Extensions of Simple Modules for Special Algebraic Groups

Sin, Peter

doi:10.1006/jabr.1994.1373

Cited by 10 publications

(18 citation statements)

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“…As usual, we list the factors in a good filtration of H°(u) ® H°(v'), narrowing down the possibilities for the Gsocles of L[v) ® L(v'). These are listed in Table III The final answer can then be obtained by straightforward calculations in all cases except the following: L^) ® L ( 8 2 ) , L(S2) ® L(S\ + 83 + 84) (which were already computed in [18]) and L(S 2 ) ® L(&2 + 83 + 84), which is handled in the next subsection along with a couple of other miscellaneous lemmas about tensor product socles which are needed for the computation of the Ext)) structures. For future reference, we also record a table (Table III.A.6) of composition factors of certain Weyl modules which will be used throughout the paper.…”

Section: J ( 2 "')mentioning

confidence: 99%

“…The goal of this chapter is to compute the extensions of simple modules for the simply connected semisimple algebraic groups of types An, B±, C\, and Di and for their Lie algebras. The calculations for F4 were carried out in [18]. Much of the effort in this chapter is directed towards the computation of the quantities E -Ext Gi (L(A°), L((i 0 )) for all pairs of 2-restricted highest weights (A°,p°) in the case where G is simply connected of type A4 or D4.…”

Section: Extensions For G Tmentioning

confidence: 99%

“…The following nonzero Ext)^ modules for D 4 were calculated in [18]: a) ExtJ,,(*,L{6 2 )) =2k® i ( 2^) © L{25 3 so that the unique composition factor of V{8 2 + 83 + 8 4 ) isomorphic to L(J 3 -f 8 4 ) must equal rad 2 (V(<5 2 + S 3 + £4)), i.e.,…”

Section: Extensions For (D 4 )Imentioning

confidence: 99%

“…The central theme will be the study of some intimate connections among the groups of types Bi, Ci and Di (and F4 when I = 4). We also give calculations for those other groups of rank 4 or less which have not already been treated elsewhere ( [1], [18]), but this is primarily for the sake of completeness.…”

Section: Introductionmentioning

confidence: 99%

“…In [18], similar isogenies for the groups of types Bi and F4 in characteristic 2 and type G2 in characteristic 3 were exploited to compute the simple module extensions. The kernel of the differential of the special isogeny for F4 also happens to be isomorphic to a quotient of a Lie algebra of type D4, and this link made it possible to draw on knowledge about D4 in making calculations about Fi.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On Representations of Algebraic Groups in Characteristic Two

Dowd

Sin

1996

Communications in Algebra

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Section: J ( 2 "')mentioning

confidence: 99%

Section: Extensions For G Tmentioning

confidence: 99%

Section: Extensions For (D 4 )Imentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Representations of Algebraic Groups in Characteristic Two

Dowd

Sin

1996

Communications in Algebra

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Bounding Cohomology for Finite Groups and Frobenius Kernels

Bendel

Nakano

Parshall

et al. 2015

Algebr Represent Theor

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Let G be a simple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p. Let σ : G → G be a strict endomorphism (i. e., the subgroup G(σ) of σ-fixed points is finite). Also, let Gσ be the scheme-theoretic kernel of σ, an infinitesimal subgroup of G. This paper shows that the degree m cohomology H m (G(σ), L) of any irreducible kG(σ)-module L is bounded by a constant depending on the root system Φ of G and the integer m. A similar result holds for the degree m cohomology of Gσ. These bounds are actually established for the degree m extension groups Ext m G(σ) (L, L ′ ) between irreducible kG(σ)-modules L, L ′ , with again a similar result holding for Gσ. In these Ext m results, of interest in their own right, the bounds depend also on L, or, more precisely, on length of the p-adic expansion of the highest weight associated to L. All bounds are, nevertheless, independent of the characteristic p. These results extend earlier work of Parshall and Scott for rational representations of algebraic groups G.We also show that one can find bounds independent of the prime for the Cartan invariants of G(σ) and Gσ, and even for the lengths of the underlying PIMs. These bounds, which depend only on the root system of G and the "height" of σ, provide in a strong way an affirmative answer to a question of Hiss, for the special case of finite groups G(σ) of Lie type in the defining characteristic.

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Bounding Ext for modules for algebraic groups, finite groups and quantum groups

Parshall

Scott

2011

Advances in Mathematics

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Given a finite root system Φ, we show that there is an integer c = c(Φ) such that dim Ext 1 G (L, L ′ ) < c, for any reductive algebraic group G with root system Φ and any irreducible rational G-modules L, L ′ . There also is such a bound in the case of finite groups of Lie type, depending only on the root system and not on the underlying field. For quantum groups, a similar result holds for Ext n , for any integer n ≥ 0, using a constant depending only on n and the root system. Weaker versions of this are proved in the algebraic and finite group cases, sufficient to give similar results for algebraic and generic cohomology. The results both use, and have consequences for, Kazhdan-Lusztig polynomials. An appendix proves a stable version, needed for small prime arguments, of Donkin's tilting module conjecture.

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Extensions of Simple Modules for Special Algebraic Groups

Cited by 10 publications

References 0 publications

On Representations of Algebraic Groups in Characteristic Two

On Representations of Algebraic Groups in Characteristic Two

Bounding Cohomology for Finite Groups and Frobenius Kernels

Bounding Ext for modules for algebraic groups, finite groups and quantum groups

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