The geometry of root elements in groups of type~$\mathrm E_6$
Algebra i analiz, Tome 23 (2011) no. 3, pp. 261-309.

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I. M. Pevzner. The geometry of root elements in groups of type~$\mathrm E_6$. Algebra i analiz, Tome 23 (2011) no. 3, pp. 261-309. http://geodesic.mathdoc.fr/item/AA_2011_23_3_a8/

[1] Bashkirov E. L., “O lineinykh gruppakh, porozhdennykh dvumya dlinnymi kornevymi podgruppami”, Sib. mat. zh., 34:2 (1993), 15–22 | MR | Zbl

[2] Bashkirov E. L., “Lineinye gruppy, soderzhaschie kornevuyu podgruppu”, Sib. mat. zh., 37:6 (1996), 1238–1255 | MR | Zbl

[3] Bashkirov E. L., “O podgruppakh polnoi lineinoi gruppy stepeni 4 nad telom kvaternionov, soderzhaschikh spetsialnuyu unitarnuyu gruppu indeksa 1”, Algebra i analiz, 13:3 (2001), 18–42 | MR | Zbl

[4] Bashkirov E. L., “Gruppa $\mathrm{Spin}_8$ i nekotorye podgruppy unitarnoi gruppy stepeni 4 nad telom kvaternionov”, Algebra i analiz, 13:3 (2001), 43–64 | MR | Zbl

[5] Bashkirov E. L., Lineinye gruppy nad telami, soderzhaschie podgruppy kvadratichnykh unipotentnykh elementov, Dokt. dic., Belorusskii gos. un-t informatiki i radioelektroniki, 2006

[6] Borel A., “Svoistva i lineinye predstavleniya grupp Shevalle”, Seminar po algebraicheskim gruppam, Mir, M., 1973, 9–59 | MR

[7] Burbaki N., Gruppy i algebry Li, Gl. IV–VI, Mir, M., 1972 | MR | Zbl

[8] Burbaki N., Gruppy i algebry Li, Gl. VII–VIII, Mir, M., 1978 | MR

[9] Vavilov N. A., “O geometrii dlinnykh kornevykh podgrupp v gruppakh Shevalle”, Vestn. Leningr. un-ta. Ser. 1, 1988:1, 8–11 | MR | Zbl

[10] Vavilov N. A., “Razlozhenie Bryua dlinnykh kornevykh poluprostykh elementov v gruppakh Shevalle”, Koltsa i moduli. Predelnye teoremy teorii veroyatnostei, 2, LGU, L., 1988, 18–39 | MR

[11] Vavilov N. A., “O podgruppakh rasschepimykh ortogonalnykh grupp nad koltsom”, Sib. mat. zh., 29:4 (1988), 31–43 | MR | Zbl

[12] Vavilov N. A., “Geometriya 1-torov v $\mathrm{GL}_n$”, Algebra i analiz, 19:3 (2007), 119–150 | MR

[13] Vavilov N. A., “Kak uvidet znaki strukturnykh konstant?”, Algebra i analiz, 19:4 (2007), 34–68 | MR

[14] Vavilov N. A., Gavrilovich M. R., “$\mathrm A_2$-dokazatelstvo strukturnykh teorem dlya grupp Shevalle tipov $\mathrm E_6$ i $\mathrm E_7$”, Algebra i analiz, 16:4 (2004), 54–87 | MR | Zbl

[15] Vavilov N. A., Gavrilovich M. R., Nikolenko S. I., “Stroenie grupp Shevalle: dokazatelstvo iz Knigi”, Zap. nauch. semin. POMI, 330, 2006, 36–76 | MR | Zbl

[16] Vavilov N. A., Luzgarev A. Yu., “Normalizator gruppy Shevalle tipa $\mathrm E_6$”, Algebra i analiz, 19:5 (2007), 37–64 | MR

[17] Vavilov N. A., Luzgarev A. Yu., Pevzner I. M., “Gruppa Shevalle tipa $\mathrm E_6$ v 27-mernom predstavlenii”, Zap. nauch. semin. POMI, 338, 2006, 5–68 | MR | Zbl

[18] Vavilov N. A., Nesterov V. V., “Geometriya mikrovesovykh torov”, Vladikavkaz. mat. zh., 10:1 (2008), 10–23 | MR

