Skip to Main Content

St. Petersburg Mathematical Journal

This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.

ISSN 1547-7371 (online) ISSN 1061-0022 (print)

The 2020 MCQ for St. Petersburg Mathematical Journal is 0.68.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Overgroups of Levi subgroups I. The case of abelian unipotent radical
HTML articles powered by AMS MathViewer

by P. Gvozdevsky
Translated by: The author
St. Petersburg Math. J. 31 (2020), 969-999
DOI: https://doi.org/10.1090/spmj/1631
Published electronically: October 27, 2020
PDF | Request permission

Abstract:

In the present paper, sandwich classification is established for the overgroups of the subsystem subgroup $E(\Delta ,R)$ of the Chevalley group $G(\Phi ,R)$ for the three types of the pair $(\Phi ,\Delta )$ (the root system and its subsystem) listed below such that the group $G(\Delta ,R)$ is (up to a torus) a Levi subgroup of the parabolic subgroup with Abelian unipotent radical. Namely, it is shown that for any overgroup $H$ of this sort, there exists a unique pair of ideals $\sigma$ of the ring $R$ with $E(\Phi ,\Delta ,R,\sigma )\le H\le N_{G(\Phi ,R)}(E(\Phi ,\Delta ,R,\sigma ))$.
References
  • Z. I. Borevich and N. A. Vavilov, Subgroups of the general linear group over a commutative ring, Dokl. Akad. Nauk SSSR 267 (1982), no. 4, 777–778 (Russian). MR 681025
  • Z. I. Borevich and N. A. Vavilov, Arrangement of subgroups containing a group of block diagonal matrices in the general linear group over a ring, Izv. Vyssh. Uchebn. Zaved. Mat. 11 (1982), 12–16 (Russian). MR 687307
  • Z. I. Borevich and N. A. Vavilov, Arrangement of subgroups in the general linear group over a commutative ring, Trudy Mat. Inst. Steklov. 165 (1984), 24–42 (Russian). Algebraic geometry and its applications. MR 752930
  • Z. I. Borevič, N. A. Vavilov, and V. Narkevič, Subgroups of the general linear group over a Dedekind ring, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 94 (1979), 13–20, 149 (Russian). Rings and modules, 2. MR 571511
  • N. A. Vavilov, Subgroups of the general linear group over a semilocal ring that contain the group of block-diagonal matrices, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. (1983), 16–21, 125 (Russian, with English summary). MR 691839
  • N. A. Vavilov, Subgroups of split orthogonal groups over a ring, Sibirsk. Mat. Zh. 29 (1988), no. 4, 31–43, 222 (Russian); English transl., Siberian Math. J. 29 (1988), no. 4, 537–547 (1989). MR 969101, DOI 10.1007/BF00969861
  • N. A. Vavilov, Subgroups of splittable classical groups, Trudy Mat. Inst. Steklov. 183 (1990), 29–42, 223 (Russian). Translated in Proc. Steklov Inst. Math. 1991, no. 4, 27–41; Galois theory, rings, algebraic groups and their applications (Russian). MR 1092012
  • N. A. Vavilov, Subgroups of split orthogonal groups over a commutative ring, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 281 (2001), no. Vopr. Teor. Predst. Algebr. i Grupp. 8, 35–59, 280 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 120 (2004), no. 4, 1501–1512. MR 1875717, DOI 10.1023/B:JOTH.0000017881.22871.49
  • N. A. Vavilov, How is one to view the signs of structure constants?, Algebra i Analiz 19 (2007), no. 4, 34–68 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 19 (2008), no. 4, 519–543. MR 2381932, DOI 10.1090/S1061-0022-08-01008-X
  • N. A. Vavilov, On subgroups of a symplectic group containing a subsystem subgroup, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 349 (2007), no. Voprosy Teorii Predstavleniĭ Algebr i Grupp. 16, 5–29, 242 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 151 (2008), no. 3, 2937–2948. MR 2742852, DOI 10.1007/s10958-008-9020-8
  • N. A. Vavilov, Numerology of quadratic equations, Algebra i Analiz 20 (2008), no. 5, 9–40 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 20 (2009), no. 5, 687–707. MR 2492358, DOI 10.1090/S1061-0022-09-01068-1
  • 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 and A. V. Stepanov, Overgroups of semisimple groups, Vestn. Samar. Gos. Univ. Estestvennonauchn. Ser. 3 (2008), 51–95 (Russian, with English and Russian summaries). MR 2473730
  • N. A. Vavilov and A. V. Shchegolev, Overgroups of subsystem subgroups in exceptional groups: levels, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 400 (2012), no. Voprosy Teorii Predstavleniĭ Algebr i Grupp. 23, 70–126, 247 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 192 (2013), no. 2, 164–195. MR 3029566, DOI 10.1007/s10958-013-1382-x
