Pipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 3 results · all verified · 3 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs; all 3 also cleared it.

Finite Reflection Length and Orthogonal Moved Spaces — Examples

1 · Prerequisites

2 · Summary

This companion is a dependency leaf: its three examples use only the theory of finite-reflection-length-and-orthogonal-moved-spaces and that page's prerequisite closure, and no other page or item depends on them. Each example is a complete finite computation.

Simple and reflection lengths of a long transposition in $S_{5}$ works in type $A_{4}$ : under the isomorphism $W \to S_{5}$ , $s_{i} \mapsto (i i + 1)$ , the reflections are exactly the transpositions, the simple length of a transposition $(i j)$ with $i < j$ is its inversion number $2 (j - i) - 1$ , and $ℓ_{T} (φ^{- 1} (σ)) = 5 - c (σ)$ is read off the fixed space of $σ$ in the sum-zero hyperplane. The long transposition $(1 5)$ therefore has $ℓ = 7$ but $ℓ_{T} = 1$ , and the longest element $w_{0} = (1 5) (2 4)$ has $ℓ = 10$ and $ℓ_{T} = 2$ , while every $5$ -cycle has $ℓ_{T} = 4 = \dim V$ ; the example also exhibits $(1 5) = (2 5) (1 2) (2 5)^{- 1}$ explicitly as a reflection.

A moved-space intersection in $A_{3}$ that is not the meet works in type $A_{3}$ inside the sum-zero hyperplane of $R^{4}$ , under the isometry $e_{s_{i}} \mapsto \frac{1}{\sqrt{2}} (e_{i} - e_{i + 1})$ : the elements $α = φ^{- 1} ((1 2) (3 4))$ and $β = φ^{- 1} ((1 4) (2 3))$ both lie below the $4$ -cycle $γ$ , and their moved spaces are the two planes ${x_{2} = - x_{1}, x_{4} = - x_{3}}$ and ${x_{2} = - x_{3}, x_{4} = - x_{1}}$ , which meet in the line $R (e_{1} - e_{2} + e_{3} - e_{4})$ . That line contains no root of $A_{3}$ , so no element of $W$ has it as moved space, the only common lower bound of $α$ and $β$ is $1$ , and the moved space of the meet, $M (1) = 0$ , is strictly smaller than the intersection $M (α) \cap M (β)$ : arbitrary subspace intersection does not compute the meet.

The Wall form and line restrictions of a plane rotation, and the necessity of a common upper bound works in the rank-two system $I_{2} (m)$ : for $A = ρ (s t) = r_{s} r_{t}$ the moved space is all of $V$ , in oriented orthonormal coordinates $S_{A} = (A - i d)^{- 1}$ is multiplication by $1 / (e^{i θ} - 1) = - \frac{1}{2} - \frac{i}{2} \cot (θ / 2)$ with $θ = 2 π / m$ , and the line restrictions $H_{L} = Π_{L} S_{A} Π_{L}$ are the scalar $- \frac{1}{2}$ , so $L \mapsto A_{L} = i d - 2 Π_{L}$ is the bijection from lines to reflections below $A$ , with $A_{L} \in ρ (W)$ exactly for the $m$ root lines. For $m = 4$ the pair $A$ and $w_{0} = A^{2}$ satisfies $M (w_{0}) = V = M (A)$ although $w_{0} ̸ \leq_{T} A$ , $A ̸ \leq_{T} w_{0}$ , and no element of $W$ lies above both; this shows that the common-upper-bound hypothesis in the rigidity statement is indispensable.

The results tested here are proved on the theory page: the Wall form, the subspace restriction and the prefix form of $\leq_{O}$ in The Wall form of an orthogonal operator, subspace restriction, and the interval structure of the orthogonal reflection-length order, the factorizations and independent normals in Root normals inside the moved space, factorizations into reflections, and independent normals, and Carter's formula together with the rigidity statement in Carter's reflection-length formula, the absolute order on a finite Coxeter group, and moved-space rigidity under a common upper bound. The examples are evidence within their computed scope and do not replace those proofs.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Simple and reflection lengths of a long transposition in $S_{5}$

Example

Let $W$ be the Coxeter group of type $A_{4}$ , with $S = {s_{1}, \dots, s_{4}}$ , reflection representation $V = R^{S}$ with positive definite Coxeter form $B$ , reflection set $T$ and lengths $ℓ$ and $ℓ_{T}$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone, Reflection length, the absolute order on a finite Coxeter group, and the moved and fixed spaces of an orthogonal operator, Carter's reflection-length formula, the absolute order on a finite Coxeter group, and moved-space rigidity under a common upper bound); fix the isomorphism $φ : W \to S_{5}$ with $s_{i} \mapsto (i i + 1)$ (Coxeter diagrams: edges, labels, components and finite type, Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (4), Inversions, inversion number, the sign $sgn (σ) = (- 1)^{inv (σ)}$ , and even and odd permutations). Then:

(i) $T = {φ^{- 1} (τ) : τ a transposition of S_{5}}$ and $ℓ_{T} (w) = 1$ for the transpositions $w$ ; for a transposition $(i j)$ with $i < j$ one has $ℓ (φ^{- 1} (i j)) = 2 (j - i) - 1$ .

