Calculations in exceptional groups, an update

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 $\mathrm E_8$.