Journal of Mathematical Sciences

, Volume 209, Issue 6, pp 922–934 | Cite as

Calculations in Exceptional Groups, an Update

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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(E7, R) in the 56-dimensional representation V. Before, a similar description was known for the group G(E6, R) in the 27-dimensional representation, whereas for the group of type E7 it was only known under the simplifying assumption that 2 ϵ R*. In particular, we compute the normalizer of G(E7, R) in GL(56, R) and establish that, as also the normalizer of the elementary subgroup E(E7, R), it coincides with the extended Chevalley group \( \overline{G}\left({\mathrm{E}}_7,R\right) \). The construction is based on the works of J. Lurie and the first author on the E7-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 E6, E7, and E8. Part of these equations not involving zero weights, the so-called square equations (or π/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 2π/3-equations) and quadratically (the π-equations). Also, we briefly discuss recent results by S.Garibaldi and R.M.Guralnick on octic invariants for E8.

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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.St.Petersburg State UniversitySt.PetersburgRussia

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