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Journal of Mathematical Sciences

, Volume 192, Issue 2, pp 164–195 | Cite as

Overgroups of Subsystem Subgroups in Exceptional Groups: Levels

  • N. A. Vavilov
  • A. V. Shchegolev
Article

An embedding of root systems ∆ ⊆ Φ determines the corresponding regular embedding G(∆, R)≤ G(Φ, R) of Chevalley groups, over an arbitrary commutative ring R. Denote by E(∆, R) the elementary subgroup of G(∆, R). In the present paper we initiate the study of intermediate subgroups H, E∆, R) ≤ HG(Φ, R), provided that Φ=E6, E7, E8, F4 or G2, and there are no roots in Φ orthogonal to all of ∆. There are 72 such pairs (Φ, Δ)$. For F4 and G2 we assume, moreover, that 2 ∈ R * or 6 ∈ R *, respectively. For all such subsystems Δ we construct the levels of intermediate subgroups. We prove that these levels are determined by certain systems of ideals in R, one for each Δ-equivalence class of roots in Φ\∆, and calculate all relations among these ideals, in each case. Bibliography: 64 titles.

Keywords

Root System Commutative Ring Chevalley Group Exceptional Group Elementary Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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