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Geodesics are exactly length-additive wall words

PDF p. 13: Proposition 3.29; PDF p. 14: Proposition 3.29

Statement

Let $(W, S)$ be a finite-rank Coxeter system with vertex set $Γ_{0}$ , and fix $J \subseteq Γ_{0}$ . In its relative chamber arrangement, let $p = ((x_{1}, I_{1}), \dots, (x_{m}, I_{m}))$ be a nonempty chamber path. For each consecutive pair let $a_{i}$ be its crossing index and $v_{a_{i}, I_{i}} \in W$ its simple wall-crossing element. Put $v_{p} = v_{a_{1}, I_{1}} \dots v_{a_{m - 1}, I_{m - 1}}$ , with the empty product equal to the identity. Write $ℓ (w)$ for the least length of a word in the simple generators representing $w$ .

The path $p$ is geodesic, meaning that its number of crossings is minimal among paths with the same endpoints, if and only if $ℓ (v_{p}) = \sum_{i = 1}^{m - 1} ℓ (v_{a_{i}, I_{i}})$ . The empty sum is zero.

Statement references

Tits cone intersections, relative roots and arrangements

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

Chamber paths, concatenation and wall words

Geodesic chamber paths

Simple wall-crossing elements and maps

Facts & Assumptions

[F1]

The relative space $E = Θ_{J}$ , cone $C o n e (J)$ and distinct relative hyperplanes $H^{J}$ are specified by Tits cone intersections, relative roots and arrangements. The space $E$ is finite-dimensional by finite rank. The relative cone is convex by Convexity, stratification and faces of the relative cone, so its interior $T = C o n e (J)^{\circ}$ is open and convex in $E$ : equal-radius balls about two interior points give equal-radius balls about their convex combinations. The arrangement is locally finite on $T$ by Local finiteness of the interior relative arrangement; hyperplanes missing $T$ do not affect its chambers or crossings. Thus the gallery interface applies. A path crosses each distinct relative hyperplane at most once exactly when it is geodesic. Each adjacency crosses a unique relative hyperplane, and an adjacency changes the side of that hyperplane and no other. The side-change assertion is established in the gallery interface's adjacency argument, step 3.1. Geodesic chamber paths Finite galleries and separating hyperplanes

[F2]

For a crossing from $(x, I)$ with index $a$ , its target has first coordinate $x v_{a, I}$ , and its relative supporting hyperplane is the trace of $H_{x α_{a}} = {f : f (x α_{a}) = 0}$ . The relevant crossing component is finite. The adjacent-wall formula supplies the common wall, and relative stratification identifies its supporting hyperplane. Simple wall-crossing elements and maps Adjacent chambers and inverse wall-crossing elements Convexity, stratification and faces of the relative cone

[F3]

For such a crossing, $ℓ (x v_{a, I}) = ℓ (x) + ℓ (v_{a, I})$ exactly when the base chamber $(e, J)$ and $(x, I)$ lie on the same side of its supporting hyperplane. Length additivity and the base-chamber half-space

[F4]

Translation by the inverse of the first coordinate of a starting label $(z, L)$ bijects its relative arrangement with that for $L$ , sends $(z, L)$ to $(e, L)$ , preserves and reflects geodesicity, and preserves every crossing index and crossing element. Translation between relative arrangements in the geodesic argument

[F5]

Word length is the minimum number of simple generators in a representing word; concatenating such words gives $ℓ (u v) \leq ℓ (u) + ℓ (v)$ and $ℓ (e) = 0$ . Coxeter-system, roots, length and weak-order conventions

Proof

Given: The finite-rank Coxeter system, relative arrangement and path in the Statement. A relative hyperplane separates two chambers when they lie in its opposite open half-spaces.

Proof technique: direct.

Goal: Prove the stated equivalence, including the one-chamber case.

1.1F1F2F3given

First suppose $p$ starts at $(e, J)$ . Write $v_{i} = v_{a_{i}, I_{i}}$ and $u_{k} = v_{1} \dots v_{k}$ for $0 \leq k \leq m - 1$ , with $u_{0} = e$ . Repeatedly applying the target formula in [F2] gives $x_{k + 1} = u_{k}$ . Let $H_{k}$ be the relative hyperplane crossed at the $k$ th adjacency. A prefix ending at $(x_{k}, I_{k})$ is geodesic exactly when it has crossed no hyperplane twice. If that prefix is geodesic, it has crossed $H_{k}$ either zero or one time; by the side-change property in [F1], it has crossed it zero times exactly when its endpoint and the base chamber lie on the same side of $H_{k}$ . Thus the prefix through the $k$ th crossing is geodesic exactly when the preceding prefix is geodesic and the side condition in [F3] holds.

2.1F3F5step 1.1

By induction on $k$ , step 1.1 and [F3] show that the prefix through crossing $k$ is geodesic exactly when $ℓ (u_{i}) = ℓ (u_{i - 1}) + ℓ (v_{i})$ for every $1 \leq i \leq k$ . The initial one-chamber prefix is geodesic and has $u_{0} = e$ . By [F5], every difference $ℓ (u_{i - 1}) + ℓ (v_{i}) - ℓ (u_{i})$ is nonnegative. Their sum is $\sum_{i = 1}^{k} ℓ (v_{i}) - ℓ (u_{k})$ , since $ℓ (u_{0}) = 0$ . Consequently all the individual equalities hold exactly when $ℓ (u_{k}) = \sum_{i = 1}^{k} ℓ (v_{i})$ . This proves the claim for paths starting at the base chamber, including $k = 0$ .

3.1F4step 2.1∎

For a path starting at $(z, L)$ , apply [F4]. The translated path starts at $(e, L)$ , is geodesic exactly when $p$ is, and has the same ordered crossing elements. Its wall word and the sum of their lengths are therefore the same as those of $p$ , including the empty-product case. Applying step 2.1 to the translated path proves the equivalence for $p$ .

Source notes

The article reference attributes the proposition. The proof uses the declared local suppliers and does not use the proposition itself as a premise.

Depends on

Used by · 3

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF pp. 13–14, Proposition 3.29