[19] Vavilov N. A., Nesterov V. V., “Pary mikrovesovykh torov v gruppe Shevalle tipa $\mathrm E_6$”, Algebra i analiz, gotovitsya k publikatsii

[20] Vavilov N. A., Nikolenko S. I., “$\mathrm A_2$-dokazatelstvo strukturnykh teorem dlya grupp Shevalle tipa $\mathrm F_4$”, Algebra i analiz, 20:4 (2008), 27–63 | MR

[21] Vavilov N. A., Pevzner I. M., “Troiki dlinnykh kornevykh podgrupp”, Zap. nauch. semin. POMI, 343, 2007, 54–83 | MR

[22] Vavilov N. A., Plotkin E. B., Stepanov A. V., “Vychisleniya v gruppakh Shevalle nad kommutativnymi koltsami”, Dokl. AN SSSR, 307:4 (1989), 788–791 | MR | Zbl

[23] Vavilov N. A., Semenov A. A., “Razlozhenie Bryua dlinnykh kornevykh torov v gruppakh Shevalle”, Zap. nauch. semin. LOMI, 175, 1989, 12–23 | Zbl

[24] Vavilov N. A., Semenov A. A., “Dlinnye kornevye poluprostye elementy v gruppakh Shevalle”, Dokl. RAN, 338:6 (1994), 725–727 | MR | Zbl

[25] Vinberg E. B., Onischik A. L., Seminar po gruppam Li i algebraicheskim gruppam, Nauka, M., 1988 | MR

[26] Zalesskii A. E., “Lineinye gruppy”, Uspekhi mat. nauk, 36:5 (1981), 57–107 | MR | Zbl

[27] Zalesskii A. E., “Lineinye gruppy”, Algebra – 4, Itogi nauki i tekhn. Sovrem. probl. mat. Fundam. napravleniya, 37, VINITI, M., 1989, 114–228 | MR | Zbl

[28] Zalesskii A. E., Serezhkin V. N., “Lineinye gruppy, porozhdennye transvektsiyami”, Izv. AN SSSR. Ser. mat., 40:1 (1976), 26–49 | MR | Zbl

[29] Kondratev A. S., “Podgruppy konechnykh grupp Shevalle”, Uspekhi mat. nauk, 41:1 (1986), 57–96 | MR | Zbl

[30] Luzgarev A. Yu., “O nadgruppakh $\mathrm E(\mathrm E_6,R)$ i $\mathrm E(\mathrm E_7,R)$ v minimalnykh predstavleniyakh”, Zap. nauch. semin. POMI, 319, 2004, 216–243 | MR | Zbl

[31] Luzgarev A. Yu., “Opisanie nadgrupp $\mathrm F_4$ v $\mathrm E_6$ nad kommutativnym koltsom”, Algebra i analiz, 20:6 (2008), 148–185 | MR

[32] Luzgarev A. Yu., Pevzner I. M., “Nekotorye fakty iz zhizni $\mathrm{GL}(5,\mathbb Z)$”, Zap. nauch. semin. POMI, 305, 2003, 153–162 | MR | Zbl

[33] Manin Yu. I., Kubicheskie formy. Algebra, geometriya, arifmetika, Nauka, M., 1972 | MR | Zbl

[34] Nesterov V. V., “Pary korotkikh kornevykh podgrupp v gruppakh Shevalle”, Dokl. RAN, 357:3 (1997), 302–305 | MR | Zbl

[35] Nesterov V. V., “Pary korotkikh kornevykh podgrupp v gruppe Shevalle tipa $\mathrm G_2$”, Zap. nauch. semin. POMI, 281, 2001, 253–273 | MR | Zbl

[36] Nesterov V. V., “Porozhdenie par korotkikh kornevykh podgrupp v gruppakh Shevalle”, Algebra i analiz, 16:6 (2004), 172–208 | MR | Zbl

[37] O'Mira O., “Lektsii o lineinykh gruppakh”, Avtomorfizmy klassicheskikh grupp, Mir, M., 1976, 57–167 | MR

[38] Rozenfeld B. A., Zamakhovskii M. P., Geometriya grupp Li. Simmetricheskie, parabolicheskie i periodicheskie prostranstva, MTsNMO, M., 2003 | MR

