Hecke category

Unitary Representations of Cyclotomic Hecke Algebras at Roots of Unity: Combinatorial Classification and BGG Resolutions

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