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Brink–Howlett generators and invariance of standard length

PDF p. 18: Corollary 4.13

Statement

The groupoid $B H$ is generated by $v_{a, I} : I \to I + a - ι_{[I, a]} (a)$ over associates $I$ and $a \in I^{c}$ with finite $W_{[I, a]}$ . Any two standard expressions of the same morphism have the same number of factors.

Facts & Assumptions

[F1]

The definitions and results used in this argument are supplied, under their stated hypotheses, by Standard expressions and their lengths; Every standard expression counts the same parallel classes.

Proof

Given: The Coxeter and groupoid data of Standard expressions and their lengths and the existence and factor-count conclusions of Every standard expression counts the same parallel classes.

Goal: Establish the assertion in the Statement.

1.1F1

By Every standard expression counts the same parallel classes, every morphism has a standard expression in the generators displayed in the statement. By the definition of a standard expression, this is a finite composable product equal to that morphism.

2.1F1step 1.1∎

Hence those generators generate the groupoid. For any two standard expressions of a fixed morphism, the factor-count conclusion of Every standard expression counts the same parallel classes makes their numbers of factors equal: both equal the expression-independent count supplied by that lemma. This applies to every Coxeter rank, including infinite rank.

Source notes

The original article Corollary 4.13 is retained as attribution only. The proof uses the standard-expression definition and the factor-count lemma as its two direct prerequisites. The factor-count lemma supplies the arbitrary-rank result through its own declared prerequisites. No claim about the original 1999 paper having been read is made.

Depends on

Used by · 5

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.

Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF pp. 18; Corollary 4.13