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Canonical Roots, Signs, and Faithful Reflections

1 · Prerequisites

2 · Summary

The canonical reflection representation of a Coxeter group presented by an abstract Coxeter matrix is not an extra hypothesis: this page proves, with no definiteness condition on the form, that $s \mapsto r_{s}$ extends to a homomorphism whose kernel is trivial and whose chamber geometry mirrors the length function. The rank-two half-space alternative and the chamber-length induction $(P_{n})$ , $(Q_{n})$ first reduces to a rank-two plane, where the two standard dihedral pictures (finite and infinite) are read off from the explicit dual action, and then runs the simultaneous induction $(P_{n})$ , $(Q_{n})$ of Davis 4.8.3 with both implication steps: every chamber $w C^{\circ}$ lies wholly on one side of every simple wall, the negative side forcing the corresponding left descent, and every $w$ admits a rank-two prefix $u$ with $w C^{\circ} \subseteq u (B_{s} \cap B_{t})$ and $ℓ (w) = ℓ_{s, t} (u) + ℓ (u^{- 1} w)$ . Only the open chamber and open half-spaces are used; equality on closed walls is never asserted.

From that induction, Root sign coherence and the action of simple reflections on positive roots completes the sign theory: every root lies in $V_{+} ∖ {0}$ or in $- V_{+} ∖ {0}$ and not in both, with no crystallographic integrality or assumed root-system axiom substituted for the real proof, and $r_{s}$ permutes $Φ_{+} ∖ {e_{s}}$ while sending $e_{s}$ to $- e_{s}$ . The root-length criterion and faithfulness of the canonical reflection representation then converts signs into lengths, $ℓ (w s) > ℓ (w) ⟺ ρ (w) e_{s} \in Φ_{+}$ , deduces that the chambers $w C^{\circ}$ are pairwise disjoint, and concludes that $ρ$ — and the dual action — is injective for every Coxeter matrix in finite rank, indefinite or degenerate $B$ included.

The geometric inversion set $N (w)$ of an element of a Coxeter group defines the geometric inversion set $N (w) = Φ_{+} \cap ρ (w)^{- 1} Φ_{-}$ with an explicit left/right convention (suffix roots for $N (w)$ , prefix roots for $N (w^{- 1})$ ) and records its step recursion, asserting no finiteness in the definition itself; finiteness and independence are supplied by The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange, which proves $∣ N (w) ∣ = ℓ (w)$ and the two root lists, identifies $\pm α$ with the conjugate reflection $t_{α}$ through faithfulness, and proves strong exchange: for $t \in T$ with $ℓ (t w) < ℓ (w)$ , one letter of any reduced expression of $w$ is deleted, and $t$ is the corresponding prefix reflection.

Earlier pages: real-forms-and-reflection-geometry supplies the Coxeter form, the reflections $r_{a}$ , the canonical homomorphism, the dual chambers and the rank-two chamber tiling, and coxeter-presentations-exchange-and-reduced-word-theorems supplies the presented group with its length function, the dihedral root recurrences, exchange and deletion, the signed action and the parabolic minimal-representative theorem. The companion canonical-roots-signs-and-faithful-reflections-examples computes the roots, inversion sets and chamber images in $I_{2} (5)$ , $A_{2}$ and infinite dihedral type. Every argument on this page is choice-free.

3 · Logical flowchart

4 · Definitions, theorems and proofs

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

The rank-two half-space alternative and the chamber-length induction $(P_{n})$ , $(Q_{n})$

Statement

Let $S$ be a finite set, $m$ a Coxeter matrix on $S$ , $W$ the group presented by $(S, m)$ with length function $ℓ$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), $V = R^{S}$ , $B$ the Coxeter form, $ρ : W \to G L (V)$ the canonical reflection homomorphism with root system $Φ$ (The canonical reflection homomorphism, roots, reflections, and the positive cone), and let $V^{*}$ be the algebraic dual with its dual action, its chamber $C$ , its interior $C^{\circ}$ and its root hyperplanes $H_{α}$ (The dual action, chambers, faces, and root hyperplanes). For $s \in S$ put $B_{s} : = {f \in V^{*} : f (e_{s}) > 0};$ then $C^{\circ} = ⋂_{s \in S} B_{s}$ and the translate $s B_{s} = {f \in V^{*} : f (e_{s}) < 0}$ is the opposite open half-space. For distinct $s, t$ let $W_{s, t} : = ⟨ s, t ⟩ \leq W$ be the standard parabolic subgroup, with intrinsic length $ℓ_{s, t}$ , which agrees with $ℓ$ on $W_{s, t}$ (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (2)). For $s \neq t$ and $k \geq 0$ let $u_{k}$ be the alternating word $s t s \dots$ of $k$ letters beginning with $s$ , so $u_{0} = 1$ .

(1) Rank-two half-space alternative. Let $s \neq t$ and $v \in W_{s, t}$ . For each $s^{'} \in {s, t}$ exactly one of $v (B_{s} \cap B_{t}) \subseteq B_{s^{'}}, v (B_{s} \cap B_{t}) \subseteq s^{'} B_{s^{'}}$ holds, and if the second holds then $ℓ_{s, t} (s^{'} v) = ℓ_{s, t} (v) - 1$ . Moreover, if $m (s, t) < \infty$ then the elements $u_{0}, \dots, u_{2 m (s, t) - 1}$ exhaust $W_{s, t}$ , with $ℓ_{s, t} (u_{k}) = \min (k, 2 m (s, t) - k)$ , and the unique element of $W_{s, t}$ having both $s$ and $t$ as left descents is $u_{m (s, t)}$ ; if $m (s, t) = \infty$ then $ℓ_{s, t} (u_{k}) = k$ for every $k \geq 0$ , the elements $u_{k}$ are pairwise distinct, and no element of $W_{s, t}$ has both $s$ and $t$ as left descents.

(2) The simultaneous induction. For $n \geq 0$ let $(P_{n}) : for all w \in W with ℓ (w) = n and all s \in S, w C^{\circ} \subseteq B_{s}, or w C^{\circ} \subseteq s B_{s} and ℓ (s w) = ℓ (w) - 1;$ $(Q_{n}) : for all w \in W with ℓ (w) = n and all s \neq t there is u \in W_{s, t} with w C^{\circ} \subseteq u (B_{s} \cap B_{t}) and ℓ (w) = ℓ_{s, t} (u) + ℓ (u^{- 1} w) .$ Then $(P_{n})$ and $(Q_{n})$ hold for every $n \geq 0$ . Explicitly: $(P_{0})$ and $(Q_{0})$ are immediate; (1) together with $(Q_{n})$ yields $(P_{n + 1})$ ; and (1) together with $(Q_{n})$ and $(P_{m})$ for all $m \leq n + 1$ yields $(Q_{n + 1})$ .

(3) Canonical factorisation. Let $w \in W$ and $s \neq t$ . Write $w = u d$ with $u \in W_{s, t}$ and $d$ the minimal representative of the right coset $W_{s, t} w$ . Then $ℓ (w) = ℓ (u) + ℓ (d)$ and $ℓ (s^{'} d) > ℓ (d)$ for $s^{'} \in {s, t}$ (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (3)), and $w C^{\circ} \subseteq u (B_{s} \cap B_{t}), ℓ (w) = ℓ_{s, t} (u) + ℓ (u^{- 1} w) .$

(4) Chamber-descent equivalence. For every $w \in W$ and $s \in S$ one has $w C^{\circ} \subseteq B_{s}$ or $w C^{\circ} \subseteq s B_{s}$ ; moreover $w C^{\circ} \subseteq s B_{s} ⟺ ℓ (s w) < ℓ (w) .$

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 its canonical basis $(e_{s})$ , the Coxeter form $B$ , the canonical reflection homomorphism $ρ$ with root system $Φ$ , the dual action on $V^{*}$ with chamber $C$ , interior $C^{\circ}$ and root hyperplanes $H_{α}$ , and for $s \neq t$ the standard parabolic subgroup $W_{s, t} = ⟨ s, t ⟩$ with intrinsic length $ℓ_{s, t}$ and alternating words $u_{k} = s t s \dots$ of $k$ letters beginning with $s$ .

[F1]

For $s \neq t$ put $c : = c (s, t)$ , equal to $\cos (π / m (s, t))$ when $m (s, t) < \infty$ and to $1$ when $m (s, t) = \infty$ ; then $B (e_{s}, e_{s}) = 1$ , $B (e_{s}, e_{t}) = - c$ , and with $P : = R e_{s} + R e_{t}$ , $P^{⊥} = {v \in V : B (v, e_{s}) = B (v, e_{t}) = 0}$ the reflections $r_{s}, r_{t}$ preserve $P$ , fix $P^{⊥}$ pointwise, and satisfy $r_{s} (e_{s}) = - e_{s}$ , $r_{s} (e_{t}) = e_{t} + 2 c e_{s}$ (The real Coxeter form, its radical, reflections, and form-preserving maps, Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order).

[F2]

The assignment $s \mapsto r_{s}$ induces the group homomorphism $ρ : W \to G L (V)$ , the root system is $Φ = {ρ (w) e_{s} : w \in W, s \in S}$ , every root satisfies $B (α, α) = 1$ , and $ρ (w s w^{- 1}) = r_{ρ (w) e_{s}}$ for all $w \in W$ , $s \in S$ (The canonical reflection homomorphism, roots, reflections, and the positive cone, Descent of the reflection representation, unit root norms, and conjugation of reflections).