[39] Semenov A. A., Razlozhenie Bryua dlinnykh kornevykh poluprostykh torov v gruppakh Shevalle, Kand. dis., SPb GU, SPb., 1991

[40] Springer T. A., “Lineinye algebraicheskie gruppy”, Algebraicheskaya geometriya – 4, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 55, VINITI, M., 1989, 5–136 | MR | Zbl

[41] Steinberg R., Lektsii o gruppakh Shevalle, Mir, M., 1975 | MR | Zbl

[42] Khamfri Dzh., Lineinye algebraicheskie gruppy, Nauka, M., 1980 | MR

[43] Khamfri Dzh., Vvedenie v teoriyu algebr Li i ikh predstavlenii, MTsNMO, M., 2003

[44] Aschbacher M., “The 27-dimensional module for $\mathrm E_6$, I”, Invent. Math., 89:1 (1987), 159–195 | DOI | MR | Zbl

[45] Aschbacher M., “The 27-dimensional module for $\mathrm E_6$, II”, J. London Math. Soc. (2), 37 (1988), 275–293 | DOI | MR | Zbl

[46] Aschbacher M., “The 27-dimensional module for $\mathrm E_6$, III”, Trans. Amer. Math. Soc., 321 (1990), 45–84 | DOI | MR | Zbl

[47] Aschbacher M., “The 27-dimensional module for $\mathrm E_6$, IV”, J. Algebra, 131 (1990), 23–39 | DOI | MR | Zbl

[48] Aschbacher M., “Some multilinear forms with large isometry groups”, Geom. Dedicata, 25:1–3 (1988), 417–465 | MR | Zbl

[49] Aschbacher M., “The geometry of trilinear forms”, Finite Geometries, Buildings and Related Topics (Pingree Park, CO, 1988), Oxford Univ. Press, Oxford, 1990, 75–84 | MR

[50] Aschbacher M., Seitz G. M., “Involutions in Chevalley groups over fields of even order”, Nagoya Math. J., 63 (1976), 1–91 | MR | Zbl

[51] Bashkirov E. L., “Some completely reducible linear groups over a quaternion division ring, containing a root subgroup”, Comm. Algebra, 31:12 (2003), 5727–5754 | DOI | MR | Zbl

[52] Bashkirov E. L., “Irreducible linear groups of degree 3 over a quaternion division ring, containing a root subgroup”, Comm. Algebra, 32:5 (2004), 1747–1761 | DOI | MR | Zbl

[53] Bashkirov E. L., “Irreducible linear groups of degree 4 over a quaternion division algebra that contain a subgroup $\mathrm{diag}(T_3(K,\Phi_0),1)$”, J. Algebra, 287:2 (2005), 319–350 | DOI | MR | Zbl

[54] Brown R., Humphries S. P., “Orbits under symplectic transvections, I”, Proc. London Math. Soc. (3), 52 (1986), 517–531 | DOI | MR | Zbl

[55] Cameron P. J., Hall J. I., “Some groups generated by transvection subgroups”, J. Algebra, 140:1 (1991), 184–209 | DOI | MR | Zbl

[56] Carter R. W., Simple groups of Lie type, Wiley, London, 1989 | MR | Zbl

[57] Chevalley C., Schafer R. D., “The exceptional simple Lie algebras $\mathrm F_4$ and $\mathrm E_6$”, Proc. Nat. Acad. Sci. USA, 36 (1950), 137–141 | DOI | MR | Zbl

[58] Cohen A. M., Cushman R. H., “Gröbner bases and standard monomial theory”, Computational Algebraic Geometry (Nice, 1992), Progr. Math., 109, Birkhäuser Boston, Boston, MA, 1993, 41–60 | MR | Zbl

[59] Cohen A. M., Liebeck M. W., Saxl J., Seitz G. M., “The local maximal subgroups of exceptional groups of Lie type, finite and algebraic”, Proc. London Math. Soc. (3), 64:1 (1992), 21–48 | DOI | MR

[60] Cooperstein B. N., “Subgroups of the group $\mathrm E_6(q)$ which are generated by root subgroups”, J. Algebra, 46:2 (1977), 355–388 | DOI | MR | Zbl

