St. Petersburg Mathematical Journal

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1547-7371(e) ISSN 1061-0022(p)

Normalizer of the Chevalley group of type ${\mathrm E}_6$

Author(s): N. A. Vavilov; A. Yu. Luzgarev
Translated by: the authors
Original publication: Algebra i Analiz, tom 19 (2007), nomer 5.
Journal: St. Petersburg Math. J. 19 (2008), 699-718.
MSC (2000): Primary 20G15
Posted: June 25, 2008
MathSciNet review: 2381940
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We consider the simply connected Chevalley group $G(\operatorname{E}_6,R)$ of type $\operatorname{E}_6$ in a 27-dimensional representation. The main goal is to establish that the following four groups coincide: the normalizer of the Chevally group $G(\operatorname{E}_6,R)$ itself, the normalizer of its elementary subgroup $E(\operatorname{E}_6,R)$ , the transporter of $E(\operatorname{E}_6,R)$ in $G(\operatorname{E}_6,R)$ , and the extended Chevalley group $\overline G(\operatorname{E}_6,R)$ . This is true over an arbitrary commutative ring , all normalizers and transporters being taken in $\operatorname{GL}(27,R)$ . Moreover, $\overline G(\operatorname{E}_6,R)$ 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 $\mathbb{Z}$ 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 $g\in\operatorname{GL}(27,R)$ must satisfy in order to belong to $\overline G(\operatorname{E}_6,R)$ . These results are instrumental in a subsequent paper of the authors, where overgroups of exceptional groups in minimal representations will be studied.

References:

1.: E. Abe, Automorphisms of Chevalley groups over commutative rings, Algebra i Analiz 5 (1993), no. 2, 74-90; English transl., St. Petersburg Math. J. 5 (1994), no. 2, 287-300. MR 1223171 (94e:20062)
2.: A. Borel, Properties and linear representations of Chevalley groups, 1970 Seminar on Algebraic Groups and Related Finite Groups (Inst. Adv. Study, Princeton, NJ, 1968/1969), Lecture Notes in Math., vol. 131, Springer, Berlin, 1970, pp. 1-55. MR 0258838 (41:3484)
3.: N. Bourbaki, Lie groups and Lie algebras. Chapters 4-6, Springer-Verlag, Berlin, 2002. MR 1890629 (2003a:17001)
4.: -, Lie groups and Lie algebras. Chapters 7-9, Springer-Verlag, Berlin, 2005. MR 2109105 (2005h:17001)
5.: N. A. Vavilov, Weight elements of Chevalley groups, Dokl. Akad. Nauk SSSR 298 (1988), no. 3, 524-527; English transl., Soviet Math. Dokl. 37 (1988), no. 1, 92-95. MR 0925952 (88m:20093)
6.: -, Subgroups of Chevalley groups containing a maximal torus, Trudy Leningrad. Mat. Obshch. 1 (1990), 64-109; English transl., Amer. Math. Soc. Transl. (2), vol. 155, Amer. Math. Soc., Providence, RI, 1993, pp. 59-100. MR 1104207 (92f:20046)
7.: -, How to see the signs of structure constants, Algebra i Analiz 19 (2007), no. 4, 34-68; English transl. in St. Petersburg Math. J. 19 (2008), no. 4. MR 2381932
8.: -, Weight elements of Chevalley groups, Algebra i Analiz (to appear). (Russian)
9.: N. A. Vavilov and M. R. Gavrilovich, An ${\mathrm A}_2$ -proof of structure theorems for Chevalley groups of types ${\mathrm E}_6$ and ${\mathrm E}_7$ , Algebra i Analiz 16 (2004), no. 4, 54-87; English transl., St. Petersburg Math. J. 16 (2005), no. 4, 649-672. MR 2090851 (2005m:20115)
10.: N. A. Vavilov, M. R. Gavrilovich, and S. I. Nikolenko, The structure of Chevalley groups: A proof from the book, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. 330 (2006), 36-76; English transl., J. Math. Sci. (N.Y.) 140 (2007), no. 5, 626-645. MR 2253566 (2007k:20108)
11.: N. A. Vavilov, A. Yu. Luzgarev, and I. M. Pevzner, A Chevalley group of type ${\mathrm E}_6$ in the -dimensional representation, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 338 (2006), 5-68; English transl. in J. Math. Sci. (N.Y.) 145 (2007), 4697-4736. MR 2354606
12.: N. A. Vavilov and S. I. Nikolenko, ${\mathrm A}_2$ -proof of structure theorems for Chevalley groups of type ${\mathrm F}_4$ , Algebra i Analiz (to appear). (Russian)
13.: N. A. Vavilov and V. A. Petrov, Overgroups of elementary symplectic groups, Algebra i Analiz 15 (2003), no. 4, 72-114; English transl., St. Petersburg Math. J. 15 (2004), no. 4, 515-543. MR 2068980 (2005d:20086)
