Chevalley groups over commutative rings: I. Elementary calculations

Vavilov, N. A.; Plotkin, Eugene

doi:10.1007/bf00047884

Cited by 102 publications

(126 citation statements)

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“…The following charming little argument is the primary novel turn of the present proof as compared with the proofs in the papers [15,65,67] Proof. 1) Take any two distinct weights µ, ν ∈ Λ such that d(λ, µ) = d(λ, ν) = 1, and consider the root element…”

Section: Proposition 6 (Extraction From a Proper Parabolic) If H Conmentioning

confidence: 95%

“…Namely, we invoke the fact that the standard commutator formulas are satisfied in G(Φ, R) for all ideals I R. In the case of exceptional groups, this result was first proved in full generality by Taddei and Vaserstein [58,61] (see also [15,65,67,43]). In principle, this theorem as well could be demonstrated by the methods of the present paper.…”

Section: Proposition 1 (Level Reduction) Assume That For Any Pair (Rmentioning

confidence: 99%

“…The references [4,21,37] may serve as excellent generic sources on Chevalley groups and their representations. All more technical definitions related to linear groups over rings can be found in [42,56] and those related to Chevalley groups over rings can be found in [25]- [30], [47,54,55,58,61], [65]- [67], and [69,70]. We refer the reader to the same sources also for comprehensive bibliography.…”

Section: §3 Basic Notationmentioning

confidence: 99%

“…Thus, it would be only natural to call these proofs a 'D 5 -proof', a 'D 6 -proof', and a 'D 8 -proof', respectively. In [65,69,56], a thorough description of the entire project can be found. We see that every time one uses the highest rank subsystem of type A l or D l among all such subsystems contained in Φ.…”

Section: §1 Introductionmentioning

confidence: 99%

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An $\mathrm {A}_2$-proof of structure theorems for Chevalley groups of types $\mathrm {E}_6$ and $\mathrm {E}_7$

Vavilov

Gavrilovich

2005

St. Petersburg Math. J.

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In the present paper we prove the main structure theorem for Chevalley groups G = G(Φ, R) of types Φ = E 6 , E 7 over a commutative ring R. More precisely, we describe subgroups in G normalized by the elementary subgroup E(Φ, R). This result is not new, since structure theorems are known for all Chevalley groups [25,27,28,30], [38]-[40], and [58,61] (see [42,65,34,56] for further references). The gist of the present paper resides not in the results themselves, but rather in the method of their proof based on the geometry of exceptional groups. We believe that this method is novel and of significant interest. Actually the Schwerpunkt of the present paper abides in a new descent procedure, which enables reduction to groups of smaller rank. This procedure is both simpler and more powerful than any other method known today. Our results on the geometry of the 27-dimensional module for Chevalley groups of type E 6 and of the 56-dimensional module for Chevalley groups of type E 7 pave the way to much more general results such as description of subgroups normalized by some of elementary matrices. Groups of types E 8 and F 4 can be handled in essentially the same style, and we intend to return to these cases in our subsequent publications. However, from a technical viewpoint the proofs in these cases are noticeably more involved because these groups do not have microweight representations. §1. IntroductionThe proofs of structure theorems are based either on reduction of the dimension of the ground ring, or on reduction of the rank of the group, or eventually on a combination of both. The early proofs for classical groups were based on reduction of rank [71], [17]-[19], and [23,24,20,3,9]; the dimension reduction proofs appeared later [60], [62]-[64], and [44, 45, 33]. At the same time for exceptional groups the first published proofs leant upon reduction of dimension (see [25,27,28,30,58,61]). The paper [30] by E. Abe and K. Suzuki was a near miss as far as the normal structure theorem for Chevalley groups is concerned (see also [25]). In fact, a somewhat weaker result was proved in [30], namely, a description of the normal subgroups of the elementary Chevalley group E(Φ, R) over a class of rings subject to a certain finiteness condition. However, as was

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Section: Proposition 6 (Extraction From a Proper Parabolic) If H Conmentioning

confidence: 95%

Section: Proposition 1 (Level Reduction) Assume That For Any Pair (Rmentioning

confidence: 99%

Section: §3 Basic Notationmentioning

confidence: 99%

Section: §1 Introductionmentioning

confidence: 99%

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An $\mathrm {A}_2$-proof of structure theorems for Chevalley groups of types $\mathrm {E}_6$ and $\mathrm {E}_7$

Vavilov

Gavrilovich

2005

St. Petersburg Math. J.

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“…The constants N αβij do not depend on ξ and η, but, in general, they may depend on the order. The integers N αβij are called the structure constants of the Chevalley group; we refer the reader to [43,72,76,81,82] for the details and further references.…”

Section: §2 Chevalley Groupsmentioning

confidence: 99%

Weight elements of Chevalley groups

Vavilov

2008

St. Petersburg Math. J.

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Abstract. The paper is devoted to a detailed study of some remarkable semisimple elements of (extended) Chevalley groups that are diagonalizable over the ground field -the weight elements. These are the conjugates of certain semisimple elements h ω (ε) of extended Chevalley groups G = G(Φ, K), where ω is a weight of the dual root system Φ ∨ and ε ∈ K * . In the adjoint case the h ω (ε)'s were defined by Chevalley himself and in the simply connected case they were constructed by Berman and Moody. The conjugates of h ω (ε) are called weight elements of type ω. Various constructions of weight elements are discussed in the paper, in particular, their action in irreducible rational representations and weight elements induced on a regularly embedded Chevalley subgroup by the conjugation action of a larger Chevalley group. It is proved that for a given x ∈ G all elements x(ε) = xh ω (ε)x −1 , ε ∈ K * , apart maybe from a finite number of them, lie in the same Bruhat coset BwB, where w is an involution of the Weyl group W = W (Φ). The elements h ω (ε) are particularly important when ω = i is a microweight of Φ ∨ . The main result of the paper is a calculation of the factors of the Bruhat decomposition of microweight elements x(ε) for the case where ω = i . It turns out that all nontrivial x(ε)'s lie in the same Bruhat coset BwB, where w is a product of reflections in pairwise strictly orthogonal roots γ 1 , . . . , γ r+s . Moreover, if among these roots r are long and s are short, then r + 2s does not exceed the width of the unipotent radical of the ith maximal parabolic subgroup in G. A version of this result was first announced in a paper by the author in Soviet Mathematics: Doklady in 1988. From a technical viewpoint, this amounts to the determination of Borel orbits of a Levi factor of a parabolic subgroup with Abelian unipotent radical and generalizes some results of Richardson, Röhrle, and Steinberg. These results are instrumental in the description of overgroups of a split maximal torus and in the recent papers by the author and V. Nesterov on the geometry of tori.

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