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Zap. Nauchn. Sem. POMI, 2009, Volume 373, Pages 48–72 (Mi znsl3573)  

This article is cited in 12 scientific papers (total in 12 papers)

Calculations in exceptional groups over rings

N. Vavilova, A. Luzgareva, A. Stepanovbc

a С.-Петербургский государственный университет, г. Санкт-Петербург, Россия
b С.-Петербургский государственный электротехнический университет, г. Санкт-Петербург, Россия
c Abdus Salam School of Mathematical Sciences at the GCU, Lahore, Pakistan

Abstract: In the present paper we discuss a major project, whose goal is to develop theoretical background and working algorithms for calculations in exceptional Chevalley groups over commutative rings. We recall some basic facts concerning calculations in groups over fields, and indicate complications arising in the ring case. Elementary calculations as such are no longer conclusive. We describe basics of calculating with elements of exceptional groups in their minimal representations, which allow to reduce calculations in the group itself to calculations in its subgroups of smaller rank. For all practical matters such calculations are much more efficient, than localisation methods. Bibl. – 147 titles.

Key words and phrases: Chevalley groups, elementary subgroups, Steinberg groups, elementary generators, Steinberg relations, Weyl modules, multilinear invariants, decomposition of unipotents, the proof from the Book, stability conditions, localisation-completion.

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English version:
Journal of Mathematical Sciences (New York), 2010, 168:3, 334–348

Document Type: Article
UDC: 512.54
Received: 10.06.2009
Language: English

Citation: N. Vavilov, A. Luzgarev, A. Stepanov, “Calculations in exceptional groups over rings”, Representation theory, dynamical systems, combinatorial methods. Part XVII, Zap. Nauchn. Sem. POMI, 373, POMI, St. Petersburg, 2009, 48–72; J. Math. Sci. (N. Y.), 168:3 (2010), 334–348

Citation in format AMSBIB
\Bibitem{VavLuzSte09}
\by N.~Vavilov, A.~Luzgarev, A.~Stepanov
\paper Calculations in exceptional groups over rings
\inbook Representation theory, dynamical systems, combinatorial methods. Part~XVII
\serial Zap. Nauchn. Sem. POMI
\yr 2009
\vol 373
\pages 48--72
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl3573}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2010
\vol 168
\issue 3
\pages 334--348
\crossref{https://doi.org/10.1007/s10958-010-9984-z}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77954758579}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. N. A. Vavilov, V. G. Kazakevich, “More variations on decomposition of transvections”, J. Math. Sci. (N. Y.), 171:3 (2010), 322–330  mathnet  crossref  mathscinet  zmath  scopus
    2. N. A. Vavilov, “Stroenie izotropnykh reduktivnykh grupp”, Tr. In-ta matem., 18:1 (2010), 15–27  mathnet
    3. N. A. Vavilov, A. Yu. Luzgarev, “Chevalley group of type $\mathrm E_7$ in the 56-dimensional representation”, J. Math. Sci. (N. Y.), 180:3 (2012), 197–251  mathnet  crossref  mathscinet  mathscinet  zmath  scopus
    4. N. A. Vavilov, S. S. Sinchuk, “Parabolic factorizations of split classical groups”, St. Petersburg Math. J., 23:4 (2012), 637–657  mathnet  crossref  mathscinet  zmath  zmath  isi  elib  elib  scopus
    5. N. A. Vavilov, “An $\mathrm A_3$-proof of the structure theorems for Chevalley groups of types $\mathrm E_6$ and $\mathrm E_7$. II. The main lemma”, St. Petersburg Math. J., 23:6 (2012), 921–942  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    6. N. A. Vavilov, A. V. Stepanov, “Linear groups over general rings. I. Generalities”, J. Math. Sci. (N. Y.), 188:5 (2013), 490–550  mathnet  crossref  mathscinet  zmath  scopus
    7. J. Math. Sci. (N. Y.), 179:6 (2011), 662–678  mathnet  crossref  mathscinet  zmath  scopus
    8. N. A. Vavilov, A. V. Shchegolev, “Overgroups of subsystem subgroups in exceptional groups: levels”, J. Math. Sci. (N. Y.), 192:2 (2013), 164–195  mathnet  crossref  mathscinet  zmath  scopus
    9. J. Math. Sci. (N. Y.), 200:6 (2014), 742–768  mathnet  crossref  mathscinet  zmath  scopus
    10. J. Math. Sci. (N. Y.), 219:3 (2016), 355–369  mathnet  crossref  mathscinet  zmath  scopus
    11. J. Math. Sci. (N. Y.), 209:6 (2015), 922–934  mathnet  crossref  mathscinet  zmath  scopus
    12. J. Math. Sci. (N. Y.), 222:4 (2017), 466–515  mathnet  crossref  mathscinet  zmath  scopus
  • Записки научных семинаров ПОМИ
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