14.: -, On overgroups EO, Algebra i Analiz 19 (2007), no. 2, 10-51; English transl., St. Petersburg Math. J. 19 (2008), no. 2, 167-195. MR 2333895 (2008d:20094)
15.: N. A. Vavilov and E. Ya. Perel'man, Polyvector representations of GL, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 338 (2006), 69-97; English transl. in J. Math. Sci. (N.Y.) 145 (2007), 4737-4750. MR 2354607
16.: È. B. Vinberg, V. V. Gorbatsevich, and A. L. Onishchik, Structure of Lie groups and Lie algebras, Itogi Nauki i Tekhniki Sovrem. Probl. Mat. Fundam. Naprav., vol. 41, Lie Groups and Lie Algebras, III, VINITI, Moscow, 1990, pp. 5-259; English transl., Encyclopaedia Math. Sci, vol. 41, Springer-Verlag, Berlin, 1994, pp. 1-248. MR 1056486 (91j:17004); MR 1349140 (96d:22001)
17.: A. Yu. Luzgarev, On overgroups of ${\mathrm E}({\mathrm E}_6,R)$ and ${\mathrm E}({\mathrm E}_7,R)$ in minimal representations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 319 (2004), 216-243; English transl., J. Math. Sci. (N.Y.) 134 (2006), no. 6, 2558-2571. MR 2117858 (2006k:20098)
18.: N. S. Semenov, An argument in favor of the Hurwitz property of ${\mathrm G}_{\hbox{\rm sc}}({\mathrm E}_6,$ F, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 305 (2003), 228-237; English transl., J. Math. Sci. (N.Y.) 130 (2005), no. 3, 4768-4773. MR 2033644
19.: T. A. Springer, Linear algebraic groups, Itogi Nauki i Tekhniki Sovrem. Probl. Mat. Fundam. Naprav., vol. 55, Algebraic Geometry, IV, VINITI, Moscow, 1989, pp. 5-136; English transl., Encyclopaedia Math. Sci., vol. 55, Springer-Verlag, Berlin, 1994, pp. 1-121. MR 1100484 (92g:20061); MR 1309681 (95g:14002)
20.: R. Steinberg, Lectures on Chevalley groups, Yale Univ., New Haven, Conn., 1968. MR 0466335 (57:6215)
21.: J. E. Humphreys, Linear algebraic groups, Grad. Texts in Math., No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773 (53:633)
22.: -, Introduction to Lie algebras and representation theory, Grad. Texts in Math., No. 9, Springer-Verlag, New York-Berlin, 1978. MR 0499562 (81b:17007)
23.: C. Chevalley, Sur certains groupes simples, Tôhoku Math. J. (2) 7 (1955), 14-66. MR 0073602 (17:457c)
24.: E. Abe, Chevalley groups over local rings, Tôhoku Math. J. (2) 21 (1969), no. 3, 474-494. MR 0258837 (41:3483)
25.: -, Chevalley groups over commutative rings, Radical Theory (Sendai, 1988), Uchida Rokakuho, Tokyo, 1989, pp. 1-23. MR 0999577 (91a:20047)
26.: -, Normal subgroups of Chevalley groups over commutative rings, Algebraic $\mathit{K}$ -Theory and Algebraic Number Theory (Honolulu, HI, 1987), Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989, pp. 1-17. MR 0991973 (91a:20046)
27.: E. Abe and J. Hurley, Centers of Chevalley groups over commutative rings, Comm. Algebra 16 (1988), no. 1, 57-74. MR 0921942 (90e:20040)
28.: E. Abe and K. Suzuki, On normal subgroups of Chevalley groups over commutative rings, Tôhoku Math. J. (2) 28 (1976), no. 2, 185-198. MR 0439947 (55:12828)
29.: M. Aschbacher, The -dimensional module for ${\mathrm E}_6$ . I-IV, Invent. Math. 89 (1987), no. 1, 159-195; J. London Math. Soc. (2) 37 (1988), 275-293; Trans. Amer. Math. Soc. 321 (1990), 45-84; J. Algebra 191 (1990), 23-39. MR 0892190 (88h:20045); MR 0928524 (89a:20041); MR 0986684 (90m:20044); MR 1054997 (91f:20049)
30.: -, Some multilinear forms with large isometry groups, Geom. Dedicata 25 (1988), no. 1-3, 417-465. MR 0925846 (89c 20067)
31.: -, The geometry of trilinear forms, Finite Geometries, Buildings and Related Topics (Pingree Park, CO, 1988), Oxford Univ. Press, New York, 1990, pp. 75-84. MR 1072156 (91j:51005)
32.: K. Atsuyama, On the embedding of the Cayley plane into the exceptional Lie group of type F, Kodai Math. Sem. Rep. 28 (1976/77), 129-134. MR 0447469 (56:5781)
33.: A. Bak and N. Vavilov, Normality for elementary subgroup functors, Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 1, 35-47. MR 1329456 (96d:20046)
34.: -, Structure of hyperbolic unitary groups. I. Elementary subgroups, Algebra Colloq. 7 (2000), no. 2, 159-196. MR 1810843 (2002b:20070)
35.: -, Cubic form parameters (to appear).
