Coxeter Graphs

A Coxeter graph is an undirected labelled graph describing a Coxeter system. Suppose a Coxeter system has Coxeter matrix \(M=(m_{ij})_{i,j=1}^n\). Then the Coxeter graph has vertices \(1,\dots,n\); whenever \(m_{ij}>2\) there is an edge connecting \(i\) and \(j\) labeled by the value of \(m_{ij}\). When \(m_{ij}=3\), the label is usually omitted.

Since \(\infty\) is not an integer, it will be represented by \(0\) in our Coxeter graphs. Clearly a Coxeter graph must be standard, i.e. its vertices must be the integers \(1,2,\dots,n\) for some \(n\). A Coxeter system is irreducible if, and only if, its Coxeter graph is connected. Two Coxeter graphs give rise to Coxeter isomorphic groups if, and only if, they are isomorphic as labelled graphs. See Chapter Graph for more information on graphs.

IsCoxeterGraph(G): GrphUnd → BoolElt

Returns true if, and only if, the graph \(G\) is the Coxeter graph of some Coxeter group.

CoxeterGraph(M): AlgMatElt → GrphUnd

CoxeterGraph(C): AlgMatElt → GrphUnd

CoxeterGraph(D): GrphDir → GrphUnd

The Coxeter graph corresponding to a Coxeter matrix \(M\), Cartan matrix \(C\), or Dynkin digraph \(D\).

CoxeterGroupOrder(G): GrphUnd → .

CoxeterGroupFactoredOrder(G): GrphUnd → .

The (factored) order of the Coxeter group with Coxeter graph \(G\).

Example: Coxeter Graph Operations

> G := PathGraph(4);
> AssignLabel(G, 1,2, 4);
> AssignLabel(G, 3,4, 4);
> IsCoxeterGraph(G);
true
%%a> assert $1;
> CoxeterGroupOrder(G);
Infinity
>
> M := SymmetricMatrix([1, 3,1, 2,5,1]);
> G := CoxeterGraph(M);
> Labels(EdgeSet(G));
[ undef, 5 ]

IsSimplyLaced(G): GrphUnd → BoolElt

Returns true if, and only if, the Coxeter graph \(G\) is simply laced, i.e. unlabelled.

Example: Coxeter Graph Properties

> G := PathGraph(2);
> IsSimplyLaced(G);
true
%%a> assert $1;
> AssignLabel(G, 1,2, 6);
> IsSimplyLaced(G);
false
%%a> assert not $1;