Real Coxeter representation and contragredient notation

Let $S$ be a finite set and $m:S\times S\to \{1,2,3,\dots \}\cup \{\mathrm{\infty }\}$ a Coxeter matrix: $m(s,s)=1$ and, for $s\mathrm{\ne }t$, $m(s,t)=m(t,s)\in \{2,3,\dots \}\cup \{\mathrm{\infty }\}$. Its presented Coxeter group is $W=⟨S\mid s^2=1,(st{)}^{m(s,t)}=1\text{ when }s\mathrm{\ne }t\text{ and }m(s,t)<\mathrm{\infty }⟩$.

Let $V={\mathbb{R}}^S$, the space of real-valued functions on $S$, and let $e_s$ be the function that is $1$ at $s$ and $0$ elsewhere. The Coxeter form $B$ is the symmetric bilinear form determined by $B(e_s,e_s)=1$, $B(e_s,e_t)=-\cos (\pi \mathrm{/}m(s,t))$ for distinct $s,t$ with finite $m(s,t)$, and $B(e_s,e_t)=-1$ when $m(s,t)=\mathrm{\infty }$. Put $r_s(v)=v-2B(v,e_s)e_s$. The canonical real reflection representation is the homomorphism $\rho :W\to \mathrm{G}\mathrm{L}(V)$ satisfying $\rho (s)=r_s$. Here $\mathrm{G}\mathrm{L}(V)$ is the group of invertible linear maps from $V$ to itself. Write $wv$ for $\rho (w)v$, ${\alpha }_s=e_s$, and call the elements of $\mathrm{\Phi }=\{w{e}_s:w\in W,s\in S\}$ roots.

Let $\mathrm{\Theta }=V^{\ast }$ be the algebraic dual, the space of all real-valued linear functionals on $V$. The contragredient action is $(wf)(v)=f(\rho (w{)}^{-1}v)$ for $w\in W$, $f\in \mathrm{\Theta }$, and $v\in V$. The closed chamber is $C=\{f\in \mathrm{\Theta }:f(e_s)\ge 0\text{ for all }s\in S\}$, and the open chamber is $C^{\circ }=\{f\in \mathrm{\Theta }:f(e_s)>0\text{ for all }s\in S\}$. For $I\subseteq S$, its face of type $I$ is $C_I=\{f\in \mathrm{\Theta }:f(e_s)=0\text{ for }s\in I,f(e_s)>0\text{ for }s\mathrm{\notin }I\}$. For $\alpha \in \mathrm{\Phi }$, the root hyperplane is $H_{\alpha }=\{f\in \mathrm{\Theta }:f(\alpha )=0\}$.

No nondegeneracy of $B$ is assumed, and $V$ is not identified with $V^{\ast }$ using $B$. These conventions apply to finite $S$; no infinite-rank extension is asserted here.

Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups

The real Coxeter form, its radical, reflections, and form-preserving maps

The canonical reflection homomorphism, roots, reflections, and the positive cone

The dual action, chambers, faces, and root hyperplanes

Linear functionals and the algebraic dual V^*=L(V,F)

The finite-dimensional form and the maps $r_s$ are defined by the specified basis values. The reflection representation descends uniquely from the presented group: its generators satisfy the defining relations, as established in Descent of the reflection representation, unit root norms, and conjugation of reflections. Thus each $\rho (w)$ is invertible and the displayed dual formula is meaningful. The latter is a left action by linear maps, and the indicated faces exist, by The dual action, the faces, and the rank-two chamber tiling. The chamber, face, root and hyperplane expressions are consequently well-defined subsets of the spaces specified in the definition.