Coxeter Graph

The presentation of $W$ obtained in theorem shows that (as an abstract group) $W$ is determined up to isomorphism by the set of integers $m(\alpha ,\beta )$, $\alpha ,\beta \in \Delta$. A convenient way to encode this information in a picture is to construct a graph $\Gamma$ with vertex set in one-to-one correspondence with $\Delta$; join a pair of vertices corresponding to $\alpha \ne \beta$ by an edge whenever $m(\alpha ,\beta )\ge 3$, and label such an edge with $m(\alpha ,\beta )$. (For a pair of vertices not joined by an edge, it is then understood that $m(\alpha ,\beta )=2$.) This labelled graph is called the Coxeter graph of $W$. It determines $W$ up to isomorphism. Since simple systems are conjugate, it does not depend on the choice of $\Delta$.

For example, the graph of $D_m$ is $\circ \stackrel{m}{--}\circ$ while the graph of $S_{n+1}$ has $n$ vertices: $\circ \stackrel{3}{--}\circ \cdots \circ \stackrel{3}{--}\circ .$

Proposition

For $i=1,2$, let $W_i$ be a finite reflection group acting on the euclidean space $V_i$. Assume $W_i$ is essential. If $W_1$ and $W_2$ have the same Coxeter graph, then there is an isometry of $V_1$ onto $V_2$ inducing an isomorphism of $W_1$ onto $W_2$. (In particular, if $V_1=V_2$, the subgroups $W_1$ and $W_2$ are conjugate in $\mathrm{O}(V)$.)

Irreducible Coxeter System

We say a Coxeter System $(W,S)$ is irreducible if the corresponding Coxeter graph $\Gamma$ is connected. We also call $\Phi$ irreducible in this case.

In general, let ${\Gamma }_1,\dots ,{\Gamma }_r$ be the connected components of $\Gamma$, and let ${\Delta }_i,S_i$ be the corresponding sets of simple roots and simple reflections. Thus if $\alpha \in {\Delta }_i$ and $\beta \in {\Delta }_j$ ($i\ne j$), we have $m(\alpha ,\beta )=2$ and therefore $s_{\alpha }{s}_{\beta }=s_{\beta }{s}_{\alpha }$. The following proposition shows that the study of finite reflection groups can be largely reduced to the case when $\Gamma$ is connected.

Proposition

Let $(W,S)$ have Coxeter graph $\Gamma$, with connected components ${\Gamma }_1,\dots ,{\Gamma }_r$, and let $S_1,\dots ,S_r$ be the corresponding subsets of $S$. Then $W$ is the direct product of the parabolic subgroups $W_{S_1},\dots ,W_{S_r}$, and each Coxeter system $(W_{S_i},S_i)$ is irreducible.

Proof. Use induction on $r$. Since the elements of $S_i$ commute with the elements of $S_j$ when $i\ne j$, it is clear that the indicated parabolic subgroups centralize each other, hence that each is normal in $W$. Moreover, the product of these subgroups contains $S$ and therefore must be all of $W$. By induction, $W_{S\setminus S_i}$ is the direct product of the remaining $W_{S_j}$, and Proposition 1.13 implies that $W_{S_i}$ intersects it trivially. So the product is direct.

Associated Matrix

We associate to a Coxeter graph $\Gamma$ with vertex set $S$ of cardinality $n$ a symmetric $n\times n$ matrix $A$ by setting $a(s,s^{\prime }):=-\cos \frac{\pi }{m(s,s^{\prime })}.$

Recall some terminology. Any symmetric $n\times n$ matrix $A=A^t$ defines a bilinear form $x^{t}Ay$ ($x,y\in {\mathbf{R}}^n$), with associated quadratic form $x^{t}Ax$. It is well known that the eigenvalues of $A$ are all real. $A$ is called positive definite if $x^{t}Ax>0$ for all $x\ne 0$, positive semidefinite if $x^{t}Ax\ge 0$ for all $x$. Equivalently, $A$ is positive definite if all its eigenvalues are (strictly) positive, positive semidefinite if all its eigenvalues are nonnegative. By abuse of language, we also say that $A$ is of positive type if it is positive semidefinite, including positive definite. For brevity, we call $\Gamma$ positive definite or positive semidefinite when the associated matrix (or bilinear form) has the corresponding property.

Subgraph of a Coxeter graph

By a subgraph of a Coxeter graph $\Gamma$ we mean a graph ${\Gamma }^{\prime }$ obtained by omitting some vertices (and adjacent edges) or by decreasing the labels on one or more edges, or both. We also say that $\Gamma$ ‘contains’ ${\Gamma }^{\prime }$. To simplify statements, we choose not to call the graph itself a subgraph.

Indecomposable Matrix

A real $n\times n$ matrix $A$ is called indecomposable if there is no partition of the index set into nonempty subsets $I,J$ such that $a_{ij}=0$ whenever $i\in I,j\in J$. Otherwise, after renumbering indices, $A$ could be written in block diagonal form. (The less exact term ‘irreducible’ is more commonly used in linear algebra texts.) It is clear that the matrix belonging to a Coxeter graph is indecomposable precisely when the graph is connected.

Proposition

Let $A$ be a real symmetric $n\times n$ matrix which is positive semidefinite and indecomposable. (In particular, the eigenvalues of $A$ are real and nonnegative.) Assume that $a_{ij}\le 0$ whenever $i\ne j$. Then:

• (a) $N:=\{x\in {\mathbf{R}}^n\mid x^{\mathrm{t}}Ax=0\}$ coincides with the nullspace of $A$ and has dimension $\le 1$.
• (b) The smallest eigenvalue of $A$ has multiplicity $1$, and has an eigenvector whose coordinates are all strictly positive.

Corollary

If $\Gamma$ is a connected Coxeter graph of positive type, then every (proper) subgraph is positive definite.