  • V. G. Kazakevich and A. K. Stavrova, Subgroups normalized by the commutator group of the Levi subgroup, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 319 (2004), no. Vopr. Teor. Predst. Algebr. i Grupp. 11, 199–215, 301–302 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 134 (2006), no. 6, 2549–2557. MR 2117857, DOI 10.1007/s10958-006-0126-6
  • A. Yu. Luzgarev, Fourth-degree invariants for $G(\textrm {E}_7,R)$ not depending on the characteristic, Vestnik St. Petersburg Univ. Math. 46 (2013), no. 1, 29–34. MR 3087164, DOI 10.3103/S106345411301007X
  • A. V. Stepanov, Nonabelian $K$-theory of Chevalley groups over rings, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 423 (2014), no. Voprosy Teorii Predstavleniĭ Algebr i Grupp. 26, 244–263 (Russian, with English summary); English transl., J. Math. Sci. (N.Y.) 209 (2015), no. 4, 645–656. MR 3480699, DOI 10.1007/s10958-015-2518-y
  • A. V. Shchegolev, Overgroups of block-diagonal subgroups of a hyperbolic unitary group over a quasifinite ring: main results, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 443 (2016), no. Voprosy Teorii Predstavleniĭ Algebr i Grupp. 29, 222–233 (Russian, with English summary); English transl., J. Math. Sci. (N.Y.) 222 (2017), no. 4, 516–523. MR 3507773
  • A. V. Shchegolev, Overgroups of an elementary block-diagonal subgroup of the classical symplectic group over an arbitrary commutative ring, Algebra i Analiz 30 (2018), no. 6, 147–199 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 30 (2019), no. 6, 1007–1041. MR 3882542, DOI 10.1090/spmj/1580
  • M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), no. 3, 469–514. MR 746539, DOI 10.1007/BF01388470
  • A. Bak, R. Hazrat, and N. Vavilov, Localization-completion strikes again: relative $K_1$ is nilpotent by abelian, J. Pure Appl. Algebra 213 (2009), no. 6, 1075–1085. MR 2498798, DOI 10.1016/j.jpaa.2008.11.014
  • 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
  • Claude Chevalley, Certains schémas de groupes semi-simples, Séminaire Bourbaki, Vol. 6, Soc. Math. France, Paris, 1995, pp. Exp. No. 219, 219–234 (French). MR 1611814
  • Michel Demazure and Peter Gabriel, Introduction to algebraic geometry and algebraic groups, North-Holland Mathematics Studies, vol. 39, North-Holland Publishing Co., Amsterdam-New York, 1980. Translated from the French by J. Bell. MR 563524
  • A. Luzgarev, Equations determining the orbit of the highest weight vector in the adjoint representation, arXiv:1401.0849.
  • 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
  • Eugene Plotkin, Andrei Semenov, and Nikolai Vavilov, Visual basic representations: an atlas, Internat. J. Algebra Comput. 8 (1998), no. 1, 61–95. MR 1492062, DOI 10.1142/S0218196798000053
  • A. Shchegolev, Overgroups of elementary block-diagonal subgroups in even unitary groups over quasi-finite rings, Ph.D. thesis, Fak. Math. Univ., Bielefeld, 2015.
  • Anastasia Stavrova, Normal structure of maximal parabolic subgroups in Chevalley groups over rings, Algebra Colloq. 16 (2009), no. 4, 631–648. MR 2547091, DOI 10.1142/S1005386709000595
  • Michael R. Stein, 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 528869, DOI 10.4099/math1924.4.77
  • Alexei Stepanov, Subring subgroups in Chevalley groups with doubly laced root systems, J. Algebra 362 (2012), 12–29. MR 2921626, DOI 10.1016/j.jalgebra.2012.04.007
  • Alexei Stepanov, Structure of Chevalley groups over rings via universal localization, J. Algebra 450 (2016), 522–548. MR 3449702, DOI 10.1016/j.jalgebra.2015.11.031
  • 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
  • Nikolai Vavilov, A third look at weight diagrams, Rend. Sem. Mat. Univ. Padova 104 (2000), 201–250. MR 1809357
  • Deng Yin Wang, Overgroups of Levi subgroups $L_\alpha \ (n(\alpha )=1)$ in Chevalley groups $G(\Phi ,F)$, Adv. Math. (China) 31 (2002), no. 2, 148–152 (Chinese, with English and Chinese summaries). MR 1912043
  • Deng Yin Wang and Shang Zhi Li, Overgroups of Levi subgroups in parabolic subgroups in Chevalley groups, Acta Math. Sinica (Chinese Ser.) 43 (2000), no. 5, 931–936 (Chinese, with English and Chinese summaries). MR 1818559
Bibliographic Information
  • P. Gvozdevsky
  • Affiliation: Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O. 29B, St. Petersburg 199178, Russia
  • Email: gvozdevskiy96@gmail.com
  • Received by editor(s): January 25, 2019
  • Published electronically: October 27, 2020
  • Additional Notes: This publication was supported by Russian Science Foundation, grant no. 17-11-01261
    2010 Mathematics Subject Classification. Primary 20G70
  • © Copyright 2020 American Mathematical Society
  • Journal: St. Petersburg Math. J. 31 (2020), 969-999
  • DOI: https://doi.org/10.1090/spmj/1631
  • MathSciNet review: 4039348