This chapter presents the functions designed for constructing and computing with incidence geometries and coset geometries.
We recall the basic definitions and notation in the field of Incidence Geometry. We refer to the Handbook of Incidence Geometry [Bue95], edited by Francis Buekenhout, or to Antonio Pasini's book Diagram Geometries [Pas94], for a more detailed overview of the subject.
Let X and I be two finite sets. Let t : X -> I be a mapping from X onto I. Let ~ be a reflexive and symmetric relation such that forall x, y ∈X, x ~y and t(x) = t(y) => x = y. The four-tuple Γ(X, ~, t, I) is what we call an Incidence Geometry in Magma. Remark that it is not a geometry in the sense of Buekenhout since we do not impose that every flag (i.e. clique of the incidence graph) of Γ must be contained in a chamber (i.e. a clique containing one element of each type). If the latter condition is satisfied, then an incidence geometry is a geometry in the sense of Buekenhout. The set X contains the elements of the geometry, while I is called the set of types. The function t is called the type function and ~ is called the incidence relation of Γ. The cardinality of I is the rank of Γ.
It is possible to construct incidence geometries from a group and some of its subgroups using an algorithm first introduced by Jacques Tits in 1962 [Tit62]. Let G be a group and let I be a finite set. Let { Gi, i ∈I } be a set of subgroups of G. Define X = { Gig, g∈G, i∈I } to be the set of elements of Γ. Define the type function as t : X -> I : Gig -> i and the incidence relation as follows: Gig ~Gjh iff Gig ∩Gjh != emptyset. The subgroups { Gi, i ∈I } are called the maximal parabolic subgroups. The subgroup ∩i∈IGi is called the Borel subgroup. Finally, the subgroups { ∩_(j∈I - {i})Gj , i ∈I } are called the minimal parabolic subgroups. These geometries are called Coset Geometries in Magma to remind the user that they are constructed from a group. Again, a coset geometry is not a geometry in the sense of Buekenhout. If every flag of the coset geometry is contained in a chamber, then it is a Buekenhout geometry. We will see that, using coset geometries, it is easy to build huge incidence geometries by giving very little data.
The category names for the incidence geometries and coset geometries are: