Abstract:
We consider the simply connected Chevalley group G(E6,R)G(E6,R) of type E6E6 in a 27-dimensional representation. The main goal is to establish that the following four groups coincide: the normalizer of the Chevally group G(E6,R)G(E6,R) itself, the normalizer of its elementary subgroup E(E6,R)E(E6,R), the transporter of E(E6,R)E(E6,R) in G(E6,R)G(E6,R), and the extended Chevalley group ¯G(E6,R)¯¯¯¯G(E6,R). This is true over an arbitrary commutative ring RR, all normalizers and transporters being taken in GL(27,R)GL(27,R). Moreover, ¯G(E6,R)¯¯¯¯G(E6,R) is characterized as the stabilizer of a system of quadrics. This result is classically known over algebraically closed fields; in the paper it is established that the corresponding scheme over Z is smooth, which implies that the above characterization is valid over an arbitrary commutative ring. As an application of these results, we explicitly list equations a matrix g∈GL(27,R) must satisfy in order to belong to ¯G(E6,R). These results are instrumental in a subsequent paper of the authors, where overgroups of exceptional groups in minimal representations will be studied.
Keywords:
Chevalley groups, elementary subgroups, normal subgroups, standard description, minimal module, parabolic subgroups, decomposition of unipotents, root elements, orbit of the highest weight vector, the proof from the Book.
Citation:
N. A. Vavilov, A. Yu. Luzgarev, “The normalizer of Chevalley groups of type E6”, Algebra i Analiz, 19:5 (2007), 37–64; St. Petersburg Math. J., 19:5 (2008), 699–718
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\paper The normalizer of Chevalley groups of type $\mathrm{E}_6$
\jour Algebra i Analiz
\yr 2007
\vol 19
\issue 5
\pages 37--64
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\jour St. Petersburg Math. J.
\yr 2008
\vol 19
\issue 5
\pages 699--718
\crossref{https://doi.org/10.1090/S1061-0022-08-01016-9}
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Linking options:
https://www.mathnet.ru/eng/aa135
https://www.mathnet.ru/eng/aa/v19/i5/p37
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Roman Lubkov, Ilia Nekrasov, “Overgroups of exterior powers of an elementary group. levels”, Linear and Multilinear Algebra, 72:4 (2024), 563
R. A. Lubkov, “Nadgruppy elementarnykh grupp v polivektornykh predstavleniyakh”, Voprosy teorii predstavlenii algebr i grupp. 40, Posvyaschaetsya pamyati Nikolaya Aleksandrovicha VAVILOVA, Zap. nauchn. sem. POMI, 531, POMI, SPb., 2024, 101–116
R. A. Lubkov, “Obratnoe razlozhenie unipotentov v polivektornykh predstavleniyakh”, Voprosy teorii predstavlenii algebr i grupp. 38, Zap. nauchn. sem. POMI, 513, POMI, SPb., 2022, 120–138
N. Vavilov, V. Migrin, “Colourings of exceptional uniform polytopes of types E6 and E7”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye metody. XXXIV, Zap. nauchn. sem. POMI, 517, POMI, SPb., 2022, 36–54
N. A. Vavilov, Z. Zhang, “Relative Centralizers of Relative Subgroups”, J Math Sci, 264:1 (2022), 4
Lubkov R., “The Reverse Decomposition of Unipotents For Bivectors”, Commun. Algebr., 49:10 (2021), 4546–4556
N. A. Vavilov, Z. Zhang, “Relative centralisers of relative subgroups”, Voprosy teorii predstavlenii algebr i grupp. 35, Zap. nauchn. sem. POMI, 492, POMI, SPb., 2020, 10–24
J. Math. Sci. (N. Y.), 243:4 (2019), 515–526
R. A. Lubkov, I. I. Nekrasov, “Explicit equations for exterior square of the general linear group”, J. Math. Sci. (N. Y.), 243:4 (2019), 583–594
M. M. Atamanova, A. Yu. Luzgarev, “Cubic forms on adjoint representations of exceptional groups”, J. Math. Sci. (N. Y.), 222:4 (2017), 370–379
J. Math. Sci. (N. Y.), 209:6 (2015), 922–934
N. A. Vavilov, A. Yu. Luzgarev, “Normaliser of the Chevalley group of type E7”, St. Petersburg Math. J., 27:6 (2016), 899–921
J. Math. Sci. (N. Y.), 219:3 (2016), 355–369
N. A. Vavilov, A. Yu. Luzgarev, “Chevalley group of type E7 in the 56-dimensional representation”, J. Math. Sci. (N. Y.), 180:3 (2012), 197–251
I. M. Pevzner, “Width of groups of type E6 with respect to root elements. II”, J. Math. Sci. (N. Y.), 180:3 (2012), 338–350
I. M. Pevzner, “The geometry of root elements in groups of type E6”, St. Petersburg Math. J., 23:3 (2012), 603–635
A. S. Ananyevskiy, N. A. Vavilov, S. S. Sinchuk, “Overgroups of E(m,R)⊗E(n,R). I”, St. Petersburg Math. J., 23:5 (2012), 819–849
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