[F3]

The formula $(w \cdot f) (v) : = f (ρ (w)^{- 1} v)$ defines a left action of $W$ on $V^{*}$ by linear maps, $C^{\circ} = {f \in V^{*} : f (e_{r}) > 0 for all r \in S}$ is nonempty, and $H_{α} = {f \in V^{*} : f (α) = 0}$ . For $s \neq t$ , in the coordinates $(y_{s}, y_{t})$ of $P^{*} = R f_{s} + R f_{t}$ with dual basis $f_{s} (e_{t}) = δ_{s t}$ , the dual generators act by $(y_{s}, y_{t}) \mapsto (- y_{s}, 2 c y_{s} + y_{t})$ for $s$ and $(y_{s}, y_{t}) \mapsto (y_{s} + 2 c y_{t}, - y_{t})$ for $t$ , each fixing the line $H_{e_{s}} \cap P^{*}$ respectively $H_{e_{t}} \cap P^{*}$ pointwise; when $m (s, t) = \infty$ the functional $f \mapsto f (e_{s} + e_{t})$ is $W_{s, t}$ -invariant (The dual action, chambers, faces, and root hyperplanes, The dual action, the faces, and the rank-two chamber tiling).

[F4]

Let $C_{P} : = {f \in P^{*} : f (e_{s}) \geq 0, f (e_{t}) \geq 0}$ and $Φ_{P} : = W_{s, t} {e_{s}, e_{t}}$ . If $m : = m (s, t) < \infty$ , then $W_{s, t}$ has order $2 m$ and the $2 m$ chambers $w C_{P}$ $(w \in W_{s, t})$ are exactly the $2 m$ closed sectors cut out in $P^{*}$ by the $m$ root lines $H_{β} \cap P^{*}$ $(β \in Φ_{P})$ ; they have pairwise disjoint interiors, their union is $P^{*}$ , $W_{s, t}$ acts simply transitively on them, and distinct chambers are separated by a root line. If $m = \infty$ , the chambers $w C_{P}$ have pairwise disjoint interiors with union ${f \in P^{*} : f (e_{s} + e_{t}) > 0} \cup {0}$ (so every chamber meets the boundary line ${f : f (e_{s} + e_{t}) = 0}$ only in $0$ ), distinct chambers are separated by a root line, and the traces of the root lines on the affine line ${f : f (e_{s} + e_{t}) = 1}$ are exactly the integers in the coordinate $y_{s}$ (The dual action, the faces, and the rank-two chamber tiling).

[F5]

There is a homomorphism $sgn : W \to {\pm 1}$ with $sgn (s) = - 1$ and $sgn (w) = (- 1)^{ℓ (w)}$ for all $w \in W$ ; hence $ℓ (w s) = ℓ (w) \pm 1$ for all $w \in W$ and $s \in S$ . Also $ℓ (s) = 1$ for every $s \in S$ , so $s \neq 1$ in $W$ . Distinct generators are distinct in $W$ , the product $s t$ has order exactly $m (s, t)$ in $W$ , the standard parabolic subgroup $W_{s, t}$ is the Coxeter system presented by the restricted matrix on ${s, t}$ with intrinsic length $ℓ_{s, t}$ equal to $ℓ$ on $W_{s, t}$ , and every right coset $W_{s, t} a$ has a unique minimal-length representative $d$ , characterized by $ℓ (s^{'} d) > ℓ (d)$ for $s^{'} \in {s, t}$ and satisfying $ℓ (u d) = ℓ (u) + ℓ (d)$ for all $u \in W_{s, t}$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness, 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).

[F6]

For $s \neq t$ and $q \geq 1$ , if $m (s, t) < \infty$ and $q \leq m (s, t)$ , or if $m (s, t) = \infty$ , then the alternating word of length $q$ beginning with $s$ has length $q$ in $W_{s, t}$ (The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness); the same holds for the alternating word beginning with $t$ .

[F7]

Induction principle on the natural numbers: if a property holds for $0$ and its validity at all $m \leq n$ implies it at $n + 1$ , then it holds for every $n \geq 0$ (The principle of mathematical induction).

Proof

technique · direct

1.1F1F3F5given

Set-up. For the rank-two calculation fix $s \neq t$ in $S$ and adopt the notation of [F1] and [F3]. For $r \in S$ the sets $B_{r} = {f \in V^{*} : f (e_{r}) > 0}$ and $r B_{r} = {f \in V^{*} : f (e_{r}) < 0}$ are disjoint and $C^{\circ} = ⋂_{r \in S} B_{r}$ ; the dual action satisfies $v (B_{s} \cap B_{t}) = {v \cdot f : f \in B_{s} \cap B_{t}}$ , and the action of $v \in W_{s, t}$ on $V^{*}$ is linear. By [F5] the intrinsic length satisfies $ℓ_{s, t} (x) = ℓ (x)$ for $x \in W_{s, t}$ .

1.2F1F3algebra

Reduction to the rank-two plane. Put $C_{P}^{\circ} : = {g \in P^{*} : g (e_{s}) > 0, g (e_{t}) > 0}$ and let $π : V^{*} \to P^{*}$ , $π (f) : = f ∣_{P}$ . Then $π$ is linear and surjective, because a functional prescribed on $P$ extends to a functional on $V$ by $0$ on the remaining basis vectors. Every $v \in W_{s, t}$ preserves $P$ and fixes $P^{⊥}$ pointwise by [F1], so for $x \in P$ and $f \in V^{*}$ one has $(v \cdot f) (x) = f (ρ (v)^{- 1} x)$ with $ρ (v)^{- 1} x \in P$ ; hence $π (v \cdot f) = v \cdot π (f)$ and $π^{- 1} (C_{P}^{\circ}) = B_{s} \cap B_{t}$ , $π^{- 1} {g : g (e_{s^{'}}) > 0} = B_{s^{'}}$ , $π^{- 1} {g : g (e_{s^{'}}) < 0} = s^{'} B_{s^{'}}$ for $s^{'} \in {s, t}$ . Consequently $v (B_{s} \cap B_{t}) \subseteq B_{s^{'}}$ if and only if $v C_{P}^{\circ} \subseteq {g : g (e_{s^{'}}) > 0}$ , and $v (B_{s} \cap B_{t}) \subseteq s^{'} B_{s^{'}}$ if and only if $v C_{P}^{\circ} \subseteq {g : g (e_{s^{'}}) < 0}$ .

1.3F4F2algebra

Sectors avoid the walls. By [F4] the interiors of the chambers $w C_{P}$ $(w \in W_{s, t})$ are convex open cones that are connected components of the complement of the union of the root lines in $P^{*}$ (respectively in ${f \in P^{*} : f (e_{s} + e_{t}) > 0}$ when $m (s, t) = \infty$ ), so each is disjoint from every root line $H_{β} \cap P^{*}$ $(β \in Φ_{P})$ , and $H_{e_{s}} \cap P^{*}$ , $H_{e_{t}} \cap P^{*}$ are root lines because $e_{s}, e_{t} \in Φ_{P}$ . An open convex set disjoint from a line lies in exactly one of the two open half-planes bounded by it. Hence for every $v \in W_{s, t}$ and $s^{'} \in {s, t}$ exactly one of $v C_{P}^{\circ} \subseteq {g : g (e_{s^{'}}) > 0}$ and $v C_{P}^{\circ} \subseteq {g : g (e_{s^{'}}) < 0}$ holds.

1.4F5F6algebra

The finite dihedral group. Assume $m : = m (s, t) < \infty$ and write $u : = s t$ . Since $u^{m} = 1$ by [F5], the set ${u_{0}, \dots, u_{2 m - 1}}$ is closed under right multiplication by $s$ and $t$ : indeed $u_{2 j} s = u_{2 j + 1}$ and $u_{2 j + 1} s = u_{2 j}$ for $2 j < 2 m$ ; $u_{2 j} t = u_{2 j - 1}$ for $j \geq 1$ ; $u_{0} t = u_{2 m - 1}$ because $(s t)^{m - 1} s = (s t)^{- 1} s = (t s) s = t$ ; and $u_{2 j + 1} t = u_{2 j + 2}$ for $2 j + 1 < 2 m$ , $u_{2 m - 1} t = 1$ . Since $W_{s, t}$ is generated by $s, t$ and contains $1$ , it equals ${u_{0}, \dots, u_{2 m - 1}}$ . The $u_{k}$ are pairwise distinct: if $u_{j} = u_{k}$ with $j < k < 2 m$ , then $1 = u_{j}^{- 1} u_{k}$ ; for $j, k$ of equal parity this equals $u^{\pm (k - j) / 2}$ with $0 < (k - j) / 2 < m$ , contradicting the exact order $m$ of $u$ from [F5], while for parities differing it equals $s u^{p}$ or $u^{p} s$ with $2 ∣ p ∣ < 2 m$ , so it would give $u^{p} = s$ , impossible because $sgn (u^{p}) = 1$ while $sgn (s) = - 1$ . Hence $∣ W_{s, t} ∣ = 2 m$ , the $u_{k}$ are distinct, $u_{k + 2 m} = u_{k}$ , and the pairs $u_{k}, u_{k + 1}$ and $u_{2 m - 1}, u_{0}$ are edges of the Cayley graph of $W_{s, t}$ for the generating set ${s, t}$ . That graph is connected, and every vertex has degree exactly $2$ because $x \neq x s$ and $x s \neq x t$ for all $x$ (as $s \neq 1$ and $s \neq t$ by [F5]); it therefore is the displayed $2 m$ -cycle, so $ℓ_{s, t} (u_{k}) = \min (k, 2 m - k)$ is the graph distance from $u_{0}$ . Moreover $s u_{k} = u_{σ (k)}$ and $t u_{k} = u_{τ (k)}$ with $σ (k) \equiv 1 - k (m o d 2 m)$ and $τ (k) \equiv - 1 - k (m o d 2 m)$ for $0 \leq k < 2 m$ : these are $s (s t)^{j} = u^{- j} s$ , $t (s t)^{j} = u^{- j} t = u^{- j - 1} s$ applied to $u_{2 j} = u^{j}$ and $u_{2 j + 1} = u^{j} s$ , together with the case $k = 0$ .

