Subgroups of Chevalley groups of types $ B_l$ and $ C_l$ containing the group over a subring, and corresponding carpets

Ya. N. Nuzhin ; A. V. Stepanov

Algebra i analiz, Tome 31 (2019) no. 4, pp. 198-224.

Résumé

This is a continuation of the study of subgroups of the Chevalley group $ G_P(\Phi ,R)$ over a ring $ R$ with root system $ \Phi $ and weight lattice $ P$ that contain the elementary subgroup $ E_P(\Phi ,K)$ over a subring $ K$ of $ R$. A. Bak and A. V. Stepanov considered recently the case of the symplectic group (simply connected group with root system $ \Phi =C_l$) in characteristic $2$. In the current article, that result is extended to the case of $ \Phi =B_l$ and for the groups with other weight lattices. Like in the Ya. N. Nuzhin's work on the case where $ R$ is an algebraic extension of a nonperfect field $ K$ and $ \Phi $ is not simply laced, the description involves carpet subgroups parametrized by two additive subgroups. In the second part of the article, the Bruhat decomposition is established for these carpet subgroups and it is proved that they have a split saturated Tits system. As a corollary, it is shown that they are simple as abstract groups.

Détail

Keywords: classical groups, subgroup lattice, carpet subgroups, Bruhat decomposition.

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