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Parallel classes of roots and finite inversion sets

Statement

Let $(W, S)$ , its ambient standard Coxeter roots $Φ = Φ^{+} ⊔ Φ^{-}$ , its root action and its length $ℓ$ have the Coxeter-data conventions. Let $B$ be the ordinary root-action groupoid on the associates of the fixed subset $J$ , with arrows $(B, x, A) : A \to B$ satisfying $x Π_{A} = Π_{B}$ . At every object $A$ use the signed copy $Φ (A) = {A} \times Φ$ , with $Φ (A)^{\pm} = {A} \times Φ^{\pm}$ and $- (A, α) = (A, - α)$ . The arrow $(B, x, A)$ acts by $(A, α) \mapsto (B, x α)$ . Thus the roots at an object are copies of the ambient Coxeter roots.

For an arrow $x : A \to B$ , define $N (x) = Φ (B)^{+} \cap x Φ (A)^{-}$ , identified with the ambient set $Φ^{+} \cap x Φ^{-}$ . For $α, β \in Φ (A)$ put $α \sim_{A} β$ if every arrow out of $A$ sends them to roots of the same sign. Call a root at $A$ real if some outgoing arrow changes its sign, and imaginary otherwise. Each $N (x)$ is a union of entire parallel classes at its codomain, all consisting of real roots. Arrow actions carry parallel classes bijectively to parallel classes. The set $C (x)$ of classes contained in $N (x)$ is finite; these are the inversion classes in the real compression of this signed root action.

External input

The imported ambient-root facts are $Φ = Φ^{+} ⊔ Φ^{-}$ and $∣ Φ^{+} \cap w Φ^{-} ∣ = ℓ (w)$ for every $w \in W$ , with no finite-rank restriction (Dyer–Gimenez, sections 1.3–1.4, (1.4.1) and (1.4.4)). The signed-copy, realness and compression terminology follows sections 1.21 and 3.1, 3.3–3.4. The class-transport and saturation arguments below are local deductions; the imported Coxeter inversion theorem is not reproved here.

Facts & Assumptions

[F1]

The objects, arrows and composition of $B$ are those of Transfer to the root-action Brink–Howlett groupoid.

[F2]

The ambient roots $Φ$ , their positive and negative subsets, the linear $W$ -action and $ℓ$ are defined in Coxeter-system, roots, length and weak-order conventions.

[F3]

Dyer–Gimenez, sections 1.3–1.4, supplies the sign decomposition and the ambient-root inversion equality recorded in External input. Sections 1.21 and 3.1, 3.3–3.4 supply the terminology used there.

Proof

Given: The Coxeter and groupoid data and signed copies specified above, with the imported ambient-root facts.

Goal: Prove class transport, saturation by real classes and finiteness of the inversion classes.

1.1F1F2F3given

The copies inherit a sign from the ambient sign decomposition. The linear root action commutes with negation, and its identity and composition laws give the stated signed action at the objects of $B$ . Equality of signs after every outgoing arrow is reflexive, symmetric and transitive, so $\sim_{A}$ is an equivalence relation. The identity arrow shows that roots in one class already have the same sign at $A$ .

2.1F1F2step 1.1

Let $y : A \to B$ . If $α \sim_{A} β$ , then every arrow $z$ out of $B$ has $z y$ out of $A$ , so $z (y α)$ and $z (y β)$ have the same sign. Conversely, apply this implication to $y^{- 1}$ . Thus $α \sim_{A} β$ if and only if $y α \sim_{B} y β$ . Applying the same argument to the inverse action shows that $y$ carries each entire class bijectively onto an entire class. Negating all images preserves equality of signs, so negation also descends to classes.

3.1F1F3step 1.1step 2.1

For $x : A \to B$ , a root $α$ belongs to $N (x)$ exactly when it is positive at $B$ and $x^{- 1} α$ is negative at $A$ . Both sign conditions are constant on a parallel class, by the identity and $x^{- 1}$ among the outgoing arrows from $B$ . Hence $N (x)$ is a union of entire classes. The arrow $x^{- 1}$ witnesses realness for every root in those classes, so imaginary roots never occur in $N (x)$ . On the quotient action, those classes are precisely the positive classes sent to negative classes by $x^{- 1}$ ; restricting to real roots leaves them unchanged. This identifies $C (x)$ with the inversion set in the real compression.

4.1F2F3step 3.1∎

Finally $∣ N (x) ∣ = ℓ (x) < \infty$ by the imported ambient-root inversion equality and the finite word length of each Coxeter element. Its image under the quotient map is $C (x)$ , so $C (x)$ is finite.

Depends on

Used by · 4

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.

External dependencies

Matthew Dyer and Harrison Gimenez, Root systems, Tits cones and imaginary cones of Brink-Howlett groupoids · Sections 1.3–1.4, especially (1.4.1) and (1.4.4); section 1.21; sections 3.1, 3.3–3.4.

The standard Coxeter roots split into positive and negative roots; the ambient root inversion set Phi_w = Phi^+ intersect w(-Phi^+) has cardinality ell(w). Section 1.21 places a copy of these roots at each Brink–Howlett object. Realness means a sign change by some outgoing arrow; parallelism compares all outgoing signs; compression quotients by parallelism.

Hypotheses: An arbitrary-rank Coxeter system with its standard based root system. The signed-copy and compression conventions apply to the specified root-action groupoid; the ambient inversion equality applies to every underlying element w in W.

Reading limits: Read HTML sections 1.3–1.4, 1.13, 1.21 and 3.1–3.4 for these statements and conventions. No PDF inspection, whole-paper reading, reconstruction of the imported inversion theorem or reading of Brink–Howlett (1999) is claimed.

Sources

Matthew Dyer and Harrison Gimenez, Root systems, Tits cones and imaginary cones of Brink-Howlett groupoids · Sections 1.3–1.4, especially (1.4.1) and (1.4.4); section 1.21; sections 3.1, 3.3–3.4.