36.: S. Berman and R. V. Moody, Extensions of Chevalley groups, Israel J. Math. 22 (1975), no. 1, 42-51. MR 0390077 (52:10903)
37.: R. B. Brown, Groups of type ${\mathrm E}_{7}$ , J. Reine Angew. Math. 236 (1969), no. 1, 79-102. MR 0248185 (40:1439)
38.: R. W. Carter, Simple groups of Lie type, Pure Appl. Math., vol. 28, Wiley, London, 1972. MR 0407163 (53:10946)
39.: A. M. Cohen and B. N. Cooperstein, The -spaces of the standard ${\mathrm E}_{6}(q)$ -module, Geom. Dedicata 25 (1988), no. 1-3, 467-480. MR 0925847 (89c:51013)
40.: A. M. Cohen and R. H. Cushman, Gröbner bases and standard monomial theory, Computational Algebraic Geometry (Nice, 1992), Progr. Math., vol. 109, Birkhäuser Boston, Boston, MA, 1993, pp. 41-60. MR 1230857 (94g:13017)
41.: A. M. Cohen and D. B. Wales, Finite subgroups of F $_4({\mathbb{C}})$ and ${\mathrm E}_6({\mathbb{C}})$ , Proc. London Math. Soc. (3) 74 (1997), 105-150. MR 1416728 (97k:20078)
42.: B. N. Cooperstein, The fifty-six-dimensional module for ${\mathrm E}_{7}$ . I. A four form for ${\mathrm E}_{7}$ , J. Algebra 173 (1995), 361-389. MR 1325780 (96f:20063)
43.: J. D. Dixon, Rigid embeddings of simple groups in the general linear group, Canad. J. Math. 29 (1977), no. 2, 384-391. MR 0435242 (55:8202)
44.: J. R. Faulkner and J. C. Ferrar, Exceptional Lie algebras and related algebraic and geometric structures, Bull. London Math. Soc. 9 (1977), 1-35. MR 0444729 (56:3079)
45.: R. S. Garibaldi, Structurable algebras and groups of type ${\mathrm E}_6$ and ${\mathrm E}_7$ , J. Algebra 236 (2001), no. 2, 651-691. MR 1813495 (2001m:20069)
46.: -, Cohomological invariants: Exceptional groups and Spin groups, with an appendix by D. W. Hoffmann, Preprint Emory Univ. Atlanta, 2006, 75 pp.
47.: R. S. Garibaldi and H. P. Peterson, Groups of outer type ${\mathrm E}_6$ with trivial Tits algebras, arXiv:math.GR/0511229v1 (09.10.2005), 29 pp.
48.: R. L. Griess, A Moufang loop, the exceptional Jordan algebra, and a cubic form in variables, J. Algebra 131 (1990), 281-293. MR 1055009 (91g:17044)
49.: A. Hahn and O. T. O'Meara, The classical groups and K-theory, Grundlehren Math. Wiss., Bd. 291, Springer-Verlag, Berlin, 1989. MR 1007302 (90i:20002)
50.: S. J. Haris, Some irreducible representations of exceptional algebraic groups, Amer. J. Math. 93 (1971), no. 1, 75-106. MR 0279103 (43:4829)
51.: R. Hazrat and N. Vavilov, of Chevalley groups are nilpotent, J. Pure Appl. Algebra 179 (2003), 99-116. MR 1958377 (2004i:20081)
52.: J.-Y. Hée, Groupes de Chevalley et groupes classiques, Seminar on Finite Groups, Vol. II, Publ. Math. Univ. Paris VII, No. 17, Univ. Paris VII, Paris, 1984, pp. 1-54. MR 0772212 (86d:20054)
53.: A. Iliev and L. Manivel, The Chow ring of the Cayley plane, Compositio Math. 141 (2005), 146-160. MR 2099773 (2005h:14008)
54.: N. Jacobson, Structure and representations of Jordan algebras, Amer. Math. Soc. Colloq. Publ., vol. 39, Amer. Math. Soc., Providence, RI, 1968. MR 0251099 (40:4330)
55.: W. Lichtenstein, A system of quadrics describing the orbit of the highest weight vector, Proc. Amer. Math. Soc. 84 (1982), no. 4, 605-608. MR 0643758 (83h:14046)
56.: J. G. M. Mars, Les nombres de Tamagawa de certains groupes exceptionnels, Bull. Soc. Math. France 94 (1966), 97-140. MR 0213363 (35:4227)
57.: H. Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4) 2 (1969), 1-62. MR 0240214 (39:1566)