1.5F3F5F6algebra

The infinite dihedral group. Assume $m (s, t) = \infty$ , write $u : = s t$ , and let $u_{k}^{'}$ be the alternating word of length $k$ beginning with $t$ . By [F5], $u$ has infinite order, and by [F6] $ℓ_{s, t} (u_{k}) = k = ℓ_{s, t} (u_{k}^{'})$ for all $k \geq 0$ . The union $X : = {u_{k} : k \geq 0} \cup {u_{k}^{'} : k \geq 0}$ is closed under right multiplication by $s$ and $t$ ( $u_{2 j} s = u_{2 j + 1}$ , $u_{2 j + 1} s = u_{2 j}$ , $u_{2 j} t = u_{2 j - 1}$ for $j \geq 1$ , $u_{0} t = u_{1}^{'}$ , $u_{2 j + 1} t = u_{2 j + 2}$ , symmetrically for $u_{k}^{'}$ ), so $X = W_{s, t}$ ; by [F6] the elements $u_{k}$ are pairwise distinct with $ℓ_{s, t} (u_{k}) = k$ for every $k \geq 0$ , and likewise the $u_{k}^{'}$ . The Cayley graph of $W_{s, t}$ for ${s, t}$ is connected, and every vertex has degree exactly $2$ (as in 1.4), so it is the two-way infinite path $\dots u_{2}^{'}, u_{1}^{'}, 1, u_{1}, u_{2} \dots$ , and the graph distance from $1$ to $u_{k}$ , or to $u_{k}^{'}$ , equals $ℓ_{s, t}$ of that element, namely $k$ . On the affine line ${f \in P^{*} : f (e_{s} + e_{t}) = 1}$ , parametrized by $y_{s}$ , the generators act by $s : y_{s} \mapsto - y_{s}$ and $t : y_{s} \mapsto 2 - y_{s}$ by [F3] (here $c = 1$ ); consequently the affine map of $u_{k}$ is $y_{s} \mapsto y_{s} - 2 j$ for $k = 2 j$ and $y_{s} \mapsto - y_{s} - 2 j$ for $k = 2 j + 1$ , and that of $u_{k}^{'}$ is $y_{s} \mapsto y_{s} + 2 j$ for $k = 2 j$ and $y_{s} \mapsto - y_{s} + 2 j + 2$ for $k = 2 j + 1$ , by induction on $k$ (each step composes with $s$ or $t$ alternately). Hence the trace of $u_{k} C_{P}^{\circ}$ is the interval $(- k, - k + 1)$ and that of $u_{k}^{'} C_{P}^{\circ}$ is $(k, k + 1)$ . Since these intervals are pairwise distinct, an element of $W_{s, t} = X$ is determined by the trace of its chamber. Writing $v_{j}$ $(j \in Z)$ for the element with trace $(j, j + 1)$ , namely $v_{j} = u_{- j}$ for $j \leq 0$ and $v_{j} = u_{j}^{'}$ for $j \geq 0$ , one therefore has $ℓ_{s, t} (v_{j}) = ∣ j ∣$ together with $s \cdot v_{j} = v_{- j - 1}$ and $t \cdot v_{j} = v_{1 - j}$ , because the affine maps send the interval $(j, j + 1)$ to $(- j - 1, - j)$ , respectively $(1 - j, 2 - j)$ .

1.6F3algebra

Signs in the infinite case. In the situation of 1.5 the sign of $y_{s}$ on $v_{j} C_{P}^{\circ}$ is the sign of $y_{s}$ on the interval $(j, j + 1)$ , hence negative exactly when $j \leq - 1$ , and the sign of $y_{t} = 1 - y_{s}$ is negative exactly when $j \geq 1$ .

1.7F3given

(P_0) and (Q_0). For $w = 1$ one has $ℓ (w) = 0$ , $w C^{\circ} = C^{\circ} \subseteq B_{r}$ for every $r \in S$ by 1.1, and choosing $u = 1$ gives $C^{\circ} \subseteq B_{s} \cap B_{t}$ and $ℓ (1) = 0 = ℓ_{s, t} (1) + ℓ (1)$ for all $s \neq t$ .

2.1F3F4step 1.3step 1.4algebra

Signs of the finite sectors. In the finite case let $C_{k} : = u_{k} C_{P}^{\circ}$ $(0 \leq k < 2 m)$ ; by [F4] these are the interiors of all $2 m$ sectors. Consecutive $u_{k}$ differ by right multiplication by a generator $r$ , so $u_{k} C_{P}$ and $u_{k} r C_{P}$ share the image under $u_{k}$ of the wall of $C_{P}$ fixed by $r$ and lie on its opposite sides, by [F3]. They are adjacent sectors. The distinct vertices of the Cayley cycle in 1.4 therefore list the sectors in cyclic order. The sign of $y_{s}$ is constant on each $C_{k}$ (no $C_{k}$ meets the line $y_{s} = 0$ by 1.3), is $+$ on $C_{0}$ , and is $-$ on $C_{1} = s C_{P}^{\circ}$ because $s$ negates $y_{s}$ by [F3]. It changes exactly at those transitions whose common boundary ray lies on the line $y_{s} = 0$ : that line contains exactly two boundary rays of the arrangement, hence exactly two transitions; and the involution $s$ maps each sector to the sector on the opposite side of $y_{s} = 0$ and fixes no sector, because $s$ swaps the two half-planes and no sector meets the wall. Hence exactly $m$ sectors lie on each side, and the negative side is the contiguous block of $m$ sectors containing $C_{1}$ but not $C_{0}$ , that is ${k : y_{s} < 0 on C_{k}} = {1, \dots, m}$ . Applying the same argument to $t$ , which is $u_{2 m - 1}$ and negates $y_{t}$ , gives ${k : y_{t} < 0 on C_{k}} = {m, \dots, 2 m - 1}$ .

2.2step 1.5step 1.6algebra

Descents in the infinite case. In the situation of 1.5, let $v = v_{j}$ be the element with trace $(j, j + 1)$ ; by 1.5, $ℓ_{s, t} (v) = ∣ j ∣$ and $ℓ_{s, t} (s v) = ∣ j + 1 ∣$ , so $ℓ_{s, t} (s v) = ℓ_{s, t} (v) - 1$ exactly when $j \leq - 1$ , which by 1.6 is exactly the side $y_{s} < 0$ ; and $ℓ_{s, t} (t v) = ∣ j - 1 ∣$ , so $ℓ_{s, t} (t v) = ℓ_{s, t} (v) - 1$ exactly when $j \geq 1$ , exactly the side $y_{t} < 0$ . In particular no element has both left descents, and $ℓ_{s, t} (u_{k}) = k$ for all $k \geq 0$ with the $u_{k}$ pairwise distinct.

2.3step 1.2step 1.3

The dichotomy in the original space. Combining 1.2 with 1.3: for every $v \in W_{s, t}$ and $s^{'} \in {s, t}$ exactly one of $v (B_{s} \cap B_{t}) \subseteq B_{s^{'}}$ and $v (B_{s} \cap B_{t}) \subseteq s^{'} B_{s^{'}}$ holds, according to the side of $H_{e_{s^{'}}} \cap P^{*}$ carrying $v C_{P}^{\circ}$ .

3.1step 1.4step 2.1algebra

Descents in the finite case. Let $m (s, t) < \infty$ and $v = u_{k}$ $(0 \leq k < 2 m)$ . By 1.4, $ℓ_{s, t} (s u_{k})$ equals the distance from $0$ to $σ (k) \equiv 1 - k$ , namely $k - 1$ when $1 \leq k \leq m$ and $2 m + 1 - k$ when $m < k < 2 m$ , and equals $1$ when $k = 0$ ; comparing with $ℓ_{s, t} (u_{k}) = \min (k, 2 m - k)$ gives $ℓ_{s, t} (s u_{k}) = ℓ_{s, t} (u_{k}) - 1$ exactly for $1 \leq k \leq m$ . Likewise $ℓ_{s, t} (t u_{k}) = ℓ_{s, t} (u_{k}) - 1$ exactly for $m \leq k \leq 2 m - 1$ , and $ℓ_{s, t} (t u_{k}) = ℓ_{s, t} (u_{k}) + 1$ for $0 \leq k < m$ . Hence, by 2.1, the left descent for $s$ occurs exactly on the side $y_{s} < 0$ and the left descent for $t$ exactly on the side $y_{t} < 0$ , and $u_{m}$ is the unique element with both descents.

