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Geodesics differ by codimension-two relations

PDF p. 3: Corollary 4.26; PDF p. 21: Corollary 4.26

Statement

Let $(W, S)$ be a finite-rank Coxeter system with diagram vertex set $Γ_{0}$ , and fix $J \subseteq Γ_{0}$ . Write $W_{L}$ for the subgroup generated by the simple reflections indexed by $L$ . Let $Θ$ be the dual of the real geometric representation, let $Θ_{J}$ be the subspace annihilating the simple roots indexed by $J$ , and let $C o n e (J)^{\circ}$ be the interior in $Θ_{J}$ of the intersection of $Θ_{J}$ with the Tits cone. The relative arrangement $H^{J}$ consists of the distinct restrictions to $Θ_{J}$ of root hyperplanes not containing $Θ_{J}$ .

A chamber is a connected component of the complement of these hyperplanes in $C o n e (J)^{\circ}$ ; chambers are adjacent when they share a wall, a codimension-one intersection of a chamber closure with an arrangement hyperplane. A chamber path is a finite nonempty sequence of adjacent chambers, and it is geodesic when its number of crossings is minimal among paths with the same endpoints. The chamber labels $(x, I)$ belong to $C h a m (J)$ : thus $W_{J} x = x W_{I}$ and $x$ has minimum Coxeter length in $x W_{I}$ .

A codimension-two relation replaces one of the two geodesic paths supplied by Codimension-two relations by the other, with the same endpoints, inside a longer path. Any two geodesic paths $p, p^{'} : (x, I) \to (y, K)$ in $(C o n e (J)^{\circ}, H^{J})$ are related by a finite sequence of codimension-two relations.

Facts & Assumptions

[F1]

The endpoint labels determine a Brink–Howlett morphism $w = x^{- 1} y : I \to K$ . A morphism $w : I \to K$ carries the simple roots indexed by $K$ onto those indexed by $I$ ; composable labels multiply from left to right. Correspondence between chamber labels and Brink–Howlett morphisms Standard expressions and their lengths

[F2]

The wall word of a geodesic is a standard expression, meaning a composable sequence of elementary wall-crossing elements whose Coxeter lengths sum to the length of their product. Conversely, each standard expression lifts uniquely from a prescribed source chamber to a geodesic, whose endpoint is obtained by right multiplication by the product. Standard expressions correspond to geodesics Standard expressions and their lengths

[F3]

Any two standard expressions for the same Brink–Howlett morphism are connected by finitely many atomic braid relations. An atomic braid relation exchanges the two standard expressions for the same two-generator morphism $q$ , associated with distinct vertices $a, b$ outside its source type and a finite relative parabolic-root difference. Atomic Matsumoto theorem Atomic braid relations

[F4]

At any chamber of that source type, the two standard expressions for $q$ lift to the two paths identified by the associated codimension-two relation. They have the same endpoint. Two-geodesic wall-crossing lemma Codimension-two relations

Proof

Given: The finite-rank Coxeter system, subset $J$ , and geodesics $p, p^{'}$ with the same labelled endpoints in the Statement.

Proof technique: direct.

Goal: Connect $p$ and $p^{'}$ by finitely many codimension-two replacements.

1.1F1F2F3given

By [F1] and [F2], the wall words of $p$ and $p^{'}$ are standard expressions for the same morphism $w = x^{- 1} y$ . By [F3], there is a finite chain of standard expressions for $w$ joining them, with consecutive expressions differing by one atomic braid relation.

2.1F2F3F4step 1.1

Lift every expression in this chain from $(x, I)$ . By [F2] each lift is a geodesic ending at $(x w, K) = (y, K)$ , and uniqueness identifies the first and last lifts with $p$ and $p^{'}$ . Consider one move, replacing a consecutive block by the other expression for $q$ . The common prefix lifts to the same intermediate chamber in both paths. Each of the two blocks is a standard expression for $q$ , so [F2] and [F4] identify its lift from that chamber with one of the two paths in the corresponding codimension-two relation. Both blocks end at the same chamber, so the unchanged suffix lifts identically after either block.

3.1step 1.1step 2.1∎

Thus each atomic move lifts to one codimension-two replacement inside the geodesic path. The finite chain from step 1.1 therefore gives a finite sequence of such replacements from $p$ to $p^{'}$ , as required.

Source notes

The article citation attributes Corollary 4.26. The proof uses the declared suppliers and does not take the article's corollary as a premise.

Depends on

Used by · 2

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF pp. 3 and 21, Corollary 4.26