(ii) For the long transposition $φ^{- 1} (1 5)$ the two lengths are $ℓ = 2 \cdot 4 - 1 = 7$ while $ℓ_{T} = 1$ ; explicitly $(1 5) = (2 5) (1 2) (2 5)^{- 1}$ is a reflection and has $7$ inversions.

(iii) Under the linear isometry $e_{s_{i}} \mapsto \frac{1}{\sqrt{2}} (e_{i} - e_{i + 1})$ of $(V, B)$ onto the hyperplane $H = {x \in R^{5} : \sum_{i} x_{i} = 0}$ with the standard inner product (Real and complex inner-product spaces and their induced length, Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces), $φ$ corresponds to the permutation action, so $F (φ^{- 1} (σ))$ corresponds to the fixed space ${x \in H : σ x = x}$ , of dimension $c (σ) - 1$ for the cycle count $c (σ)$ (fixed points included). Since $\dim M = \dim V - \dim F$ (In finite dimension, $W^{⊥ ⊥} = W$ and $\dim W + \dim W^{⊥} = \dim V$ ) and $ℓ_{T} = \dim M$ , one has $ℓ_{T} (φ^{- 1} (σ)) = \dim H - (c (σ) - 1) = 5 - c (σ)$ for every $σ \in S_{5}$ . In particular $ℓ_{T} (φ^{- 1} (1 5)) = 1$ ; for the longest element $w_{0} = φ^{- 1} ((1 5) (2 4))$ one has $ℓ (w_{0}) = 10$ while $ℓ_{T} (w_{0}) = 5 - 3 = 2$ (the reversal has three cycles), and every $5$ -cycle has $ℓ_{T} = 4 = \dim V$ .

Facts & Assumptions

Given: The type- $A_{4}$ Coxeter datum $W, S, V, B, ρ, Φ, T$ and the isomorphism $φ : W \to S_{5}$ with $s_{i} \mapsto (i i + 1)$ ; a permutation $σ \in S_{5}$ acts on $R^{5}$ by permuting coordinates.

[F1]

$s_{i} \mapsto (i i + 1)$ extends to an isomorphism $φ : W \to S_{5}$ , $ℓ (w) = inv (φ (w))$ for the inversion number, and $S$ generates $W$ . Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification

[F2]

$ρ (s_{i}) = r_{e_{s_{i}}}$ , $T = {w s w^{- 1} : w \in W, s \in S}$ , $Φ = {ρ (w) e_{s} : w \in W, s \in S}$ , and $B (e_{s}, e_{t}) = - \cos (π / m (s, t))$ with $m (s_{i}, s_{j}) = 3$ for adjacent and $2$ for non-adjacent generators of $A_{4}$ . The canonical reflection homomorphism, roots, reflections, and the positive cone The real Coxeter form, its radical, reflections, and form-preserving maps

[F3]

$ℓ_{T} (w) = \dim M (w) = \dim V - \dim F (w)$ , $M (w) = F (w)^{⊥}$ and $V = M (w) \oplus F (w)$ ; every line of $V$ is the moved space of a unique reflection of the orthogonal group of $(V, B)$ . Carter's reflection-length formula, the absolute order on a finite Coxeter group, and moved-space rigidity under a common upper bound The Wall form of an orthogonal operator, subspace restriction, and the interval structure of the orthogonal reflection-length order

[F4]

$F (w) = \ker (ρ (w) - i d)$ and $M (w) = im (ρ (w) - i d)$ for $w \in W$ , and $T$ is the reflection set in which $ℓ_{T}$ is computed. Reflection length, the absolute order on a finite Coxeter group, and the moved and fixed spaces of an orthogonal operator

[F5]

The inversion number of a permutation is the number of pairs $a < b$ with $σ (a) > σ (b)$ . Inversions, inversion number, the sign $sgn (σ) = (- 1)^{inv (σ)}$ , and even and odd permutations

[F6]

Each $ρ (s_{i})$ is an inner-product-preserving involution with normal $e_{s_{i}}$ . An orthogonal operator with moved line $L$ equals $i d - 2 Π_{L}$ and fixes $L^{⊥}$ pointwise. Descent of the reflection representation, unit root norms, and conjugation of reflections (1), (2), (4), The Wall form of an orthogonal operator, subspace restriction, and the interval structure of the orthogonal reflection-length order (3).

[F7]

Write $γ = π / 2$ for the smallest positive cosine zero; $0 < γ < 2$ , and cosine is strictly decreasing on $[0, 2]$ . Also $\cos (π / 2) = 0$ , $\cos (π - x) = - \cos x$ , and $\cos (2 x) = 2 \cos^{2} x - 1$ . Pi as twice the smallest positive zero of cosine Cosine has a smallest positive zero, lying strictly between zero and two Sine is positive and cosine is strictly decreasing on (0,2), with cos 2 at most -1/3 Quarter-turn values and shifts by pi/2 and pi Double-angle and quadratic power-reduction identities Parity and the Pythagorean identity for sine and cosine

