Extensions for finite Chevalley groups I

Bendel, Christopher P.; Nakano, Daniel K.; Pillen, Cornelius

doi:10.1016/s0001-8708(03)00091-4

Cited by 12 publications

(17 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, in that case, there are no self-extensions by [AJL,BNP2]. For r = 1, we apply Theorem 2.1(c) and obtain…”

Section: We Now Present Some Other Consequences Of Theorem 53(a)mentioning

confidence: 99%

“…The methods used in the following proposition are similar to those in Sections 4.5 and 4.6 of [BNP2].…”

Section: In This Section We Determine Conditions Under Which Extmentioning

confidence: 99%

“…The proof uses the same strategy as the proof of Proposition 4.6 in [BNP2], although the details differ slightly.…”

Section: In This Section We Determine Conditions Under Which Extmentioning

confidence: 99%

“…Using the notation of Section 2.1, we can now prove a generalization of Corollary 3.3 in [BNP2] and give sufficient conditions to insure that Ext…”

Section: Extensions For Finite Chevalley Groups 4445mentioning

confidence: 99%

“…The simple modules for G are labeled by the set X(T ) + and are given by the correspondence λ → L(λ) = soc G H 0 (λ). A complete set of nonisomorphic simple G r -modules and simple G (F q Finally, we make use of the following sets of weights and modules, which were used in [BNP1,BNP2]. For each s with 0 ≤ s ≤ p − 1, let π s,r = {λ ∈ X(T ) + : λ + ρ, α ∨ 0 < 2sp r ρ, α ∨ 0 }.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Extensions for finite Chevalley groups II

Bendel

Nakano

Pillen

2002

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. Let G be a semisimple simply connected algebraic group defined and split over the field Fp with p elements, let G(Fq) be the finite Chevalley group consisting of the Fq-rational points of G where q = p r , and let Gr be the rth Frobenius kernel. The purpose of this paper is to relate extensions between modules in Mod(G(Fq)) and Mod(Gr) with extensions between modules in Mod(G). Among the results obtained are the following: for r > 2 and p ≥ 3(h − 1), the G(Fq)-extensions between two simple G(Fq)-modules are isomorphic to the G-extensions between two simple p r -restricted G-modules with suitably "twisted" highest weights. For p ≥ 3(h − 1), we provide a complete characterization of H 1 (G(Fq), H 0 (λ)) where H 0 (λ) = ind G B λ and λ is p r -restricted. Furthermore, for p ≥ 3(h − 1), necessary and sufficient bounds on the size of the highest weight of a G-module V are given to insure that the restriction mapFinally, it is shown that the extensions between two simple p r -restricted G-modules coincide in all three categories provided the highest weights are "close" together.

show abstract

“…However, in that case, there are no self-extensions by [AJL,BNP2]. For r = 1, we apply Theorem 2.1(c) and obtain…”

Section: We Now Present Some Other Consequences Of Theorem 53(a)mentioning

confidence: 99%

“…The methods used in the following proposition are similar to those in Sections 4.5 and 4.6 of [BNP2].…”

Section: In This Section We Determine Conditions Under Which Extmentioning

confidence: 99%

“…The proof uses the same strategy as the proof of Proposition 4.6 in [BNP2], although the details differ slightly.…”

Section: In This Section We Determine Conditions Under Which Extmentioning

confidence: 99%

“…Using the notation of Section 2.1, we can now prove a generalization of Corollary 3.3 in [BNP2] and give sufficient conditions to insure that Ext…”

Section: Extensions For Finite Chevalley Groups 4445mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Extensions for finite Chevalley groups II

Bendel

Nakano

Pillen

2002

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

Extensions for finite Chevalley groups III: Rational and generic cohomology

Bendel

Nakano

Pillen

2014

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

Abstract. Let G be a connected reductive algebraic group and B be a Borel subgroup defined over an algebraically closed field of characteristic p > 0. In this paper, the authors study the existence of generic G-cohomology and its stability with rational G-cohomology groups via the use of methods from the authors' earlier work. New results on the vanishing of G and B-cohomology groups are presented. Furthermore, vanishing ranges for the associated finite group cohomology of G(Fq) are established which generalize earlier work of Hiller, in addition to stability ranges for generic cohomology which improve on seminal work of Cline, Parshall, Scott and van der Kallen.

show abstract

Second cohomology for finite groups of Lie type

2012

Journal of Algebra

View full text Add to dashboard Cite

Let $G$ be a simple, simply-connected algebraic group defined over $\mathbb{F}_p$. Given a power $q = p^r$ of $p$, let $G(\mathbb{F}_q) \subset G$ be the subgroup of $\mathbb{F}_q$-rational points. Let $L(\lambda)$ be the simple rational $G$-module of highest weight $\lambda$. In this paper we establish sufficient criteria for the restriction map in second cohomology $H^2(G,L(\lambda)) \rightarrow H^2(G(\mathbb{F}_q),L(\lambda))$ to be an isomorphism. In particular, the restriction map is an isomorphism under very mild conditions on $p$ and $q$ provided $\lambda$ is less than or equal to a fundamental dominant weight. Even when the restriction map is not an isomorphism, we are often able to describe $H^2(G(\mathbb{F}_q),L(\lambda))$ in terms of rational cohomology for $G$. We apply our techniques to compute $H^2(G(\mathbb{F}_q),L(\lambda))$ in a wide range of cases, and obtain new examples of nonzero second cohomology for finite groups of Lie type.Comment: 29 pages, GAP code included as an ancillary file. Rewritten to include the adjoint representation in types An, B2, and Cn. Corrections made to Theorem 3.1.3 and subsequent dependent results in Sections 3-4. Additional minor corrections and improvements also implemente

show abstract

Extensions for finite Chevalley groups I

Cited by 12 publications

References 13 publications

Extensions for finite Chevalley groups II

Extensions for finite Chevalley groups II

Extensions for finite Chevalley groups III: Rational and generic cohomology

Second cohomology for finite groups of Lie type

Contact Info

Product

Resources

About