Geodesic
Chevalley group of type $\mathrm E_7$ in the 56-dimensional representation
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 20, Tome 386 (2011), pp. 5-99

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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. 
@article{ZNSL_2011_386_a0,
     author = {N. A. Vavilov and A. Yu. Luzgarev},
     title = {Chevalley group of type $\mathrm E_7$ in the 56-dimensional representation},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--99},
     publisher = {mathdoc},
     volume = {386},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_386_a0/}
}
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N. A. Vavilov; A. Yu. Luzgarev. Chevalley group of type $\mathrm E_7$ in the 56-dimensional representation. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 20, Tome 386 (2011), pp. 5-99. http://geodesic.mathdoc.fr/item/ZNSL_2011_386_a0/