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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
\Bibitem{VavLuz11} \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 \mathnet{http://mi.mathnet.ru/znsl3908} \transl \jour J. Math. Sci. (N. Y.) \yr 2012 \vol 180 \issue 3 \pages 197--251 \crossref{https://doi.org/10.1007/s10958-011-0641-y} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84855684282} 

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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
2. 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
3. J. Math. Sci. (N. Y.), 219:3 (2016), 355–369
4. J. Math. Sci. (N. Y.), 209:6 (2015), 922–934
5. 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
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