[61] Cooperstein B. N., “The geometry of root subgroups in exceptional groups, I”, Geom. Dedicata, 8 (1979), 317–381 | DOI | MR | Zbl

[62] Cooperstein B. N., “Geometry of long root subgroups in groups of Lie type”, The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, CA, 1979), Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, RI, 1980, 243–248 | MR

[63] Cooperstein B. N., “Subgroups of exceptional groups of Lie type generated by long root elements, I”, J. Algebra, 70:1 (1981), 270–282 | DOI | MR | Zbl

[64] Cooperstein B. N., “The geometry of root subgroups in exceptional groups, II”, Geom. Dedicata, 15 (1983), 1–45 | DOI | MR | Zbl

[65] Cuypers H., “Symplectic geometries, transvection groups and modules”, J. Combin. Theory Ser. A, 65 (1994), 39–59 | DOI | MR | Zbl

[66] Cuypers H., “The geometry of $k$-transvection groups”, J. Algebra, 300:2 (2006), 455–471 | DOI | MR | Zbl

[67] Cuypers H., Steinbach A. I., “Linear transvection groups and embedded polar spaces”, Invent. Math., 137:1 (1999), 169–198 | DOI | MR | Zbl

[68] Cuypers H., Steinbach A. I., “Special linear groups generated by transvections and embedded projective spaces”, J. London Math. Soc. (2), 64:3 (2001), 576–594 | DOI | MR | Zbl

[69] Deriziotis D. I., Fakiolas A. P., “The maximal tori in the finite Chevalley groups of type $\mathrm E_6$, $\mathrm E_7$ and $\mathrm E_8$”, Comm. Algebra, 19:3 (1991), 889–903 | DOI | MR | Zbl

[70] Di Martino L., Vavilov N. A., “$(2,3)$-generation of $\mathrm{SL}(n,q)$. I: Cases $n=5,6,7$”, Comm. Algebra, 22:4 (1994), 1321–1347 | DOI | MR | Zbl

[71] Di Martino L., Vavilov N. A., “$(2,3)$-generation of $\mathrm{SL}(n,q)$. II: Cases $n\ge8$”, Comm. Algebra, 24:2 (1996), 487–515 | DOI | MR | Zbl

[72] Gilkey P., Seitz G. M., “Some representations of exceptional Lie algebras”, Geom. Dedicata, 25:1–3 (1988), 407–416 | MR | Zbl

[73] Kantor W. M., “Subgroups of classical groups generated by long root elements”, Trans. Amer. Math. Soc., 248:2 (1979), 347–379 | DOI | MR | Zbl

[74] Kantor W. M., “Generation of linear groups”, The Geometric Vein: Coxeter Festschrift, Springer, New York–Berlin, 1981, 497–509 | MR

[75] Li Shang Zhi, “Irreducible subgroups of $\mathrm{SL}(n,K)$ generated by root subgroups”, Geom. Dedicata, 31 (1989), 41–44 | DOI | MR | Zbl

[76] Liebeck M. W., Seitz G. M., “Subgroups generated by root elements in groups of Lie type”, Ann. of Math. (2), 139 (1994), 293–361 | DOI | MR | Zbl

[77] Matsumoto H., “Sur les sous-groupes arithmétiques des groupes semi-simples déployés”, Ann. Sci. École Norm. Sup. (4), 2:1 (1969), 1–62 | MR | Zbl

[78] Parker Ch., Röhrle G. E., Miniscule representations, Preprint No 72, Univ. Bielefeld, 1993, 12 pp.

[79] Plotkin E. B., Semenov A. A., Vavilov N. A., “Visual basic representations: an atlas”, Internat. J. Algebra Comput., 8:1 (1998), 61–95 | DOI | MR | Zbl

[80] Röhrle G. E., “On extraspecial parabolic subgroups”, Linear Algebraic Groups and their Representations (Los Angeles, CA, 1992), Contemp. Math., 153, Amer. Math. Soc., Providence, RI, 1993, 143–155 | MR

[81] Springer T. A., Linear algebraic groups, Progr. Math., 9, Birkhäuser Boston, Inc., Boston, MA, 1998 | MR | Zbl

