On support varieties and the Humphreys conjecture in type A

Hardesty, William

doi:10.1016/j.aim.2018.01.023

Cited by 6 publications

(7 citation statements)

References 21 publications

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“…(10.3) V uq(H) (M q ) |Uq(H) = N C (H) by (10.2). Now by [Ha,Proposition 6.2.1] 1 , under our assumptions (M q ) |Uq(H) is a tilting object in Rep(U q (H)); hence this object is the "quantization" of the tilting module M |H in Rep(H) by the process considered in Remark 7.3. For ν a dominant weight for H, we denote by T H (ν), resp.…”

Section: Exotic Parity Sheaves and Tilting Modulesmentioning

confidence: 94%

“…By [Ha,Lemma 3.2.1 and its proof] this implies that T(w · p 0) is a direct summand in a tilting object of the form Θ r1 •· · ·•Θ r k (T(w 12 · p 0)) for some r 1 , . .…”

mentioning

confidence: 88%

“…We can assume that λ belongs to W · p 0. Then by [Ha,Proposition 3.1.3] we have T λ µ (T(µ)) ∼ = T(λ), and T µ λ (T(λ)) is a (non-empty) sum of copies of T(µ). Since translation functors can be described as tensoring with a tilting module and then taking a direct summand (see [Ja1,Remark 7.6(1)]), the claim follows.…”

Section: Cells and Weight Cellsmentioning

confidence: 99%

“…By Lemma 10.1, T q (w 12 · p 0) = T(w 12 · p 0) q and thus by [B1] we have V uq (T(w 12 · p 0) q ) = O C min (see §8.4). From Figure 2, we can see that for any w ∈ c min we have w 12 w, where is the weak order defined in [Ha,§2.2].…”

Section: Exotic Parity Sheaves and Tilting Modulesmentioning

confidence: 99%

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On the Humphreys Conjecture on Support Varieties of Tilting Modules

Achar

Hardesty

Riche

2019

Transformation Groups

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Let G be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic p, assumed to be larger than the Coxeter number. The "support variety" of a G-module M is a certain closed subvariety of the nilpotent cone of G, defined in terms of cohomology for the first Frobenius kernel G 1 . In the 1990s, Humphreys proposed a conjectural description of the support varieties of tilting modules; this conjecture has been proved for G = SLn in earlier work of the second author.In this paper, we show that for any G, the support variety of a tilting module always contains the variety predicted by Humphreys, and that they coincide (i.e., the Humphreys conjecture is true) when p is sufficiently large. We also prove variants of these statements involving "relative support varieties."1.2. The Humphreys conjecture. We fix a Borel subgroup B ⊂ G and a maximal torus T ⊂ G, denote by X the character lattice of T , and let X + ⊂ X be the subset of dominant weights (for the choice of positive roots such that B is the P.A.

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Section: Exotic Parity Sheaves and Tilting Modulesmentioning

confidence: 94%

“…By [Ha,Lemma 3.2.1 and its proof] this implies that T(w · p 0) is a direct summand in a tilting object of the form Θ r1 •· · ·•Θ r k (T(w 12 · p 0)) for some r 1 , . .…”

mentioning

confidence: 88%

Section: Cells and Weight Cellsmentioning

confidence: 99%

Section: Exotic Parity Sheaves and Tilting Modulesmentioning

confidence: 99%

See 2 more Smart Citations

On the Humphreys Conjecture on Support Varieties of Tilting Modules

Achar

Hardesty

Riche

2019

Transformation Groups

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“…The lower closure Alc(w) meets X + if and only if w ∈ f W ext . We also note that V g (T(λ)) = V g (T(µ)) for any w ∈ f W ext and λ, µ ∈ Alc(w) (see [An,Proposition 8] or [Ha,Proposition 3.1.3]).…”

Section: Cohomology and Support Varietiesmentioning

confidence: 99%

Conjectures on tilting modules and antispherical $p$-cells

Achar,

Hardesty,

Riche

2018

Preprint

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For quantum groups at a root of unity, there is a web of theorems (due to Bezrukavnikov and Ostrik, and relying on work of Lusztig) connecting the following topics: (i) tilting modules; (ii) vector bundles on nilpotent orbits; and (iii) Kazhdan-Lusztig cells in the affine Weyl group. In this paper, we propose a (partly conjectural) analogous picture for reductive algebraic groups over fields of positive characteristic, inspired by a conjecture of Humphreys.

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Generalized negligible morphisms and their tensor ideals

Heidersdorf

Wenzl

2022

Sel. Math. New Ser.

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We introduce a generalization of the notion of a negligible morphism and study the associated tensor ideals and thick ideals. These ideals are defined by considering deformations of a given monoidal category $${\mathcal {C}}$$ C over a local ring R. If the maximal ideal of R is generated by a single element, we show that any thick ideal of $${\mathcal {C}}$$ C admits an explicitly given modified trace function. As examples we consider various Deligne categories and the categories of tilting modules for a quantum group at a root of unity and for a semisimple, simply connected algebraic group in prime characteristic. We prove an elementary geometric description of the thick ideals in quantum type A and propose a similar one in the modular case.

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On support varieties and the Humphreys conjecture in type A

Abstract: Let G be a reductive algebraic group scheme defined over Fp and let G1 denote the Frobenius kernel of G. To each finite-dimensional G-module M , one can define the support variety VG 1 pM q, which can be regarded as a G-stable closed subvariety of the nilpotent cone.

Cited by 6 publications

References 21 publications

On the Humphreys Conjecture on Support Varieties of Tilting Modules

On the Humphreys Conjecture on Support Varieties of Tilting Modules

Conjectures on tilting modules and antispherical $p$-cells

Generalized negligible morphisms and their tensor ideals

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