Overgroups of $\text \textrm {F}_4$ in $\text \textrm {E}_6$ over commutative rings
HTML articles powered by AMS MathViewer
- by
A. Yu. Luzgarev
Translated by: the author - St. Petersburg Math. J. 20 (2009), 955-981
- DOI: https://doi.org/10.1090/S1061-0022-09-01080-2
- Published electronically: October 2, 2009
- PDF | Request permission
Abstract:
Overgroups of the elementary Chevalley group of type $\mathrm {F}_4$ in the Chevalley group of type $\mathrm {E}_6$ over an arbitrary commutative ring are described.References
- Armand Borel, Properties and linear representations of Chevalley groups, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 1–55. MR 0258838
- Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR 1890629, DOI 10.1007/978-3-540-89394-3
- N. Vavilov, Weight elements of Chevalley groups, Algebra i Analiz 20 (2008), no. 1, 34–85 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 20 (2009), no. 1, 23–57. MR 2411968, DOI 10.1090/S1061-0022-08-01036-4
- N. A. Vavilov and M. R. Gavrilovich, $A_2$-proof of structure theorems for Chevalley groups of types $E_6$ and $E_7$, Algebra i Analiz 16 (2004), no. 4, 54–87 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 16 (2005), no. 4, 649–672. MR 2090851, DOI 10.1090/S1061-0022-05-00871-X
- 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. (POMI) 330 (2006), no. Vopr. Teor. Predst. Algebr. i Grupp. 13, 36–76, 271–272 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 140 (2007), no. 5, 626–645. MR 2253566, DOI 10.1007/s10958-007-0003-y
- N. A. Vavilov and A. Yu. Luzgarev, The normalizer of Chevalley groups of type $E_6$, Algebra i Analiz 19 (2007), no. 5, 37–64 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 19 (2008), no. 5, 699–718. MR 2381940, DOI 10.1090/S1061-0022-08-01016-9
- N. A. Vavilov, A. Yu. Luzgarev, and I. M. Pevzner, A Chevalley group of type $E_6$ in the 27-dimensional representation, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 338 (2006), no. Vopr. Teor. Predst. Algebr. i Grupp. 14, 5–68, 261 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 145 (2007), no. 1, 4697–4736. MR 2354606, DOI 10.1007/s10958-007-0304-1
- N. A. Vavilov and S. I. Nikolenko, $A_2$-proof of structure theorems for a Chevalley group of type $F_4$, Algebra i Analiz 20 (2008), no. 4, 27–63 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 20 (2009), no. 4, 527–551. MR 2473743, DOI 10.1090/S1061-0022-09-01060-7
- N. A. Vavilov and V. A. Petrov, On supergroups of $\textrm {EO}(2l,R)$, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 272 (2000), no. Vopr. Teor. Predst. Algebr i Grupp. 7, 68–85, 345–346 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 116 (2003), no. 1, 2917–2925. MR 1811793, DOI 10.1023/A:1023442407926
- N. A. Vavilov and V. A. Petrov, On overgroups of $\textrm {Ep}(2l,R)$, Algebra i Analiz 15 (2003), no. 4, 72–114 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 15 (2004), no. 4, 515–543. MR 2068980, DOI 10.1090/S1061-0022-04-00820-9
- N. A. Vavilov and V. A. Petrov, On overgroups of $\textrm {EO}(n,R)$, Algebra i Analiz 19 (2007), no. 2, 10–51 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 19 (2008), no. 2, 167–195. MR 2333895, DOI 10.1090/S1061-0022-08-00992-8
- A. Yu. Luzgarev, On overgroups of $E(E_6,R)$ and $E(E_7,R)$ in minimal representations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 319 (2004), no. Vopr. Teor. Predst. Algebr. i Grupp. 11, 216–243, 302 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 134 (2006), no. 6, 2558–2571. MR 2117858, DOI 10.1007/s10958-006-0127-5
- Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335
- C. Chevalley, Sur certains groupes simples, Tohoku Math. J. (2) 7 (1955), 14–66 (French). MR 73602, DOI 10.2748/tmj/1178245104
- Eiichi Abe, Chevalley groups over commutative rings, Radical theory (Sendai, 1988) Uchida Rokakuho, Tokyo, 1989, pp. 1–23. MR 999577
- Eiichi Abe and Kazuo Suzuki, On normal subgroups of Chevalley groups over commutative rings, Tohoku Math. J. (2) 28 (1976), no. 2, 185–198. MR 439947, DOI 10.2748/tmj/1178240833
- Michael Aschbacher, The $27$-dimensional module for $E_6$. I, Invent. Math. 89 (1987), no. 1, 159–195. MR 892190, DOI 10.1007/BF01404676
- Michael Aschbacher, Some multilinear forms with large isometry groups, Geom. Dedicata 25 (1988), no. 1-3, 417–465. Geometries and groups (Noordwijkerhout, 1986). MR 925846, DOI 10.1007/BF00191936
- Anthony Bak, Nonabelian $K$-theory: the nilpotent class of $K_1$ and general stability, $K$-Theory 4 (1991), no. 4, 363–397. MR 1115826, DOI 10.1007/BF00533991