[82] Springer T. A., Veldkamp F. D., Octonions, Jordan algebras and exceptional groups, Springer-Verlag, Berlin, 2000 | MR | Zbl

[83] Steinbach A. I., Untergruppen von klassischen Gruppen, die von Transvektionen oder Siegel–Transvektionen erzeugt werden, Ph.-D. Thesis, Univ. of Gießen, 1995

[84] Steinbach A. I., “Subgroups of classical groups generated by transvections or Siegel transvections, I”, Geom. Dedicata, 68 (1997), 281–322 | DOI | MR | Zbl

[85] Steinbach A. I., “Subgroups isomorphic to $\mathrm G_2(L)$ in orthogonal groups”, J. Algebra, 205:1 (1998), 77–90 | DOI | MR | Zbl

[86] Steinbach A. I., Groups of Lie type generated by long root elements in $\mathrm F_4(K)$, Habilitationsschrift, Gießen, 2000

[87] Steinbach A. I., “Subgroups of the Chevalley groups of type $\mathrm F_4(K)$ arising from a polar space”, Adv. Geom., 3 (2003), 73–100 | DOI | MR | Zbl

[88] Timmesfeld F. G., “Groups generated by $k$-transvections”, Invent. Math., 100 (1990), 167–206 | DOI | MR | Zbl

[89] Timmesfeld F. G., “Groups generated by $k$-root subgroups”, Invent. Math., 106 (1991), 575–666 | DOI | MR | Zbl

[90] Timmesfeld F. G., “Groups generated by $k$-root subgroups – a survey”, Groups, Combinatorics and Geometry (Durham–1990), London Math. Soc. Lecture Note Ser., 165, Cambridge Univ. Press, Cambridge, 1992, 183–204 | MR | Zbl

[91] Timmesfeld F. G., “Moufang planes and the groups $\mathrm E_6^K$ and $\mathrm{SL}_2(K)$, $K$ a Cayley division algebra”, Forum Math., 6:2 (1994), 209–231 | DOI | MR | Zbl

[92] Timmesfeld F. G., “Subgroups generated by root elements of groups generated by $k$-root subgroups”, Geom. Dedicata, 49 (1994), 293–321 | DOI | MR | Zbl

[93] Timmesfeld F. G., “Abstract root subgroups and quadratic action”, With an appendix by A. E. Zalesskii, Adv. Math., 142:1 (1999), 1–150 | DOI | MR | Zbl

[94] Timmesfeld F. G., Abstract root subgroups and simple groups of Lie type, Monogr. in Math., 95, Birkhäuser Verlag, Basel, 2001 | MR | Zbl

[95] Tits J., “Sur les constantes de structure et le théorème d'existence des algèbres de Lie semi-simples”, Inst. Hautes Études Sci. Publ. Math., 31 (1966), 21–58 | DOI | MR

[96] Tits J., Building of spherical type and finite $BN$-pairs, Lecture Notes in Math., 386, Springer-Verlag, Berlin, 1974 | MR | Zbl

[97] Vavilov N. A., “Structure of Chevalley groups over commutative rings”, Nonassociative Algebras and Related Topics (Hiroshima, 1990), World Sci. Publ., River Edge, NJ, 1991, 219–335 | MR | Zbl

[98] Vavilov N. A., “A third look at weight diagrams”, Rend. Sem. Mat. Univ. Padova, 104 (2000), 201–250 | MR | Zbl

[99] Vavilov N. A., “Do it yourself structure constants for Lie algebras of type $E_l$”, Zap. nauch. semin. POMI, 281, 2001, 60–104 | MR | Zbl

[100] Vavilov N. A., “An $\mathrm A_3$-proof of structure theorems for Chevalley groups of types $\mathrm E_6$ and $\mathrm E_7$”, Internat J. Algebra Comput., 17:5–6 (2007), 1283–1298 | DOI | MR | Zbl

[101] Vavilov N. A., Plotkin E. B., “Chevalley groups over commutative rings. I: Elementary calculations”, Acta Appl. Math., 45 (1996), 73–113 | DOI | MR | Zbl

[102] Wagner A., “Groups generated by elations”, Abh. Math. Sem. Univ. Hamburg, 41 (1974), 190–205 | DOI | MR | Zbl