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 \((G_x, x \in C)\) 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 \(C\).

Example: Constructors (ex-cfd076)#

Taking back the last example for incidence geometries, we can convert the Neumaier geometry into a coset geometry by typing the following command (neumaier is the Neumaier geometry constructed above):

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

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This means the conversion has been done successfully. So \(cg\) is the coset geometry corresponding to neumaier.

> 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 @}

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