Smith-Treumann theory and the linkage principle

Riche, Simon; Williamson, Geordie

doi:10.48550/arxiv.2003.08522

Cited by 2 publications

(3 citation statements)

References 18 publications

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“…We are given to understand that J. Ciappara has recently proved a result similar to our main theorem via a completely different method. His proof relies on the Smith-Treumann theory of [RW2], and is still in preparation. We thank G. The project realized in this paper was conceived during the workshop Modular Representation Theory organized in Oxford by the Clay Mathematical Institute.…”

Section: The Hecke Category and Representations Of The Universal Cent...mentioning

confidence: 99%

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Hecke action on the principal block

Bezrukavnikov¹,

Riche²

2020

Preprint

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In this paper we construct an action of the affine Hecke category on the principal block of representations of a simply-connected semisimple algebraic group over an algebraically closed field of characteristic bigger than the Coxeter number. This confirms a conjecture of G. Williamson and the second author, and provides a new proof of the tilting character formula in terms of antispherical p-Kazhdan-Lusztig polynomials. * (1) reg and g * (1) . We deduce a canonical isomorphism

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Section: The Hecke Category and Representations Of The Universal Cent...mentioning

confidence: 99%

“…Later the "combinatorial" consequence of this conjecture (the tilting character formula) was proved for general G (in fact, in two very different ways, see [AMRW2] and [RW2]), but these proofs use other tools, and none of them imply the original categorical conjecture. The main result of the present paper is a construction of this Hecke category action.…”

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confidence: 99%

Hecke action on the principal block

Bezrukavnikov¹,

Riche²

2020

Preprint

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“…We give a practical (but slow) algorithm for the calculation of ℎ , , relying on the technique of localisation developed in the work of Elias and Williamson. In the anti-spherical case, the algorithm makes use of ideas from Arikhipov-Bezrukavnikov [AB], Riche-Williamson [RW20], and Libedinsky-Williamson [LW17].…”

Section: Introductionmentioning

confidence: 99%

Calculating the $p$-canonical basis of Hecke algebras

Gibson¹,

Jensen²,

Williamson³

2022

Preprint

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We describe an algorithm for computing the -canonical basis of the Hecke algebra,or one of its antispherical modules. The algorithm does not operate in the Hecke category directly, but rather uses a faithful embedding of the Hecke category inside a semisimple category to build a "model" for indecomposable objects and bases of their morphism spaces. Inside this semisimple category, objects are sequences of Coxeter group elements, and morphisms are (sparse) matrices over a fraction field, making it quite amenable to computations. This strategy works for the full Hecke category over any base field, but in the antispherical case we must instead work over Z ( ) and use an idempotent lifting argument to deduce the result for a field of characteristic > 0.We also describe a less sophisticated algorithm which is much more suited to the case of finite groups. We provide complete implementations of both algorithms in the MAGMA computer algebra system.by Jensen to generate data leading to his revised form of the billiards conjecture [Jen21]. The simpler algorithm presented at the end of this paper was used to generate the tables in the appendix of [EJG21].The algorithm is programmed using the computer algebra package MAGMA [BCP97]. The source code for the main algorithm (as well as a simpler algorithm appropriate for finite groups) is available online at https://github.com/joelgibson/ASLoc.Organisation of the paper is as follows. In Section 2 we fix notation and briefly recall the definitions of Hecke algebra, realisation, root system, Hecke category, and antispherical category. In Section 3 we review localisation for the Hecke and antispherical categories, our key tool for computations. Finally, the main algorithm is stated in Section 4, and a simpler algorithm which works particularly well for finite groups of small rank is stated in Section 5.

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Smith-Treumann theory and the linkage principle

Cited by 2 publications

References 18 publications

Hecke action on the principal block

Hecke action on the principal block

Calculating the $p$-canonical basis of Hecke algebras

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