Smith–Treumann theory and the linkage principle

Riche, Simon; Williamson, Geordie

doi:10.1007/s10240-022-00134-y

Cited by 10 publications

(37 citation statements)

References 43 publications

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“…In 2020 Riche and Williamson 〈51〉 showed that the decomposition matrices of the symmetric groups, Iwahori–Hecke algebras and

q

‐Schur algebras are given by the

p

‐canonical basis , generalising earlier work that had restrictions on

p

〈1〉. Within James' lifetime, this gave a solution to the main problem that he worked on throughout his career.…”

Section: Mathematicsmentioning

confidence: 94%

Gordon Douglas James, 1945–2020

Mathas

2022

Bulletin of London Math Soc

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It is said that every generation of mathematicians rewrites the representation theory of the symmetric groups in their own words. Gordon James literally wrote the book on the representation theory of the symmetric groups for mathematicians in the late twentieth century and early twenty-first century, putting a particular focus on representations over finite fields. He proved many of the foundational results in the field. He was one of the first to realise, and to capitalise on, the fundamental connections between the symmetric groups, the finite general linear groups, the Iwahori-Hecke algebras and the quantised Schur algebras. The James Conjecture [34] was a central focus of research in representation theory for 30 years, before finally being shown to be false 〈58〉. (Citations of the form [A] refer to Gordon James' papers, with other papers in the literature being cited as 〈B〉.)In writing this article I reached out to many colleagues. With friends, family and students alike, a common theme was the description of Gordon as being patient and kind, with many saying that he had a mischievous sense of humour. He was an incredibly sharp and gifted mathematician who knew everything there was to know about the symmetric groups and their representation theory, and he played a pivotal role in developing much of this theory. EARLY LIFEGordon was born in Newcastle-upon-Tyne on 31 December 1945, and he grew up in Polegate, near Eastbourne. Gordon's father, Douglas, was an accountant. His mother, Lily (who was known as Toni), was the head of the language department at Hailsham School in Sussex. Toni was fluent in French and German and she picked up new languages easily -a skill that skipped a generation with Gordon. Like his mother, Gordon was an excellent teacher. From a young age, Gordon was identified as being talented at 'arithmetic', so his parents sent him to Eastbourne College as a day student for secondary school from 1959 to 1963. Like many

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“…In 2020 Riche and Williamson 〈51〉 showed that the decomposition matrices of the symmetric groups, Iwahori–Hecke algebras and

q

‐Schur algebras are given by the

p

‐canonical basis , generalising earlier work that had restrictions on

p

〈1〉. Within James' lifetime, this gave a solution to the main problem that he worked on throughout his career.…”

Section: Mathematicsmentioning

confidence: 94%

Gordon Douglas James, 1945–2020

Mathas

2022

Bulletin of London Math Soc

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“…There may be a conjecture for the characters of tilting modules of quantum groups that includes ` h, along the lines of [11,Section 8.1] and [25,Theorem 1.6]. Ideally, the conjecture would relate tilting characters for the quantum group at a root of unity to singular, antispherical Kazhdan-Lusztig polynomials.…”

Section: Potential Applicationsmentioning

confidence: 99%

Web calculus and tilting modules in type $C_2$

Bodish¹

2023

Quantum Topol.

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Using Kuperberg's web calculus (1996), and following Elias and Libedinsky, we describe a "light leaves" algorithm to construct a basis of morphisms between arbitrary tensor products of fundamental representations for sp 4 (and the associated quantum group). Our argument has very little dependence on the base field. As a result, we prove that when OE2 q ¤ 0, the Karoubi envelope of the C 2 web category is equivalent to the category of tilting modules for the divided powers quantum group U A q .sp 4 /.

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“…Also key is that the forgetful functor is an equivalence of categories; this is shown in [RW22, Lemma 5.2].…”

Section: Geometric Ingredientsmentioning

confidence: 99%

“…After the tilting and irreducible character formulas were established by other means [AMRW19] for , the following year saw two major developments. The expansion of Smith–Treumann theory (as initiated by Treumann [Tre19] and Leslie and Lonergan [LL21]) by Riche and Williamson [RW22], yielding a geometric proof of the linkage principle and establishing the tilting character formulas in all characteristics. The resolution of the categorical conjecture in full generality by Bezrukavnikov and Riche [BR22]. Their approach is essentially coherent, making use of localisation theorems in positive characteristic and a new bimodule-theoretic realisation of found by Abe [Abe21].…”

Section: Introductionmentioning

confidence: 99%

“…This realisation induces a graded right -module equivalence between the antispherical quotient of and , the category of Iwahori–Whittaker parity complexes on . Through an understanding of the morphism spaces between parity objects, as provided by [RW22, § 7], we show that the graded action of descends further to the Smith quotient of . Up to graded shift, the indecomposable object acts on by the functor induced by the composite , where is an affine simple reflection and is the natural morphism between certain partial affine flag varieties.…”

Section: Introductionmentioning

confidence: 99%

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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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Smith–Treumann theory and the linkage principle

Cited by 10 publications

References 43 publications

Gordon Douglas James, 1945–2020

Gordon Douglas James, 1945–2020

Web calculus and tilting modules in type $C_2$

Hecke category actions via Smith–Treumann theory

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