Coordinatization structures for generalized quadrangles and glued near hexagons

A generalized admissible triple is a triple T = (L, X,∆), where X is a set of size s + 1 ≥ 2, L is a Steiner system S(2, s + 1, st + 1), t ≥ 1, with point-set P and ∆ is a very nice map from P × P to the group Sym(X) of all permutations of the set X. Generalized admissible triples can be used to coordinatize generalized quadrangles with a regular spread and glued near hexagons. The idea of coordinatizing these incidence structures in this way has led to several breakthrough results in the theory of generalized quadrangles and near polygons. Generalized admissible triples allow to give unified constructions for several classes of generalized quadrangles, they allow to classify all generalized quadrangles of order 5 with a center of symmetry and they also allow to characterize the symplectic generalized quadrangle W (q) as this generalized quadrangle of order q having a hyperbolic line consisting of only regular points. In this chapter, we give a survey of the theory of generalized admissible triples and of many of its interesting applications.