Geodesic
Reduction theorems for triples of short root subgroups in Chevalley groups
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 29, Tome 443 (2016), pp. 106-132 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

In the present paper we prove the reduction theorems for triple short root unipotent subgroups in Chevalley groups of type $\mathrm B_\ell$ and $\mathrm C_\ell$. The main result roughly speaking is the following. Any subgroup generated by a triple of subgroups in question (apart from one case) is conjugate to a subgroup of $$ G(\mathrm B_4,K)U(\mathrm B_5,K)\quad\mathrm{or}\quad G(\mathrm C_4,K)U(\mathrm C_5,K), $$ respectively. 
@article{ZNSL_2016_443_a9,
     author = {V. V. Nesterov},
     title = {Reduction theorems for triples of short root subgroups in {Chevalley} groups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {106--132},
     year = {2016},
     volume = {443},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_443_a9/}
}
TY  - JOUR
AU  - V. V. Nesterov
TI  - Reduction theorems for triples of short root subgroups in Chevalley groups
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2016
SP  - 106
EP  - 132
VL  - 443
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2016_443_a9/
LA  - ru
ID  - ZNSL_2016_443_a9
ER  - 
%0 Journal Article
%A V. V. Nesterov
%T Reduction theorems for triples of short root subgroups in Chevalley groups
%J Zapiski Nauchnykh Seminarov POMI
%D 2016
%P 106-132
%V 443
%U http://geodesic.mathdoc.fr/item/ZNSL_2016_443_a9/
%G ru
%F ZNSL_2016_443_a9
V. V. Nesterov. Reduction theorems for triples of short root subgroups in Chevalley groups. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 29, Tome 443 (2016), pp. 106-132. http://geodesic.mathdoc.fr/item/ZNSL_2016_443_a9/

[1] N. Burbaki, Gruppy i algebry Li, gl. IV–VI, Mir, M., 1972 | MR

[2] N. A. Vavilov, “O geometrii dlinnykh kornevykh podgrupp v gruppakh Shevalle”, Vestnik LGU, ser. 1, 1988, no. 1, 8–11 | MR | Zbl

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

[4] N. A. Vavilov, “Vzaimnoe raspolozhenie dlinnoi i korotkoi kornevykh podgrupp v gruppe Shevalle”, Vestnik LGU, ser. 1, 1989, no. 1, 3–7 | MR | Zbl

[5] N. A. Vavilov, “Podgruppy grupp Shevalle, soderzhaschie maksimalnyi tor”, Trudy Leningr. Mat. Obschestva, 1, 1990, 64–109 | MR

[6] N. A. Vavilov, “Vesovye elementy grupp Shevalle”, Algebra i analiz, 20 (2008), 34–85 | MR | Zbl

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

[8] N. A. Vavilov, A. A. Semenov, “Dlinnye kornevye tory v gruppakh Shevalle”, Algebra i analiz, 24:3 (2012), 22–83 | MR | Zbl

[9] E. B. Dynkin, “Poluprostye podalgebry poluprostykh algebr Li”, Matem. sb., 30(72):2 (1952), 349–462 | MR | Zbl

[10] A. E. Zalesskii, “Lineinye gruppy”, Uspekhi mat. nauk, 36:5 (1981), 56–107 | MR | Zbl

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

[12] V. V. Nesterov, “Raspolozhenie dlinnoi i korotkoi kornevykh podgrupp v gruppe Shevalle tipa $\mathrm G_2$”, Zap. nauchn. semin. POMI, 272, 2000, 273–285 | MR | Zbl

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

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

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

[16] M. Aschbacher, G. M. Seitz, “Involutions in Chevalley groups over fields of even order”, Nagoya Math. J., 63 (1976), 1–91 ; “Corrections”, Nagoya Math. J., 72 (1978), 135–136 | MR | Zbl | MR | Zbl

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

[18] R. W. Carter, Simple groups of Lie type, Pure Appl. Math., 28, Wiley, London et al., 1972 | MR

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

[20] B. N. Cooperstein, “The geometry of root subgroups in exceptional groups”, Geometria dedicata, 8:3 (1979), 317–381 ; “II”, Geometria dedicata, 15:1 (1983), 1–45 | DOI | MR | Zbl | DOI | MR | Zbl

[21] B. N. Cooperstein, “Geometry of long root subgroups in groups of Lie type”, Proc. Symp. Pure Math., 37 (1980), 243–248 | DOI | MR | Zbl

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

[23] L. Di Martino, N. A. Vavilov, “(2,3)-generation of $\mathrm{SL}(n,q)$. I”, Comm. Algebra, 22:4 (1994), 1321–1347 ; “II”, Comm. Algebra, 24:2 (1996), 487–515 | DOI | MR | Zbl | DOI | MR | Zbl

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

[25] Li Shang Zhi, “Maximal subgroups containing root subgroups in finite classical groups”, Kexue Tongbao, 29:1 (1984), 14–18 | MR | Zbl

[26] Li Shang Zhi, “Maximal subgroups in $\mathrm P\Omega(n,F,Q)$ with root subgroups”, Sci. Sinica Ser. A, 28:8 (1985), 826–838 | MR | Zbl

[27] Li Shang Zhi, “Maximal subgroups containing short root subgroups in $\mathrm{PSp}(2n,\mathbb F)$”, Acta Math. Sinica, New ser., 3:1 (1987), 82–91 | DOI | MR | Zbl

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

[29] B. S. Stark, “Some subgroups of $\Omega(V)$ generated by groups of root type 1”, Illinois J. Math., 17:4 (1973), 584–607 | MR | Zbl

[30] B. S. Stark, “Some subgroups of $\Omega(V)$ generated by groups of root type”, J. Algebra, 17:1 (1974), 33–41 | DOI | MR

[31] B. S. Stark, “Irreducible subgroups of orthogonal groups generated by groups of root type 1”, Pacific J. Math., 53:2 (1974), 611–625 | DOI | MR | Zbl

[32] D. I. Stewart, The reductive subgroups of $\mathrm F_4$, Mem. Amer. Math. Soc., 223, no. 1049, 2013 | MR

[33] F. G. Timmesfeld, “Groups generated by root involutions. I”, J. Algebra, 33 (1975), 75–134 | DOI | MR | Zbl

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

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

[36] F. G. Timmesfeld, “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