\(C^+\)-Groups#
The concept of C\(^+\)-groups has been introduced by Fernandes, Leemans and Weiss [M. E. Fernandes and Weiss, n.d.] to generalise the study of rotational subgroups of Coxeter groups and study chiral geometries.
Let \(R\) be the set of generators of a group \(G\). Suppose \(R:=\{\alpha_1, \ldots, \alpha_{r-1}\}\). Define \(\alpha_0:=1_{G^+}\) and \(\alpha_{ij} := \alpha_i^{-1}\alpha_j\) for all \(0\leq i,j \leq r-1\). Let \(G_I := \langle \alpha_{ij} \mid i,j \in I\rangle\) for \(I\subseteq \{0, \ldots, r-1\}\).
If the pair \((G,R)\) satisfies the condition that \(\forall I, J \subseteq \{0, \ldots, r-1\}\), with \(\;|I|, |J| \geq 2, G_I \cap G_J = G_{I\cap J}\), called the intersection property IP\(^+\) (obtained in analogy with the intersection property of C-groups keeping only those equalities that involve subsets \(I\) and \(J\) of cardinality at least two), we say that \((G,R)\) is a C\(^+\)-group.
It follows immediately from the intersection property IP\(^+\), that \(R\) is an independent generating set for \(G\), that means that \(\alpha_i \not\in \langle \alpha_j : j \neq i\rangle\).
A group \(G\) whose set of generators satisfies the intersection property IP\(^+\) is called a C\(^+\)-group.
The B-diagram of \(G\) is a graph whose vertices are \(\alpha_0 := Id(G)\) and \(\alpha_i = G.i\) for each generator \(G.i\) of \(G\). Two vertices are joined by an edge provided the order of \(\alpha_i^{-1}\alpha_j\) is at least 3. Moreover, this edge is labelled with the order of \(\alpha_i^{-1}\alpha_j\).
- HasIntersectionPropertyPlus(G): GrpPerm -> BoolElt#
- IsCPlusGroup(G): GrpPerm -> BooElt#
Given a permutation group \(G\), determines whether or not it is a C\(^+\)-group, that is whether or not its set of generators \(G.1, \ldots, G.n\) satisfy the intersection property IP\(^+\).
- CosetGeometryFromCPlusGroup(G): GrpPerm -> CosetGeom#
Given a C\(^+\)-group \(G\), construct the coset geometry whose maximal parabolic subgroups are either generated by all but one generator of \(G\), or the generators \(G.1^{-1}G.i\) with \(i=2, \ldots, n\) where \(n\) is the number of generators of \(G\). The constructed geometry is of rank \(n+1\).
- BDiagram(G): GrpPerm -> GrphUnd#
Given a C\(^+\)-group \(G\), construct the B-diagram of \(G\).
- Example: cgrouptocosetgeometry (ex-4866f0)#
\(C^+\)-groups can be used to construct coset geometries as well. The user has to be careful that the corresponding geometry will have its rotation group acting on it. Let us again construct the 5-simplex using its rotation subgroup that is the alternating group \(A_6\).
> Gp := sub<Alt(6)|(1,2,3),(1,2)(3,4),(1,2)(4,5),(1,2)(5,6)>; > IsCPlusGroup(Gp); true > d,v,e := BDiagram(Gp); > for x in e do print x,Label(x); end for; {1, 2} 3 {2, 3} 3 {3, 4} 3 {4, 5} 3 > cg:=CosetGeometryFromCPlusGroup(Gp); > IsThin(cg); false
Above we see that the coset geometry constructed is not considered thin by Magma in terms of groups. This is because the stabilizer of a flag of rank 4 is the identity in \(A_6\). To get the group \(S_6\) acting, we have to convert the coset geometry into an incidence geometry and convert that incidence geometry in a coset geometry again. The latter one will have the automorphism group of the underlying incidence geometry acting on it, that is \(S_6\).
> ig:=IncidenceGeometry(cg); > ok,cg2:=CosetGeometry(ig); > ok; true > IsThin(cg2); true > IsRC(cg2); true > IsFTGeometry(cg2); true > d,v,e:=Diagram(cg2); > for x in v do print x,Label(x); end for; 1 [ 1, 15 ] 2 [ 1, 6 ] 3 [ 1, 15 ] 4 [ 1, 6 ] 5 [ 1, 20 ] > for x in e do print x,Label(x); end for; {1, 2} [ 3, 3, 3 ] {1, 3} [ 2, 2, 2 ] {1, 4} [ 2, 2, 2 ] {1, 5} [ 3, 3, 3 ] {2, 3} [ 2, 2, 2 ] {2, 4} [ 2, 2, 2 ] {2, 5} [ 2, 2, 2 ] {3, 4} [ 3, 3, 3 ] {3, 5} [ 3, 3, 3 ] {4, 5} [ 2, 2, 2 ]
The interest of C\(^+\)-group is in that they permit also to construct chiral geometries. This is described in more details in [M. E. Fernandes and Weiss, n.d.].