58.: V. A. Petrov, Overgroups of unitary groups, -Theory 29 (2003), 147-174. MR 2028500 (2004m: 20094)
59.: V. Petrov, N. Semenov, and K. Zainoulline, Zero cycles on a twisted Cayley plane, arXiv:math. AG/0508200v2 (22.09.2005), 15 pp. Canad. Math. Bull. 51 (2008), 114-124. MR 2384744
60.: E. B. Plotkin, On the stability of the -functor for Chevalley groups of type ${\mathrm E}_7$ , J. Algebra 210 (1998), 67-85. MR 1656415 (99k:20099)
61.: E. B. Plotkin, A. A. Semenov, and N. A. Vavilov, Visual basic representations: An atlas, Internat. J. Algebra Comput. 8 (1998), no. 1, 61-95. MR 1492062 (98m:17010)
62.: G. M. Seitz, The maximal subgroups of classical algebraic groups, Mem. Amer. Math. Soc. 67 (1987), no. 365. MR 0888704 (88g:20092)
63.: T. A. Springer, The projective octave plane. I, II, Indag. Math. 22 (1960), 74-101. MR 0126196 (23:A3492)
64.: -, Characterization of a class of cubic forms, Indag. Math. 24 (1962), 259-265. MR 0138661 (25:2104)
65.: -, On the geometric algebra of the octave planes, Indag. Math. 24 (1962), 451-468. MR 0142045 (25:5439)
66.: -, Some groups of type ${\mathrm E}_7$ , Nagoya Math. J. 182 (2006), 259-284. MR 2235344 (2007b:20096)
67.: T. A. Springer and F. D. Veldkamp, Octonions, Jordan algebras and exceptional groups, Springer-Verlag, Berlin, 2000. MR 1763974 (2001f:17006)
68.: M. R. Stein, Generators, relations and coverings of Chevalley groups over commutative rings, Amer. J. Math. 93 (1971), no. 4, 965-1004. MR 0322073 (48:437)
69.: -, Stability theorems for $K_{1}$ , $K_{2}$ and related functors modeled on Chevalley groups, Japan. J. Math. (N.S.) 4 (1978), no. 1, 77-108. MR 0528869 (81c:20031)
70.: A. V. Stepanov and N. A. Vavilov, Decomposition of transvections: A theme with variations, -Theory 19 (2000), 109-153. MR 1740757 (2000m:20076)
71.: G. Taddei, Normalité des groupes élémentaires dans les groupes de Chevalley sur un anneau, Applications of Algebraic -Theory to Algebraic Geometry and Number Theory, Part II (Boulder, Colo., 1983), Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 693-710. MR 0862660 (88a:20054)
72.: L. N. Vaserstein, On normal subgroups of Chevalley groups over commutative rings, Tôhoku Math. J. (2) 38 (1986), no. 2, 219-230. MR 0843808 (87k:20081)
73.: N. A. Vavilov, Structure of Chevalley groups over commutative rings, Nonassociative Algebras and Related Topics (Hiroshima, 1990), World Sci. Publ., River Edge, NJ, 1991, pp. 219-335. MR 1150262 (92k:20090)
74.: -, A third look at weight diagrams, Rend. Sem. Mat. Univ. Padova 104 (2000), 201-250. MR 1809357 (2001i:20099)
75.: -, An ${\mathrm A}_3$ -proof of structure theorems for Chevalley groups of types ${\mathrm E}_6$ and ${\mathrm E}_7$ , Internat. J. Algebra Comput. 17 (2007), no. 5-6, 1283-1298. MR 2355697
76.: N. A. Vavilov and E. B. Plotkin, Chevalley groups over commutative rings. I. Elementary calculations, Acta Appl. Math. 45 (1996), no. 1, 73-113. MR 1409655 (97h:20056)
77.: W. C. Waterhouse, Introduction to affine group schemes, Grad. Texts in Math., vol. 66, Springer-Verlag, New York-Berlin, 1979. MR 0547117 (82e:14003)
78.: -, Automorphisms of quotients of $\Pi$ GL, Pacific J. Math. 102 (1982), 221-233. MR 0682053 (84m:20050)
79.: -, Automorphisms of $\det(X_{ij})$ : The group scheme approach, Adv. Math. 65 (1987), no. 2, 171-203. MR 0900267 (88k:14025)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|