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Length additivity and the base-chamber half-space

PDF p. 11: Lemma 3.20(2)

Statement

Let $(W, S)$ be a finite-rank Coxeter system with simple generators $s_{i}$ and simple roots $α_{i}$ indexed by $Γ_{0}$ . Use its standard geometric representation, root signs, word length $ℓ$ , dual space $Θ$ , and contragredient action. For $L \subseteq Γ_{0}$ , write $W_{L} = ⟨ s_{i} : i \in L ⟩$ . Fix $J \subseteq Γ_{0}$ and a label $(x, I) \in C h a m (J)$ , so $W_{J} x = x W_{I}$ and $x$ has minimum length in $x W_{I}$ . The label represents the relative chamber $x C_{I}$ , where $C_{I}$ consists of dual vectors vanishing on $α_{i}$ for $i \in I$ and positive on the other simple roots.

Choose $a \notin I$ , let $D = [I, a]$ be the component containing $a$ of the diagram induced on $I \cup {a}$ , and suppose $W_{D}$ is finite. Put $v_{a, I} = w_{D ∖ {a}} w_{D}$ , using the longest elements of the indicated finite parabolics. Suppose the pair $ω_{a, I} (x, I)$ is a label adjacent to $(x, I)$ . Then $ℓ (x v_{a, I}) = ℓ (x) + ℓ (v_{a, I})$ if and only if the base chamber $C_{J}$ and $x C_{I}$ lie in the same open half-space of $H_{x α_{a}} = {f \in Θ : f (x α_{a}) = 0}$ .

Statement references

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

Reflecting hyperplanes, fundamental chamber and Tits cone

Algebraic chamber-label set

Simple wall-crossing elements and maps

Chambers, walls and adjacency

Separation by a hyperplane

Labelling theorem for relative chambers

Facts & Assumptions

[F1]

Roots have exactly one sign, and for $N (w) = {β \in Φ^{+} : w β \in Φ^{-}}$ one has $∣ N (w) ∣ = ℓ (w) = ℓ (w^{- 1})$ . Root signs, inversions and descent translation

[F2]

A minimal representative $x$ of $x W_{I}$ satisfies $ℓ (x s_{i}) > ℓ (x)$ for every $i \in I$ ; the root-length criterion then gives $x α_{i} \in Φ^{+}$ . Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification The root-length criterion and faithfulness of the canonical reflection representation

[F3]

For $Q = D$ and $Q = E = D ∖ {a}$ , the standard subsystem has root set $Φ_{Q} = Φ \cap span {α_{i} : i \in Q}$ with the inherited signs, by part 2 of Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives. The longest elements $w_{D}, w_{E}$ are involutions and send their respective positive subsystem roots to negative roots. Finite parabolic longest elements and disconnected factors

[F4]

The inversion roots of a reduced expression of an element of $W_{D}$ belong to $Φ_{D}$ ; their number is its length. Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification The inversion formula |N(w)|=ℓ(w), the root-reflection dictionary and strong exchange

[F5]

The label $(x, I)$ represents $x C_{I}$ in the relative space $Θ_{J} = {f : f (α_{j}) = 0 for j \in J}$ , and the base label $(1, J)$ represents $C_{J}$ . The dual action obeys $(x f) (β) = f (x^{- 1} β)$ . Labelling theorem for relative chambers Coxeter-system, roots, length and weak-order conventions

Proof

Given: The finite-rank Coxeter data, label, vertex $a$ , finite component $D$ , and adjacency in the Statement. Put $E = D ∖ {a}$ and $v = w_{E} w_{D}$ .

Proof technique: direct.

Goal: Prove the equivalence between length additivity and the stated same-half-space condition.

1.1F1F3F4given

Inversions of the crossing element. Every generator indexed by $E$ changes only its own simple-root coordinate. Thus $w_{E}$ preserves the coefficient of $α_{a}$ . If $β \in Φ_{D}^{+}$ has positive $α_{a}$ coefficient, then $w_{E} β$ is a root with positive $α_{a}$ coefficient and hence is positive by [F1]. If that coefficient is zero, [F3] identifies $β$ with a root of $Φ_{E}^{+}$ , and $w_{E} β$ is negative. Since $w_{D}$ exchanges the two signs in $Φ_{D}$ and $v^{- 1} = w_{D} w_{E}$ , it follows that $N (v^{- 1}) \cap Φ_{D} = Φ_{D}^{+} ∖ Φ_{E}^{+}$ . By [F4] all inversions of $v^{- 1}$ lie in $Φ_{D}$ , so this is the entire inversion set. In particular $α_{a} \in N (v^{- 1})$ .

2.1F1F2step 1.1

Length criterion. For arbitrary $u, t \in W$ , partitioning positive roots by the signs of $t^{- 1} β$ and $u β$ gives $ℓ (u t) = ℓ (u) + ℓ (t) - 2 ∣ N (u) \cap N (t^{- 1}) ∣$ . Indeed, the roots in the intersection are precisely the cancellations between inversions of $t$ and those of $u$ , after the sign-preserving bijection induced by $t$ on the other sign classes; all sets involved are finite by [F1]. Consequently $ℓ (x v) = ℓ (x) + ℓ (v)$ exactly when $x$ sends every root in $N (v^{- 1})$ to a positive root. By [F2], $x α_{i}$ is positive for $i \in E$ . If $x α_{a}$ is positive, every positive root supported on $D$ is a nonzero nonnegative combination of simple roots whose images are positive, so its image is positive. Conversely, $α_{a}$ itself belongs to $N (v^{- 1})$ by 1.1. Therefore length additivity is equivalent to $x α_{a} \in Φ^{+}$ .

3.1F1F5step 2.1given∎

Translate to half-spaces. For $f \in C_{I}$ , the dual-action formula gives $(x f) (x α_{a}) = f (α_{a}) > 0$ . Hence the linear functional $g \mapsto g (x α_{a})$ is nonzero on $Θ_{J}$ , because $x C_{I} \subseteq Θ_{J}$ by [F5]. Write $x α_{a}$ in the simple-root basis. By [F1] its nonzero coefficients all have the same sign, and nonvanishing on $Θ_{J}$ means at least one coefficient outside $J$ is nonzero. Every $g \in C_{J}$ vanishes on the $J$ coordinates and is strictly positive on the others, so $g (x α_{a}) > 0$ exactly when $x α_{a}$ is positive, and is negative otherwise. Thus $C_{J}$ and $x C_{I}$ are in the same open half-space of $H_{x α_{a}}$ exactly when $x α_{a}$ is positive. Combining this with step 2.1 proves the assertion.

Source notes

The article citation records attribution of the retained claim. The proof above uses the declared suppliers and does not invoke the article's claim as a premise.

Depends on

Used by · 1

Geodesics are exactly length-additive wall words

Dependency tree · 2 levels

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 11, Lemma 3.20(2)