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Algebra i Analiz, 2007, Volume 19, Issue 4, Pages 34–68 (Mi aa126)  

This article is cited in 25 scientific papers (total in 25 papers)

Research Papers

Can one see the signs of structure constants?

N. A. Vavilov

St. Petersburg State University, Department of Mathematics and Mechanics

Abstract: It is described how one can see the signs of action structure constants directly in the weight diagram of microweight and adjoint representations for groups of types $\mathrm{E}_6$$\mathrm{E}_7$ and $\mathrm{E}_8$. This generalizes the results of the preceding paper, “A third look at weight diagrams”, where a similar algorithm was discussed for microweight representations of $\mathrm{E}_6$ and $\mathrm{E}_7$. The proofs are purely combinatorial and can be viewed as an elementary construction of Lie algebras and Chevalley groups of types $\mathrm{E}_l$ .

Keywords: Microweight representation, adjoint representation, weight diagram, structure constants.

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English version:
St. Petersburg Mathematical Journal, 2008, 19:4, 519–543

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MSC: 20G05
Received: 06.11.2006

Citation: N. A. Vavilov, “Can one see the signs of structure constants?”, Algebra i Analiz, 19:4 (2007), 34–68; St. Petersburg Math. J., 19:4 (2008), 519–543

Citation in format AMSBIB
\by N.~A.~Vavilov
\paper Can one see the signs of structure constants?
\jour Algebra i Analiz
\yr 2007
\vol 19
\issue 4
\pages 34--68
\jour St. Petersburg Math. J.
\yr 2008
\vol 19
\issue 4
\pages 519--543

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    This publication is cited in the following articles:
    1. N. A. Vavilov, M. R. Gavrilovich, “$\mathrm A_2$-proof of structure theorem for Chevaller groups of type $\mathrm E_6$ and $\mathrm E_7$”, St. Petersburg Math. J., 16:4 (2005), 649–672  mathnet  crossref  mathscinet  zmath
    2. N. A. Vavilov, A. Yu. Luzgarev, I. M. Pevzner, “Chevalley group of type $\mathrm E_6$ in the 27-dimensional representation”, J. Math. Sci. (N. Y.), 145:1 (2007), 4697–4736  mathnet  crossref  mathscinet  zmath  elib  elib
    3. N. A. Vavilov, E. Ya. Perelman, “Polyvector representations of $\operatorname{GL}_n$”, J. Math. Sci. (N. Y.), 145:1 (2007), 4737–4750  mathnet  crossref  mathscinet  zmath
    4. N. A. Vavilov, M. R. Gavrilovich, S. I. Nikolenko, “Structure of Chevalley groups: the proof from the Book”, J. Math. Sci. (N. Y.), 140:5 (2007), 626–645  mathnet  crossref  mathscinet  zmath  elib  elib
    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, 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
    7. 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
    8. N. A. Vavilov, “Numerology of square equations”, St. Petersburg Math. J., 20:5 (2009), 687–707  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. N. A. Vavilov, A. A. Semenov, “Long root tori in Chevalley groups”, St. Petersburg Math. J., 24:3 (2013), 387–430  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    18. 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
    19. Dietrich H., Faccin P., de Graaf W.A., “Computing with Real Lie Algebras: Real Forms, Cartan Decompositions, and Cartan Subalgebras”, J. Symbolic Comput., 56 (2013), 27–45  crossref  mathscinet  zmath  isi
    20. J. Math. Sci. (N. Y.), 219:3 (2016), 355–369  mathnet  crossref  mathscinet
    21. J. Math. Sci. (N. Y.), 209:6 (2015), 922–934  mathnet  crossref
    22. N. A. Vavilov, A. Yu. Luzgarev, “Normaliser of the Chevalley group of type $\mathrm E_7$”, St. Petersburg Math. J., 27:6 (2016), 899–921  mathnet  crossref  mathscinet  isi  elib
    23. Sinchuk S., “on Centrality of K-2 For Chevalley Groups of Type E-l”, J. Pure Appl. Algebr., 220:2 (2016), 857–875  crossref  mathscinet  zmath  isi  elib
    24. 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
    25. Geck M., “Minuscule Weights and Chevalley Groups”, Finite Simple Groups: Thirty Years of the Atlas and Beyond, Contemporary Mathematics, 694, eds. Bhargava M., Guralnick R., Hiss G., Lux K., Tiep PH., Amer Mathematical Soc, 2017, 159–176  crossref  mathscinet  isi  scopus
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