- Anthony Bak and Nikolai Vavilov, Normality for elementary subgroup functors, Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 1, 35–47. MR 1329456, DOI 10.1017/S0305004100073436
- Stephen Berman and Robert V. Moody, Extensions of Chevalley groups, Israel J. Math. 22 (1975), no. 1, 42–51. MR 390077, DOI 10.1007/BF02757272
- R. H. Dye, Interrelations of symplectic and orthogonal groups in characteristic two, J. Algebra 59 (1979), no. 1, 202–221. MR 541675, DOI 10.1016/0021-8693(79)90157-1
- Roger H. Dye, On the maximality of the orthogonal groups in the symplectic groups in characteristic two, Math. Z. 172 (1980), no. 3, 203–212. MR 581439, DOI 10.1007/BF01215085
- R. H. Dye, Maximal subgroups of $\textrm {GL}_{2n}(K)$, $\textrm {SL}_{2n}(K)$, $\textrm {PGL}_{2n}(K)$ and $\textrm {PSL}_{2n}(K)$ associated with symplectic polarities, J. Algebra 66 (1980), no. 1, 1–11. MR 591244, DOI 10.1016/0021-8693(80)90110-6
- Roozbeh Hazrat, Dimension theory and nonstable $K_1$ of quadratic modules, $K$-Theory 27 (2002), no. 4, 293–328. MR 1962906, DOI 10.1023/A:1022623004336
- Roozbeh Hazrat and Nikolai Vavilov, $K_1$ of Chevalley groups are nilpotent, J. Pure Appl. Algebra 179 (2003), no. 1-2, 99–116. MR 1958377, DOI 10.1016/S0022-4049(02)00292-X
- Hong You, Overgroups of symplectic group in linear group over commutative rings, J. Algebra 282 (2004), no. 1, 23–32. MR 2095570, DOI 10.1016/j.jalgebra.2004.07.036
- Hong You, Overgroups of classical groups over commutative rings in linear group, Sci. China Ser. A 49 (2006), no. 5, 626–638. MR 2250893, DOI 10.1007/s11425-006-0626-3
- Oliver King, On subgroups of the special linear group containing the special orthogonal group, J. Algebra 96 (1985), no. 1, 178–193. MR 808847, DOI 10.1016/0021-8693(85)90045-6
- Oliver King, On subgroups of the special linear group containing the special unitary group, Geom. Dedicata 19 (1985), no. 3, 297–310. MR 815209, DOI 10.1007/BF00149370
- Shang Zhi Li, Overgroups of $\textrm {SU}(n,K,f)$ or $\Omega (n,K,Q)$ in $\textrm {GL}(n,K)$, Geom. Dedicata 33 (1990), no. 3, 241–250. MR 1050412, DOI 10.1007/BF00181331
- Shang Zhi Li, Overgroups of a unitary group in $\textrm {GL}(2,K)$, J. Algebra 149 (1992), no. 2, 275–286. MR 1172429, DOI 10.1016/0021-8693(92)90016-F
- Shang Zhi Li, Overgroups in $\textrm {GL}(n,F)$ of a classical group over a subfield of $F$, Algebra Colloq. 1 (1994), no. 4, 335–346. MR 1301157
- Hideya Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4) 2 (1969), 1–62 (French). MR 240214
- Viktor Petrov, Overgroups of unitary groups, $K$-Theory 29 (2003), no. 3, 147–174. MR 2028500, DOI 10.1023/B:KTHE.0000006934.95243.91
- Michael R. Stein, Generators, relations and coverings of Chevalley groups over commutative rings, Amer. J. Math. 93 (1971), 965–1004. MR 322073, DOI 10.2307/2373742
- Giovanni Taddei, Normalité des groupes élémentaires dans les groupes de Chevalley sur un anneau, Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983) Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 693–710 (French). MR 862660, DOI 10.1090/conm/055.2/1862660
- Jacques Tits, Systèmes générateurs de groupes de congruence, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 9, Ai, A693–A695 (French, with English summary). MR 424966
- Leonid N. Vaserstein, On normal subgroups of Chevalley groups over commutative rings, Tohoku Math. J. (2) 38 (1986), no. 2, 219–230. MR 843808, DOI 10.2748/tmj/1178228489
- Nikolai 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
- Nikolai Vavilov, A third look at weight diagrams, Rend. Sem. Mat. Univ. Padova 104 (2000), 201–250. MR 1809357
- Nikolai Vavilov, An $A_3$-proof of structure theorems for Chevalley groups of types $E_6$ and $E_7$, Internat. J. Algebra Comput. 17 (2007), no. 5-6, 1283–1298. MR 2355697, DOI 10.1142/S0218196707003998
Bibliographic Information
- A. Yu. Luzgarev
- Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospekt 28, Petrodvorets, 198504 St. Petersburg, Russia
- Email: mahalex@yandex.ru
- Received by editor(s): February 12, 2007
- Published electronically: October 2, 2009
- Additional Notes: Supported by the joint program “Mikhail Lomonosov” of DAAD and the Russian Ministry of Education, and by INTAS (grant no. 03-51-3251)
- © Copyright 2009 American Mathematical Society
- Journal: St. Petersburg Math. J. 20 (2009), 955-981
- MSC (2000): Primary 20H25
- DOI: https://doi.org/10.1090/S1061-0022-09-01080-2
- MathSciNet review: 2530897