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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 PCA-2014. 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 characteristic-free 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 so-called 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

Document Type: Article
UDC: 512.5
Received: 26.11.2014
Language: English

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 177--195
\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 922--934
\crossref{https://doi.org/10.1007/s10958-015-2538-7}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84939447980}


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