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

This article is cited in 24 scientific papers (total in 24 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

Bibliographic databases:

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
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\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
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2381932}
\zmath{https://zbmath.org/?q=an:1203.20041}
\transl
\jour St. Petersburg Math. J.
\yr 2008
\vol 19
\issue 4
\pages 519--543
\crossref{https://doi.org/10.1090/S1061-0022-08-01008-X}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. N. A. Vavilov, M. R. Gavrilovich, “$\mathrm A_2$-dokazatelstvo strukturnykh teorem dlya grupp Shevalle tipov $\mathrm E_6$ i $\mathrm E_7$”, Algebra i analiz, 16:4 (2004), 54–87  mathnet  mathscinet  zmath; 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  crossref
    2. N. A. Vavilov, A. Yu. Luzgarev, I. M. Pevzner, “Gruppa Shevalle tipa $\mathrm E_6$ v 27-mernom predstavlenii”, Voprosy teorii predstavlenii algebr i grupp. 14, Zap. nauchn. sem. POMI, 338, POMI, SPb., 2006, 5–68  mathnet  mathscinet  zmath  elib; 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  crossref  elib
    3. N. A. Vavilov, E. Ya. Perelman, “Polivektornye predstavleniya $\operatorname{GL}_n$”, Voprosy teorii predstavlenii algebr i grupp. 14, Zap. nauchn. sem. POMI, 338, POMI, SPb., 2006, 69–97  mathnet  mathscinet  zmath; N. A. Vavilov, E. Ya. Perelman, “Polyvector representations of $\operatorname{GL}_n$”, J. Math. Sci. (N. Y.), 145:1 (2007), 4737–4750  crossref
    4. N. A. Vavilov, M. R. Gavrilovich, S. I. Nikolenko, “Stroenie grupp Shevalle: Dokazatelstvo iz Knigi”, Voprosy teorii predstavlenii algebr i grupp. 13, Zap. nauchn. sem. POMI, 330, POMI, SPb., 2006, 36–76  mathnet  mathscinet  zmath  elib; 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  crossref  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, “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
    7. 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
    8. 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
    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. N. A. Vavilov, A. A. Semenov, “Dlinnye kornevye tory v gruppakh Shevalle”, Algebra i analiz, 24:3 (2012), 22–83  mathnet  mathscinet  zmath  elib; N. A. Vavilov, A. A. Semenov, “Long root tori in Chevalley groups”, St. Petersburg Math. J., 24:3 (2013), 387–430  crossref  isi  elib
    18. 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
    19. Dietrich H., Faccin P., de Graaf W.A., “Computing with Real Lie Algebras: Real Forms, Cartan Decompositions, and Cartan Subalgebras”, J. Symb. Comput., 56 (2013), 27–45  crossref  mathscinet  zmath  isi
    20. 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
    21. J. Math. Sci. (N. Y.), 209:6 (2015), 922–934  mathnet  crossref
    22. N. A. Vavilov, A. Yu. Luzgarev, “Normalizator gruppy Shevalle tipa $\mathrm E_7$”, Algebra i analiz, 27:6 (2015), 57–88  mathnet  mathscinet  elib; 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  crossref  isi
    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, “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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