4.1step 1.4step 1.5step 3.1step 2.2step 2.3

Part (1). Let $v \in W_{s, t}$ and $s^{'} \in {s, t}$ . The dichotomy is 2.3, and its two alternatives correspond to the signs of $y_{s^{'}}$ on $v C_{P}^{\circ}$ . If $m (s, t) < \infty$ then $v = u_{k}$ for a unique $k$ by 1.4 and $ℓ_{s, t} (u_{k}) = \min (k, 2 m - k)$ ; the negative side alternative holds exactly for $k \in {1, \dots, m}$ when $s^{'} = s$ and for $k \in {m, \dots, 2 m - 1}$ when $s^{'} = t$ by 2.1, and exactly there $s^{'} v$ shortens by 3.1; the elements $u_{0}, \dots, u_{2 m - 1}$ exhaust $W_{s, t}$ , and the unique double-descent element is $u_{m}$ . If $m (s, t) = \infty$ then every $v$ has a trace $(j, j + 1)$ and $ℓ_{s, t} (u_{k}) = k$ with the $u_{k}$ pairwise distinct by 2.2; the negative side alternative for $s^{'}$ holds exactly for $j \leq - 1$ (respectively $j \geq 1$ ), and exactly there $s^{'} v$ shortens by 2.2, and no element has both descents. This proves all clauses of (1) in both cases.

5.1step 4.1F3F5algebra

The conditional step for $(P)$ . Fix $n \geq 0$ and suppose only $(Q_{n})$ . If $S$ is empty there is no $s$ to check. If $S = {s}$ then $W = {1, s}$ by $s^{2} = 1$ and [F5], and the only positive-length chamber is $s C^{\circ} \subseteq s B_{s}$ , with $ℓ (s) = 1$ , proving $(P_{n + 1})$ directly. Now suppose $∣ S ∣ \geq 2$ . First $(Q_{n})$ and (1) imply $(P_{n})$ : for any $a$ of length $n$ and generator $r$ , choose $q \neq r$ and use $(Q_{n})$ to write $a = u d$ , with $a C^{\circ} \subseteq u (B_{r} \cap B_{q})$ and $n = ℓ_{r, q} (u) + ℓ (d)$ . The rank-two dichotomy gives the positive alternative or the negative alternative with $ℓ_{r, q} (r u) = ℓ_{r, q} (u) - 1$ ; in the latter case $ℓ (r a) \leq ℓ_{r, q} (r u) + ℓ (d) = n - 1$ , hence equality by [F5]. This is $(P_{n})$ . Let $w \in W$ with $ℓ (w) = n + 1$ and let $s \in S$ ; choose $t$ with $w = t w^{'}$ and $ℓ (w^{'}) = n$ . If $s = t$ , then $(P_{n})$ applied to $w^{'}$ gives $w^{'} C^{\circ} \subseteq B_{s}$ : the second alternative would give $ℓ (s w^{'}) = ℓ (w^{'}) - 1 = n - 1$ , whereas $ℓ (s w^{'}) = ℓ (w) = n + 1$ here. Hence $w C^{\circ} = t (w^{'} C^{\circ}) \subseteq t B_{s} = s B_{s}$ and $ℓ (s w) = ℓ (w^{'}) = n = ℓ (w) - 1$ , the second alternative of $(P_{n + 1})$ . If $s \neq t$ , apply $(Q_{n})$ to $w^{'}$ : there is $u \in W_{s, t}$ with $w^{'} C^{\circ} \subseteq u (B_{s} \cap B_{t})$ and $ℓ (w^{'}) = ℓ_{s, t} (u) + ℓ (u^{- 1} w^{'})$ . Put $v : = t u \in W_{s, t}$ . If $v (B_{s} \cap B_{t}) \subseteq B_{s}$ , then $w C^{\circ} \subseteq t u (B_{s} \cap B_{t}) \subseteq B_{s}$ . Otherwise $v (B_{s} \cap B_{t}) \subseteq s B_{s}$ and $ℓ_{s, t} (s v) = ℓ_{s, t} (v) - 1$ by (1) from 4.1, so with $d : = u^{- 1} w^{'}$ one has $w C^{\circ} \subseteq s B_{s}$ and $ℓ (s w) = ℓ (s t u d) \leq ℓ_{s, t} (s t u) + ℓ (d) = ℓ_{s, t} (v) - 1 + ℓ (d) \leq ℓ_{s, t} (u) + ℓ (d) = ℓ (w^{'}) = ℓ (w) - 1$ , where $ℓ_{s, t} (t u) \leq 1 + ℓ_{s, t} (u)$ follows by prepending $t$ to a shortest word for $u$ . The reverse bound $ℓ (s w) \geq ℓ (w) - 1$ follows from [F5], so $ℓ (s w) = ℓ (w) - 1$ , the second alternative of $(P_{n + 1})$ .

6.1step 5.1F5algebra

The conditional step for $(Q)$ . Suppose $(Q_{n})$ and $(P_{m})$ for all $m \leq n + 1$ , as in the second conditional implication of (2), and let $w \in W$ with $ℓ (w) = n + 1$ and $s \neq t$ . By [F5] factor $w = u d$ with $u \in W_{s, t}$ and $d$ the minimal representative of $W_{s, t} w$ ; then $ℓ (w) = ℓ (u) + ℓ (d)$ and $ℓ (s^{'} d) > ℓ (d)$ for $s^{'} \in {s, t}$ . Since $ℓ (d) \leq n + 1$ , $(P_{ℓ (d)})$ is available: it is part of the stated supposition. Applying it to $d$ and $s^{'} \in {s, t}$ , the second alternative would give $ℓ (s^{'} d) = ℓ (d) - 1$ , contradicting $ℓ (s^{'} d) > ℓ (d)$ ; hence $d C^{\circ} \subseteq B_{s} \cap B_{t}$ . Therefore $w C^{\circ} = u (d C^{\circ}) \subseteq u (B_{s} \cap B_{t})$ and $ℓ (w) = ℓ (u) + ℓ (d) = ℓ_{s, t} (u) + ℓ (u^{- 1} w)$ , which is $(Q_{n + 1})$ .

7.1step 1.7step 5.1step 6.1F7

The induction. $(P_{0})$ and $(Q_{0})$ hold by 1.7, and 5.1 and 6.1 show that the validity of $(P_{m})$ and $(Q_{m})$ for all $m \leq n$ implies $(P_{n + 1})$ and $(Q_{n + 1})$ , for every $n \geq 0$ . By the induction principle [F7], $(P_{n})$ and $(Q_{n})$ hold for every $n \geq 0$ .

8.1step 7.1F5algebra

Part (3). Let $w \in W$ and $s \neq t$ , and factor $w = u d$ with $u \in W_{s, t}$ and $d$ the minimal representative of $W_{s, t} w$ . By [F5], $ℓ (w) = ℓ (u) + ℓ (d)$ and $ℓ (s^{'} d) > ℓ (d)$ for $s^{'} \in {s, t}$ ; since $ℓ (d) \leq ℓ (w)$ , $(P_{ℓ (d)})$ from 7.1 applies to $d$ , and the second alternative would give $ℓ (s^{'} d) = ℓ (d) - 1$ , a contradiction; hence $d C^{\circ} \subseteq B_{s} \cap B_{t}$ and $w C^{\circ} = u (d C^{\circ}) \subseteq u (B_{s} \cap B_{t})$ , with $ℓ (w) = ℓ (u) + ℓ (d) = ℓ_{s, t} (u) + ℓ (u^{- 1} w)$ because $u^{- 1} w = d$ and $ℓ_{s, t} (u) = ℓ (u)$ .

8.2step 7.1

Forward direction of (4). Let $w \in W$ and $s \in S$ . By $(P_{ℓ (w)})$ from 7.1, $w C^{\circ} \subseteq B_{s}$ , or $w C^{\circ} \subseteq s B_{s}$ and $ℓ (s w) = ℓ (w) - 1$ ; in particular $w C^{\circ} \subseteq s B_{s}$ implies $ℓ (s w) < ℓ (w)$ .

9.1step 8.2step 7.1step 4.1step 8.1∎

Converse direction of (4) and conclusion. Suppose $ℓ (s w) < ℓ (w)$ and put $w^{'} : = s w$ , so that $ℓ (w^{'}) = ℓ (w) - 1$ and $w = s w^{'}$ . By $(P_{ℓ (w^{'})})$ from 7.1 applied to $w^{'}$ , either $w^{'} C^{\circ} \subseteq B_{s}$ , or $w^{'} C^{\circ} \subseteq s B_{s}$ and $ℓ (s w^{'}) = ℓ (w^{'}) - 1$ ; the second alternative would give $ℓ (w) = ℓ (s w^{'}) = ℓ (w^{'}) - 1 = ℓ (w) - 2$ , which is impossible, so $w^{'} C^{\circ} \subseteq B_{s}$ . Applying the linear bijection $f \mapsto s \cdot f$ of [F3] gives $w C^{\circ} = s (w^{'} C^{\circ}) \subseteq s B_{s}$ . Together with 8.2 this is the equivalence (4); and (1) is 4.1, (2) is 7.1 and (3) is 8.1, so all four clauses of the statement hold.

TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Root sign coherence and the action of simple reflections on positive roots

Statement

Let $S$ be a finite set, $m$ a Coxeter matrix, $W$ the presented group (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), $V = R^{S}$ with Coxeter form $B$ , the reflections $r_{a}$ (The real Coxeter form, its radical, reflections, and form-preserving maps), the canonical reflection homomorphism $ρ : W \to G L (V)$ , the root system $Φ = {ρ (w) e_{s}}$ , the reflections $T$ , and the positive cone $V_{+} = {\sum_{s} λ_{s} e_{s} : λ_{s} \geq 0}$ (The canonical reflection homomorphism, roots, reflections, and the positive cone); every root satisfies $B (α, α) = 1$ (Descent of the reflection representation, unit root norms, and conjugation of reflections (3)). Let $C, C^{\circ}$ be the chamber and its interior of the dual action and put $B_{s} : = {f \in V^{*} : f (e_{s}) > 0}$ (The dual action, chambers, faces, and root hyperplanes).

(1) Sign criterion for the cone. For $v \in V$ : $v \in V_{+} ∖ {0}$ if and only if $f (v) > 0$ for every $f \in C^{\circ}$ .

(2) Roots have a sign. Every root $α \in Φ$ lies in $V_{+} ∖ {0}$ or in $- V_{+} ∖ {0}$ , and not in both. Hence, with $Φ_{+} : = Φ \cap V_{+}, Φ_{-} : = Φ \cap (- V_{+}),$ one has $Φ = Φ_{+} ⊔ Φ_{-}$ , $Φ_{-} = - Φ_{+}$ , $e_{s} \in Φ_{+}$ for every $s \in S$ , and for every $w \in W$ and $s \in S$ $ρ (w) e_{s} \in Φ_{+} ⟺ w^{- 1} C^{\circ} \subseteq B_{s}, ρ (w) e_{s} \in Φ_{-} ⟺ w^{- 1} C^{\circ} \subseteq s B_{s} .$ In particular $f (α) > 0$ for all $f \in C^{\circ}$ when $α \in Φ_{+}$ , and $f (α) < 0$ for all $f \in C^{\circ}$ when $α \in Φ_{-}$ .

(3) Simple reflections act on positive roots. For every $s \in S$ , $r_{s} (Φ_{+} ∖ {e_{s}}) = Φ_{+} ∖ {e_{s}}, r_{s} e_{s} = - e_{s}, r_{s} Φ = Φ;$ equivalently $r_{s} Φ_{+} = (Φ_{+} ∖ {e_{s}}) \cup {- e_{s}}$ .

Facts & Assumptions

Given: a finite set $S$ , a Coxeter matrix $m$ , the presented group $W$ with its universal property, the space $V = R^{S}$ with its canonical basis $(e_{s})_{s \in S}$ , the Coxeter form $B$ , the canonical reflection homomorphism $ρ$ with root system $Φ$ and reflection set $T$ , the positive cone $V_{+}$ and the negative cone $- V_{+}$ , and the dual action on $V^{*}$ with chamber $C$ , interior $C^{\circ}$ and root hyperplanes $H_{α}$ .

[F1]

For every $w \in W$ and $s \in S$ , exactly one of $w C^{\circ} \subseteq B_{s}$ and $w C^{\circ} \subseteq s B_{s}$ holds, and in the second case $ℓ (s w) = ℓ (w) - 1$ ; equivalently, $w C^{\circ} \subseteq s B_{s}$ if and only if $ℓ (s w) < ℓ (w)$ (The rank-two half-space alternative and the chamber-length induction $(P_{n})$ , $(Q_{n})$ ).

[F2]

The reflections $r_{a}$ are defined for $B (a, a) \neq 0$ by $r_{a} (v) = v - \frac{2 B (v, a)}{B (a, a)} a$ ; one has $B (e_{s}, e_{s}) = 1$ and $r_{s} (e_{s}) = - e_{s}$ , and the assignment $s \mapsto r_{s}$ induces the homomorphism $ρ : W \to G L (V)$ (The real Coxeter form, its radical, reflections, and form-preserving maps, Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order).

[F3]

The root system is $Φ = {ρ (w) e_{s} : w \in W, s \in S}$ , every root satisfies $B (α, α) = 1$ , the set $Φ$ is invariant under every $ρ (w)$ , and $ρ (w) e_{s} = - ρ (w s) e_{s}$ for all $w \in W$ , $s \in S$ (The canonical reflection homomorphism, roots, reflections, and the positive cone, Descent of the reflection representation, unit root norms, and conjugation of reflections).

[F4]

The dual action is given by $(w \cdot f) (v) = f (ρ (w)^{- 1} v)$ ; the closed chamber is $C = {f \in V^{*} : f (e_{s}) \geq 0 for all s}$ , its interior is $C^{\circ} = {f \in V^{*} : f (e_{s}) > 0 for all s}$ and is nonempty, $B_{s} = {f \in V^{*} : f (e_{s}) > 0}$ and $s B_{s} = {f \in V^{*} : f (e_{s}) < 0}$ are disjoint open half-spaces, and the dual basis functionals $f_{s} \in V^{*}$ satisfy $f_{s} (e_{t}) = δ_{s t}$ and $f (e_{s}) \geq 0$ for every $f \in C$ (The dual action, chambers, faces, and root hyperplanes, The dual action, the faces, and the rank-two chamber tiling).

[F5]

On the finite-dimensional real vector space $V$ with basis $(e_{s})$ : every $v \in V$ has unique coordinates $v = \sum_{s} v (s) e_{s}$ with $v (s) \in R$ , evaluation $f \mapsto f (v)$ is linear in $v$ for fixed $f \in V^{*}$ , and sums and nonnegative multiples of elements of $V_{+}$ lie in $V_{+}$ (Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis, Linear combination of a finite list, and the span $span (S)$ as the smallest linear subspace containing $S$ , Linear functionals and the algebraic dual $V^{*} = L (V, F)$ ).

Proof

technique · direct

1.1F2F3F4given

Set-up. Every $f \in C^{\circ}$ satisfies $f (e_{s}) > 0$ for all $s \in S$ , so that $C^{\circ} \subseteq B_{s}$ and $C^{\circ} \subseteq B_{s} \cap B_{t}$ for all $s \neq t$ ; moreover $C^{\circ} \neq \emptyset$ . Every root is of the form $ρ (w) e_{s}$ with $B (ρ (w) e_{s}, ρ (w) e_{s}) = 1$ , hence nonzero, and $- ρ (w) e_{s} = ρ (w s) e_{s}$ is again a root, so $Φ = - Φ$ .

1.2F4F5given

The sign criterion, forward direction. Let $v = \sum_{s} λ_{s} e_{s} \in V_{+} ∖ {0}$ , so all $λ_{s} \geq 0$ and some $λ_{r} > 0$ . For $f \in C^{\circ}$ linearity gives $f (v) = \sum_{s} λ_{s} f (e_{s}) \geq λ_{r} f (e_{r}) > 0$ , since all summands are nonnegative and the summand at $r$ is positive.

1.3F4F5algebra

The sign criterion, converse direction. Let $v \notin V_{+} ∖ {0}$ . If $v = 0$ then $f (v) = 0$ for every $f$ , so assume $v \notin V_{+}$ ; then some coordinate $λ_{r} = v (r)$ is negative. Let $g : = f_{r} + ε \sum_{s} f_{s}$ with $ε > 0$ so small that $ε ∣ \sum_{s} v (s) ∣ < ∣ λ_{r} ∣$ . Then $g (e_{t}) = δ_{r t} + ε > 0$ for every $t \in S$ , so $g \in C^{\circ}$ , and $g (v) = λ_{r} + ε \sum_{s} v (s) < 0$ . Hence some element of $C^{\circ}$ evaluates $v$ negatively, and the criterion of (1) holds in both directions.

2.1step 1.2step 1.3

The equivalence of the two criteria. By 1.2 and 1.3, for every $v \in V$ : $v \in V_{+} ∖ {0}$ if and only if $f (v) > 0$ for every $f \in C^{\circ}$ .

