Relative complexified hyperplane complement

PDF p. 13: (3.26) and following formula

Let $(W,S)$ be a Coxeter system with finite $S$, and let $J\subseteq S$. In the real dual representation space $\mathrm{\Theta }$, let ${\mathrm{\Theta }}_J$ be the subspace on which the simple roots indexed by $J$ vanish. Let $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}(J)$ be the intersection of the Tits cone with ${\mathrm{\Theta }}_J$, and let $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}(J{)}^{\circ }$ be its interior in ${\mathrm{\Theta }}_J$. Let ${\mathcal{H}}^J$ be the set of distinct hyperplanes obtained by restricting to ${\mathrm{\Theta }}_J$ those root hyperplanes that do not contain ${\mathrm{\Theta }}_J$.

The relative complexified hyperplane complement is the space

${\mathcal{X}}_J^{\mathbb{C}}=(\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}(J{)}^{\circ }\times \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}(J{)}^{\circ })\setminus {\bigcup }_{H\in {\mathcal{H}}^J}(H\times H)\mathrm{.}$

It has the subspace topology inherited from the product topology. If $W$ is finite, a case also called spherical type, then $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}(J{)}^{\circ }={\mathrm{\Theta }}_J$. Under the real-linear identification $(x,y)\mapsto x\otimes 1+y\otimes i$ of ${\mathrm{\Theta }}_J\times {\mathrm{\Theta }}_J$ with ${\mathrm{\Theta }}_J{\otimes }_{\mathbb{R}}\mathbb{C}$, the space is

$({\mathrm{\Theta }}_J{\otimes }_{\mathbb{R}}\mathbb{C})\setminus {\bigcup }_{H\in {\mathcal{H}}^J}(H{\otimes }_{\mathbb{R}}\mathbb{C})\mathrm{.}$

Coxeter-system, roots, length and weak-order conventions

Tits cone intersections, relative roots and arrangements

Basic Tits-cone properties

Complexified complements in an open convex cone

The product set ∏_i ∈ I Xᵢ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space

Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace

The relative-cone definition supplies the finite-dimensional space ${\mathrm{\Theta }}_J$, its relative topology, the set $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}(J{)}^{\circ }$ and the collection of actual linear hyperplanes ${\mathcal{H}}^J$. Thus the displayed union and its complement are well-defined subsets of the product, whose product and subspace topologies are specified by The product set ∏_i ∈ I Xᵢ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space and Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace, respectively.

If $W$ is finite, Basic Tits-cone properties gives $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}=\mathrm{\Theta }$. Hence $\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}(J)={\mathrm{\Theta }}_J$, and its interior in ${\mathrm{\Theta }}_J$ is all of ${\mathrm{\Theta }}_J$. The root set is finite because, by Coxeter-system, roots, length and weak-order conventions, it is the union of the $W$-orbits of the finitely many simple roots; consequently ${\mathcal{H}}^J$ is finite. Its members are linear hyperplanes and meet ${\mathrm{\Theta }}_J$ at zero, so they form a locally finite arrangement there. The complexification identification in Complexified complements in an open convex cone sends each $H\times H$ to $H{\otimes }_{\mathbb{R}}\mathbb{C}$, and therefore sends the product complement to the displayed complexified complement.

The article is cited for attribution of the definition and spherical-type formulation. The well-definedness and identification above use the listed mathematical suppliers.