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The root-length criterion and faithfulness of the canonical reflection representation

Statement

Let $S$ be a finite set, $m$ a Coxeter matrix, $W$ the presented group with length function $ℓ$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), $V = R^{S}$ with Coxeter form $B$ and canonical reflection homomorphism $ρ : W \to G L (V)$ with root system $Φ$ (The canonical reflection homomorphism, roots, reflections, and the positive cone), and let $Φ_{+}, Φ_{-}$ be the positive and negative roots and $C^{\circ}$ the open chamber of the dual action (Root sign coherence and the action of simple reflections on positive roots).

(1) Root-length criterion. For all $w \in W$ and $s \in S$ , $ℓ (w s) > ℓ (w) ⟺ ρ (w) e_{s} \in Φ_{+}, ℓ (w s) < ℓ (w) ⟺ ρ (w) e_{s} \in Φ_{-} .$

(2) Disjoint chambers. If $w C^{\circ} \cap C^{\circ} \neq \emptyset$ for some $w \in W$ , then $w = 1$ . Equivalently, the chambers $w C^{\circ}$ ( $w \in W$ ) are pairwise disjoint, that is, $C^{\circ}$ is prefundamental for the $W$ -action on $V^{*}$ in the sense that $w C^{\circ} \cap C^{\circ} \neq \emptyset$ forces $w = 1$ for every $w \in W$ .

(3) Faithfulness. The homomorphism $ρ$ is injective, the dual action $W \to G L (V^{*})$ is injective, and for every $w \neq 1$ there is $s \in S$ with $ρ (w) e_{s} \in Φ_{-}$ .

Facts & Assumptions

Given: a finite set $S$ , a Coxeter matrix $m$ , the presented group $W$ with length function $ℓ$ , the space $V = R^{S}$ with Coxeter form $B$ , the canonical reflection homomorphism $ρ$ with root system $Φ = Φ_{+} ⊔ Φ_{-}$ , and the dual action on $V^{*}$ with open chamber $C^{\circ}$ and half-spaces $B_{s} = {f \in V^{*} : f (e_{s}) > 0}$ , $s B_{s} = {f \in V^{*} : f (e_{s}) < 0}$ .

[F1]

For every $w \in W$ and $s \in S$ : the chamber $w C^{\circ}$ satisfies $w C^{\circ} \subseteq B_{s}$ or $w C^{\circ} \subseteq s B_{s}$ , and $w C^{\circ} \subseteq s B_{s}$ if and only if $ℓ (s w) < ℓ (w)$ ; for $s \neq t$ the rank-two half-space alternative and the induction $(P_{n})$ , $(Q_{n})$ hold (The rank-two half-space alternative and the chamber-length induction $(P_{n})$ , $(Q_{n})$ ).

[F2]

For every $w \in W$ and $s \in S$ : $ρ (w) e_{s} \in Φ_{-} ⟺ w^{- 1} C^{\circ} \subseteq s B_{s}$ and $ρ (w) e_{s} \in Φ_{+} ⟺ w^{- 1} C^{\circ} \subseteq B_{s}$ ; moreover $C^{\circ} \subseteq B_{s}$ and $B_{s} \cap s B_{s} = \emptyset$ for every $s$ , and $C^{\circ} \neq \emptyset$ (Root sign coherence and the action of simple reflections on positive roots, The dual action, chambers, faces, and root hyperplanes).

[F3]

For all $w \in W$ and $s \in S$ one has $ℓ (w s) = ℓ (w) \pm 1$ , and $ℓ$ is invariant under inversion: $ℓ (w^{- 1}) = ℓ (w)$ ; every element $w \neq 1$ has a reduced expression $w = s_{1} \dots s_{n}$ with $n = ℓ (w) \geq 1$ , and then $w = (s_{1} \dots s_{n - 1}) s_{n}$ has $ℓ (w s_{n}) = ℓ (w) - 1$ (Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action, Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification).

[F4]

The dual action $(w \cdot f) (v) = f (ρ (w)^{- 1} v)$ is a left action of $W$ on $V^{*}$ by linear bijections, with $ρ (w)^{- 1} = ρ (w^{- 1})$ for every $w$ (The dual action, chambers, faces, and root hyperplanes, The dual action, the faces, and the rank-two chamber tiling).

