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Finite parabolic longest elements and disconnected factors

Statement

Let $(W, S)$ be a Coxeter system with $S$ finite and length function $ℓ$ . For $I \subseteq S$ , write $W_{I} = ⟨ I ⟩$ , $V_{I} = span {e_{i} : i \in I}$ and $Φ_{I} = {w e_{i} : w \in W_{I}, i \in I}$ in the canonical real reflection representation. Put $Φ_{I}^{+} = Φ_{I} \cap Φ_{+}$ and $Φ_{I}^{-} = - Φ_{I}^{+}$ .

If $W_{I}$ is finite, it has a unique element $w_{I}$ of maximum length, $w_{I}^{2} = 1$ , and $w_{I} Φ_{I}^{+} = Φ_{I}^{-}$ . There is a permutation $ι_{I}$ of $I$ such that $w_{I} e_{i} = - e_{ι_{I} (i)}$ for every $i \in I$ .

For the connected components $I_{1}, \dots, I_{k}$ of the Coxeter diagram induced on $I$ , the factors $W_{I_{j}}$ commute elementwise, their multiplication map is an isomorphism from their direct product onto $W_{I}$ , and their lengths add. If $W_{I}$ is finite, each factor is finite and $w_{I} = \prod_{j} w_{I_{j}}$ , including the empty product when $I = \emptyset$ .

In particular, let $a \in S ∖ I$ , put $K = I \cup {a}$ , and suppose $W_{K}$ is finite. If $H$ is a union of connected components common to the induced diagrams on $I$ and $K$ , write $J = I ∖ H$ and $L = K ∖ H$ . Then $w_{I} w_{K} = w_{J} w_{L}$ . Thus common disconnected finite factors cancel.

Facts & Assumptions

[F1]

The Coxeter system $(W, S)$ is presented by its Coxeter matrix $m$ , its length $ℓ (w)$ is the least number of letters from $S$ representing $w$ , and $W_{I} = ⟨ I ⟩$ is the standard parabolic subgroup (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups). For every $I \subseteq S$ , $(W_{I}, I)$ is a Coxeter system for the restricted matrix $m ∣_{I \times I}$ and its intrinsic length is the restriction of $ℓ$ (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification, part 2).

[F2]

A finite Coxeter system has a unique longest element, which is an involution, sends its positive roots to its negative roots and sends each simple root to the negative of a simple root according to a permutation of the generators (The longest element as the opposition of the chamber, and longest elements of finite parabolics, part 1).

[F3]

For the connected components of a finite Coxeter diagram, the corresponding standard parabolic subgroups form a direct product by multiplication, commute elementwise, and their lengths add (Disconnected diagrams, direct products, and comparison of invariant forms, part 1).

[F4]

On the simple-root basis of $V = R^{S}$ , the Coxeter form satisfies $B (e_{s}, e_{s}) = 1$ , $B (e_{s}, e_{t}) = - \cos (π / m (s, t))$ for distinct $s, t$ with $m (s, t) < \infty$ , and $B (e_{s}, e_{t}) = - 1$ when $m (s, t) = \infty$ ; the reflection with normal $e_{s}$ is $r_{e_{s}} (v) = v - 2 B (v, e_{s}) e_{s}$ (The real Coxeter form, its radical, reflections, and form-preserving maps, parts 1–3). The canonical homomorphism satisfies $ρ (s) = r_{e_{s}}$ , and $w v$ denotes $ρ (w) v$ (The canonical reflection homomorphism, roots, reflections, and the positive cone, part 1). Moreover $Φ_{I} = Φ \cap V_{I}$ and its positive and negative roots are the intersections with the ambient positive and negative roots (Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives, part 2).

[F5]

The Coxeter diagram has vertex set $S$ and an edge between distinct $s, t$ exactly when $m (s, t) \geq 3$ , with the Coxeter label on that edge. Its induced diagram on $I$ is the diagram of the restricted matrix $m ∣_{I \times I}$ , and its components are the connected components of the underlying graph (Coxeter diagrams: edges, labels, components and finite type, parts 1 and 2).

[F6]

In the canonical representation put $Φ = {w e_{s} : w \in W, s \in S}$ and $V_{+} = {\sum_{s \in S} λ_{s} e_{s} : λ_{s} \geq 0}$ . The ambient sign convention is $Φ_{+} = Φ \cap V_{+}$ and $Φ_{-} = Φ \cap (- V_{+}) = - Φ_{+}$ , with $Φ = Φ_{+} ⊔ Φ_{-}$ (Root sign coherence and the action of simple reflections on positive roots, part 2).

Proof

Given: The finite-rank Coxeter system, canonical representation and subsets in the statement. A connected component means a maximal connected vertex set in the indicated induced Coxeter diagram; distinct components have no diagram edges between them.

Goal: Establish the longest-element, product and cancellation assertions in the statement.

1.1F1F2F4F6

By [F1], $(W_{I}, I)$ is a Coxeter system with the ambient length restricted to $W_{I}$ . The entries of $B ∣_{V_{I} \times V_{I}}$ in [F4] depend only on $m ∣_{I \times I}$ , so this restricted form is the Coxeter form of $(W_{I}, I)$ . For $i \in I$ and $v \in V_{I}$ , the formula $r_{e_{i}} (v) = v - 2 B (v, e_{i}) e_{i}$ stays in $V_{I}$ and is exactly the reflection for that restricted form. Thus the restricted ambient action and the intrinsic canonical action agree on each generator in $I$ , hence on every word representing an element of $W_{I}$ by [F1]. Consequently its canonical representation is the restriction to $V_{I}$ . The nonnegative-coordinate convention in [F6] agrees on $V_{I}$ with the intrinsic convention, so its positive and negative roots are $Φ_{I}^{+}$ and $Φ_{I}^{-}$ . When $W_{I}$ is finite, applying [F2] to this system gives the unique maximum-length element $w_{I}$ , its involution and root-sign properties, and the permutation $ι_{I}$ with $w_{I} e_{i} = - e_{ι_{I} (i)}$ . This includes $I = \emptyset$ , for which the group and permutation are trivial and the root assertions are vacuous.

2.1F1F3F5step 1.1

By [F5], the diagram induced on $I$ is the diagram of the restricted Coxeter matrix. Apply [F3] to the Coxeter system $(W_{I}, I)$ from [F1] with this diagram. The component subgroups commute and multiply isomorphically onto $W_{I}$ , with additive intrinsic lengths; [F1] identifies these lengths with ambient lengths. If $W_{I}$ is finite, each subgroup $W_{I_{j}}$ is finite. Every element of $W_{I}$ has a unique product decomposition, and its length is the sum of the component lengths. The product of the component longest elements therefore has maximum length in $W_{I}$ ; uniqueness from step 1.1 identifies it with $w_{I}$ . The empty-component case gives the identity.

3.1F5step 1.1step 2.1algebra∎

By [F5], since $H$ is a union of components in both induced diagrams, there are no diagram edges between $H$ and $J$ , or between $H$ and $L$ . The component formula of step 2.1 gives $w_{I} = w_{H} w_{J}$ and $w_{K} = w_{H} w_{L}$ , with $w_{H}$ commuting with both $w_{J}$ and $w_{L}$ . All these parabolics are finite because they are subgroups of finite $W_{K}$ . Hence $w_{I} w_{K} = w_{H} w_{J} w_{H} w_{L} = w_{H}^{2} w_{J} w_{L} = w_{J} w_{L}$ , using $w_{H}^{2} = 1$ from step 1.1. This proves the stated cancellation and completes the proof.

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