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 Zap. Nauchn. Sem. POMI, 2012, Volume 400, Pages 70–126 (Mi znsl5612)

Overgroups of subsystem subgroups in exceptional groups: levels

N. A. Vavilov, A. V. Shchegolev

Saint-Petersburg State University, Saint-Petersburg, Russia

Abstract: An embedding of root systems $\Delta\subseteq\Phi$ determines the corresponding regular embedding $G(\Delta,R)\le G(\Phi,R)$ of Chevalley groups, over an arbitrary commutative ring $R$. Denote by $E(\Delta,R)$ the elementary subgroup of $G(\Delta,R)$. In the present paper we initiate the study of intermediate subgroups $H$, $E(\Delta,R)\le H\le G(\Phi,R)$, provided that $\Phi=\mathrm{E_6,E_7,E_8,F}_4$ or $\mathrm G_2$, and there are no roots in $\Phi$ orthogonal to all of $\Delta$. There are 72 such pairs $(\Phi,\Delta)$. For $\mathrm F_4$ and $\mathrm G_2$ we assume, moreover, that $2\in R^*$ or $6\in R^*$, respectively. For all such subsystems $\Delta$ we construct the levels of intermediate subgroups. We prove that these levels are detemined by certain systems of ideals in $R$, one for each $\Delta$-equivalence class of roots in $\Phi\setminus\Delta$, and calculate all relations among these ideals, in each case.

Key words and phrases: exceptional Chevalley groups, subsystem subgroups, levels, root elements, Chevalley commutator formula, shapes of roots.

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English version:
Journal of Mathematical Sciences (New York), 2013, 192:2, 164–195

Bibliographic databases:

Document Type: Article
UDC: 513.6

Citation: N. A. Vavilov, A. V. Shchegolev, “Overgroups of subsystem subgroups in exceptional groups: levels”, Problems in the theory of representations of algebras and groups. Part 23, Zap. Nauchn. Sem. POMI, 400, POMI, St. Petersburg, 2012, 70–126; J. Math. Sci. (N. Y.), 192:2 (2013), 164–195

Citation in format AMSBIB
\Bibitem{VavShc12} \by N.~A.~Vavilov, A.~V.~Shchegolev \paper Overgroups of subsystem subgroups in exceptional groups: levels \inbook Problems in the theory of representations of algebras and groups. Part~23 \serial Zap. Nauchn. Sem. POMI \yr 2012 \vol 400 \pages 70--126 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl5612} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3029566} \transl \jour J. Math. Sci. (N. Y.) \yr 2013 \vol 192 \issue 2 \pages 164--195 \crossref{https://doi.org/10.1007/s10958-013-1382-x} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84884989403} 

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This publication is cited in the following articles:
1. N. A. Vavilov, “Decomposition of unipotents for $\mathrm E_6$ and $\mathrm E_7$: 25 years after”, Voprosy teorii predstavlenii algebr i grupp. 27, Zap. nauchn. sem. POMI, 430, POMI, SPb., 2014, 32–52    ; J. Math. Sci. (N. Y.), 219:3 (2016), 355–369
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