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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A finite connected component gives the opposite relative chamber

Statement

Let $(W, S)$ be a Coxeter system with finite generating set $S$ , in its canonical real representation and dual chamber arrangement, and fix $J \subseteq S$ . Let $(x, I) \in C h a m (J)$ and $a \in S ∖ I$ with finite $W_{D}$ , where $D = [I, a]$ . Set $L = I + a : = I \cup {a}$ , $b = ι_{D} (a)$ , $K = L ∖ {b}$ , and $v = w_{D ∖ {a}} w_{D}$ . Then $(x v, K) \in C h a m (J)$ , its chamber has the closed facet $\overline{x C_{L}}$ , and it lies on the opposite side of its supporting hyperplane from $x C_{I}$ . If this facet is the wall of a specified adjacent chamber, that chamber has label $(x v, K)$ .

Facts & Assumptions

[F1]

Component opposition gives the simple-root bijection, fixes $C_{L}$ , and gives the strict opposite-side evaluation, by Component opposition transports the retained simple roots.

[F2]

The algebraic label conditions are Algebraic chamber-label set and their geometric bijection is Labelling theorem for relative chambers.

[F3]

Positive images of simple roots are equivalent to absence of right descents, by The root-length criterion and faithfulness of the canonical reflection representation; absence of right descents indexed by a subset is equivalent to minimality in the left coset of the form $x W_{I}$ , by Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification, part (3).

[F4]

Chamber facets are translated coordinate faces, by Convexity, stratification and faces of the relative cone.

[F5]

The component $D = [I, a]$ is defined by Connected component notation [I,a]. The longest elements and opposition permutation are defined by Longest parabolic elements and opposition permutation for every finite standard parabolic, including the empty one. Since $W_{D ∖ {a}} \leq W_{D}$ , both longest elements used in $v$ are defined.

[F6]

Walls are codimension-one intersections of chamber closures with arrangement hyperplanes; distinct chambers are adjacent when they share a wall, by Chambers, walls and adjacency.

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.

1.1F1F2F3F5

Both longest elements in $v$ are defined by [F5]. Since $x$ is minimal in $x W_{I}$ , [F3] gives $x α_{i} \in Φ^{+}$ for every $i \in I$ . The bijection in [F1] therefore gives $x v α_{k} \in Φ^{+}$ for every $k \in K$ . A second application of [F3] shows that $x v$ is minimal in $x v W_{K}$ . Conjugation by $v$ carries each generating reflection indexed by $K$ to a generating reflection indexed by $I$ , bijectively, since it carries their simple roots bijectively. Hence $v W_{K} v^{- 1} = W_{I}$ . It follows that $x v W_{K} (x v)^{- 1} = x W_{I} x^{- 1} = W_{J}$ . Both label conditions hold.

2.1F1F2F4step 1.1

By [F2] this label represents the relative chamber $x v C_{K}$ . Its facet at $b$ is $x v C_{K + b} = x v C_{L} = x C_{L}$ by [F1, F4]. The same face is the facet at $a$ of $x C_{I}$ . For $f \in C_{K}$ , the functional $x v f$ has negative value on $x α_{a}$ by [F1], whereas every point of $x C_{I}$ has positive value there. Thus the two labelled chambers have the claimed common closed facet and opposite sides.

3.1F2F4F6step 2.1∎

Use the wall and adjacency convention in [F6]. For a facet of a chamber orthant, choose a point in its relative interior that belongs to the arrangement domain. In a sufficiently small neighbourhood, the only active chamber inequality is its facet inequality. Consequently at most one chamber on each side can have that whole facet: two would overlap in an open half-neighbourhood. Applied to the specified adjacent chamber and the chamber in step 2.1, this proves equality on the opposite side. Label uniqueness in [F2] then proves the final assertion. This argument is needed only when the facet is an actual adjacency wall inside the domain.

Depends on

Used by · 3

Dependency tree · 2 levels

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