Verification

technique · direct

1.1F1F2F3F6F7

Put $c : = \cos (π / 3)$ . By [F7], $0 < π / 3 < γ < 2$ gives $c > \cos γ = 0$ , while $2 c^{2} - 1 = \cos (2 π / 3) = - c$ , so $(2 c - 1) (c + 1) = 0$ and $c = \frac{1}{2}$ ; also $\cos (π / 2) = 0$ by [F7]. Put $v_{i} : = \frac{1}{\sqrt{2}} (e_{i} - e_{i + 1}) \in H$ for $1 \leq i \leq 4$ . Then $⟨ v_{i}, v_{i} ⟩ = 1$ , $⟨ v_{i}, v_{i + 1} ⟩ = - \frac{1}{2}$ and $⟨ v_{i}, v_{j} ⟩ = 0$ for $∣ i - j ∣ \geq 2$ , matching $B (e_{s_{i}}, e_{s_{j}}) = - \cos (π / m (s_{i}, s_{j}))$ by [F2] and [F7]; since the $v_{i}$ are linearly independent (their coordinates in the order $(e_{1}, \dots, e_{5})$ are $(a, - a + b, - b + c, - c + d, - d) / \sqrt{2}$ for $a v_{1} + b v_{2} + c v_{3} + d v_{4}$ ) and every $x \in H$ equals $\sqrt{2} \sum_{i = 1}^{4} (x_{1} + \dots + x_{i}) v_{i}$ , they form a basis of $H$ , so $\dim H = 4$ and the map $Θ : V \to H$ with $Θ (e_{s_{i}}) = v_{i}$ is a linear isometry onto $H$ . For each $i$ the transposition $(i i + 1)$ preserves $H$ , reverses $v_{i}$ and fixes ${v_{i}}^{⊥} \cap H$ pointwise, so it is the reflection with normal line $R v_{i}$ ; the image $Θ ρ (s_{i}) Θ^{- 1}$ is by [F2] and [F6] likewise an inner-product-preserving involution of $H$ that reverses $v_{i}$ and fixes ${v_{i}}^{⊥} \cap H$ pointwise, so the two agree on all of $H$ . Since $φ$ is an isomorphism and $S$ generates $W$ [F1], the homomorphisms $Θ ρ Θ^{- 1}$ and $σ \mapsto σ ∣_{H}$ agree on $S$ , hence everywhere: $Θ ρ (w) Θ^{- 1} = φ (w) ∣_{H}$ for all $w \in W$ . The permutation action on $H$ is faithful, because if $σ$ acts trivially then $σ (e_{i} - e_{k}) = e_{i} - e_{k}$ for all $i \neq k$ , and choosing $k \notin {i, σ (i)}$ forces $σ (i) = i$ . Therefore for $t = w s w^{- 1} \in T$ the permutation $φ (t)$ has moved line $Θ R ρ (w) e_{s} = R (e_{p} - e_{q})$ for some $p \neq q$ and fixes the orthogonal hyperplane pointwise, so $φ (t) (p q)^{- 1}$ acts trivially on $H$ and $φ (t) = (p q)$ ; conversely every transposition $(p q) = σ (1 2) σ^{- 1}$ equals $φ (u s_{1} u^{- 1})$ for $u : = φ^{- 1} (σ)$ , so $T = {φ^{- 1} (τ) : τ a transposition}$ .

1.2F5algebra

For a permutation $σ \in S_{5}$ the inversion number of the transposition $(i j)$ , $i < j$ , is $2 (j - i) - 1$ : the pairs $a < b$ with $(i j) a > (i j) b$ are exactly the $j - i - 1$ pairs $(i, k)$ with $i < k < j$ , the single pair $(i, j)$ , and the $j - i - 1$ pairs $(k, j)$ with $i < k < j$ . In particular the transposition $(1 5)$ has $2 \cdot 4 - 1 = 7$ inversions, and the reversal $(1 5) (2 4)$ inverts every one of the $(\binom{5}{2}) = 10$ pairs, so it has $10$ inversions, the maximal value; in $S_{5}$ this is the longest element.

2.1step 1.1F3F4

By step 1.1 the fixed space of $ρ (w)$ corresponds to ${x \in H : σ x = x}$ for $σ = φ (w)$ , and $ℓ_{T} (w) = 4 - \dim {x \in H : σ x = x}$ by [F3]. The fixed space of $σ$ in $R^{5}$ is spanned by the incidence vectors of its cycles, so it has dimension $c (σ)$ and its intersection with $H$ is defined by the single equation $\sum_{C} ∣ C ∣ a_{C} = 0$ on the cycle coefficients $a_{C}$ . Every $∣ C ∣$ is positive, so fixing one cycle lets its coefficient be solved uniquely from the other $c (σ) - 1$ coefficients; the intersection therefore has dimension $c (σ) - 1$ ; hence $ℓ_{T} (w) = 4 - (c (σ) - 1) = 5 - c (σ)$ , with $c (σ)$ the number of cycles of $σ$ (fixed points included). In particular a transposition has $c = 4$ and $ℓ_{T} = 1$ , the long transposition $(1 5)$ has $c = 4$ and $ℓ_{T} = 1$ , the reversal $(1 5) (2 4)$ has $c = 3$ and $ℓ_{T} = 2$ , and a $5$ -cycle has $c = 1$ and $ℓ_{T} = 4 = \dim V$ .

