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Labelling theorem for relative chambers

PDF p. 9: Theorem 3.9; (3.10); PDF p. 10: Theorem 3.9; (3.10)

Statement

The map $(x, I) \mapsto x C_{I}$ is a bijection from $C h a m (J)$ to the chambers of $(C o n e (J)^{\circ}, H^{J})$ .

Statement references

Algebraic chamber-label set

Tits cone intersections, relative roots and arrangements

Chambers, walls and adjacency

Independent relative chamber labelling input

Facts & Assumptions

[F1]

The domain is the set of pairs $(x, I)$ with $W_{J} x = x W_{I}$ and $x$ of minimum length in $x W_{I}$ . Algebraic chamber-label set

[F2]

The relative space is $Θ_{J}$ , the cone is $C o n e \cap Θ_{J}$ , its interior is taken in $Θ_{J}$ , and its hyperplanes are the distinct traces $H_{α} \cap Θ_{J}$ with $Θ_{J} \subseteq̸ H_{α}$ . Chambers are connected components of the complement; restriction omits traces missing the domain. Tits cone intersections, relative roots and arrangements Chambers, walls and adjacency

[F3]

The independent labelling theorem applies to every finite-rank Coxeter system and arbitrary $J$ , and maps exactly the pairs in [F1] bijectively to the chambers of the relative-interior arrangement by $(x, I) \mapsto x C_{I}$ . Independent relative chamber labelling input

[F4]

The full-cone and relative-interior chamber sets coincide. Their chambers are the full-dimensional translates of the coordinate faces used here. Interior restriction preserves chambers and adjacency Convexity, stratification and faces of the relative cone

Proof

Given: The finite-rank geometric convention of Tits cone intersections, relative roots and arrangements, with arbitrary Coxeter type and arbitrary $J \subseteq Γ_{0}$ . Neither $W$ nor $W_{J}$ is required to be finite. Goal: Prove the displayed bijection in these conventions.

1.1F2F3

Identify the notation in [F3] with that of [F2]: its $E_{J}$ is $Θ_{J}$ , its $U$ is $C o n e$ , its $T^{\circ}$ is $C o n e (J)^{\circ}$ , and its proper root traces are precisely $H^{J}$ . Both constructions omit traces missing this interior and use the induced Euclidean topology. Thus their chambers, defined as connected components, agree as subsets.

1.2F1F3F4

The label conditions in [F3] are exactly those in [F1], with the same standard parabolics and length function. Its face $C_{I}$ has zero coordinates on $I$ and strictly positive coordinates off $I$ , which is the coordinate-face convention in [F4]. In particular the map in [F3] is exactly the displayed map, rather than a map to a different choice of chamber representatives.

2.1F3F4step 1.1step 1.2∎

Apply the bijection in [F3] using the identifications in steps 1.1–1.2. This proves existence and uniqueness of the label of every target chamber. The full-cone interpretation, if used, agrees by [F4]. Since [F3] places no sphericity restriction on $W$ or $W_{J}$ , none is introduced in this application.

Source notes

The article attributes Theorem 3.9 to [24, Theorem 1.15]. The independently inspected author-hosted Iyama–Wemyss draft used by the completed labelling supplier numbers the applicable result Theorem 1.12, with labels in Definition 1.8 (printed pp. 6–7 / PDF pp. 30–31). Its §1.1 starts with an $n \times n$ Coxeter matrix and vertex set ${1, \dots, n}$ , while §1.2 permits the Coxeter group to be infinite. The number difference is recorded as a source-version difference; no claim about an unread version is made. The present proof applies the completed supplier and does not reconstruct the independent source proof.

The article's §2.2 follows [24] for its dual representation and refers to Vinberg [31]; Proposition 3.6 explicitly uses a finite-dimensional vector space. Vinberg's paper has not been independently read here. Although Theorem 3.9 has no explicit rank qualifier, its geometric objects in this collection already have the finite-rank convention in the relative-cone definition. The displayed statement is retained with that convention, preserving all Coxeter types and all vertex subsets, including nonspherical ones. The arbitrary-rank algebraic definition of the label set remains available; it alone supplies no infinite-rank chamber topology. An extension to infinitely many simple generators with a specified topology is not established here. This scope boundary is internal applicability evidence, not an original-article defect judgment.

Depends on

Used by · 18

Dependency tree · 2 levels

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF pp. 9, 10; Theorem 3.9; (3.10)