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Failure of local finiteness on the full cone

PDF p. 8: §3.1, opening paragraph

Statement

Let (W,S) be a Coxeter system with S finite, in its canonical real Coxeter representation. Let Φ+ be its positive roots, Hα={f∈V∗:f(α)=0}, and Cone the union of the translates of the closed fundamental chamber. For J=∅, the relative cone is Cone and the relative hyperplanes are the distinct root hyperplanes H={Hα:α∈Φ+}. If W is infinite, H is infinite and the arrangement (Cone,H) is not locally finite at the origin.

Here local finiteness at a point means that some neighborhood of the point in the subspace topology on Cone meets only finitely many distinct members of H.

Facts & Assumptions

[F1]

For w∈W, its word length ℓ(w) is the least length of a word in the finite generating set S representing it. Its inversion set N(w)={α∈Φ+:wα∈−Φ+} has cardinality ℓ(w), and distinct positive roots define distinct full hyperplanes. Root signs, inversions and descent translation

[F2]

The root hyperplanes are linear, 0∈Cone, and for J=∅ the relative cone and hyperplanes are Cone and H, respectively. Real Coxeter representation and contragredient notation Tits cone geometry with its exact interior boundary Tits cone intersections, relative roots and arrangements

[F3]

The arrangement is locally finite at 0 precisely if some neighborhood of 0 in Cone meets only finitely many distinct hyperplanes in H. Linear hyperplane arrangements and local finiteness

Proof

Given: The finite-rank Coxeter system and the notation in the Statement, with W infinite and J=∅.

Proof technique: direct.

Goal: Prove that there are infinitely many distinct hyperplanes and that no neighborhood of 0 in the full cone meets only finitely many of them.

1.1F1given

Unbounded lengths. For each nonnegative integer n, only finitely many words in the finite alphabet S have length at most n. Their values therefore form a finite set. Since W is infinite, some element has length greater than n. By [F1], its inversion set contains more than n positive roots. Hence Φ+ is infinite.

2.1F1F2F3step 1.1∎

Hyperplanes through the origin. By [F1], distinct positive roots yield distinct hyperplanes, so H is infinite. By [F2], every member contains 0, and 0 lies in Cone. Consequently every neighborhood of 0 in Cone meets every member of H. By [F3], the arrangement is not locally finite at 0. This proves the example.

Source notes

The article states this example in the opening paragraph of §3.1. The proof above uses the root-inversion and arrangement suppliers, rather than the article's assertion as a premise.

Depends on

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Sources