3.1step 1.1step 1.2step 2.1F1F3∎

Collecting the results: by step 1.1 the reflections of $W$ are exactly the $φ^{- 1} (τ)$ with $τ$ a transposition, and each has $ℓ_{T} = 1$ by step 2.1, which is claim (i)'s first part, while claim (i)'s second part is the inversion count of step 1.2. For the long transposition, $ℓ (φ^{- 1} (1 5)) = 7$ by steps 1.2 and [F1] and $ℓ_{T} (φ^{- 1} (1 5)) = 1$ by step 2.1, and $φ^{- 1} (1 5) \in T$ because $(1 5) = (2 5) (1 2) (2 5)^{- 1}$ exhibits $(1 5)$ as $φ (u s_{1} u^{- 1})$ for the element $u : = φ^{- 1} (2 5)$ ; this is claim (ii). Claim (iii)'s dimension formula, the values $ℓ (w_{0}) = 10$ and $ℓ_{T} (w_{0}) = 2$ , and $ℓ_{T} = 4$ for every $5$ -cycle are steps 1.2, 2.1 and [F1].

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

The Wall form and line restrictions of a plane rotation, and the necessity of a common upper bound

Example

Let $m \geq 3$ , $c = \cos (π / m)$ , $V = R e_{s} + R e_{t}$ with the positive definite Coxeter form $B$ of the rank-two system $I_{2} (m)$ , and let $A : = ρ (s t) = r_{s} r_{t}$ (The real Coxeter form, its radical, reflections, and form-preserving maps, Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order (3), The canonical reflection homomorphism, roots, reflections, and the positive cone, Classification of finite Coxeter systems, including the H and dihedral families); let $ℓ_{T}$ , $M$ and $\leq_{O}$ be as in Reflection length, the absolute order on a finite Coxeter group, and the moved and fixed spaces of an orthogonal operator and The Wall form of an orthogonal operator, subspace restriction, and the interval structure of the orthogonal reflection-length order, and let $T$ , $Φ$ be the reflection set and root system (The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange (1)). Use faithfulness of The root-length criterion and faithfulness of the canonical reflection representation (3) to identify $W$ with $ρ (W)$ when writing $ℓ$ , $ℓ_{T}$ and $\leq_{T}$ for these operators. Then:

(i) $tr A = 2 \cos (2 π / m) \neq 2 = tr i d$ , so $1$ is not an eigenvalue of $A$ , $F (A) = 0$ and $M (A) = V$ . In oriented orthonormal coordinates adapted to $A$ it is the rotation by $θ = 2 π / m$ , and

$S_{A} = (A - i d)^{- 1} is multiplication by \frac{1}{e^{i θ} - 1} = - \frac{1}{2} - \frac{i}{2} \cot \frac{θ}{2},$

so the Wall form $χ_{A} (u, v) = B (S_{A} u, v)$ satisfies $χ_{A} (u, v) + χ_{A} (v, u) = - B (u, v)$ with symmetric part $- \frac{1}{2} B$ .

(ii) For every line $L \subseteq V$ the operator $H_{L} = Π_{L} S_{A} Π_{L}$ is the scalar $- \frac{1}{2}$ on $L$ , so $A_{L} = i d - 2 Π_{L}$ is the reflection of the plane with normal line $L$ , $A_{L} \leq_{O} A$ , and $L \mapsto A_{L}$ is a bijection from the lines of $V$ onto the reflections $B \leq_{O} A$ . Moreover $A_{L}$ lies in $ρ (W)$ if and only if $L$ is one of the $m$ root lines $R β$ , $β \in Φ$ ; so for a line $L$ that is not a root line, $A_{L}$ is an orthogonal reflection whose moved space is $L$ but $A_{L} \notin ρ (W)$ .

(iii) Take $m = 4$ and put $w_{0} : = A^{2}$ (the rotation by $π$ , the longest element of $W = I_{2} (4)$ ). Then $M (A) = M (w_{0}) = V$ , so $M (w_{0}) \subseteq M (A)$ , while

$ℓ_{T} (w_{0}^{- 1} A) = ℓ_{T} (A^{- 1} w_{0}) = ℓ_{T} (A) = 2,$

so $w_{0} ̸ \leq_{T} A$ and $A ̸ \leq_{T} w_{0}$ ; in fact $A$ and $w_{0}$ have no common upper bound in $W$ , since every $δ \in W$ has $ℓ_{T} (δ) = \dim M (δ) \leq 2$ (Carter's reflection-length formula, the absolute order on a finite Coxeter group, and moved-space rigidity under a common upper bound (1)), so a common upper bound would have to equal both $A$ and $w_{0}$ . Hence the implication $M (α) \subseteq M (β) \Rightarrow α \leq_{T} β$ fails without the common-upper-bound hypothesis of the rigidity theorem.

Facts & Assumptions

Given: The rank-two datum $m \geq 3$ , $c = \cos (π / m)$ , $V = R e_{s} + R e_{t}$ , $B$ , $ρ$ , $T$ , $Φ$ and the elements $A = ρ (s t) = r_{s} r_{t}$ , $w_{0} = A^{2}$ ; $ℓ_{T}$ , $M$ , $F$ , $Π_{L}$ , $H_{L}$ , $A_{L}$ are as in Reflection length, the absolute order on a finite Coxeter group, and the moved and fixed spaces of an orthogonal operator and The Wall form of an orthogonal operator, subspace restriction, and the interval structure of the orthogonal reflection-length order.

