Frames and bases of Lie incidence geometries
The authors consider Lie incidence geometries \(\Gamma=({\mathcal P},{\mathcal L})\) of types \(B_{n,n}\) and \(C_{n,1}\) over a field of characteristic not two, \(A_{n,k}\), \(D_{n,1}\), \(D_{n,n}\), \(E_{6,1}\) or \(E_{7,1}\) and show that a subset \(X\subset {\mathcal P}\) is an apartment of \(\Gamma\) if and only if one of the following conditions is satisfied: (i) The induced point collinearity graphs induced on a fixed apartment \(A\) and on \(X\) are isomorphic and \(X\) is \(e\)-independent for the standard embedding \(e:\Gamma\to\text{ PG}(V)\); (ii) \(X\) spans \(\Gamma\) (and hence is a basis) and the induced graph on \(X\) is isomorphic to an apartment. \textit{M. Ronan} and \textit{S. Smith} [J. Algebra 96, 319-346 (1985; Zbl 0604.20043)] essentially proved that in the above cases an apartment spans the geometry. Their unified proof however relies on the theory of buildings of spherical type. In the paper under review the authors deal with the geometries indendently and provide more elementary proofs. The authors define for a projective embedding \(e:\Gamma\to\text{ PG}(V)\) the concepts of \(e\)-independence of a point set and a basis. They then show that for each of the above types the automorphism group \(Aut(\Gamma)\) is transitive on frames, that is, sets of \(e\)-independent points on which the subgraph of an apartment of the geometry is induced. In the final section of the paper two examples are presented that illustrate that \(e\)-independence is necessary in the above mentioned characterization of apartments.
scientific article; zbMATH DE number 1106600