3.1step 2.1F1F3algebra

Roots have a sign. Fix $w \in W$ and $s \in S$ . For $f \in V^{*}$ one has $(w^{- 1} \cdot f) (e_{s}) = f (ρ (w^{- 1})^{- 1} e_{s}) = f (ρ (w) e_{s})$ ; by [F1] applied to the element $w^{- 1}$ either $w^{- 1} C^{\circ} \subseteq B_{s}$ or $w^{- 1} C^{\circ} \subseteq s B_{s}$ . In the first case $f (ρ (w) e_{s}) > 0$ for every $f \in C^{\circ}$ , so $ρ (w) e_{s} \in V_{+} ∖ {0}$ by 2.1 and hence in $Φ_{+}$ ; in the second case $f (ρ (w) e_{s}) < 0$ for every $f \in C^{\circ}$ , so $- f (ρ (w) e_{s}) = f (- ρ (w) e_{s}) > 0$ for every $f \in C^{\circ}$ , whence $- ρ (w) e_{s} \in V_{+} ∖ {0}$ by 2.1 and $ρ (w) e_{s} \in - V_{+} ∖ {0}$ , so it lies in $Φ_{-}$ . The two alternatives are exclusive because $V_{+} \cap (- V_{+}) = {0}$ while $ρ (w) e_{s} \neq 0$ by 1.1. Since every root is some $ρ (w) e_{s}$ and $- ρ (w) e_{s} = ρ (w s) e_{s}$ is a root, this gives $Φ = Φ_{+} ⊔ Φ_{-}$ and $Φ_{-} = - Φ_{+}$ ; taking $w = 1$ gives $e_{s} = ρ (1) e_{s} \in V_{+} ∖ {0}$ and hence $e_{s} \in Φ_{+}$ . Reading the two cases displayed above as equivalences with 2.1 gives $ρ (w) e_{s} \in Φ_{+} ⟺ w^{- 1} C^{\circ} \subseteq B_{s}$ and $ρ (w) e_{s} \in Φ_{-} ⟺ w^{- 1} C^{\circ} \subseteq s B_{s}$ , and with 2.1 the last sentence of (2) follows.

4.1step 3.1F2F3algebra∎

Simple reflections act on positive roots. Fix $s \in S$ . That $r_{s} e_{s} = - e_{s}$ is [F2]. Let $α \in Φ_{+} ∖ {e_{s}}$ . Since $r_{s} = ρ (s)$ and $Φ$ is $ρ (W)$ -invariant by [F3], $r_{s} α \in Φ$ , so by 3.1 either $r_{s} α \in Φ_{+}$ or $r_{s} α \in Φ_{-}$ . Suppose $r_{s} α \in Φ_{-}$ , that is $r_{s} α \in - V_{+} ∖ {0}$ , and write $r_{s} α = α - 2 B (α, e_{s}) e_{s}$ by [F2] with $B (e_{s}, e_{s}) = 1$ . In coordinates: $(r_{s} α) (r) = α (r)$ for $r \neq s$ and $(r_{s} α) (s) = α (s) - 2 B (α, e_{s})$ . Since $α \in V_{+} ∖ {0}$ all coordinates $α (r)$ are nonnegative and not all vanish, and since $r_{s} α \in - V_{+} ∖ {0}$ all its coordinates are nonpositive; for $r \neq s$ both statements apply to $α (r)$ , so $α (r) = 0$ for all $r \neq s$ , that is $α = α (s) e_{s}$ with $α (s) > 0$ . Then $1 = B (α, α) = α (s)^{2} B (e_{s}, e_{s}) = α (s)^{2}$ , so $α (s) = 1$ and $α = e_{s}$ , contradicting the choice of $α$ . Hence $r_{s} α \in Φ_{+}$ ; moreover $r_{s} α \neq e_{s}$ , because $r_{s} e_{s} = - e_{s}$ and $r_{s}$ is an involution, so $r_{s} α = e_{s}$ would give $α = - e_{s} \notin Φ_{+}$ . Thus $r_{s} (Φ_{+} ∖ {e_{s}}) \subseteq Φ_{+} ∖ {e_{s}}$ , and applying the involution $r_{s}$ once more gives equality. Finally $r_{s} Φ = Φ$ : indeed $r_{s} = ρ (s)$ maps $Φ$ into $Φ$ by [F3], and $r_{s}^{2} = i d$ by [F2], so this map is a bijection of $Φ$ ; consequently $r_{s} Φ_{+} = (Φ_{+} ∖ {e_{s}}) \cup {- e_{s}}$ . This is (3), while (1) is 2.1 and (2) is 3.1.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-10-08Open item page →

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, $ρ (w) e_{s} \in Φ_{-}$ .

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

The geometric inversion set $N (w)$ of an element of a Coxeter group

Definition

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 $ρ : W \to G L (V)$ (The canonical reflection homomorphism, roots, reflections, and the positive cone), and let $Φ = Φ_{+} ⊔ Φ_{-}$ be the signed root system (Root sign coherence and the action of simple reflections on positive roots).

(1) Definition. For $w \in W$ the inversion set of $w$ is $N (w) : = {α \in Φ_{+} : ρ (w) α \in Φ_{-}} = Φ_{+} \cap ρ (w)^{- 1} Φ_{-} .$ Its elements are the roots inverted by $w$ . This is a subset of $Φ_{+}$ , defined before any finiteness or independence assertion; by construction $ρ (w) N (w) \subseteq Φ_{-}$ , and $N (w)$ is determined by the linear map $ρ (w)$ .

(2) Elementary identities. $N (1) = \emptyset$ ; for every $w \in W$ $N (w^{- 1}) = - ρ (w) N (w),$ and consequently $∣ N (w^{- 1}) ∣ = ∣ N (w) ∣$ whenever one of the two sets is finite. For every $u \in W$ and $s \in S$ , with $s N (u) : = {ρ (s) β : β \in N (u)}$ , $ℓ (u s) > ℓ (u) ⟹ N (u s) = {e_{s}} ⊔ s N (u), ℓ (u s) < ℓ (u) ⟹ N (u s) = s (N (u) ∖ {e_{s}});$ in the first case $s N (u) \subseteq Φ_{+} ∖ {e_{s}}$ and $e_{s} \notin N (u)$ , so the union is disjoint.

The identities are elementary: $N (1) = \emptyset$ because $ρ (1) = {i d}_{V}$ would force $α \in Φ_{+} \cap Φ_{-}$ for $α \in N (1)$ , and $Φ_{+}, Φ_{-}$ are disjoint. For the second identity let $α \in Φ_{+}$ : then $α \in N (w^{- 1})$ means $ρ (w)^{- 1} α \in Φ_{-}$ , and with $β : = - ρ (w)^{- 1} α$ one has $β \in Φ_{+}$ and $ρ (w) β = - α \in Φ_{-}$ , so $β \in N (w)$ and $α = - ρ (w) β$ ; conversely $α = - ρ (w) β$ for some $β \in N (w)$ gives $α \in Φ_{+}$ because $Φ_{-} = - Φ_{+}$ , and $ρ (w)^{- 1} α = - β \in Φ_{-}$ , so $α \in N (w^{- 1})$ . Since $β \mapsto - ρ (w) β$ is a bijection, the cardinality statement follows. For the recursion, let $β \in Φ_{+}$ ; then $ρ (u s) β = ρ (u) ρ (s) β$ , and $ρ (s)$ maps $Φ_{+} ∖ {e_{s}}$ bijectively onto itself while $ρ (s) e_{s} = - e_{s}$ (Root sign coherence and the action of simple reflections on positive roots (3)), so $β \mapsto ρ (s) β$ is a bijection of $Φ_{+} ∖ {e_{s}}$ that carries $N (u s) ∖ {e_{s}}$ onto $N (u) ∖ {e_{s}}$ . By the root-length criterion (The root-length criterion and faithfulness of the canonical reflection representation (1)) one has $e_{s} \in N (u) ⟺ ρ (u) e_{s} \in Φ_{-} ⟺ ℓ (u s) < ℓ (u)$ and $e_{s} \in N (u s) ⟺ ρ (u) ρ (s) e_{s} = - ρ (u) e_{s} \in Φ_{-} ⟺ ρ (u) e_{s} \in Φ_{+} ⟺ ℓ (u s) > ℓ (u)$ . If $ℓ (u s) > ℓ (u)$ this gives $e_{s} \in N (u s)$ , $e_{s} \notin N (u)$ and hence $N (u s) = {e_{s}} ⊔ s N (u)$ with $s N (u) \subseteq Φ_{+} ∖ {e_{s}}$ ; if $ℓ (u s) < ℓ (u)$ it gives $e_{s} \notin N (u s)$ , $e_{s} \in N (u)$ and hence $N (u s) = s (N (u) ∖ {e_{s}})$ .

(3) Convention on reduced words. We keep the left action $ρ$ of $W$ on $V$ . If $w = s_{1} \dots s_{n}$ is a reduced expression, then the suffix roots $ρ (s_{i + 1} \dots s_{n})^{- 1} e_{s_{i}}$ $(1 \leq i \leq n)$ are elements of $N (w)$ , and the prefix roots $ρ (s_{1} \dots s_{i - 1}) e_{s_{i}}$ are elements of $N (w^{- 1})$ . That these lists are exactly $N (w)$ and $N (w^{- 1})$ , that their elements are pairwise distinct positive roots, and that in particular $∣ N (w) ∣ = ℓ (w)$ , is proved in The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange ↗; no finiteness or independence beyond the identities of (2) is asserted here.

