Elementary Invariants

Points(D): IncGeom -> SetIndx
Elements(D): IncGeom -> SetIndx: Given an incidence geometry \(D\), return the set of elements of \(D\). These elements are the points of the incidence graph of \(D\).

Types(D): IncGeom -> SetIndx: Given an incidence geometry \(D\), return the set of types of \(D\).

Types(C): CosetGeom -> SetIndx: Given a coset geometry \(C\), return the set of types of \(C\).

Rank(D): IncGeom -> RngIntElt: Given an incidence geometry \(D\), return the rank of \(D\), i.e. the cardinality of the set of types.

Rank(C): CosetGeom -> RngIntElt: Given a coset geometry \(C\), return the rank of \(C\).

IncidenceGraph(D): IncGeom -> GrphUnd, GrphVertSet, GrphEdgeSet

Given an incidence geometry \(D\), return the incidence graph of \(D\), its vertex set, and its edge set.

We remark that this function is not implemented for coset geometries but we may convert a coset geometry into an incidence geometry using IncidenceGeometry and then compute its incidence graph.

Group(C): CosetGeom -> GrpPerm: Given a coset geometry \(C\), return the group from which \(C\) is constructed.

MaxParabolics(C): CosetGeom -> SetIndx
MaximalParabolics(C): CosetGeom -> SetIndx: Given a coset geometry \(C\), return an indexed set containing the maximal parabolics of \(C\).

MinParabolics(C): CosetGeom -> SetIndx
MinimalParabolics(C): CosetGeom -> SetIndx: Given a coset geometry \(C\), return an indexed set containing the minimal parabolics of \(C\).

Borel(C): CosetGeom -> GrpPerm
BorelSubgroup(C): CosetGeom -> GrpPerm: Given a coset geometry \(C\), return the Borel subgroup of \(C\), i.e. the intersection of all maximal parabolic subgroups of \(C\).

Kernel(C): CosetGeom -> GrpPerm: Given a coset geometry \(C\), return a permutation group which is its kernel, i.e. the subgroup of the Borel subgroup of \(C\) that fixes all elements of the geometry \(C\).

Kernels(C): CosetGeom -> SeqEnum: Given a coset geometry \(C\), return a sequence containing the \(i\)-kernel \(K_i\) of each maximal parabolic subgroup \(G_i\) of \(C\). The \(i\)-kernel of the subgroup \(G_i\) is the subgroup consisting of all the elements of \(G_i\) that fix all the elements of the residue of \(G_i\).

Quotient(C, K): CosetGeom, GrpPerm -> CosetGeom: Given a coset geometry \(C = (G; (G_i)_{i\in I})\) and a permutation group \(K\), return the coset geometry \((G/K; (G_i/K)_{i\in I})\) provided that \(K\) is a normal subgroup of \(G\) and of all the maximal parabolic subgroups of \(C\).

Elementary Invariants#

Elementary Invariants