Characterization of some subgraphs of point-collinearity graphs of building geometries
The author gives a beautiful characterization of the shadows of apartments of residues in the point-line truncations of \(J\)-Grassmannians of buildings. In fact, these are characterized by two conditions on a given connected induced subgraph of the point graph of the point-line truncation (a local one and one involving circuits and planes of the \(J\)-Grassmannian); roughly the former controls the thickness (or rather the thinness) and the latter controls the diameter and girth of the rank 2 residues of the subgraph. A number of other results is presented, such as the theorem that the shadow of any residue in the point-line truncation of a J-Grassmannian is a convex subspace. The usefulness of the main theorem is illustrated with a number of examples. The reviewer likes to stress that the results proved in this paper hold for any \(J\) (of course, \(J\) must meet every connected component of the diagram). As such, this paper is a marvellous contribution to the theory of building Grassmannians.
scientific article; zbMATH DE number 5154718