[F1]

In the ordered basis $(e_{s}, e_{t})$ one has $B (e_{s}, e_{s}) = B (e_{t}, e_{t}) = 1$ and $B (e_{s}, e_{t}) = - c$ with $c = \cos (π / m)$ , and $B ∣_{V}$ is positive definite for finite $m$ ; the product $A = r_{s} r_{t}$ has the matrix $(\begin{array}{cc} 4 c^{2} - 1 & - 2 c \\ 2 c & - 1 \end{array})$ of determinant $1$ , trace $2 \cos (2 π / m)$ and order $m$ (that is, $A^{m} = {i d}_{V}$ and $A^{k} \neq {i d}_{V}$ for $0 < k < m$ ). The real Coxeter form, its radical, reflections, and form-preserving maps Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order

[F2]

The type $I_{2} (m)$ , $3 \leq m < \infty$ , is the finite-type diagram with two vertices joined by a single edge labelled $m$ ; the corresponding Coxeter system has exactly the two simple reflections $s, t$ , and $ρ (s t) = r_{s} r_{t}$ . Classification of finite Coxeter systems, including the H and dihedral families Coxeter diagrams: edges, labels, components and finite type The canonical reflection homomorphism, roots, reflections, and the positive cone

[F3]

The Wall form lemma holds on the positive definite plane $(V, B)$ : (1) $M (X) = F (X)^{⊥}$ and $V = M (X) \oplus F (X)$ ; (2) $χ_{X} (u, v) = B ((X - i d) ∣_{M (X)}^{- 1} u, v)$ satisfies $χ_{X} (u, v) + χ_{X} (v, u) = - B (u, v)$ , is nondegenerate, and has symmetric part $- \frac{1}{2} B$ ; (3) for a line $L \subseteq M (X)$ the operator $H_{L}$ with $B (H_{L} u, v) = χ_{X} (u, v)$ on $L$ satisfies $H_{L} + H_{L}^{*} = - {i d}_{L}$ and is invertible, $A_{L} = i d + H_{L}^{- 1}$ on $L$ and $i d$ on $L^{⊥}$ has $M (A_{L}) = L$ , and the reflections of the orthogonal group of $V$ are exactly the maps $i d - 2 Π_{L}$ for lines $L$ ; (4) $A_{L} \leq_{O} X$ , and $U \mapsto X_{U}$ is a bijection from the subspaces of $M (X)$ onto ${Y : Y \leq_{O} X}$ with inverse $Y \mapsto M (Y)$ , order-preserving and order-reflecting for inclusion and $\leq_{O}$ . The Wall form of an orthogonal operator, subspace restriction, and the interval structure of the orthogonal reflection-length order

[F4]

$ℓ_{T} (w) = \dim M (w)$ for all $w \in W$ . Carter's reflection-length formula, the absolute order on a finite Coxeter group, and moved-space rigidity under a common upper bound

[F5]

Every nonzero moved space of an element of $W$ contains a root; for every root $β$ the reflection with normal $β$ equals $ρ (t_{β})$ with $t_{β} \in T$ ; the map ${\pm α : α \in Φ} \to T$ , ${\pm α} \mapsto t_{α}$ , is a bijection, so the root lines $R α$ are in bijection with $T$ ; and $ρ$ is injective. Root normals inside the moved space, factorizations into reflections, and independent normals The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange The root-length criterion and faithfulness of the canonical reflection representation

[F6]

Trigonometry: $π$ is twice the smallest positive zero of $\cos$ , which lies in $(0, 2)$ , so $0 < π < 4$ , $\cos (π / 2) = 0$ and $\cos$ has no zero in $[0, π / 2)$ ; $\sin x > 0$ for $0 < x \leq 2$ and $\cos$ is strictly decreasing on $[0, 2]$ ; $\sin^{2} x + \cos^{2} x = 1$ ; $\sin 2 x = 2 \sin x \cos x$ and $\cos 2 x = 1 - 2 \sin^{2} x$ ; $e^{i θ} = \cos θ + i \sin θ$ ; and $\cot = \cos / \sin$ where defined. Pi as twice the smallest positive zero of cosine Cosine has a smallest positive zero, lying strictly between zero and two Sine is positive and cosine is strictly decreasing on (0,2), with cos 2 at most -1/3 Parity and the Pythagorean identity for sine and cosine Double-angle and quadratic power-reduction identities $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ Tangent, cotangent, secant, and cosecant on their exact natural domains

[F7]

$u \leq_{T} v$ means $ℓ_{T} (v) = ℓ_{T} (u) + ℓ_{T} (u^{- 1} v)$ , and $B \leq_{O} X$ means $\dim M (X) = \dim M (B) + \dim M (B^{- 1} X)$ . Reflection length, the absolute order on a finite Coxeter group, and the moved and fixed spaces of an orthogonal operator

Verification

technique · direct

1.1F1F2F6

By [F1] the matrix of $A$ in the basis $(e_{s}, e_{t})$ has determinant $1$ and trace $2 \cos (2 π / m)$ , and $A$ has order $m$ ; the trace differs from $tr i d = 2$ because $1 - \cos (2 π / m) = 2 \sin^{2} (π / m) > 0$ by [F6], as $π / m \in (0, π / 3] \subset (0, 2]$ gives $\sin (π / m) > 0$ .

