Described here are constructions of finite geometries, with special emphasis on generalized quadrangles, using first a strange subset \({\mathcal C}\) of flags of a known finite geometry and then defining new objects by a process of sequentially taking collections of new flags in various residues according to certain rules. These rules can be thought of as a game played on a Dynkin diagram, and the diagram itself can be thought of as a playing field.
In Section 2 the rules of the game are given and the construction of the geometry \(\Gamma ({\mathcal C})\) is described in detail, illustrated by several constructions of known geometries.
In Section 3 the parameters associated with the construction of \(\Gamma ({\mathcal C})\) are determined by the number of flags opposite a given flag in some residue of a building. Section 4 concerns automorphisms of the geometry \(\Gamma ({\mathcal C})\), with special emphasis on generalized quadrangles.
Constructions based on buildings of type \(C_n\) or \(D_n\) are considered in Section 5. It appears that when considering only flags of size one, the constructions can be divided into two classes: the polar space constructions and the dual polar space constructions.
Sections 6-9 are devoted to the polar space constructions. It is shown that starting from a building \(\Delta\) of type \(C_n\) or \(D_n\), a geometry \(\Gamma ({\mathcal C})\) is a generalized quadrangle if and only if all elements of \({\mathcal C}\) (which are totally singular spaces of \(\Delta)\) pairwise intersect at the same space \(F_0\) and the set \(\overline {\mathcal C}\) corresponding to \({\mathcal C}\) in \(\text{Res}_\Delta (F_0)= \overline {\Delta}\) is a partial \(m\)-system satisfying the BLT property, that is, all elements of \(\overline {\mathcal C}\) are totally singular \(m\)-spaces of \(\overline {\Delta}\) which are pairwise opposite, such that each singular line of \(\overline {\Delta}\) which is not contained in an element of \(\overline {\mathcal C}\) intersects at most two elements of \(\overline {\mathcal C}\). Any partial \(m\)-system satisfying the BLT property which has the right size and consists of spaces of the right dimension, defines a generalized quadrangle by the construction described in Section 2. Several restrictions on the parameters are proved which eliminate many cases in the polar space construction. Most of the known quadrangles are seen to be of type \(\Gamma ({\mathcal C})\), and by the theory of translation generalized quadrangles many parameter sets can be excluded. A summary of the surviving low rank cases is given in the form of a table for each type of polar space. Finally, we investigate when \(\overline {\mathcal C}\) possibly is an \(m\)-system. There are five interesting “sporadic” cases and one infinite class.
Schemata based on dual polar spaces are considered in Section 10. Here it is shown that \(\Gamma ({\mathcal C})\) can never be a generalized polygon except in the trivial rank 2 case. Schemata based on half-spin geometries are the topic of Section 11. Here again it is proved that no interesting case appears. Section 12 contains a few remarks on constructions involving exceptional diagrams.