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

This article is cited in 22 scientific papers (total in 22 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}
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\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}
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    This publication is cited in the following articles:
    1. N. A. Vavilov, N. P. Kharchev, “Orbits of subsystem stabilisers”, J. Math. Sci. (N. Y.), 145:1 (2007), 4751–4764  mathnet  crossref  mathscinet  zmath  elib
    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, “Can one see the signs of structure constants?”, St. Petersburg Math. J., 19:4 (2008), 519–543  mathnet  crossref  mathscinet  zmath  isi
    4. 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  mathnet  crossref  mathscinet  zmath  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, “$\mathrm A_2$-proof of structure theorems for Chevalley groups of type $\mathrm F_4$”, St. Petersburg Math. J., 20:4 (2009), 527–551  mathnet  crossref  mathscinet  zmath  isi  elib
    7. N. A. Vavilov, “Numerology of square equations”, St. Petersburg Math. J., 20:5 (2009), 687–707  mathnet  crossref  mathscinet  zmath  isi
    8. A. Yu. Luzgarev, “Overgroups of $\mathrm{F}_4$ in $\mathrm{E}_6$ over commutative rings”, St. Petersburg Math. J., 20:6 (2009), 955–981  mathnet  crossref  mathscinet  zmath  isi
    9. J. Math. Sci. (N. Y.), 168:3 (2010), 334–348  mathnet  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, “Chevalley group of type $\mathrm E_7$ in the 56-dimensional representation”, J. Math. Sci. (N. Y.), 180:3 (2012), 197–251  mathnet  crossref
    13. 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  mathnet  crossref
    14. I. M. Pevzner, “The geometry of root elements in groups of type $\mathrm E_6$”, St. Petersburg Math. J., 23:3 (2012), 603–635  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    15. 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  mathnet  crossref  mathscinet  isi  elib  elib
    16. 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  isi  elib  elib
    17. 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  mathnet  crossref  mathscinet
    18. J. Math. Sci. (N. Y.), 219:3 (2016), 355–369  mathnet  crossref  mathscinet
    19. J. Math. Sci. (N. Y.), 209:6 (2015), 922–934  mathnet  crossref
    20. I. M. Pevzner, “Width of extraspecial unipotent radical with respect to root elements”, J. Math. Sci. (N. Y.), 219:4 (2016), 598–603  mathnet  crossref  mathscinet
    21. M. M. Atamanova, A. Yu. Luzgarev, “Cubic forms on adjoint representations of exceptional groups”, J. Math. Sci. (N. Y.), 222:4 (2017), 370–379  mathnet  crossref  mathscinet
    22. I. M. Pevzner, “Suschestvovanie kornevoi podgruppy, kotoruyu dannyi element perevodit v protivopolozhnuyu”, Voprosy teorii predstavlenii algebr i grupp. 32, Zap. nauchn. sem. POMI, 460, POMI, SPb., 2017, 190–202  mathnet
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