LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Adjacent relative chambers force a finite crossing component

Statement

In the finite-rank canonical Coxeter representation, fix $J$ and let $(x, I), (y, K)$ label adjacent relative chambers $x C_{I}, y C_{K} \subseteq Θ_{J}$ . Let $a \notin I$ be the index of their common wall, and put $L = I + a = I \cup {a}$ and $u = y^{- 1} x$ . Then $Φ_{L}^{+} ∖ Φ_{I}^{+} \subseteq N (u), ∣ Φ_{L} ∖ Φ_{I} ∣ \leq 2 ℓ (u) .$ In particular $W_{D}$ is finite, where $D = [I, a]$ is the vertex set of the connected component containing $a$ in the Coxeter diagram induced on $I + a$ .

Facts & Assumptions

[F1]

The common wall is the facet at $a$ and the two chambers are on opposite sides of its supporting hyperplane, by Common relative facets have one enlarged standard subset.

[F2]

Roots have one sign, dual evaluation obeys $(x f) (β) = f (x^{- 1} β)$ , and $∣ N (u) ∣ = ℓ (u)$ , by Root signs, inversions and descent translation.

[F3]

The roots of a standard parabolic are exactly the ambient roots supported on its vertex subset, by Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives, part (2).

[F4]

Finiteness of $Φ_{I + a} ∖ Φ_{I}$ implies finiteness of $W_{[I, a]}$ by Finite root difference forces finite connected parabolic.

[F5]

For $M \subseteq S$ , $Θ_{M}$ is the subspace annihilating the simple roots indexed by $M$ , and $C_{M}$ consists of its points strictly positive on all remaining simple roots. A relative chamber $x C_{I} \subseteq Θ_{J}$ has $∣ I ∣ = ∣ J ∣$ , and its wall at $a$ has supporting relative hyperplane $H_{x α_{a}} \cap Θ_{J}$ , where $H_{γ} = {q : q (γ) = 0}$ . These coordinate-face and wall descriptions are supplied by Convexity, stratification and faces of the relative cone.

[F6]

The notation $[I, a]$ denotes the vertices of the connected component containing $a$ in the Coxeter diagram induced on $I + a$ , by Connected component notation [I,a].

Proof

Given: The finite-rank canonical Coxeter representation, root basis and dual action, with the notation and hypotheses in the Statement. Goal: prove the stated assertions.

Proof technique: direct.

1.1F2F3F5given

Let $β \in Φ_{L}^{+} ∖ Φ_{I}^{+}$ . By [F2, F3], its simple-root coefficients are nonnegative, supported on $I + a$ , and its coefficient $c_{a}$ at $a$ is strictly positive. Thus $f (β) = c_{a} f (α_{a})$ for every $f \in Θ_{I}$ . By [F5], the positive coordinate orthant $C_{I}$ spans $Θ_{I}$ , so $x C_{I} \subseteq Θ_{J}$ implies $x Θ_{I} \subseteq Θ_{J}$ . Since $∣ I ∣ = ∣ J ∣$ , the subspaces have the same finite dimension and $x Θ_{I} = Θ_{J}$ . Dual evaluation now gives $q (x β) = c_{a} q (x α_{a})$ for every $q \in Θ_{J}$ .

2.1F1F2F5step 1.1

For $q = x f$ with $f \in C_{I}$ , [F5] gives $q (x α_{a}) = f (α_{a}) > 0$ . The common wall's supporting hyperplane is exactly $H_{x α_{a}} \cap Θ_{J}$ by [F5]. Consequently [F1] puts all of $y C_{K}$ in the half-space where $q (x α_{a}) < 0$ . Since $c_{a} > 0$ , step 1.1 shows that evaluation on $x β$ is positive on $x C_{I}$ and negative on $y C_{K}$ .

3.1F2F5step 2.1choose

Choose $g \in C_{K}$ , which is nonempty since one may prescribe value zero on the basis roots indexed by $K$ and value one on the others, by [F5]. Then $(y g) (x β) = g (y^{- 1} x β) < 0$ by step 2.1 and [F2]. The root $u β$ cannot be positive: a positive root has nonnegative simple-root coefficients, whereas $g$ has nonnegative values on every simple root by [F5], so its evaluation would be nonnegative. Thus $u β$ is negative by [F2], and $β \in N (u)$ .

4.1F2F3F4F6step 3.1∎

The positive part $Φ_{L}^{+} ∖ Φ_{I}^{+}$ of the root difference has at most $ℓ (u)$ members by [F2] and step 3.1. By [F2, F3], negation is a bijection from its positive part to its negative part, and the two parts are disjoint. Hence $∣ Φ_{L} ∖ Φ_{I} ∣ \leq 2 ℓ (u)$ . Apply [F4] with the component $D$ of [F6] to obtain finiteness of $W_{D}$ . This uses a finite inversion set, regardless of the size or type of $W_{I}$ or $W_{L}$ .

Depends on

Used by · 1

Adjacent chambers and inverse wall-crossing elements

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.