Properties of Incidence Geometries and Coset Geometries

Let us recall definitions of the properties described in this section.

An incidence geometry Gamma is flag--transitive if for every two flags x, y of the same type of Gamma, there exists an element g of Aut(Gamma) such that g(x) = y. We also say that Aut(Gamma) acts flag--transitively in this case.

Moreover, it is a flag--transitive geometry if it contains at least one chamber.

A coset geometry Gamma(G; (G_i)_(i in I)) is flag--transitive if for every two flags x, y of the same type of Gamma, there exists an element g of G such that g(x) = y. It is then a flag--transitive geometry since the set { (G_i)_(i in I) } is a chamber of Gamma.

IsFTGeometry(D) : IncGeom -> BoolElt

Given an incidence geometry D, return true iff the automorphism group of D acts flag--transitively on D and D has at least one chamber.

IsFTGeometry(C) : CosetGeom -> BoolElt

Given a coset geometry C, return true iff the group of C acts flag--transitively on C.

IsFirm(D) : IncGeom -> BoolElt

Given a flag--transitive incidence geometry D, return true iff every flag of D is contained in at least two chambers.

IsFirm(C) : CosetGeom -> BoolElt

Given a flag--transitive coset geometry C, return true iff every flag of C is contained in at least two chambers.

IsThin(D) : IncGeom -> BoolElt

Given a flag--transitive incidence geometry D, return true iff every flag of D is contained in exactly two chambers.

IsThin(C) : CosetGeom -> BoolElt

Given a flag--transitive coset geometry C, return true iff every flag of C is contained in exactly chambers.

IsThick(D) : IncGeom -> BoolElt

Given a flag--transitive incidence geometry D, return true iff every flag of D is contained in exactly three chambers.

IsThick(C) : CosetGeom -> BoolElt

Given a flag--transitive coset geometry C, return true iff every flag of C is contained in exactly three chambers.

IsResiduallyConnected(D) : IncGeom -> BoolElt

IsRC(D) : IncGeom -> BoolElt

Given a flag--transitive incidence geometry D, return true iff every residue of rank at least two of D has a connected incidence graph.

IsResiduallyConnected(C) : CosetGeom -> BoolElt

IsRC(C) : CosetGeom -> BoolElt

Given a flag--transitive coset geometry C, return true iff every residue of rank at least two of C has a connected incidence graph.

IsGraph(D) : IncGeom -> GrphUnd

Given an incidence geometry D, tests if this incidence geometry corresponds to a graph: D must be of rank two and such that for one of the two types, say e, all elements of this type are incident with exactly two elements of the other type. Elements of type e then correspond to edges of an undirected graph and elements of the other type to the vertices of that graph.

IsGraph(C) : CosetGeom -> GrphUnd

Given a coset geometry C, tests if this geometry corresponds to a graph: C must be of rank two and one of the two maximal parabolic subgroups, say G_e, must contain the Borel subgroup as a subgroup of index 2. In that case, the cosets of G_e correspond to edges of a graph and the cosets of the other maximal parabolic subgroup correspond to the vertices of this graph.