Some observations on Karoubian complete strongly exceptional posets on the projective homogeneous varieties

Kaneda, Masaharu; Ye, Jiachen

doi:10.48550/arxiv.0911.2568

Cited by 3 publications

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“…s jr yields w = y −1 r w I , as desired. (5.4) Remark: This is a generalization of [KY,1.5], [K09] and [K12, 3.5]. In case λ = 0 we constructed for G of rank at most 2 [KY] or in case G = GL n+1 (k) and P is maximal parabolic such that G/P ≃ P n for any n ∈ N [K09] a Karoubian complete strongly exceptional sequence {E w | w ∈ W I } for the bounded derived category of coherent O G C /P C -modules out of G 1 Mod(L((w • 0) 0 ), soc ℓ(w)+1 ∇P (0)), where G C and P C are the groups over the complex number field corresponding to G and P , respectively.…”

Section: Thusmentioning

confidence: 97%

“…On the other hand, there is another intertwining homomorphism φ (4.7) Remark: This is a generalization of [KY,1.4] and [K09], where we found for G of rank at most 2 or in case G = GL n+1 (k) with P a maximal parabolic such that G/P ≃ P n for any n ∈ N that ℓℓ( ∇P (λ)) = ℓ(w I ) + 1 for p-regular λ ∈ Λ P . In fact, for G/P ≃ P n we computed ℓℓ( ∇P (λ)) for any λ ∈ Λ P in [K09, 2.3] dispensing with the Lusztig conjecture.…”

Section: Thus Fix a P-regular Weight λ And Putmentioning

confidence: 99%

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Loewy series of parabolically induced -Verma modules

Abe

Kaneda

2014

J. Inst. Math. Jussieu

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Assuming the Lusztig conjecture on the irreducible characters for reductive algebraic groups in positive characteristic p, which is now a theorem for large p, we show that the modules for their Frobenius kernels induced from the simple modules of p-regular highest weights for their parabolic subgroups are rigid and determine their Loewy series.Let G be a reductive algebraic group over an algebraically closed field k of positive characteristic p, P a parabolic subgroup of G, T a maximal torus of P , and G 1 (resp. P 1 ) the Frobenius kernel of G (resp. P ). In this paper we study the structure of G 1 Tmodules induced from the simple P 1 T -modules of p-regular highest weights. Thus our study goes parallel to parabolically induced Verma modules in characteristic 0. In case P is a Borel subgroup of G, assuming Lusztig's conjecture for the irreducible characters for G 1 T , which is now a theorem for large p thanks to [AJS] . We now show that the parabolically induced modules are also rigid and describe their Loewy series.To go into more details, let B be a Borel subgroup of P containing T , Λ the character group of B, R ⊂ Λ the root system of G relative to T , and R + the positive system of R such that the roots of B are −R + . We let R s denote the set of simple roots, and I a subset of R s such that the root subgroups U α of G associated to α ∈ I generate P together with B. Denote by∇ P the induction functor from the category of P 1 T -modules to the category of G 1 T -modules, and letL P (λ) denote the simple P 1 T -module of highest weight λ ∈ Λ. Our object of study is∇ P (L P (λ)). After stating some generalities in § §1 and 2, we specialize into the case where λ is p-regular, i.e., if α ∨ is the coroot of each root α and if ρ =

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Section: Thusmentioning

confidence: 97%

Section: Thus Fix a P-regular Weight λ And Putmentioning

confidence: 99%

Loewy series of parabolically induced -Verma modules

Abe

Kaneda

2014

J. Inst. Math. Jussieu

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“…Remark 6.2. Decomposition of F n * O Sp 4 /B for n = 1 was previously obtained in [20] (see also [19]) using results of [5] about the structure of G 1 T-socle series of the induced G 1 B-module ∇(0) = Ind G 1 B B k. Again, their methods are different from the one presented here. 6.4.…”

Section: 2mentioning

confidence: 99%

“…(1) G/B -mod (see [17]). Representation-theoretic approach to G 1 B-structure of F * O G/B is worked out in [20] and is based on the study of G 1 T (or G 1 B)-structure of the induced G 1 B-module ∇(0) := Ind G 1 B B χ 0 (the so-called Humphreys-Verma module). It is generally believed (see loc.cit.)…”

Section: Introductionmentioning

confidence: 99%

The Frobenius morphism on flag varieties, I

Samokhin

2014

Preprint

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In this paper, given a semisimple algebraic group G of rank 2, we construct a special semiorthogonal decomposition in the derived category of coherent sheaves on the flag variety G/B. These decompositions are defined over the localization ZS, where S is the set of bad primes for G, while their block structure is compatible with the Bruhat order on Schubert varieties. The nonstandard t-structures on D b (G/B) defined by these decompositions are self-dual with respect to the duality RHom G/B (−, ω 1 2 G/B ) given by the square root of the canonical sheaf of G/B. For the groups of classical type, this allows to construct an explicit decomposition of the higher Frobenii pushforward bundles F n * O G/B into a direct sum of indecomposable bundles. When p > h, the Coxeer number of the corresponding group, this set of indecomposable bundles forms a full exceptional collection in D b (G/B) defined over ZS.

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“…Our approach to finding Frobenius summands is different from the methods used in some earlier papers (e.g. [32,34] and [48]) and also from [61] which uses the results from [4,5] (see §1.5).…”

Section: That This Exceptionalmentioning

confidence: 99%

The Frobenius morphism in invariant theory

Raedschelders

Špenko

Bergh

2019

Advances in Mathematics

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Let R be the homogeneous coordinate ring of the Grassmannian G = Gr(2, n) defined over an algebraically closed field k of characteristic p ≥ max{n − 2, 3}. In this paper we give a description of the decomposition of R, considered as graded R p r -module, for r ≥ 2. This is a companion paper to [16], where the case r = 1 was treated, and taken together, our results imply that R has finite F-representation type (FFRT). Though it is expected that all rings of invariants for reductive groups have FFRT, ours is the first non-trivial example of such a ring for a group which is not linearly reductive. As a corollary, we show that the ring of differential operators D k (R) is simple, that G has global finite F-representation type (GFFRT) and that R provides a noncommutative resolution for R p r .

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Some observations on Karoubian complete strongly exceptional posets on the projective homogeneous varieties

Cited by 3 publications

References 0 publications

Loewy series of parabolically induced -Verma modules

Loewy series of parabolically induced -Verma modules

The Frobenius morphism on flag varieties, I

The Frobenius morphism in invariant theory

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