|
Let us recall definitions of the properties described in this section.
An incidence geometry Γ is flag--transitive if for
every two flags x, y of the same type of Γ, there exists an
element g of Aut(Γ) such that g(x) = y. We also say that
Aut(Γ) acts flag--transitively in this case.
Moreover, it is a flag--transitive geometry if it contains at least
one chamber.
A coset geometry Γ(G; (Gi)i∈I) is flag--transitive
if for every two flags x, y of the same type of Γ, there
exists an element g of G such that g(x) = y. It is then a
flag--transitive geometry since the set { (Gi)i∈I
} is a chamber of Γ.
Given an incidence geometry D, return true if and only if
the automorphism group of D acts flag--transitively on D and
D has at least one chamber.
Given a coset geometry C, return true if and only if the group
of C acts flag--transitively on C.
IsFirm(X) : CosetGeom -> BoolElt
Given either a coset geometry or an incidence geometry X that
is flag transitive, return true if and only if every flag
of X is contained in at least two chambers.
IsThin(X) : IncGeom -> BoolElt
Given either a coset geometry or an incidence geometry X that
is flag transitive, return true if and only if every flag
of X is contained in exactly two chambers.
IsThick(X) : IncGeom -> BoolElt
Given either a coset geometry or an incidence geometry X that
is flag transitive, return true if and only if every flag
of the geometry is contained in exactly three chambers.
IsRC(X) : IncGeom -> BoolElt
IsResiduallyConnected(X) : CosetGeom -> BoolElt
IsRC(X) : CosetGeom -> BoolElt
Given either a coset geometry or an incidence geometry X that
is flag transitive, return true if and only if every
residue of rank at least two of X has a connected incidence
graph.
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.
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 Ge, must contain the Borel subgroup as a subgroup
of index 2. In that case, the cosets of Ge correspond to edges
of a graph and the cosets of the other maximal parabolic subgroup
correspond to the vertices of this graph.
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
|