On a system of basis invariants of the group \(E_ 7\)
Suppose a rectangular Cartesian coordinate system \(Ox_i\), \(i=\overline{1,7}\), is defined in the real Euclidean space \(E_7\). The hyperplanes of symmetry of the Gosset polyhedron \(3_{21}\) that are invariant relative to the group \(E_7\) can be defined by the normalized equations \(\eta_s=0\), \(s=\overline{1,63}\). The basis invariants of the group \(E_7\) have degrees \(n_i=2,6,8,10,12,14,18\). \textit{V. F. Ignatenko} [Ukr. Geom. Sb. 23, 50-56 (1980; Zbl 0475.14007)] has proved the following result. Theorem. The polynomials \(P_{2s}=\sum^{63}_{s=1}\eta^{n_i}_s\), \(i=\overline{1,7}\), form a system of basis invariants of the group \(E_7\). In this paper we give a new proof of this theorem (using a computer).
scientific article; zbMATH DE number 1026968
scientific article