1.2F1F3F6

Put $u : = (e_{s} + e_{t}) / \sqrt{2 - 2 c}$ and $\tilde{w} : = (e_{t} - e_{s}) / \sqrt{2 + 2 c}$ ; the square roots are nonzero because $2 - 2 c = 4 \sin^{2} (π / (2 m)) > 0$ and $2 + 2 c = 4 \cos^{2} (π / (2 m)) > 0$ , since $0 < π / (2 m) \leq π / 6 < 2$ and $\cos$ has no zero in $[0, π / 2)$ by [F6]. A direct computation with [F1] gives $B (u, u) = B (\tilde{w}, \tilde{w}) = 1$ and $B (u, \tilde{w}) = 0$ , so $(u, \tilde{w})$ is an oriented orthonormal basis, and computing $A u = \cos θ u + \sin θ \tilde{w}$ , $A \tilde{w} = - \sin θ u + \cos θ \tilde{w}$ with $θ : = 2 π / m$ : indeed $B (A u, u) = 2 c^{2} - 1 = \cos θ$ , $B (A u, \tilde{w}) = 2 c \sin (π / m) = \sin θ$ , $B (A \tilde{w}, u) = - \sin θ$ and $B (A \tilde{w}, \tilde{w}) = 2 c^{2} - 1 = \cos θ$ , using $1 - c^{2} = \sin^{2} (π / m)$ and the double-angle identities of [F6]. Hence in these coordinates $A$ is the rotation by $θ = 2 π / m$ , and $1$ is not an eigenvalue of $A$ : the matrix of $A - i d$ has determinant $(\cos θ - 1)^{2} + \sin^{2} θ = 2 - 2 \cos θ = 4 \sin^{2} (θ / 2) \neq 0$ , since $θ / 2 = π / m \in (0, 2]$ . Therefore $F (A) = \ker (A - i d) = 0$ and $M (A) = F (A)^{⊥} = V$ by F3. Moreover $S_{A} = (A - i d)^{- 1}$ , computed from the displayed rotation matrix, is $\frac{1}{2} (\begin{array}{cc} - 1 & \sin θ / (1 - \cos θ) \\ - \sin θ / (1 - \cos θ) & - 1 \end{array}) = - \frac{1}{2} i d - \frac{1}{2} \cot (θ / 2) J$ with $J = (\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array})$ , because $\sin θ / (1 - \cos θ) = \cot (θ / 2)$ by the double-angle identities; under the identification of the oriented plane with $C$ , the matrix $J$ is multiplication by $i$ , so $S_{A}$ is multiplication by $- \frac{1}{2} - \frac{i}{2} \cot (θ / 2)$ , and $(e^{i θ} - 1) (- \frac{1}{2} - \frac{i}{2} \cot (θ / 2)) = 1$ by the displayed identity $e^{i θ} = \cos θ + i \sin θ$ together with $\sin^{2} θ + \cos^{2} θ = 1$ and $\sin θ / (1 - \cos θ) = \cot (θ / 2)$ , so $S_{A}$ is multiplication by $1 / (e^{i θ} - 1)$ . The identity $χ_{A} (u, v) + χ_{A} (v, u) = - B (u, v)$ with symmetric part $- \frac{1}{2} B$ is F3.

2.1step 1.2F1F3F4F5F7

Part (iii). Take $m = 4$ , so $θ = π / 2$ and the rotation matrix of step 1.2 gives $A^{2} = (\begin{array}{cc} - 1 & 0 \\ 0 & - 1 \end{array}) = - {i d}_{V}$ ; hence $M (w_{0}) = im (- 2 i d) = V = M (A)$ for $w_{0} : = A^{2}$ , so $M (w_{0}) \subseteq M (A)$ . By [F4] and step 1.2, $ℓ_{T} (A) = \dim M (A) = 2$ and $ℓ_{T} (w_{0}) = \dim M (w_{0}) = 2$ ; also $w_{0}^{- 1} A = A^{- 2} A = A^{- 1}$ and $A^{- 1} w_{0} = A^{- 1} A^{2} = A$ , and $A^{- 1} = A^{3}$ has moved space $V$ because $F (A^{- 1}) = F (A) = 0$ , so $ℓ_{T} (w_{0}^{- 1} A) = ℓ_{T} (A^{- 1} w_{0}) = ℓ_{T} (A) = 2$ . Hence $ℓ_{T} (A) = 2 \neq 4 = ℓ_{T} (w_{0}) + ℓ_{T} (w_{0}^{- 1} A)$ and $ℓ_{T} (w_{0}) = 2 \neq 4 = ℓ_{T} (A) + ℓ_{T} (A^{- 1} w_{0})$ , so $w_{0} ̸ \leq_{T} A$ and $A ̸ \leq_{T} w_{0}$ by [F7]. If some $δ \in W$ satisfied $A \leq_{T} δ$ and $w_{0} \leq_{T} δ$ , then $2 = ℓ_{T} (A) \leq ℓ_{T} (δ)$ and $ℓ_{T} (δ) = \dim M (δ) \leq \dim V = 2$ by [F4], so [F7] gives $ℓ_{T} (δ) = ℓ_{T} (A) + ℓ_{T} (A^{- 1} δ) = 2$ and hence $ℓ_{T} (A^{- 1} δ) = 0$ ; then $A^{- 1} δ = 1$ , since $ℓ_{T} (x) = \dim M (x) = 0$ forces $ρ (x) = i d$ and $x = 1$ by [F5], so $δ = A$ , contradicting $w_{0} ̸ \leq_{T} A$ ; thus $A$ and $w_{0}$ have no common upper bound. Finally $w_{0} = A^{2}$ is the unique longest element. Since $s A s = A^{- 1}$ and $t = A^{- 1} s$ , every word reduces to $A^{k}$ or $A^{k} s$ with $0 \leq k \leq 3$ . The four rotations are distinct by the order of $A$ ; the four $A^{k} s$ are distinct by cancellation, and the two lists cannot overlap: overlap would give $s = A^{j}$ , whereas $s$ has moved dimension one by [F3] and [F5], and $A^{j}$ has moved dimension zero for $j = 0$ and two for $j = 1, 2, 3$ by the displayed rotation matrices. These eight elements are $1, s, t, s t, t s, s t s, t s t, s t s t$ , because $A s = s t s$ , $A^{3} s = t$ , and $A^{2} s = t s t$ (the relation $A^{4} = 1$ gives $s t s t = t s t s$ ). A word of length at most three reduces, by cancelling adjacent equal generators, to one of the first seven words; these are distinct from $A^{2} = s t s t$ . Thus $ℓ (A^{2}) = 4$ , and every other element has length at most three.

