Algebraic representations and constructible sheaves

Williamson, Geordie

doi:10.1007/s11537-017-1646-1

Cited by 36 publications

(11 citation statements)

References 98 publications

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“…To be not too small depends on the case and on the author. For example, in Lusztig conjecture for SL n (F p ), in Lusztig's original formulation the bound was p > 2n − 3 and the bound in Kato's version was p > n. In James's conjecture for S n , the bound was p > √ n. A new paradigm has emerged in the last few years by the work of Williamson and his collaborators (see [1,21,40,48,49]). Now we know that p-Kazhdan-Lusztig polynomials are central objects of study in modular representation theory of Lie type objects.…”

Section: A New Paradigmmentioning

confidence: 99%

Blob algebra approach to modular representation theory

Libedinsky

Plaza

2020

Proc. London Math. Soc.

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Two decades ago, Martin and Woodcock made a surprising and prophetic link between statistical mechanics and representation theory. They observed that the decomposition numbers of the blob algebra (that appeared in the context of transfer matrix algebras) are Kazhdan–Lusztig polynomials in type A∼1. In this paper, we take that observation far beyond its original scope. We conjecture that for A∼n there is an equivalence of categories between the characteristic p diagrammatic Hecke category and a ‘blob category’ that we introduce (using certain quotients of KLR algebras called generalized blob algebras). Using alcove geometry we prove the ‘graded degree’ part of this equivalence for all n and all prime numbers p. If our conjecture was verified, it would imply that the graded decomposition numbers of the generalized blob algebras in characteristic p give the p‐Kazhdan–Lusztig polynomials in type A∼n. We prove this for A∼1, the only case where the p‐Kazhdan–Lusztig polynomials are known. This paper relies extensively on color figures. Some references to color may not be meaningful in the printed version, and we refer the reader to the online version which includes the color figures.

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Section: A New Paradigmmentioning

confidence: 99%

Blob algebra approach to modular representation theory

Libedinsky

Plaza

2020

Proc. London Math. Soc.

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“…In this section we introduce the (extended) affine Weyl group and the corresponding Hecke algebras. We refer to [Wil17] and [Kno05] for more details.…”

Section: Definition Of the Pre-canonical Basesmentioning

confidence: 99%

Pre-canonical bases on affine Hecke algebras

Libedinsky¹,

Patimo²,

Plaza³

2021

Preprint

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“…Let G denote a split simple and simply connected algebraic group over a field k of characteristic p. We fix a Borel subgroup and maximal torus T ⊂ B ⊂ G. We will try to follow the notation of [24]. In particular: 27 Figure 1.…”

Section: Generational Philosophymentioning

confidence: 99%

Billiards and Tilting Characters for SL<sub>3</sub>

Lusztig¹,

Williamson²

2018

SIGMA

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We formulate a conjecture for the second generation characters of indecomposable tilting modules for SL 3 . This gives many new conjectural decomposition numbers for symmetric groups. Our conjecture can be interpreted as saying that these characters are governed by a discrete dynamical system ("billiards bouncing in alcoves"). The conjecture implies that decomposition numbers for symmetric groups display (at least) exponential growth.

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Algebraic representations and constructible sheaves

Cited by 36 publications

References 98 publications

Blob algebra approach to modular representation theory

Blob algebra approach to modular representation theory

Pre-canonical bases on affine Hecke algebras

Billiards and Tilting Characters for SL<sub>3</sub>

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