Abstract:
We consider the simply connected Chevalley group G(E6,R) of type E6 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) itself, the normalizer of its elementary subgroup E(E6,R), the transporter of E(E6,R) in G(E6,R), and the extended Chevalley group ¯G(E6,R). This is true over an arbitrary commutative ring R, all normalizers and transporters being taken in GL(27,R). Moreover, ¯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
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\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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Elena Bunina, “Automorphisms of Chevalley groups over commutative rings”, Communications in Algebra, 52:6 (2024), 2313
Anneleen De Schepper, Jeroen Schillewaert, Hendrik Van Maldeghem, Magali Victoor, “Construction and characterisation of the varieties of the third row of the Freudenthal–Tits magic square”, Geom Dedicata, 218:1 (2024)
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 $\mathrm{E}_6$ and $\mathrm{E}_7$”, 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 $\mathrm E_7$”, 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 $\mathrm E_7$ in the 56-dimensional representation”, J. Math. Sci. (N. Y.), 180:3 (2012), 197–251
I. M. Pevzner, “Width of groups of type $\mathrm E_6$ 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 $\mathrm E_6$”, St. Petersburg Math. J., 23:3 (2012), 603–635
A. S. Ananyevskiy, N. A. Vavilov, S. S. Sinchuk, “Overgroups of $E(m,R)\otimes E(n,R)$. I”, St. Petersburg Math. J., 23:5 (2012), 819–849
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