Embeddings and hyperplanes of the Lie incidence geometry of type \(E_{7,1}\)
A Lie incidence geometry is a point-line geometry \(\Gamma=({\mathcal P},{\mathcal L})\) whose points are the cosets of a maximal parabolic subgroup \(P\) of a group of Lie type and whose lines are the cosets of the parabolic subgroup corresponding to the collection of all nodes in the Dynkin diagram adjacent to the unique node corresponding to \(P\). A (projective) embedding \(e\) of \(\Gamma\) is a one-to-one map from \({\mathcal P}\) into the point set of some projective space \({\mathbf P}(V)\) over a vector space \(V\) such that \(e(L)\) is a projective line for each line \(L\in{\mathcal L}\) and such that the image points \(e({\mathcal P})\) span \({\mathbf P}(V)\). A geometric hyperplane \(H\) of \(\Gamma\), that is, a proper subspace which meets each line non-trivially, is said to arise from \(e\) if \(H=e^{-1}(H')\) for a projective hyperplane \(H'\) of \({\mathbf P}(V)\). If every geometric hyperplane of a Lie incidence geometry \(\Gamma\) arises from an embedding \(e\), then \(e\) is relatively universal, that is, \(e\) is a source in the subcategory of all projective embeddings of \(\Gamma\) possessing a morphism to \(e\). The question whether or not every geometric hyperplane of a Lie incidence geometry arises from an embedding has been answered in the affirmative for a large class of Lie incidence geometries by the author and others in a series of papers. For example, projective spaces, polar spaces of rank at least 2, Grassmann spaces of type \(A_{n,k}\) over a field, spin geometries over finite fields of odd order, half-spin geometries, geometries of type \(D_{n,n}\) and the exceptional geometries of type \(E_{6,1}\) are known to fall into this class. In the paper under review the author adds Lie incidence geometries of type \(E_{7,1}\) to this list. More precisely, he shows that for any relative universal embeddding \(e\) every geometric hyperplane arises from \(e\). Furthermore, such an embedding is unique and hence absolutely universal in the category of all projective embeddings and their morphisms. Using a construction of \textit{M. Ronan} [Eur. J. Comb. 8, 179-185 (1987; Zbl 0624.51007)] a relatively universal embedding can be constructed from the standard 56-dimensional module for \(E_7(K)\), \(K\) a field. After briefly discussing some approaches to showing his result, the author settles for the circuitry method for parapolar spaces. This method is based on results from M. Ronan's seminal paper [loc. cit.] and has been successfully exploited by \textit{B. Cooperstein} and the author in Geom. Dedicata 64, No. 1, 17-40 (1997; Zbl 0868.51006). If in such a space every circuit in the collinearity graph induced on \({\mathcal P} - {\mathcal H}\) for any geometric hyperplane \({\mathcal H}\) is a sum of circuits each one of which lies in some symplecton (i.e., a convex polar subspace of rank at least two), then for any embedding \(e\), every geometric hyperplane arises from the embedding \(\hat e\) universal for \(e\). Using special properties of the \(E_{7,1}\) geometry, in particular, the possible relationships between two distinct symplecta, and after some reductions, the author derives in a number of steps a contradiction to the assumption that there is a curcuit of minimal length that cannot be reduced as required.
scientific article; zbMATH DE number 1053897