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Relative cones can avoid the interior of the full cone

PDF p. 9: Remark 3.7

Statement

Let $(W, S)$ be a Coxeter system with $S$ finite, with canonical real representation on $V$ , simple roots $(e_{s})_{s \in S}$ and dual space $Θ = V^{*}$ as in Real Coxeter representation and contragredient notation. Let $J \subseteq S$ and $W_{J} = ⟨ J ⟩$ . Write $Θ_{J} = {f \in Θ : f (e_{j}) = 0 for every j \in J}$ , let $C o n e$ be the union of the $W$ -translates of the closed fundamental chamber, and put $C o n e (J) = C o n e \cap Θ_{J}$ as in Tits cone intersections, relative roots and arrangements. The superscript $\circ$ denotes interior in the indicated ambient space: $Θ$ for $C o n e$ and $Θ_{J}$ for $C o n e (J)$ .

Although $C o n e (J) \subseteq C o n e$ , the relative cone can be disjoint from ${C o n e}^{\circ}$ . Indeed, if $W_{J}$ is infinite, then $C o n e (J) \cap {C o n e}^{\circ} = \emptyset$ , and hence $C o n e (J)^{\circ}$ is also disjoint from ${C o n e}^{\circ}$ . Consequently the relative interior arrangement need not be a restriction of the full interior arrangement, so its local finiteness does not immediately follow by restricting the latter. The general proof of relative local finiteness also applies to $J = \emptyset$ , where it gives the full interior arrangement.

Proof

Given: The finite-rank Coxeter system and notation in the Statement.

Goal: Prove the disjointness assertion, exhibit an instance, and explain the arrangement comparison.

[F1] For $j \in S$ , the simple reflection acts on $V$ by $r_{j} (v) = v - 2 B (v, e_{j}) e_{j}$ , and the dual action is $(w f) (v) = f (w^{- 1} v)$ . The simple reflections are involutions. Real Coxeter representation and contragredient notation

[F2] The closed fundamental chamber meets every $W$ -orbit in the Tits cone. A point of its face with zero-coordinate set $I$ has stabilizer $W_{I}$ and lies in the full Tits-cone interior exactly when $W_{I}$ is finite. The cone is the union of the translates of the chamber. Tits cone geometry with its exact interior boundary

[F3] The relative cone is the intersection specified in the Statement. For $J = \emptyset$ , its relative interior and relative hyperplanes give the full interior and full root-hyperplane arrangement. Tits cone intersections, relative roots and arrangements

[F4] The relative local-finiteness proposition and its proof apply for every subset $J \subseteq S$ , including the empty subset. Local finiteness of the interior relative arrangement

[F5] An arrangement consists of a convex underlying set and a set of linear hyperplanes meeting it. Its restriction has a convex underlying subset of that set and retains the hyperplanes meeting the subset. Local finiteness is the open-neighbourhood condition on finitely many distinct hyperplanes. Linear hyperplane arrangements and local finiteness

Proof technique: direct.

1.1F1given

Let $f \in Θ_{J}$ . For $j \in J$ and $v \in V$ , [F1] gives $(j f) (v) = f (r_{j} v) = f (v) - 2 B (v, e_{j}) f (e_{j}) = f (v)$ . Thus every generator in $J$ fixes $f$ , so $W_{J}$ is contained in the stabilizer of $f$ .

2.1F2F3step 1.1

The action of each $w \in W$ on $Θ$ is an invertible linear map, hence a homeomorphism. Since the union defining $C o n e$ is $W$ -invariant, its interior is also $W$ -invariant. If $f \in {C o n e}^{\circ}$ , [F2] gives $w$ with $w f$ in the closed chamber. This point is interior and lies in a face of some type $I$ ; [F2] makes its stabilizer $W_{I}$ finite. The stabilizer of $f$ is conjugate to that of $w f$ , so it too is finite. Therefore, if $W_{J}$ is infinite, step 1.1 rules out every point of $Θ_{J}$ from ${C o n e}^{\circ}$ . Intersecting with $C o n e$ proves the claimed disjointness, and the same holds for the relative interior.

3.1F1F3F4F5step 2.1∎

Such a case exists: take $S = J = {s, t}$ with Coxeter exponent $m (s, t) = \infty$ . The assignments $s (n) = - n$ and $t (n) = 1 - n$ define involutions of $Z$ , so the presentation gives an action of $W$ on $Z$ . Their product acts as translation by $- 1$ , and therefore has infinite order; $W_{J} = W$ is infinite. Also $Θ_{J} = {0}$ and $0 \in C o n e$ , so $C o n e (J) = {0}$ and its interior in $Θ_{J}$ is nonempty. Step 2.1 makes it disjoint from the full interior. By [F5], restriction requires its underlying set to be a subset of the original underlying set. The nonempty set $C o n e (J)^{\circ} = {0}$ is disjoint from ${C o n e}^{\circ}$ , so this relative arrangement cannot be obtained by restricting an arrangement whose underlying space is ${C o n e}^{\circ}$ . Finally, [F3] and [F4] give the empty-subset specialization of the general relative proof.

Source notes

Remark 3.7 of the article is cited for attribution. The disjointness argument above uses the stated stabilizer and interior criteria.

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 9, Remark 3.7