Tits cone intersections, relative roots and arrangements

PDF p. 2: §1.1.2; PDF p. 7: §3.1, equation (3.3)

Let ${\mathrm{\Gamma }}_0$ be finite, with Coxeter group $W$, real simple-root basis $({\alpha }_i{)}_{i\in {\mathrm{\Gamma }}_0}$, root set $\mathrm{\Phi }={\mathrm{\Phi }}^{+}\bigsqcup {\mathrm{\Phi }}^{-}$ and dual space $\mathrm{\Theta }$. For $J\subseteq {\mathrm{\Gamma }}_0$, let ${\mathrm{\Theta }}_J=\{f\in \mathrm{\Theta }:f({\alpha }_j)=0\text{ for every }j\in J\}$ and let ${\mathrm{\Phi }}_J=\{w{\alpha }_j:w\in W_J,j\in J\}$, where $W_J$ is the subgroup generated by the simple reflections indexed by $J$. Write $H_{\alpha }=\{f\in \mathrm{\Theta }:f(\alpha )=0\}$ and $\mathcal{H}=\{H_{\alpha }:\alpha \in \mathrm{\Phi }\}$, and let $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}$ be the Tits cone, the union of the $W$-translates of the closed fundamental chamber.

The Tits cone intersection, or $J$-cone, is $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}(J)=\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}\cap {\mathrm{\Theta }}_J$. Its relative interior $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}(J{)}^{\circ }$ is its interior in ${\mathrm{\Theta }}_J$, equipped with the topology induced from the coordinate Euclidean topology on $\mathrm{\Theta }$.

The relative roots, or $J$-roots, and their positive and negative parts are ${\mathrm{\Phi }}^J=\{\alpha \in \mathrm{\Phi }:{\mathrm{\Theta }}_J\subseteq ̸H_{\alpha }\}=\mathrm{\Phi }\setminus {\mathrm{\Phi }}_J,{\mathrm{\Phi }}_{\pm }^J={\mathrm{\Phi }}^J\cap {\mathrm{\Phi }}^{\pm }\mathrm{.}$ The relative arrangement is the set of distinct hyperplanes in ${\mathrm{\Theta }}_J$ ${\mathcal{H}}^J=\{H\cap {\mathrm{\Theta }}_J:H\in \mathcal{H},{\mathrm{\Theta }}_J\subseteq ̸H\}=\{H_{\alpha }\cap {\mathrm{\Theta }}_J:\alpha \in {\mathrm{\Phi }}_{+}^J\}\mathrm{.}$ Different positive relative roots may give the same restricted hyperplane. When $J=\mathrm{\varnothing }$, the pair $(\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}(J{)}^{\circ },{\mathcal{H}}^J)$ is $({\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}}^{\circ },\mathcal{H})$, the Coxeter arrangement in the interior of the Tits cone.

Parabolic coordinate subspaces and fundamental faces

Reflecting hyperplanes, fundamental chamber and Tits cone

Standard parabolic subgroups and parabolic roots

Annihilators, relative hyperplanes and complexification

Parabolic root subsystems and unique coset minima

Root signs, inversions and descent translation

The coordinate subspace and parabolic roots are defined by Parabolic coordinate subspaces and fundamental faces and Standard parabolic subgroups and parabolic roots, and the Tits cone, root hyperplanes and ambient topology by Reflecting hyperplanes, fundamental chamber and Tits cone. The relative interior is therefore well-defined in the induced topology, as specified by Annihilators, relative hyperplanes and complexification.

Put $V_J=\mathrm{span}\{{\alpha }_j:j\in J\}$. By Annihilators, relative hyperplanes and complexification, ${\mathrm{\Theta }}_J\subseteq H_{\alpha }$ exactly when $\alpha \in V_J$. By Parabolic root subsystems and unique coset minima, ${\mathrm{\Phi }}_J=\mathrm{\Phi }\cap V_J$. These two facts prove the displayed equality defining ${\mathrm{\Phi }}^J$, without assuming $W_J$ finite. The same linear-interface proposition shows that each included intersection is a hyperplane of ${\mathrm{\Theta }}_J$.

By Root signs, inversions and descent translation, every root is positive or the negative of a positive root. Since $H_{\alpha }=H_{-\alpha }$, and the condition that ${\mathrm{\Theta }}_J$ is not contained in the hyperplane is unchanged by sign, every member of the first description of ${\mathcal{H}}^J$ occurs in the second; the reverse inclusion is immediate. The arrangement is a set, so equal restricted hyperplanes are counted once.

For a concrete coincidence, take ${\mathrm{\Gamma }}_0=\{1,2\}$ with $m(s_1,s_2)=3$ (type $A_2$) and $J=\{1\}$. The form and reflection formula in Coxeter-system, roots, length and weak-order conventions give $B({\alpha }_2,{\alpha }_1)=-\cos (\pi \mathrm{/}3)=-1\mathrm{/}2$ and $s_1{\alpha }_2={\alpha }_2+{\alpha }_1$. Thus ${\alpha }_2$ and $\beta ={\alpha }_1+{\alpha }_2$ are distinct roots with nonnegative simple-root coordinates, so both are positive. By Parabolic coordinate subspaces and fundamental faces, every $f\in {\mathrm{\Theta }}_J$ has $f({\alpha }_1)=0$, hence $f(\beta )=f({\alpha }_2)$. The functional specified on the basis by $f({\alpha }_1)=0$ and $f({\alpha }_2)=1$ belongs to ${\mathrm{\Theta }}_J$, so neither restricted evaluation is zero and both roots are relative. Their restricted hyperplanes coincide: $H_{{\alpha }_2}\cap {\mathrm{\Theta }}_J=H_{\beta }\cap {\mathrm{\Theta }}_J=\{0\}$, since a functional vanishing on both basis vectors is zero.

If $J=\mathrm{\varnothing }$, then ${\mathrm{\Theta }}_J=\mathrm{\Theta }$ and ${\mathrm{\Phi }}_J=\mathrm{\varnothing }$. Every root hyperplane is proper, so ${\mathcal{H}}^J=\mathcal{H}$; the two cone interiors agree because they are taken in the same space. All these conclusions also cover the empty vertex set.

The cited article supplies attribution for the formulation. The equalities and well-definedness above use the listed mathematical suppliers, not the article claim as its own proof.