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Classical Matsumoto assertion recalled in the article

PDF p. 2: §1.2.1; PDF p. 19: §4.2, final paragraph

Statement

Let $(W, S)$ be a Coxeter system as in Coxeter-system, roots, length and weak-order conventions, with $S$ possibly infinite. A reduced expression for $w \in W$ is a finite word in $S$ representing $w$ whose number of letters is the least possible, denoted $ℓ (w)$ . For distinct $s, t \in S$ with $m (s, t) < \infty$ , a braid move replaces a consecutive alternating word of length $m (s, t)$ beginning with $s$ by the alternating word of the same length beginning with $t$ , or conversely. Any two reduced expressions of $w$ are connected by a finite sequence of braid moves.

Statement references

Coxeter-system, roots, length and weak-order conventions

Facts & Assumptions

[F1]

For a finite subset $J \subseteq S$ , the Coxeter group $W_{J}^{*}$ presented by the restricted matrix maps injectively to $W$ by sending each generator to its namesake, and its intrinsic length equals ambient length on its image. Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank

[F2]

In a Coxeter system with finite generating set, any two reduced expressions of the same element are connected by finitely many braid moves. Matsumoto's theorem: braid connectivity of reduced expressions, with singleton detection in dihedral subgroups

[F3]

The Coxeter matrix, its presented group, and word length have the conventions stated in Coxeter-system, roots, length and weak-order conventions.

Proof

Given: A Coxeter system $(W, S)$ and two reduced expressions for the same element $w \in W$ , with the conventions in the Statement.

Given: The two reduced expressions in the Facts & Assumptions.

Proof technique: reduction to finite rank.

Goal: Connect the two expressions by finitely many braid moves in $S$ .

1.1F1F3givenconstruct

Let $J$ be the union of the letters occurring in the two expressions. It is finite because each expression is a finite word. Both words represent elements of the restricted Coxeter group $W_{J}^{*}$ whose images in $W$ equal $w$ . By injectivity in [F1], they represent the same element of $W_{J}^{*}$ . Their common word length is $ℓ (w)$ ; the length equality in [F1] therefore makes both expressions reduced in $W_{J}^{*}$ . This also covers the empty-word case.

2.1F2F3step 1.1∎

Apply [F2] to the finite generating set $J$ and these two reduced expressions. Every move uses a pair in $J$ and the restricted matrix entry, which is the same as the corresponding entry of $m$ on $S$ . Thus the resulting finite sequence consists of braid moves in the original system and connects the given expressions.

Source notes

The article recalls the classical assertion. The proof above uses the separately supplied finite-rank theorem and finite-parabolic embedding, not the article's assertion as a premise.

Depends on

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF pp. 2 and 19; §1.2.1 and §4.2, final paragraph