# 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*(*E*_{7}, *R*) in the 56-dimensional representation *V*. Before, a similar description was known for the group *G*(*E*_{6}, *R*) in the 27-dimensional representation, whereas for the group of type E_{7} it was only known under the simplifying assumption that 2 ϵ *R*^{*}. In particular, we compute the normalizer of *G*(*E*_{7}, *R*) in GL(56, *R*) and establish that, as also the normalizer of the elementary subgroup *E*(E_{7}, *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 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 E_{6}, E_{7}, and E_{8}. 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 2*π*/3-equations) and quadratically (the *π*-equations). Also, we briefly discuss recent results by S.Garibaldi and R.M.Guralnick on octic invariants for E_{8}.

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