LemmaStatement: AI-generatedProof: AI-generated

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AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The full-cone and interior arrangements have identical chambers

Statement

Let $(W, S)$ be a finite-rank Coxeter system in its canonical real representation $V = R^{S}$ , with basis $(α_{s})$ , dual $Θ$ , root system $Φ$ , contragredient action, and Tits cone $U$ . For $I \subseteq S$ put $E_{I} = {f : f (α_{i}) = 0 (i \in I)}$ and $C_{I} = {f \in E_{I} : f (α_{s}) > 0 (s \notin I)}$ . Fix $J \subseteq S$ and put $E = E_{J}$ , $T = U \cap E$ , and $T^{\circ} = {int}_{E} T$ , the set of points admitting an open ball in $E$ contained in $T$ . Write $H_{α} = {f \in Θ : f (α) = 0}$ and $Φ^{J} = {α \in Φ : E \subseteq̸ H_{α}}$ . Let $A = {E \cap H_{α} : α \in Φ^{J}}$ , $D = ⋃_{K \in A} K$ , and $A^{\circ} = {K \in A : K \cap T^{\circ} \neq \emptyset}$ . Give $E$ its Euclidean topology and all its subsets their induced subspace topologies.

The identity identifies $T ∖ D$ and $T^{\circ} ∖ D$ as topological spaces, and hence identifies their connected components. Thus Iyama–Wemyss $J$ -chambers, the sets $x C_{I} \subseteq E_{J}$ with $∣ I ∣ = ∣ J ∣$ , are exactly the chambers of the restricted arrangement $(T^{\circ}, A^{\circ})$ . Traces missing $T^{\circ}$ may be discarded without changing this conclusion.

Statement references

Deleting relative root hyperplanes removes all cone-boundary points

Locally finite arrangements and relative face terminology

The relative root deletion set is the restricted-hyperplane union

Tits cone geometry with its exact interior boundary

External input

Iyama–Wemyss, Definition 1.6 and its following sentence, supplies this premise: Under the statement's hypotheses, the sets $x C_{I} \subseteq E_{J}$ with $∣ I ∣ = ∣ J ∣$ are precisely the connected components of $(U \cap E_{J}) ∖ ⋃_{α \in Φ^{J}} H_{α}$ .

Facts & Assumptions

[F1]

The two complements are equal by Deleting relative root hyperplanes removes all cone-boundary points.

[F2]

Chambers are connected components with the subspace topology, as defined in Locally finite arrangements and relative face terminology. An arrangement has a convex domain and linear hyperplanes meeting it; restriction keeps precisely those meeting the smaller convex domain.

[F3]

Each member of $A$ is a hyperplane in $E$ , and $T ∖ D = T ∖ ⋃_{α \in Φ^{J}} H_{α}$ , by The relative root deletion set is the restricted-hyperplane union.

[F4]

The cone $U$ is convex and contains $0$ , with the representation and root-hyperplane conventions of Tits cone geometry with its exact interior boundary.

[F5]

The chamber-component identification recorded in External input is imported directly from Iyama–Wemyss, Definition 1.6 and its following sentence.

Proof

Given: The objects and hypotheses of the statement. Goal: Establish its conclusion.

1.1F2F3F4given

By [F4], $T = U \cap E$ is convex: a convex combination of two points lies in both $U$ and the linear subspace $E$ . Also $0 \in T$ . By [F3], the members of $A$ are linear hyperplanes in $E$ ; each meets $T$ at $0$ . Thus $(T, A)$ is an arrangement in the sense of [F2].

1.2F1F2algebra

Both complements inherit their topology from $E$ . Successive subspace topologies agree with the direct subspace topology: if $B \subseteq A \subseteq E$ , an open set of $B$ induced through $A$ has the form $B \cap (A \cap O) = B \cap O$ for an open $O \subseteq E$ , and conversely. By [F1], the two complements are the same subset of $E$ , hence the same topological space.

2.1F2step 1.1givenalgebra

Use a coordinate Euclidean norm $∥ \cdot ∥$ on $E$ and write $B_{E} (x, r) = {u \in E : ∥ u - x ∥ < r}$ . The interior $T^{\circ}$ is convex. Indeed, for $x, y \in T^{\circ}$ , choose $r > 0$ such that both $E$ -balls $B_{E} (x, r)$ and $B_{E} (y, r)$ lie in $T$ . For $0 \leq t \leq 1$ and $z \in E$ with $∥ z ∥ < r$ , convexity of $T$ gives $(1 - t) (x + z) + t (y + z) = (1 - t) x + t y + z \in T$ . Hence $B_{E} ((1 - t) x + t y, r) \subseteq T$ and $(1 - t) x + t y \in T^{\circ}$ . The assertion is vacuous if the interior is empty. Every member of $A^{\circ}$ meets $T^{\circ}$ by its definition, so $(T^{\circ}, A^{\circ})$ is precisely the restriction defined in [F2].

3.1F2F3F5step 2.1step 1.2∎

By [F3], the full-cone complement is exactly the complement in [F5], so its connected components are the source $J$ -chambers. Connectedness and maximal connected subsets are preserved by the identity in step 1.2, so the components of $T^{\circ} ∖ D$ are the same subsets. A trace missing $T^{\circ}$ deletes no point there, and consequently $T^{\circ} ∖ D = T^{\circ} ∖ ⋃_{K \in A^{\circ}} K$ . By [F2] and step 2.1, the components of this last space are the chambers of the restricted arrangement. This proves both assertions.

Depends on

Used by · 4

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Osamu Iyama and Michael Wemyss, Tits Cone Intersections and Applications (author-hosted draft) · §§1.1, 1.3–1.4; Definitions 1.4–1.6 and the sentence immediately after Definition 1.6; printed pp. 3–6, especially p. 6 (PDF pp. 27–30, especially p. 30).

Under the statement's hypotheses, the sets $xC_I\subseteq E_J$ with $|I|=|J|$ are precisely the connected components of $(U\cap E_J)\setminus\bigcup_{\alpha\in\Phi^J}H_\alpha$.

Hypotheses: Finite Coxeter matrix; canonical real reflection representation with B(alpha_i,alpha_j)=-cos(pi/m_ij), its contragredient dual action and Tits cone; arbitrary fixed vertex subset J. Our I renames the secondary subset in Definition 1.6; our E_J is its Theta_J. No sphericity assumption.

Reading limits: Read the web-extracted text of §§1.1, 1.3–1.4, including Definitions 1.3–1.6 and the following component assertion. PDF screenshots were unavailable. Imported only the stated identification; did not review its proof or infer local finiteness.

Sources

Osamu Iyama and Michael Wemyss, Tits Cone Intersections and Applications (author-hosted draft) · §§1.1, 1.3–1.4; Definitions 1.4–1.6 and the sentence immediately after Definition 1.6; printed pp. 3–6, especially p. 6 (PDF pp. 27–30, especially p. 30).