Constructions of Polygons from Buildings

Described here are constructions of finite geometries, with special emphasis on generalized quadrangles, using first a strange subset <# 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 the playing field. In § 2 the rules of the game are given and the construction of the geometry r ( ^ ) is described in detail, illustrated by several constructions of known geometries. In § 3 the parameters associated with the construction of FC^) are determined by the number of flags opposite a given flag in some residue of a building. Section 4 concerns automorphisms of the geometry r (^ ) , with special emphasis on generalized quadrangles. Constructions based on buildings of type Cn or Dn are considered in § 5. It appears that when considering only flags of size 1, 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 A of type Cn or Dn, a geometry r ( ^ ) is a generalized quadrangle if and only if all elements of <# (which are totally singular spaces of A) pairwise intersect at the same space Fo and the set <# corresponding to ^ in ResA(F0) = A is a partial m-system satisfying the BLT property. That is, all elements of <£ are totally singular m-spaces of A which are pairwise opposite, having the property that each singular line of A which is not contained in an element of % intersects at most two elements of %. 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 § 2. Several restrictions on the parameters are proved which eliminate many cases in the polar space construction. Section 9 is a section on existence and non-existence questions with respect to the polar space construction. Most of the known quadrangles are seen to be of type r ( ^ ) , 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 the possibility of % being an m-system. There are five interesting 'sporadic' cases and one infinite class.