The Tits cone, its interior, and the negative-root set of a functional

Let $S$ be a finite set, $m$ a Coxeter matrix on $S$, $W$ the presented group with length function $\mathrm{\ell }$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), $V={\mathbb{R}}^S$ with Coxeter form $B$ and reflections $r_a$ (The real Coxeter form, its radical, reflections, and form-preserving maps), $\rho :W\to \mathrm{G}\mathrm{L}(V)$ the canonical reflection homomorphism with root system $\mathrm{\Phi }$ and positive cone $V_{+}$ (The canonical reflection homomorphism, roots, reflections, and the positive cone), and let $V^{\ast }$ be the algebraic dual with its dual action, closed chamber $C$, faces $C_I$ and root hyperplanes $H_{\alpha }$ in the notation of The dual action, chambers, faces, and root hyperplanes. Let $\mathrm{\Phi }={\mathrm{\Phi }}_{+}\bigsqcup {\mathrm{\Phi }}_{-}$ be the signed root system and $C^{\circ }={\bigcap }_{s\in S}\{f\in V^{\ast }:f(e_s)>0\}$ the open chamber (Root sign coherence and the action of simple reflections on positive roots).

(1) The chamber system. For $w\in W$ put $wC:=\{w\cdot f:f\in C\}$. The sets $wC$ are the chambers of $(W,S)$ and $C$ is the fundamental chamber; for $w\in W$ and $s\in S$ the hyperplane $w{H}_{e_s}$ is a wall of the chamber $wC$.

(2) The Tits cone. The Tits cone is $U:={\bigcup }_{w\in W}wC\subseteq V^{\ast }\mathrm{.}$ This definition asserts no convexity or closedness of $U$, face-intersection properties, or local finiteness. Convexity is proved in The finite-negativity criterion, the reduction step, and convexity of the Tits cone ↗, chamber intersections in Chamber collisions, point stabilizers, and the intersection rule, and local finiteness of chambers and walls only at points of $U^{\circ }$ in The interior of the Tits cone, finite parabolic stabilizers, and local finiteness (3). Immediate from the definition, the group law and the left-action property of the dual action (The dual action, the faces, and the rank-two chamber tiling (1)): $C\subseteq U$, $0\in U$ (because $0\in C$), and $w^{\mathrm{\prime }}U=U$ for every $w^{\mathrm{\prime }}\in W$.

(3) The finite-dimensional topology and the interior. Since $S$ is finite, $f\mapsto (f(e_s){)}_{s\in S}$ is a linear bijection $V^{\ast }\to {\mathbb{R}}^S$ (The dual family $(b^{\ast }{)}_{b\in B}$ associated to a Hamel basis $B$, defined by $b^{\ast }(c)={\delta }_{bc}$, The dual family of a finite basis is a basis of the dual space, with the same dimension, Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis). If $S\mathrm{\ne }\mathrm{\varnothing }$, pulling back the metric $d_{\mathrm{\infty }}$ of ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it along this bijection gives the metric $d(f,g):={\max }_{s\in S}\mathrm{\mid }f(e_s)-g(e_s)\mathrm{\mid }$ on $V^{\ast }$ (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement); if $S=\mathrm{\varnothing }$ then $V^{\ast }=\{0\}$ is a singleton and $d$ is its unique metric, $d(0,0)=0$.

A second basis enters only when $S\mathrm{\ne }\mathrm{\varnothing }$. If $(e_t^{\mathrm{\prime }})$ is a second basis with $e_t^{\mathrm{\prime }}={\sum }_{s\in S}{a}_{ts}{e}_s$, then ${\max }_t\mathrm{\mid }f(e_t^{\mathrm{\prime }})\mathrm{\mid }\le C{\max }_s\mathrm{\mid }f(e_s)\mathrm{\mid }$ and ${\max }_s\mathrm{\mid }f(e_s)\mathrm{\mid }\le C^{\mathrm{\prime }}{\max }_t\mathrm{\mid }f(e_t^{\mathrm{\prime }})\mathrm{\mid }$ with $C={\max }_t{\sum }_s\mathrm{\mid }{a}_{ts}\mathrm{\mid }$ and $C^{\mathrm{\prime }}$ the analogous constant of the inverse matrix, whose entries are finite because $S$ is finite; so a second coordinate system gives an equivalent norm, hence the same open sets and the same interior (For $n\ge 1$ all norms on ${\mathbb{R}}^n$ are equivalent). The interior of the Tits cone is $U^{\circ }:=\mathrm{int}U,$ the interior of $U$ for this ordinary finite-dimensional topology (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space). Thus a neighborhood of $f$ is a set containing some ball $\{g:d(f,g)<\epsilon \}$, $\epsilon >0$, and compact means compact for this topology (Open cover, subcover, compact metric space, and compact subset of a metric space).

(4) The negative-root set of a functional. For $f\in V^{\ast }$ put $\mathrm{Neg}(f):=\{\alpha \in {\mathrm{\Phi }}_{+}:f(\alpha )<0\}\subseteq {\mathrm{\Phi }}_{+}\mathrm{.}$ This attaches a set of positive roots to a functional; it is not the inversion set $N(w)$ attached to a group element (The geometric inversion set $N(w)$ of an element of a Coxeter group). No finiteness of $\mathrm{Neg}(f)$ is asserted by this definition: the equivalence "$f\in U$ if and only if $\mathrm{Neg}(f)$ is finite" is the theorem The finite-negativity criterion, the reduction step, and convexity of the Tits cone ↗.