The p-canonical basis for Hecke
                    algebras

Jensen, Lars Thorge; Williamson, Geordie

doi:10.1090/conm/683/13719

Cited by 40 publications

(24 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, we recall the definition of the p-canonical basis and its elementary properties (see [JW17]). Instead of a based root datum, we use a generalized Cartan matrix as input, but all the results from [JW17] still hold in this slightly more general setting. Let k be a field of characteristic p 0.…”

Section: The P-canonical Basismentioning

confidence: 99%

See 1 more Smart Citation

The ABC of p-cells

Jensen

2020

Sel. Math. New Ser.

View full text Add to dashboard Cite

Parallel to the very rich theory of Kazhdan-Lusztig cells in characteristic 0, we try to build a similar theory in positive characteristic. We study cells with respect to the p-canonical basis of the Hecke algebra of a crystallographic Coxeter system (see [JW17]). Our main technical tool are the star-operations introduced by Kazhdan-Lusztig in [KL79] which have interesting numerical consequences for the p-canonical basis. As an application, we explicitely describe p-cells in finite type A (i.e. for symmetric groups) using the Robinson-Schensted correspondence. Moreover, we show that Kazhdan-Lusztig cells in finite types B and C decompose into p-cells for p > 2.

show abstract

Section: The P-canonical Basismentioning

confidence: 99%

“…Consider the local intersection forms I xy,xz of xy at xz (resp. I y,z of y at z) and the matrices representing them with respect to the light leaves bases (see [JW17,§3]). For two subexpressions e, f of y expressing z we get in k H ≮xz ⊗ R k:…”

Section: Proof Of Theorem 39 and Corollary 310mentioning

confidence: 99%

The ABC of p-cells

Jensen

2020

Sel. Math. New Ser.

View full text Add to dashboard Cite

show abstract

“…There is an algorithm to calculate the p-canonical basis, involving the generators and relations presentation of the Hecke category discussed earlier. This algorithm is described in detail in [JW17,§3]. Remark 2.24.…”

Section: The Hecke Categorymentioning

confidence: 99%

Parity Sheaves and the Hecke Category

Williamson

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

Self Cite

View full text Add to dashboard Cite

IntroductionOne of the first theorems of representation theory is Maschke's theorem: any representation of a finite group over a field of characteristic zero is semi-simple. This theorem is ubiquitous throughout mathematics. (We often use it without realising it; for example, when we write a function of one variable as the sum of an odd and an even function.) The next step is Weyl's theorem: any finite-dimensional representation of a compact Lie group is semi-simple 1 . It is likewise fundamental: for the circle group Weyl's theorem is closely tied to the theory of Fourier series.Beyond the theorems of Maschke and Weyl lies the realm of where semi-simplicity fails. Non semi-simple phenomena in representation theory were first encountered when studying the modular (i.e. characteristic p) representations of finite groups. This theory is the next step beyond the classical theory of the character table, and is important in understanding the deeper structure of finite groups. A second example (of fundamental importance throughout mathematics from number theory to mathematical physics) occurs when studying the infinite-dimensional representation theory of semi-simple Lie groups and their p-adic counterparts.Throughout the history of representation theory, geometric methods have played an important role. Over the last forty years, the theory of intersection cohomology and perverse sheaves has provided powerful new tools. To any complex reductive group is naturally associated several varieties (e.g. unipotent and nilpotent orbits and their closures, the flag variety and its Schubert varieties, the affine Grassmannian and its Schubert varieties . . . ). In contrast to the group itself, these varieties are often singular. The theory of perverse sheaves provides a collection of constructible complexes of sheaves (intersection cohomology sheaves) on such varieties, and the "IC data" associated to intersection cohomology sheaves (graded dimensions of stalks, total cohomology, . . . ) appears throughout Lie theory.The first example of the power of this theory is the Kazhdan-Lusztig conjecture (a theorem of Beilinson-Bernstein and Brylinski-Kashiwara), which expresses the character of a simple highest weight module over a complex semi-simple Lie algebra in terms of IC data of Schubert varieties in the flag variety. This theorem is an important first step towards understanding the irreducible representations of semisimple Lie groups. A second example is Lusztig's theory of character sheaves, 1 Weyl first proved his theorem via integration over the group to produce an invariant Hermitian form. To do this he needed the theory of manifolds. One can view his proof as an early appearance of geometric methods in representation theory.

show abstract

“…is a basis which only depends on the characteristic p of k, the p-canonical basis (see [Wil12,JW]). Let us write Throughout this paper we will say that p occurs as torsion in SL n if there exists…”

Section: We Denote By [B] the Split Grothendieck Group Of B (Ie [B]mentioning

confidence: 99%

Schubert calculus and torsion explosion

Williamson

2016

J. Amer. Math. Soc.

Self Cite

104

View full text Add to dashboard Cite

Abstract. We observe that certain numbers occurring in Schubert calculus for SLn also occur as entries in intersection forms controlling decompositions of Soergel bimodules in higher rank. These numbers grow exponentially. This observation gives many counter-examples to the expected bounds in Lusztig's conjecture on the characters of simple rational modules for SLn over fields of positive characteristic. Our examples also give counter-examples to the James conjecture on decomposition numbers for symmetric groups.

show abstract

The p-canonical basis for Hecke algebras

Cited by 40 publications

References 46 publications

The ABC of p-cells

The ABC of p-cells

Parity Sheaves and the Hecke Category

Schubert calculus and torsion explosion

Contact Info

Product

Resources

About