Some computations involving simple Lie algebras

Burgoyne, N.; Williamson, C.

doi:10.1145/800204.806283

Cited by 20 publications

(38 citation statements)

References 4 publications

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“…By the general representation theory (Gilkey and Seitz [25]; Burgoyne and Williamson [10]) we have that dim V λ = µ<λ W W µ m µ , where µ ranges over the weights subdominant to λ (that is, λ − µ is a positive integral combination of positive roots and µ is a positive integral combination of the λ i ), m µ is the multiplicity of the weight µ, W is the Weyl group of G = Sym 3 in this case), and W µ is the stabilizer of µ in W . Now a result of Suprunenko [55], as quoted and extended by Premet [47] shows that m µ = 0 for any subdominant weight µ.…”

Section: Visible Flatness For Irreducible Modules Of Groups With Bn-pairmentioning

confidence: 99%

The Depth of Invariant Rings and Cohomology

Kemper

2001

Journal of Algebra

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Let G be a finite group acting linearly on a vector space V over a field K of positive characteristic p and let P ≤ G be a Sylow p-subgroup. Ellingsrud and Skjelbred [Compositio Math. 41 (1980), 233-244] proved the lower boundfor the depth of the invariant ring, with equality if G is a cyclic p-group. Let us call the pair G V flat if equality holds in the above. In this paper we use cohomological methods to obtain information about the depth of invariant rings and in particular to study the question of flatness. For G of order not divisible by p 2 it ensues that G V is flat if and only if H 1 G K V = 0 or dim V P + 2 ≥ dim V . We obtain a formula for the depth of the invariant ring in the case that G permutes a basis of V and has order not divisible by p 2 . In this situation G V is usually not flat. Moreover, we introduce the notion of visible flatness of pairs G V and prove that this implies flatness. For example, the groups SL 2 q SO 3 q SU 3 q Sz q , and R q with many interesting representations in defining characteristic are visibly flat. In particular, if G = SL 2 q and V is the space of binary forms of degree n or a direct sum of such spaces, then G V is flat for all q = p r with the exception of a finite number of primes p. Along the way, we obtain results about the Buchsbaum property of invariant rings and about the depth of the cohomology modules H i G K V . We also determine the support of the positive cohomology H + G K V as a module over K V G . In the Appendix, the visibly flat pairs G V of a group with BN-pair and an irreducible representation are classified.

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Section: Visible Flatness For Irreducible Modules Of Groups With Bn-pairmentioning

confidence: 99%

The Depth of Invariant Rings and Cohomology

Kemper

2001

Journal of Algebra

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“…The author does not know any details of the program [2], but in view of the remarks above and what is explained in [1] and [2], it seems that our algorithm is crucially more efficient, particularly when p is small.…”

Section: Comparing the Two Methodsmentioning

confidence: 95%

“…The method described in [1] and [2] uses commutation rules in ~ and brings p in only at the end. So it yields simultaneously information in any characteristic, but the computations become accordingly laborious.…”

Section: Comparing the Two Methodsmentioning

confidence: 99%

Computation of simple characters of a Chevalley group

Koppinen

1986

BIT

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“…In the case when all roots of • have the same length (the 'simply laced' case), only these two possibilities occur. Thus, in this case, the coefficients in the Chevalley formula are completely determined by the structure constants in the corresponding Lie algebra (and, therefore, can be found in the tables [36,60,87,88,142], see the details below). For the general case, the coefficients Naoij are expressed via the constants M,~i, mentioned in Section 2.…”

Section: Chevalley Commutator Formulamentioning

confidence: 99%

Chevalley groups over commutative rings: I. Elementary calculations

1996

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This is the first in a series of papers dedicated to the structure of Chevalley groups over commutative rings. The goal of this series is to systematically develop methods of calculations in Chevalley groups over rings, based on the use of their minimal modules. As an application, we give new direct proofs for normality of the elementary subgroup, description of normal subgroups and similar results due to E. Abe, G. Taddei, L. N. Vaserstein, and others, as well as some generalizations. In this first part we outline the whole project, reproduce construction of Chevalley groups and their elementary subgroups, recall familiar facts about the elementary calculations in these groups, and fix a specific choice of the structure constants. (1991): 20G35, 20G15. Mathematics Subject Classifications

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Some computations involving simple Lie algebras

Cited by 20 publications

References 4 publications

The Depth of Invariant Rings and Cohomology

The Depth of Invariant Rings and Cohomology

Computation of simple characters of a Chevalley group

Chevalley groups over commutative rings: I. Elementary calculations

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