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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
\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
\jour J. Math. Sci. (N. Y.)
\yr 2010
\vol 168
\issue 3
\pages 334--348

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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
    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, “Gruppa Shevalle tipa $\mathrm E_7$ v 56-mernom predstavlenii”, Voprosy teorii predstavlenii algebr i grupp. 20, Zap. nauchn. sem. POMI, 386, POMI, SPb., 2011, 5–99  mathnet; 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  crossref
    4. N. A. Vavilov, S. S. Sinchuk, “Parabolicheskie faktorizatsii rasschepimykh klassicheskikh grupp”, Algebra i analiz, 23:4 (2011), 1–30  mathnet  mathscinet  zmath  elib; N. A. Vavilov, S. S. Sinchuk, “Parabolic factorizations of split classical groups”, St. Petersburg Math. J., 23:4 (2012), 637–657  crossref  isi  elib
    5. N. A. Vavilov, “$\mathrm A_3$-dokazatelstvo strukturnykh teorem dlya grupp Shevalle tipov $\mathrm E_6$$\mathrm E_7$. II. Osnovnaya lemma”, Algebra i analiz, 23:6 (2011), 1–31  mathnet  mathscinet  elib; 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  crossref  isi  elib
    6. N. A. Vavilov, A. V. Stepanov, “Lineinye gruppy nad obschimi koltsami I. Obschie mesta”, Voprosy teorii predstavlenii algebr i grupp. 22, Zap. nauchn. sem. POMI, 394, POMI, SPb., 2011, 33–139  mathnet  mathscinet; N. A. Vavilov, A. V. Stepanov, “Linear groups over general rings. I. Generalities”, J. Math. Sci. (N. Y.), 188:5 (2013), 490–550  crossref
    7. J. Math. Sci. (N. Y.), 179:6 (2011), 662–678  mathnet  crossref
    8. N. A. Vavilov, A. V. Schegolev, “Nadgruppy subsystem subgroups v isklyuchitelnykh gruppakh: urovni”, Voprosy teorii predstavlenii algebr i grupp. 23, Zap. nauchn. sem. POMI, 400, POMI, SPb., 2012, 70–126  mathnet  mathscinet; N. A. Vavilov, A. V. Shchegolev, “Overgroups of subsystem subgroups in exceptional groups: levels”, J. Math. Sci. (N. Y.), 192:2 (2013), 164–195  crossref
    9. J. Math. Sci. (N. Y.), 200:6 (2014), 742–768  mathnet  crossref
    10. N. A. Vavilov, “Decomposition of unipotents for $\mathrm E_6$ and $\mathrm E_7$: 25 years after”, Voprosy teorii predstavlenii algebr i grupp. 27, Zap. nauchn. sem. POMI, 430, POMI, SPb., 2014, 32–52  mathnet  mathscinet; J. Math. Sci. (N. Y.), 219:3 (2016), 355–369  crossref
    11. J. Math. Sci. (N. Y.), 209:6 (2015), 922–934  mathnet  crossref
    12. R. Hazrat, N. Vavilov, Z. Zhang, “The commutators of classical groups”, Voprosy teorii predstavlenii algebr i grupp. 29, Zap. nauchn. sem. POMI, 443, POMI, SPb., 2016, 151–221  mathnet  mathscinet; J. Math. Sci. (N. Y.), 222:4 (2017), 466–515  crossref
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