2.2step 1.2F3

Part (ii), first half. Let $L \subseteq V$ be a line; since $M (A) = V$ by step 1.2, $L \subseteq M (A)$ , so the operator $H_{L} = Π_{L} S_{A} Π_{L}$ of F3 is defined. As $L$ is one-dimensional, $H_{L}$ is multiplication by a real scalar $λ$ , and the identity $H_{L} + H_{L}^{*} = - {i d}_{L}$ of F3 gives $2 λ = - 1$ , so $λ = - \frac{1}{2}$ ; hence $A_{L} = i d + H_{L}^{- 1}$ acts on $L$ as $- {i d}_{L}$ and on $L^{⊥}$ as the identity, that is $A_{L} = i d - 2 Π_{L}$ . By F3 $M (A_{L}) = L$ , and $A_{L} \leq_{O} A$ by F3. The assignment $L \mapsto A_{L}$ is a bijection from the lines of $V$ onto the reflections $B \leq_{O} A$ : F3 gives a bijection $U \mapsto A_{U}$ from the subspaces of $M (A) = V$ onto ${B : B \leq_{O} A}$ whose inverse is $B \mapsto M (B)$ and which satisfies $M (A_{U}) = U$ , and the one-dimensional subspaces of $V$ are the lines, while the elements $B \leq_{O} A$ with $\dim M (B) = 1$ are exactly the reflections $B \leq_{O} A$ (the elements $B = i d$ and $B = A$ correspond to $U = 0$ and $U = V$ and are not reflections, as $\dim M (A) = 2$ ).

2.3step 1.2F1F5

$T$ has exactly $m$ elements. Since $A = s t$ one has $s A = t$ and $A^{- 1} = t s$ , and the set $W_{0} : = {A^{j}, A^{j} s : 0 \leq j < m}$ contains $1$ , $A$ , $s = A^{0} s$ and $t = A^{m - 1} s$ and is closed under multiplication and inversion, since $s A^{j} s = A^{- j}$ for every $j$ : it is a subgroup of $W$ containing $s$ and $t$ , so $W_{0} = W$ . Conjugation by the elements of $W_{0}$ gives $A^{k} s A^{- k} = A^{2 k} s$ , $A^{k} t A^{- k} = A^{2 k - 1} s$ and $(A^{k} s) t (A^{k} s)^{- 1} = A^{2 k + 1} s$ , so every element of $T$ lies in ${A^{j} s : 0 \leq j < m}$ , while conversely $A^{2 k} s = A^{k} s A^{- k}$ and $A^{2 k - 1} s = A^{k} t A^{- k}$ are conjugates of $s$ and $t$ ; hence $T = {A^{j} s : 0 \leq j < m}$ , and these $m$ elements are distinct because $A$ has order $m$ by [F1]. By the bijection ${\pm α : α \in Φ} \to T$ of [F5] the plane has exactly $m$ root lines, and the lines $R (e_{s} + a e_{t})$ for $a \in R$ are pairwise distinct, so there are infinitely many lines and some are not root lines.

3.1step 2.2F5

Part (ii), second half. If $L = R β$ for a root $β \in Φ$ , then by [F5] the reflection $i d - 2 Π_{L}$ with normal $β$ equals $r_{β} = ρ (t_{β}) \in ρ (W)$ ; by step 2.2 this operator is $A_{L}$ , so $A_{L} \in ρ (W)$ . Conversely suppose $A_{L} \in ρ (W)$ , say $A_{L} = ρ (w)$ ; then $\dim M (w) = \dim M (A_{L}) = 1$ by step 2.2, so the nonzero moved space $M (w) = L$ contains a root $β$ by [F5], and $L = R β$ is a root line. Hence $A_{L}$ lies in $ρ (W)$ exactly when $L$ is one of the $m$ root lines, and for every other line $L$ the operator $A_{L}$ is an orthogonal reflection with moved space $L$ such that $A_{L} \notin ρ (W)$ .