For the two membership statements put $w_{i - 1} : = s_{1} \dots s_{i - 1}$ and $α_{i} : = ρ (s_{i + 1} \dots s_{n})^{- 1} e_{s_{i}}$ . Since $ℓ (w_{i - 1} s_{i}) = i > ℓ (w_{i - 1}) = i - 1$ by reducedness of the expression, the root-length criterion gives $γ_{i} : = ρ (w_{i - 1}) e_{s_{i}} \in Φ_{+}$ and also, applied to the reversed reduced expression of $w^{- 1} = s_{n} \dots s_{1}$ , gives $α_{i} \in Φ_{+}$ . Moreover $(s_{1} \dots s_{n}) (s_{i + 1} \dots s_{n})^{- 1} = s_{1} \dots s_{i}$ , so $ρ (w) α_{i} = ρ (s_{1} \dots s_{i}) e_{s_{i}} = - ρ (w_{i - 1}) e_{s_{i}} = - γ_{i} \in Φ_{-}$ , which shows $α_{i} \in N (w)$ ; and $(s_{n} \dots s_{1}) (s_{1} \dots s_{i - 1}) = s_{n} \dots s_{i}$ , so $ρ (w^{- 1}) γ_{i} = ρ (s_{n} \dots s_{i}) e_{s_{i}} = - ρ (s_{n} \dots s_{i + 1}) e_{s_{i}} = - α_{i} \in Φ_{-}$ , which shows $γ_{i} \in N (w^{- 1})$ .

TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange

Statement

Let $S$ be a finite set, $m$ a Coxeter matrix, $W$ the presented group with length $ℓ$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), $V = R^{S}$ with Coxeter form $B$ , canonical reflection homomorphism $ρ$ , root system $Φ = Φ_{+} ⊔ Φ_{-}$ and reflection set $T = {w s w^{- 1} : w \in W, s \in S}$ (The canonical reflection homomorphism, roots, reflections, and the positive cone, Root sign coherence and the action of simple reflections on positive roots); every root has $B$ -norm one.

(1) The root-reflection dictionary. For $α \in Φ$ choose $w \in W$ , $s \in S$ with $α = ρ (w) e_{s}$ and put $t_{α} : = w s w^{- 1} \in T$ . Then: (i) $t_{α}$ is independent of the chosen representation of $α$ ; (ii) $ρ (t_{α}) = r_{α}$ , the reflection with normal $α$ ; and $t_{ρ (w) α} = w t_{α} w^{- 1}$ for all $w \in W$ , $α \in Φ$ ; (iii) $t_{- α} = t_{α}$ , and for $α, β \in Φ$ one has $t_{α} = t_{β}$ if and only if $α = \pm β$ ; (iv) the induced map ${\pm α : α \in Φ} \to T$ is a bijection; so is the map $Φ_{+} \to T$ , $α \mapsto t_{α}$ .

(2) Inversion formula. For every $w \in W$ one has $∣ N (w) ∣ = ℓ (w)$ (with $N$ as in The geometric inversion set $N (w)$ of an element of a Coxeter group); and for every reduced expression $w = s_{1} \dots s_{n}$ , $N (w) = {ρ (s_{i + 1} \dots s_{n})^{- 1} e_{s_{i}} : 1 \leq i \leq n}, N (w^{- 1}) = {ρ (s_{1} \dots s_{i - 1}) e_{s_{i}} : 1 \leq i \leq n},$ the displayed elements being pairwise distinct positive roots.

(3) Strong exchange. Let $w \in W$ and $t \in T$ satisfy $ℓ (t w) < ℓ (w)$ , and let $w = s_{1} \dots s_{n}$ be a reduced expression. Then there is a unique $i \in {1, \dots, n}$ with $t w = s_{1} \dots \hat{s_{i}} \dots s_{n}, t = r_{i} : = s_{1} \dots s_{i - 1} s_{i} s_{i - 1} \dots s_{1};$ moreover, if $α \in Φ_{+}$ is the positive root with $t = t_{α}$ , then $α \in N (w^{- 1})$ and $α = ρ (s_{1} \dots s_{i - 1}) e_{s_{i}}$ .

Facts & Assumptions

Given: a finite set $S$ , a Coxeter matrix $m$ , the presented group $W$ with length $ℓ$ , the space $V = R^{S}$ with Coxeter form $B$ , the canonical reflection homomorphism $ρ$ with root system $Φ = Φ_{+} ⊔ Φ_{-}$ , the reflections $r_{a}$ , and the reflection set $T = {w s w^{- 1} : w \in W, s \in S}$ .

[F1]

For all $w \in W$ and $s \in S$ one has $ρ (w s w^{- 1}) = r_{ρ (w) e_{s}}$ , and every root $α$ satisfies $B (α, α) = 1$ (Descent of the reflection representation, unit root norms, and conjugation of reflections).

[F2]

The canonical reflection homomorphism $ρ$ is injective (The root-length criterion and faithfulness of the canonical reflection representation).

[F3]

For $a \in V$ with $B (a, a) \neq 0$ the reflection $r_{a}$ satisfies $r_{a} (a) = - a$ , fixes every $v$ with $B (v, a) = 0$ pointwise, preserves $B$ , and $\ker B (-, a)$ has dimension $\dim V - 1$ ; since $B (a, a) \neq 0$ one has $R a \cap \ker B (-, a) = {0}$ , so $V = R a \oplus \ker B (-, a)$ and the $(- 1)$ -eigenspace of $r_{a}$ is exactly $R a$ (Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order).

[F4]

The reflection set carries the right action $(ε, r) \cdot s = U_{s} (ε, r) = (ε \cdot (- 1)^{δ (s, r)}, s r s)$ of $W$ on ${\pm 1} \times T$ , and $(ε, r) \cdot w = (ε η (r, w), w^{- 1} r w)$ for a well-defined sign $η (r, w) \in {\pm 1}$ depending only on $w$ and $r$ ; for a reduced word with prefix reflections $r_{i}$ one has $n (r) \in {0, 1}$ , the map $i \mapsto r_{i}$ is injective, and $Φ (w) : = {r_{1}, \dots, r_{k}} = {r \in T : η (r, w) = - 1}$ is independent of the reduced expression and has cardinality $ℓ (w)$ (The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness).

[F5]

The inversion set is $N (w) = {α \in Φ_{+} : ρ (w) α \in Φ_{-}}$ ; it satisfies $N (1) = \emptyset$ , $N (w^{- 1}) = - ρ (w) N (w)$ , and the step recursion: for $u \in W$ , $s \in S$ with $ℓ (u s) > ℓ (u)$ one has $N (u s) = {e_{s}} ⊔ s N (u)$ with $s N (u) \subseteq Φ_{+} ∖ {e_{s}}$ and $e_{s} \notin N (u)$ , while for $ℓ (u s) < ℓ (u)$ one has $N (u s) = s (N (u) ∖ {e_{s}})$ (The geometric inversion set $N (w)$ of an element of a Coxeter group).

[F6]

$ρ$ is a group homomorphism with $ρ (1) = {i d}_{V}$ and $ρ (u v) = ρ (u) ρ (v)$ ; $ρ (s) e_{s} = - e_{s}$ ; and $Φ = Φ_{+} ⊔ Φ_{-}$ with $Φ_{-} = - Φ_{+}$ and $e_{s} \in Φ_{+}$ (Monoid homomorphism and group homomorphism, Root sign coherence and the action of simple reflections on positive roots).

[F7]

Root-length criterion: for all $x \in W$ and $s \in S$ one has $ℓ (x s) > ℓ (x)$ if and only if $ρ (x) e_{s} \in Φ_{+}$ (The root-length criterion and faithfulness of the canonical reflection representation).

[F8]

Induction principle on the natural numbers (The principle of mathematical induction).

Proof

technique · direct

1.1F1F4F6given

Set-up. Fix a reduced expression $w = s_{1} \dots s_{n}$ of an element $w \in W$ , write $w_{j} : = s_{1} \dots s_{j}$ and $r_{i} : = w_{i - 1} s_{i} w_{i - 1}^{- 1} \in T$ for the prefix reflections, and note $ρ (w_{j}) = ρ (s_{1}) \dots ρ (s_{j})$ by [F6] and $ρ (r_{i}) = ρ (w_{i - 1} s_{i} w_{i - 1}^{- 1}) = r_{ρ (w_{i - 1}) e_{s_{i}}}$ .

1.2F5F6F8given

The inversion formula (2). We prove by induction on $m$ the assertion: for every $x$ with $ℓ (x) = m$ and every reduced expression $x = s_{1} \dots s_{m}$ one has $N (x) = {ρ (s_{i + 1} \dots s_{m})^{- 1} e_{s_{i}} : 1 \leq i \leq m}$ with pairwise distinct elements, and $∣ N (x) ∣ = m$ . For $m = 0$ this is $N (1) = \emptyset$ by [F5]. For the step let $x = u s_{m}$ with $u : = s_{1} \dots s_{m - 1}$ , so $ℓ (u) = m - 1$ and $ℓ (u s_{m}) = m > ℓ (u)$ ; the first case of the step recursion of [F5] gives $N (x) = {e_{s_{m}}} ⊔ s_{m} N (u)$ , the union being disjoint with $s_{m} N (u) \subseteq Φ_{+} ∖ {e_{s_{m}}}$ . By induction $N (u) = {ρ (s_{i + 1} \dots s_{m - 1})^{- 1} e_{s_{i}} : 1 \leq i \leq m - 1}$ with pairwise distinct elements; since $ρ (s_{m}) ρ (s_{i + 1} \dots s_{m - 1})^{- 1} = ρ (s_{m} s_{m - 1} \dots s_{i + 1}) = ρ (s_{i + 1} \dots s_{m})^{- 1}$ for $i \leq m - 1$ by [F6], one has $s_{m} N (u) = {ρ (s_{i + 1} \dots s_{m})^{- 1} e_{s_{i}} : 1 \leq i \leq m - 1}$ , and the term $i = m$ with the empty product contributes $e_{s_{m}}$ . Hence $N (x) = {ρ (s_{i + 1} \dots s_{m})^{- 1} e_{s_{i}} : 1 \leq i \leq m}$ with pairwise distinct elements, and $∣ N (x) ∣ = 1 + ∣ N (u) ∣ = m$ . Applying the same formula to the reversed reduced expression $x^{- 1} = s_{m} \dots s_{1}$ and using $ρ (s_{i - 1} \dots s_{1})^{- 1} = ρ (s_{1} \dots s_{i - 1})$ gives $N (x^{- 1}) = {ρ (s_{1} \dots s_{i - 1}) e_{s_{i}} : 1 \leq i \leq m}$ , again with pairwise distinct elements. The induction principle [F8] gives (2) for every element.

