Chevalley group of type $\mathrm E_6$ in the 27-dimensional representation

The present paper is devoted to a detailed computer study of the action of Chevalley group $G(\mathrm E_6,R)$ on the minimal module $V(\varpi_1)$. Our main objectives are an explicit choice and tabulation of the signs of structure constants for this action, compatible with the choice of positive Chevalley base, construction of multilinear invariants and equations on the matrix entries of matrices from $G(\mathrm E_6,R)$ in this representation, and explicit tabulation of root elements.