Conversion Functions

In this section we describe the functions that are available to convert incidence geometries and coset geometries in other objects.

IncidenceGeometry(C) : CosetGeom -> IncGeom

Construct the incidence geometry IG from the coset geometry C. This is done using Tits' algorithm described in the introduction of this chapter . The function returns one value: the incidence geometry IG.

CosetGeometry(D) : IncGeom -> BoolElt, CosetGeom

Convert the incidence geometry D into a coset geometry.

If D is an incidence geometry that can be converted into a coset geometry, the coset geometry isomorphic to it is constructed in the following way. The group G of the coset geometry CG is the automorphism group of D. Magma determines a chamber C of D, that is a clique of the incidence graph of D containing one element of each type. To every element x in C, Magma associates a subgroup G_x which is the stabilizer of x in G. The subgroups (G_x, x ∈C) are the maximal parabolic subgroups of CG. In order to obtain a coset geometry combinatorially isomorphic to the incidence geometry we started with, the group G must be transitive on every rank two truncation of D. If this condition is satisfied, the function returns a boolean set to the value true and the coset geometry CG. Otherwise, the function returns false.

Graph(D) : IncGeom -> GrphUnd

If IsGraph(D) returns true, this function construct the undirected graph corresponding to the incidence geometry D.

Graph(C) : CosetGeom -> GrphUnd

If IsGraph(C) returns true, this function construct the undirected graph corresponding to the coset geometry C.

> ok,cg := CosetGeometry(neumaier);
> ok;
true

> cg;
Coset geometry cg with 4 types
Group:
Permutation group acting on a set of cardinality 200
Order = 126000 = 2^4 * 3^2 * 5^3 * 7

Maximal Parabolic Subgroups:
Permutation group acting on a set of cardinality 200
Order = 2520 = 2^3 * 3^2 * 5 * 7
Permutation group acting on a set of cardinality 200
Order = 2520 = 2^3 * 3^2 * 5 * 7
Permutation group acting on a set of cardinality 200
Order = 2520 = 2^3 * 3^2 * 5 * 7
Permutation group acting on a set of cardinality 200
Order = 2520 = 2^3 * 3^2 * 5 * 7

Type Set:
{@ 1, 2, 3, 4 @}