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Zap. Nauchn. Sem. POMI, 2006, Volume 338, Pages 5–68 (Mi znsl165)  

This article is cited in 21 scientific papers (total in 21 papers)

Chevalley group of type $\mathrm E_6$ in the 27-dimensional representation

N. A. Vavilov, A. Yu. Luzgarev, I. M. Pevzner

Saint-Petersburg State University

Abstract: The present paper is devoted to a detailed computer study of the action of Chevalley group $G(\mathrm E_6,R)$ on the minimal module $V(\varpi_1)$. Our main objectives are an explicit choice and tabulation of the signs of structure constants for this action, compatible with the choice of positive Chevalley base, construction of multilinear invariants and equations on the matrix entries of matrices from $G(\mathrm E_6,R)$ in this representation, and explicit tabulation of root elements.

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English version:
Journal of Mathematical Sciences (New York), 2007, 145:1, 4697–4736

Bibliographic databases:

UDC: 512.5
Received: 09.11.2006

Citation: N. A. Vavilov, A. Yu. Luzgarev, I. M. Pevzner, “Chevalley group of type $\mathrm E_6$ in the 27-dimensional representation”, Problems in the theory of representations of algebras and groups. Part 14, Zap. Nauchn. Sem. POMI, 338, POMI, St. Petersburg, 2006, 5–68; J. Math. Sci. (N. Y.), 145:1 (2007), 4697–4736

Citation in format AMSBIB
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\by N.~A.~Vavilov, A.~Yu.~Luzgarev, I.~M.~Pevzner
\paper Chevalley group of type $\mathrm E_6$ in the 27-dimensional representation
\inbook Problems in the theory of representations of algebras and groups. Part~14
\serial Zap. Nauchn. Sem. POMI
\yr 2006
\vol 338
\pages 5--68
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl165}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2354606}
\zmath{https://zbmath.org/?q=an:1124.20032}
\elib{http://elibrary.ru/item.asp?id=9305287}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2007
\vol 145
\issue 1
\pages 4697--4736
\crossref{https://doi.org/10.1007/s10958-007-0304-1}
\elib{http://elibrary.ru/item.asp?id=13553073}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-34547520972}


