Localized calculus for the Hecke category

Elias, Ben; Williamson, Geordie

doi:10.48550/arxiv.2011.05432

Cited by 8 publications

(23 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Example 1. 16 Let W be the affine symmetric group S 3 with generators s 1 , s 2 , s 3 and let P be the maximal finite parabolic generated by s and s 3 . The Coxeter graph and Bruhat graphs are depicted in Fig.…”

Section: Example 111mentioning

confidence: 99%

The modular Weyl–Kac character formula

2022

View full text Add to dashboard Cite

We classify and explicitly construct the irreducible graded representations of anti-spherical Hecke categories which are concentrated one degree. Each of these homogeneous representations is one-dimensional and can be cohomologically constructed via a BGG resolution involving every (infinite dimensional) standard representation of the category. We hence determine the complete first row of the inverse parabolic p-Kazhdan–Lusztig matrix for an arbitrary Coxeter group and an arbitrary parabolic subgroup. This generalises the Weyl–Kac character formula to all Coxeter systems (and their parabolics) and proves that this generalised formula is rigid with respect to base change to fields of arbitrary characteristic.

show abstract

Section: Example 111mentioning

confidence: 99%

The modular Weyl–Kac character formula

2022

View full text Add to dashboard Cite

show abstract

“…There is a monoidal localisation functor Λ : ℋ k → Std sending the generating object of ℋ to the object (id, ), and the generating morphisms of ℋ (up-dots, down-dots, trivalent vertices, and 2 -valent vertices) to matrices over : these matrices are calculated explicitly in [EW20]. This gives a right action of ℋ on Std , where • = ⊗ Λ( ) for ∈ Std and ∈ ℋ .…”

Section: Localisation For the Hecke Categorymentioning

confidence: 99%

Calculating the $p$-canonical basis of Hecke algebras

Gibson¹,

Jensen²,

Williamson³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…In the proof, we use localized calculus. Ideas of localized calculus are found in [EW16,Abe19] and more systematic treatment recently appeared in [EW20].…”

Section: Localized Calculusmentioning

confidence: 99%

“…The two elements π x and π y are not the same in general. By [EW20, (7.9), (7.10)], [EW20,(6.11), (6.12)]. The realization is even-balanced if and only if ξ = 1.…”

Section: A Homomorphism Between Bott-samuelson Bimodulesmentioning

confidence: 99%