1.3F1F2F6algebra

The dictionary (i) and (ii). Let $α \in Φ$ , written as $α = ρ (w) e_{s} = ρ (w^{'}) e_{s^{'}}$ . Then $ρ (w s w^{- 1}) = r_{ρ (w) e_{s}} = r_{α} = r_{ρ (w^{'}) e_{s^{'}}} = ρ (w^{'} s^{'} w^{' - 1})$ by [F1], and injectivity of $ρ$ from [F2] gives $w s w^{- 1} = w^{'} s^{'} w^{' - 1}$ ; so $t_{α}$ is independent of the representation, which is (i). For (ii), $ρ (t_{α}) = ρ (w s w^{- 1}) = r_{ρ (w) e_{s}} = r_{α}$ ; and writing $α = ρ (x) e_{s}$ one has $ρ (w) α = ρ (w x) e_{s}$ , so $t_{ρ (w) α} = (w x) s (w x)^{- 1} = w (x s x^{- 1}) w^{- 1} = w t_{α} w^{- 1}$ .

1.4F4algebra

The shorteners are exactly the prefix reflections. Keep the reduced expression of 1.1 and let $r \in T$ . If $r = r_{i}$ is a prefix reflection of the word, then $r_{i} w = w_{i - 1} s_{i} w_{i - 1}^{- 1} w_{i - 1} s_{i} s_{i + 1} \dots s_{n} = w_{i - 1} s_{i + 1} \dots s_{n} = s_{1} \dots \hat{s_{i}} \dots s_{n}$ , so $ℓ (r_{i} w) < n = ℓ (w)$ ; and by [F4] the set ${r_{1}, \dots, r_{n}}$ equals $Φ (w) = {r : η (r, w) = - 1}$ and its members are pairwise distinct. Conversely let $r \in T$ with $ℓ (r w) < ℓ (w)$ and suppose $η (r, w) = + 1$ . The formula of [F4] implies the cocycle identity $η (x, u v) = η (x, u) η (u^{- 1} x u, v)$ for all $x \in T$ , $u, v \in W$ : indeed $(ε, x) \cdot (u v) = ((ε, x) \cdot u) \cdot v$ , and comparing first coordinates of the displayed formula gives the identity. With $x = r$ , $u = r$ , $v = w$ this gives $η (r, r w) = η (r, r) η (r, w) = η (r, r)$ ; also $η (r, 1) = 1$ , since the action of $1$ is the identity. We claim $η (r, r) = - 1$ . Write $r = u s u^{- 1}$ with $s \in S$ , $u \in W$ , so that $u^{- 1} r u = s$ ; applying the action formula of [F4] successively along the word $u s u^{- 1}$ to $(ε, r)$ gives $(ε, r) \cdot u = (ε η (r, u), u^{- 1} r u) = (ε η (r, u), s)$ , then $U_{s} (ε η (r, u), s) = (- ε η (r, u), s)$ by the definition of $U_{s}$ in [F4], and then $(ε, r) \cdot (u s u^{- 1}) = (- ε η (r, u) η (s, u^{- 1}), r)$ . Comparing with $(ε, r) \cdot r = (ε η (r, r), r)$ from [F4] yields $η (r, r) = - η (r, u) η (s, u^{- 1})$ . Applying the cocycle identity with $x = s$ , $u = u^{- 1}$ , $v = u$ gives $η (s, 1) = η (s, u^{- 1}) η (u s u^{- 1}, u)$ , that is $1 = η (s, u^{- 1}) η (r, u)$ ; hence $η (r, r) = - η (r, u)^{2} = - 1$ under the supposition $η (r, w) = + 1$ . Therefore $η (r, r w) = - 1$ , so $r \in Φ (r w) = {x \in T : η (x, r w) = - 1}$ ; by [F4] applied to the shorter element $r w$ , the element $r$ is one of the prefix reflections of a reduced expression of $r w$ , and the first paragraph of this step applied to that element gives $ℓ (r \cdot r w) < ℓ (r w)$ , that is $ℓ (w) < ℓ (r w)$ , contradicting $ℓ (r w) < ℓ (w)$ . Hence $η (r, w) = - 1$ and $r \in Φ (w) = {r_{1}, \dots, r_{n}}$ . Thus ${r \in T : ℓ (r w) < ℓ (w)} = {r_{1}, \dots, r_{n}}$ .

2.1F1F3F6step 1.1step 1.3algebra

The dictionary (iii) and (iv). Let $α, β \in Φ$ . Since $ρ (s) e_{s} = - e_{s}$ by [F6], the root $- α$ has the representation $- α = - ρ (w) e_{s} = ρ (w s) e_{s}$ , so $t_{- α} = (w s) s (w s)^{- 1} = w s w^{- 1} = t_{α}$ . If $t_{α} = t_{β}$ , then $r_{α} = ρ (t_{α}) = ρ (t_{β}) = r_{β}$ by 1.3 (ii); by [F3] and $B (α, α) = 1$ of [F1] the $(- 1)$ -eigenspace of $r_{α}$ is $R α$ , and that of $r_{β}$ is $R β$ , so $R α = R β$ and $β = λ α$ with $λ \neq 0$ ; then $1 = B (β, β) = λ^{2} B (α, α) = λ^{2}$ , so $λ = \pm 1$ . Conversely $t_{α} = t_{\pm α}$ by the first part. Hence $t_{α} = t_{β}$ if and only if $β = \pm α$ , which is (iii). For (iv), the map $[α] \mapsto t_{α}$ on classes ${\pm α}$ is well defined by (i) and (iii) and injective by (iii); it is surjective because every element of $T$ is of the form $w s w^{- 1} = t_{ρ (w) e_{s}}$ by 1.1. Hence it is a bijection. Since $Φ = Φ_{+} ⊔ Φ_{-}$ with $Φ_{-} = - Φ_{+}$ by [F6], every class ${α, - α}$ contains exactly one positive root; composing $α \mapsto [α]$ with the class bijection $[α] \mapsto t_{α}$ gives a bijection $Φ_{+} \to T$ , $α \mapsto t_{α}$ .

3.1step 1.1step 1.2step 1.4step 2.1F7algebra∎

Strong exchange (3). Let $w = s_{1} \dots s_{n}$ be a reduced expression and $t \in T$ with $ℓ (t w) < ℓ (w)$ . By 1.4 the shorteners of $w$ are exactly the prefix reflections $r_{1}, \dots, r_{n}$ , which are pairwise distinct; hence $t = r_{i}$ for a unique $i \in {1, \dots, n}$ , and $t w = r_{i} w = s_{1} \dots \hat{s_{i}} \dots s_{n}$ by the computation in 1.4. For the root clause let $α \in Φ_{+}$ satisfy $t = t_{α}$ . By 1.1, $r_{i} = w_{i - 1} s_{i} w_{i - 1}^{- 1} = t_{ρ (w_{i - 1}) e_{s_{i}}}$ ; since $ℓ (w_{i - 1} s_{i}) = i > ℓ (w_{i - 1})$ , the criterion [F7] gives $ρ (w_{i - 1}) e_{s_{i}} \in Φ_{+}$ . Both $α$ and $ρ (w_{i - 1}) e_{s_{i}}$ are positive roots with the same $t$ -image $t$ , so (iii) from 2.1 gives $α = ρ (w_{i - 1}) e_{s_{i}} = ρ (s_{1} \dots s_{i - 1}) e_{s_{i}}$ . Finally, the prefix formula of (2) from 1.2 shows that this root lies in $N (w^{- 1})$ . This proves (3), while (1) is 1.3 with 2.1 and (2) is 1.2.

5 · Examples, counterexamples and false statements

None yet.

Canonical Roots, Signs, and Faithful Reflections

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

The rank-two half-space alternative and the chamber-length induction (Pn)(Pn​), (Qn)(Qn​)

Statement

Facts & Assumptions

Proof

Root sign coherence and the action of simple reflections on positive roots

Statement

Facts & Assumptions

Proof

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

Statement

Facts & Assumptions

Proof

The geometric inversion set N(w)N(w) of an element of a Coxeter group

Definition

The inversion formula ∣N(w)∣=ℓ(w)∣N(w)∣=ℓ(w), the root-reflection dictionary and strong exchange

Statement

Facts & Assumptions

Proof

5 · Examples, counterexamples and false statements

Sources

The rank-two half-space alternative and the chamber-length induction $(P_{n})$ , $(Q_{n})$

The geometric inversion set $N (w)$ of an element of a Coxeter group

The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange