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Zap. Nauchn. Sem. POMI, 2011, Volume 386, Pages 5–99 (Mi znsl3908)  

This article is cited in 5 scientific papers (total in 5 papers)

Chevalley group of type $\mathrm E_7$ in the 56-dimensional representation

N. A. Vavilova, A. Yu. Luzgarevb

a St. Petersburg State University, St. Petersburg, Russia
b Einstein Institute of Mathematics, Hebrew University of Jerusalem

Abstract: The present paper is devoted to a detailed computer study of the action of Chevalley group $G(\mathrm E_7,R)$ on the 56-dimensional minimal module $V(\varpi_7)$. Our main objectives are an explicit choice and tabulation of the signs of structure constants for this action, compatible with a given choice of a positive Chevalley base, construction of multilinear invariants and of the equations, satisfied by the matrix entries of matrices from $G(\mathrm E_7,R)$ in this representation, and explicit tabulation of root elements. These calculations are performed in four numberings of weights: the natural one, as well as those compatible with the $\mathrm A_6$-branching, the $\mathrm D_6$-branching, and the $\mathrm E_6$-branching. Similar tables for the action of Chevalley group $G(\mathrm E_6,R)$ on the 27-dimensional minimal module $V(\varpi_1)$ were published in our joint paper with Igor Pevzner. Bibl. 142 titles.

Key words and phrases: Chevalley groups, exceptional groups, microweight representations, structure constants, invariant forms, root elements.

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English version:
Journal of Mathematical Sciences (New York), 2012, 180:3, 197–251

Document Type: Article
UDC: 512.5
Received: 24.11.2010

Citation: N. A. Vavilov, A. Yu. Luzgarev, “Chevalley group of type $\mathrm E_7$ in the 56-dimensional representation”, Problems in the theory of representations of algebras and groups. Part 20, Zap. Nauchn. Sem. POMI, 386, POMI, St. Petersburg, 2011, 5–99; J. Math. Sci. (N. Y.), 180:3 (2012), 197–251

Citation in format AMSBIB
\by N.~A.~Vavilov, A.~Yu.~Luzgarev
\paper Chevalley group of type $\mathrm E_7$ in the 56-dimensional representation
\inbook Problems in the theory of representations of algebras and groups. Part~20
\serial Zap. Nauchn. Sem. POMI
\yr 2011
\vol 386
\pages 5--99
\publ POMI
\publaddr St.~Petersburg
\jour J. Math. Sci. (N. Y.)
\yr 2012
\vol 180
\issue 3
\pages 197--251

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    This publication is cited in the following articles:
    1. N. A. Vavilov, “Stroenie izotropnykh reduktivnykh grupp”, Tr. In-ta matem., 18:1 (2010), 15–27  mathnet
    2. 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
    3. 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
    4. J. Math. Sci. (N. Y.), 209:6 (2015), 922–934  mathnet  crossref
    5. 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
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