Proof

technique · direct

1.1F1F2F4given

Set-up. Recall from [F2] that $C^{\circ} \subseteq B_{s}$ and $B_{s} \cap s B_{s} = \emptyset$ for every $s \in S$ , and that the two sign equivalences of the root system hold; from [F4] that every $w$ acts on $V^{*}$ as a bijection with inverse the action of $w^{- 1}$ ; and from [F1] that for every $x \in W$ and every generator $s$ exactly one of $x C^{\circ} \subseteq B_{s}$ , $x C^{\circ} \subseteq s B_{s}$ holds.

1.2F1F2F3algebra

The two-sided root-length criterion. Fix $w \in W$ and $s \in S$ . By [F2], $ρ (w) e_{s} \in Φ_{-}$ if and only if $w^{- 1} C^{\circ} \subseteq s B_{s}$ ; by [F1] applied to the element $w^{- 1}$ this holds if and only if $ℓ (s w^{- 1}) < ℓ (w^{- 1})$ . Since $(s w^{- 1})^{- 1} = w s$ and $ℓ$ is inversion invariant, [F3] gives $ℓ (s w^{- 1}) = ℓ (w s)$ and $ℓ (w^{- 1}) = ℓ (w)$ , so $ρ (w) e_{s} \in Φ_{-} ⟺ ℓ (w s) < ℓ (w)$ . Finally $ℓ (w s) = ℓ (w) \pm 1$ by [F3] and $Φ = Φ_{+} ⊔ Φ_{-}$ by [F2], so this is equivalent to the statement that $ℓ (w s) > ℓ (w)$ if and only if $ρ (w) e_{s} \in Φ_{+}$ ; both claims of (1) follow.

1.3F1F2F3F4algebra

Disjoint chambers. Suppose $w C^{\circ} \cap C^{\circ} \neq \emptyset$ and $w \neq 1$ . Choose a reduced expression of $w$ and let $s$ be its first letter, so that $w = s w^{'}$ with $ℓ (w^{'}) = ℓ (w) - 1$ by [F3]. The intersection point lies in $C^{\circ} \subseteq B_{s}$ , so $w C^{\circ}$ meets $B_{s}$ , whence $w C^{\circ} \subseteq̸ s B_{s}$ because $B_{s} \cap s B_{s} = \emptyset$ ; applying the bijection $f \mapsto s \cdot f$ from [F4] and using that its square is the identity gives $w^{'} C^{\circ} \subseteq̸ B_{s}$ . By [F1] applied to $w^{'}$ , therefore $w^{'} C^{\circ} \subseteq s B_{s}$ and $ℓ (s w^{'}) = ℓ (w^{'}) - 1$ ; but $s w^{'} = w$ , so $ℓ (w) = ℓ (w^{'}) - 1 = ℓ (w) - 2$ , a contradiction. Hence $w = 1$ ; and the stated equivalence holds because $v C^{\circ} \cap w C^{\circ} \neq \emptyset$ if and only if $(w^{- 1} v) C^{\circ} \cap C^{\circ} \neq \emptyset$ by [F4].

2.1step 1.2step 1.3F3F4∎

Faithfulness. If $ρ (w) = {i d}_{V}$ , then $ρ (w)^{- 1} = {i d}_{V}$ as well, so $(w \cdot f) (v) = f (ρ (w)^{- 1} v) = f (v)$ for all $f \in V^{*}$ and $v \in V$ by [F4]; hence $w \cdot f = f$ for every $f$ and $w C^{\circ} = C^{\circ}$ meets $C^{\circ}$ , so 1.3 gives $w = 1$ . If the dual action of $w$ is the identity, then for any $f \in C^{\circ}$ one has $f = w \cdot f \in w C^{\circ} \cap C^{\circ}$ , which is nonempty, and 1.3 again gives $w = 1$ . Finally let $w \neq 1$ and let $s$ be the last letter of a reduced expression of $w$ , so that $ℓ (w s) = ℓ (w) - 1$ by [F3]; by the criterion of 1.2, $ρ (w) e_{s} \in Φ_{-}$ .

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Sources

Michael W. Davis, The Geometry and Topology of Coxeter Groups (Princeton University Press, 2008; author's complete institutional PDF) (standard reference, not scraped)
Anders Bjorner and Francesco Brenti, Combinatorics of Coxeter Groups (Graduate Texts in Mathematics 231, Springer 2005; author-hosted full PDF) (standard reference, not scraped)