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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, N. P. Kharchev, “Orbity stabilizatora podsistem”, Voprosy teorii predstavlenii algebr i grupp. 14, Zap. nauchn. sem. POMI, 338, POMI, SPb., 2006, 98–124  mathnet  mathscinet  zmath  elib; N. A. Vavilov, N. P. Kharchev, “Orbits of subsystem stabilisers”, J. Math. Sci. (N. Y.), 145:1 (2007), 4751–4764  crossref
    2. N. A. Vavilov, I. M. Pevzner, “Triples of long root subgroups”, J. Math. Sci. (N. Y.), 147:5 (2007), 7005–7020  mathnet  crossref  mathscinet  elib  elib
    3. N. A. Vavilov, “Kak uvidet znaki strukturnykh konstant?”, Algebra i analiz, 19:4 (2007), 34–68  mathnet  mathscinet  zmath; N. A. Vavilov, “Can one see the signs of structure constants?”, St. Petersburg Math. J., 19:4 (2008), 519–543  crossref  isi
    4. N. A. Vavilov, A. Yu. Luzgarev, “Normalizator gruppy Shevalle tipa $\mathrm{E}_6$”, Algebra i analiz, 19:5 (2007), 37–64  mathnet  mathscinet  zmath; N. A. Vavilov, A. Yu. Luzgarev, “The normalizer of Chevalley groups of type $\mathrm{E}_6$”, St. Petersburg Math. J., 19:5 (2008), 699–718  crossref  isi
    5. Vavilov N., “An $A_3$-proof of structure theorems for Chevalley groups of types $E_6$ and $E_7$”, Internat. J. Algebra Comput., 17:5-6 (2007), 1283–1298  crossref  mathscinet  zmath  isi  elib
    6. N. A. Vavilov, S. I. Nikolenko, “$\mathrm A_2$-dokazatelstvo strukturnykh teorem dlya gruppy Shevalle tipa $\mathrm F_4$”, Algebra i analiz, 20:4 (2008), 27–63  mathnet  mathscinet  zmath  elib; N. A. Vavilov, S. I. Nikolenko, “$\mathrm A_2$-proof of structure theorems for Chevalley groups of type $\mathrm F_4$”, St. Petersburg Math. J., 20:4 (2009), 527–551  crossref  isi
    7. N. A. Vavilov, “Numerologiya kvadratnykh uravnenii”, Algebra i analiz, 20:5 (2008), 9–40  mathnet  mathscinet  zmath; N. A. Vavilov, “Numerology of square equations”, St. Petersburg Math. J., 20:5 (2009), 687–707  crossref  isi
    8. A. Yu. Luzgarëv, “Opisanie nadgrupp $\mathrm F_4$ v $\mathrm E_6$ nad kommutativnym koltsom”, Algebra i analiz, 20:6 (2008), 148–185  mathnet  mathscinet  zmath; A. Yu. Luzgarev, “Overgroups of $\mathrm{F}_4$ in $\mathrm{E}_6$ over commutative rings”, St. Petersburg Math. J., 20:6 (2009), 955–981  crossref  isi
    9. N. Vavilov, A. Luzgarev, A. Stepanov, “Calculations in exceptional groups over rings”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye metody. XVII, Zap. nauchn. sem. POMI, 373, POMI, SPb., 2009, 48–72  mathnet; J. Math. Sci. (N. Y.), 168:3 (2010), 334–348  crossref
    10. N. A. Vavilov, “Some more exceptional numerology”, J. Math. Sci. (N. Y.), 171:3 (2010), 317–321  mathnet  crossref
    11. N. A. Vavilov, “Stroenie izotropnykh reduktivnykh grupp”, Tr. In-ta matem., 18:1 (2010), 15–27  mathnet
    12. 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
    13. I. M. Pevzner, “Shirina grupp tipa $\mathrm E_6$ otnositelno mnozhestva kornevykh elementov. II”, Voprosy teorii predstavlenii algebr i grupp. 20, Zap. nauchn. sem. POMI, 386, POMI, SPb., 2011, 242–264  mathnet; I. M. Pevzner, “Width of groups of type $\mathrm E_6$ with respect to root elements. II”, J. Math. Sci. (N. Y.), 180:3 (2012), 338–350  crossref
    14. I. M. Pevzner, “Geometriya kornevykh elementov v gruppakh tipa $\mathrm E_6$”, Algebra i analiz, 23:3 (2011), 261–309  mathnet  mathscinet  zmath  elib; I. M. Pevzner, “The geometry of root elements in groups of type $\mathrm E_6$”, St. Petersburg Math. J., 23:3 (2012), 603–635  crossref  isi  elib
    15. I. M. Pevzner, “Shirina grupp tipa $\mathrm E_6$ otnositelno mnozhestva kornevykh elementov. I”, Algebra i analiz, 23:5 (2011), 155–198  mathnet  mathscinet  elib; I. M. Pevzner, “Width of groups of type $\mathrm E_6$ with respect to root elements. I”, St. Petersburg Math. J., 23:5 (2012), 891–919  crossref  isi  elib
    16. 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
    17. I. M. Pevzner, “Shirina gruppy $\mathrm{GL}(6,K)$ otnositelno mnozhestva kvazikornevykh elementov”, Voprosy teorii predstavlenii algebr i grupp. 26, Zap. nauchn. sem. POMI, 423, POMI, SPb., 2014, 183–204  mathnet  mathscinet; I. M. Pevzner, “Width of $\mathrm{GL}(6,K)$ with respect to quasi-root elements”, J. Math. Sci. (N. Y.), 209:4 (2015), 600–613  crossref
    18. 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
    19. J. Math. Sci. (N. Y.), 209:6 (2015), 922–934  mathnet  crossref
    20. I. M. Pevzner, “Shirina ekstraspetsialnogo unipotentnogo radikala otnositelno mnozhestva kornevykh elementov”, Voprosy teorii predstavlenii algebr i grupp. 28, Zap. nauchn. sem. POMI, 435, POMI, SPb., 2015, 168–177  mathnet  mathscinet
    21. M. M. Atamanova, A. Yu. Luzgarev, “Kubicheskie formy na prisoedinennykh predstavleniyakh isklyuchitelnykh grupp”, Voprosy teorii predstavlenii algebr i grupp. 29, Zap. nauchn. sem. POMI, 443, POMI, SPb., 2016, 9–23  mathnet  mathscinet
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