
Zap. Nauchn. Sem. POMI, 2015, Volume 432, Pages 177–195
(Mi znsl6117)




Calculations in exceptional groups, an update
A. Luzgarev^{}, N. Vavilov^{} ^{} St. Petersburg State University, St. Petersburg, Russia
Abstract:
This paper is a slightly expanded text of our talk at the PCA2014. There, we announced two recent results, concerning explicit polynomial equations defining exceptional Chevalley groups in microweight or adjoint representations. One of these results is an explicit characteristicfree description of equations on the entries of a matrix from the simply connected Chevalley group $G(\mathrm E_7,R)$ in the $56$dimensional representation $V$. Before, similar description was known for the group $G(\mathrm E_6,R)$ in the $27$dimensional representation, whereas for the group of type $\mathrm E_7$ it was only known under the simplifying assumption that $2\in R^*$. In particular, we compute the normalizer of $G(\mathrm E_7,R)$ in $\mathrm{GL}(56,R)$ and establish that, as also the normalizer of the elementary subgroup $E(\mathrm E_7,R)$, it coincides with the extended Chevalley group $\bar G(\mathrm E_7,R)$. The construction is based on the works of J.Lurie and the first author on the $\mathrm E_7$invariant quartic forms on $V$. Another major new result is a complete description of quadratic equations defining the highest weight orbit in the adjoint representations of Chevalley groups of types $\mathrm E_6$, $\mathrm E_7$ and $\mathrm E_8$. Part of these equations not involving zero weights, the socalled square equations (or $\pi/2$equations) were described by the second author. Recently, the first author succeeded in completing these results, explicitly listing also the equations involving zero weight coordinates linearly (the $2\pi/3$equations) and quadratically (the $\pi$equations). Also, we briefly discuss recent results by S. Garibaldi and R. M. Guralnick on octic invariants for $\mathrm E_8$.
Key words and phrases:
Chevalley groups, elementary subgroups, exceptional groups, multilinear invariants, microweight representation, adjoint representation, highest weight orbit.
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English version:
Journal of Mathematical Sciences (New York), 2015, 209:6, 922–934
UDC:
512.5 Received: 26.11.2014
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Citation:
A. Luzgarev, N. Vavilov, “Calculations in exceptional groups, an update”, Representation theory, dynamical systems, combinatorial methods. Part XXIV, Zap. Nauchn. Sem. POMI, 432, POMI, St. Petersburg, 2015, 177–195; J. Math. Sci. (N. Y.), 209:6 (2015), 922–934
Citation in format AMSBIB
\Bibitem{LuzVav15}
\by A.~Luzgarev, N.~Vavilov
\paper Calculations in exceptional groups, an update
\inbook Representation theory, dynamical systems, combinatorial methods. Part~XXIV
\serial Zap. Nauchn. Sem. POMI
\yr 2015
\vol 432
\pages 177195
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6117}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2015
\vol 209
\issue 6
\pages 922934
\crossref{https://doi.org/10.1007/s1095801525387}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2s2.084939447980}
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