A Dynkin digraph is a directed labelled graph describing a crystallographic Cartan matrix C=(cij)i, j=1n. The Dynkin digraph has vertices 1, ..., n; whenever cij<0 there is an edge from i to j labeled by the value -cij. When cij= - 1, the label is usually omitted.
In the literature, the term Dynkin diagram is used, but here this will be reserved for a printed display of the Dynkin digraph (or Coxeter graph) corresponding to a finite or affine Coxeter group (see Section Finite and Affine Coxeter Groups below). For convenience, Dynkin digraphs have labelled edges rather than multiple edges.
Clearly a Dynkin digraph must be standard, i.e. its vertices must be the integers 1, 2, ..., n for some n. A Dynkin digraph has an edge from i to j if, and only if, it has an edge from j to i (although the labels may be different). Hence strong and weak connectivity are equivalent for these graphs. The Coxeter system is irreducible if, and only if, the Dynkin digraph is connected. Two Dynkin digraphs give rise to Cartan equivalent Cartan matrices if they are isomorphic as labelled digraphs. See Chapter GRAPHS for more information on digraphs.
Note that functions are not given for computing the Dynkin digraph of a Coxeter matrix or Coxeter graph, since a particular choice of crystallographic Cartan matrix is required.
Returns true if, and only if, the digraph D is the Dynkin digraph of some crystallographic Cartan matrix.
The Dynkin digraph of the crystallographic Cartan matrix C.
The (factored) order of the Coxeter group with Dynkin digraph D.
The fundamental group of the Dynkin digraph D, i.e. Zn/Γ where n is the degree of D and Γ is the lattice generated by the rows of the corresponding Cartan matrix. The natural mapping ZntoZn/Γ is the second returned value.
Returns true if, and only if, the Dynkin digraph D is simply laced, i.e. unlabelled.
> D := Digraph< 4 | <1,{2,3,4}>, <2,{1}>, <3,{1}>, <4,{1}> >;
> AssignLabel(D, 1,2, 2);
> AssignLabel(D, 1,3, 5);
> IsDynkinDigraph(D);
true
> CartanMatrix(D);
[ 2 -2 -5 -1]
[-1 2 0 0]
[-1 0 2 0]
[-1 0 0 2]
> FundamentalGroup(D);
Abelian Group isomorphic to Z/2 + Z/8
Defined on 2 generators
Relations:
2*$.1 = 0
8*$.2 = 0
Mapping from: Standard Lattice of rank 4 and degree 4 to Abelian Group
isomorphic to Z/2 + Z/8
Defined on 2 generators
Relations:
2*$.1 = 0
8*$.2 = 0