Chevalley group of type $\mathrm E_7$ in the 56-dimensional representation

The present paper is devoted to a detailed computer study of the action of Chevalley group $G(\mathrm E_7,R)$ on the 56-dimensional minimal module $V(\varpi_7)$. Our main objectives are an explicit choice and tabulation of the signs of structure constants for this action, compatible with a given choice of a positive Chevalley base, construction of multilinear invariants and of the equations, satisfied by the matrix entries of matrices from $G(\mathrm E_7,R)$ in this representation, and explicit tabulation of root elements. These calculations are performed in four numberings of weights: the natural one, as well as those compatible with the $\mathrm A_6$-branching, the $\mathrm D_6$-branching, and the $\mathrm E_6$-branching. Similar tables for the action of Chevalley group $G(\mathrm E_6,R)$ on the 27-dimensional minimal module $V(\varpi_1)$ were published in our joint paper with Igor Pevzner. Bibl. 142 titles.