Coxeter diagrams: edges, labels, components and finite type

Let $S$ be a finite set and let $m:S\times S\to \{1,2,3,\dots \}\cup \{\mathrm{\infty }\}$ be a Coxeter matrix on $S$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), with presented group $W$, length function $\mathrm{\ell }$ and standard parabolics $W_T=⟨s:s\in T⟩$ for $T\subseteq S$ (The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups, Group and abelian group).

(1) The diagram. The Coxeter diagram $\mathrm{\Gamma }(W,S)$ of $(W,S)$ has vertex set $S$, and two distinct vertices $s\mathrm{\ne }t$ are joined by an edge exactly when $m(s,t)\ge 3$; that edge carries the label $m(s,t)\in \{3,4,5,\dots \}\cup \{\mathrm{\infty }\}$. By convention the label $3$ is omitted (an unlabelled edge has label $3$) and no edge is drawn when $m(s,t)=2$. Thus $\mathrm{\Gamma }$ is a finite simple graph whose edges are labelled in $\{3,4,\dots \}\cup \{\mathrm{\infty }\}$ (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree), and $m$ is recovered from the pair $(S,\mathrm{\Gamma })$ by $m(s,s)=1$; $m(s,t)=m(t,s)=2$ if $\{s,t\}$ is not an edge; $m(s,t)=3$ if $\{s,t\}$ is an unlabelled edge; and $m(s,t)=$ the label otherwise. In particular $m\mapsto \mathrm{\Gamma }$ is injective on the Coxeter matrices on $S$. For $T\subseteq S$ the subdiagram ${\mathrm{\Gamma }}_T$ is the induced labelled graph on $T$, i.e. the diagram of the restricted matrix $m{\mathrm{\mid }}_{T\times T}$.

(2) Graph-theoretic vocabulary. A cycle of $\mathrm{\Gamma }$ is a cycle of the underlying simple graph (Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges); a path (or chain) is a diagram whose underlying graph is a path. $\mathrm{\Gamma }$ is connected when its underlying graph is connected (Connected graphs and connected components defined by the existence of vertex paths); its components are the connected components of the underlying graph, and their vertex sets are nonempty and partition $S$. For $s\in S$ the neighbours of $s$ are $N(s):=\{t\in S:t\mathrm{\ne }s,m(s,t)\ge 3\}$ and the degree of $s$ is $\mathrm{\mid }N(s)\mathrm{\mid }$ (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).

(3) Irreducibility. $(W,S)$ is irreducible when $\mathrm{\Gamma }$ is connected, and reducible otherwise. (For $S=\mathrm{\varnothing }$ the diagram is empty; the trivial system is not called irreducible.)

(4) Finite type. $(W,S)$ - and, by extension, $\mathrm{\Gamma }$ - is of finite type, or spherical, when the group $W$ is finite. This is a property of $W$ itself: no list of diagrams is part of the definition, and no positivity, definiteness or nondegeneracy of any bilinear form, and no geometric realization, is asserted here.

(5) Invariance and abstentions. By the conventions of Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, $m(s,t)$ is the order of $st$ in $W$; consequently an isomorphism of Coxeter systems $(W,S)\to (W^{\mathrm{\prime }},S^{\mathrm{\prime }})$ (a group isomorphism carrying $S$ onto $S^{\mathrm{\prime }}$) matches the two diagrams, so connectedness, irreducibility and finite type depend only on the isomorphism type of $(W,S)$. It is not asserted here that $W$ is the direct product of the standard parabolics of its components - that is the content of Disconnected diagrams, direct products, and comparison of invariant forms - nor that a diagram of finite type is one of the diagrams classified in Classification of finite Coxeter systems, including the H and dihedral families.

• The diagram is a complete record of the matrix. The reconstruction rule displayed in (1) is a dictionary, not a theorem about groups: it lists the value of $m$ on every ordered pair of vertices in terms of $(S,\mathrm{\Gamma })$, and therefore shows simultaneously that distinct Coxeter matrices on $S$ give distinct labelled graphs, and that the induced labelled subgraph on $T$ is the diagram of the restricted matrix $m{\mathrm{\mid }}_{T\times T}$.
• Conventions used consistently on this page. Labels lie in $\{3,4,\dots \}\cup \{\mathrm{\infty }\}$; a label $3$ edge is drawn unlabelled; a pair with $m(s,t)=2$ is not joined at all. Consequently the edge set of $\mathrm{\Gamma }$ is $\{\{s,t\}:s\mathrm{\ne }t,m(s,t)\ge 3\}$, and $N(s)$ of (2) is exactly the open neighbourhood of $s$ in the underlying simple graph.
• What finite type does not mean here. In (4) "spherical" is a synonym for finiteness of $W$ only. The equivalence with positive definiteness of the Coxeter form is a theorem (Finiteness criterion: W is finite exactly when the Coxeter form is positive definite ↗), and the explicit list of finite type diagrams is the content of Classification of finite Coxeter systems, including the H and dihedral families; neither is built into the definition.