4.1step 1.1step 1.2step 2.1step 2.2step 2.3step 3.1∎

Collecting the verified claims: (i) is steps 1.1 and 1.2; (ii) is steps 2.2, 2.3 and 3.1; (iii) is step 2.1. In particular the converse implication $M (α) \subseteq M (β) \Rightarrow α \leq_{T} β$ of the rigidity theorem fails in $W = I_{2} (4)$ when no common upper bound is available: $α : = w_{0}$ and $β : = A$ satisfy $M (α) = V = M (β)$ by step 1.2, while $w_{0} ̸ \leq_{T} A$ and $A$ and $w_{0}$ have no common upper bound in $W$ , both by step 2.1.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

A moved-space intersection in $A_{3}$ that is not the meet

Example

Let $W$ be the Coxeter group of type $A_{3}$ , with $S = {s_{1}, s_{2}, s_{3}}$ , $V = R^{S}$ with positive definite Coxeter form, reflection set $T$ and lengths $ℓ_{T}, ℓ$ , and let $φ : W \to S_{4}$ , $s_{i} \mapsto (i i + 1)$ , be the type-A isomorphism (Coxeter diagrams: edges, labels, components and finite type, The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone, Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (4), Reflection length, the absolute order on a finite Coxeter group, and the moved and fixed spaces of an orthogonal operator, Carter's reflection-length formula, the absolute order on a finite Coxeter group, and moved-space rigidity under a common upper bound). Put

$γ = φ^{- 1} (1 2 3 4), α = φ^{- 1} ((1 2) (3 4)), β = φ^{- 1} ((1 4) (2 3)) .$

Then:

(i) $ℓ_{T} (γ) = 3$ , $ℓ_{T} (α) = ℓ_{T} (β) = 2$ , and $α \leq_{T} γ$ , $β \leq_{T} γ$ : indeed $α γ = φ^{- 1} (2 4)$ and $β γ = φ^{- 1} (1 3)$ are reflections, so $γ = α \cdot α^{- 1} γ$ and $γ = β \cdot β^{- 1} γ$ display the rank additivity $3 = 2 + 1$ .

(ii) Under the isometry $e_{s_{i}} \mapsto \frac{1}{\sqrt{2}} (e_{i} - e_{i + 1})$ of $V$ with $H = {x \in R^{4} : \sum_{i} x_{i} = 0}$ and the standard inner product, one has

$M (α) = {x \in H : x_{2} = - x_{1}, x_{4} = - x_{3}}, M (β) = {x \in H : x_{2} = - x_{3}, x_{4} = - x_{1}},$

and $M (α) \cap M (β) = R (e_{1} - e_{2} + e_{3} - e_{4})$ is a line containing no root of $A_{3}$ .

(iii) No element of $W$ has moved space $M (α) \cap M (β)$ : a nonzero moved space of an element of $W$ contains a root (Root normals inside the moved space, factorizations into reflections, and independent normals (1)), while this line contains none. Moreover the greatest common lower bound of $α$ and $β$ in $(W, \leq_{T})$ is $1$ : any common lower bound $τ$ satisfies $M (τ) \subseteq M (α) \cap M (β)$ (Carter's reflection-length formula, the absolute order on a finite Coxeter group, and moved-space rigidity under a common upper bound (2)(iv)), so $τ = 1$ by the same root-existence clause; hence the moved space of the meet, $M (1) = 0$ , is strictly smaller than the intersection of the two moved spaces, and arbitrary subspace intersection does not compute the meet.

Facts & Assumptions

Given: The type- $A_{3}$ Coxeter datum $W, S, V, B, ρ, Φ, T$ and the isomorphism $φ : W \to S_{4}$ with $s_{i} \mapsto (i i + 1)$ ; $ℓ_{T}$ , $M$ , $F$ are as in Reflection length, the absolute order on a finite Coxeter group, and the moved and fixed spaces of an orthogonal operator, and $σ \in S_{4}$ acts on $R^{4}$ by permuting coordinates.

[F1]

$s_{i} \mapsto (i i + 1)$ extends to an isomorphism $φ : W \to S_{4}$ , and $S$ generates $W$ . Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification

[F2]

$ℓ_{T} (w) = \dim M (w) = \dim V - \dim F (w)$ , and $M (w) = F (w)^{⊥}$ with $V = M (w) \oplus F (w)$ . Carter's reflection-length formula, the absolute order on a finite Coxeter group, and moved-space rigidity under a common upper bound The Wall form of an orthogonal operator, subspace restriction, and the interval structure of the orthogonal reflection-length order

[F3]

If $M (w) \neq 0$ for $w \in W$ , then $M (w)$ contains a root of $Φ$ . Root normals inside the moved space, factorizations into reflections, and independent normals

[F4]

$u \leq_{T} v$ implies $M (u) \subseteq M (v)$ . Carter's reflection-length formula, the absolute order on a finite Coxeter group, and moved-space rigidity under a common upper bound

[F5]

$ρ (s_{i}) = r_{e_{s_{i}}}$ , and each line of $V$ is the moved space of a unique reflection of the orthogonal group of $(V, B)$ . The canonical reflection homomorphism, roots, reflections, and the positive cone The Wall form of an orthogonal operator, subspace restriction, and the interval structure of the orthogonal reflection-length order

[F6]

$Φ = {ρ (w) e_{s} : w \in W, s \in S}$ , $T = {w s w^{- 1} : w \in W, s \in S}$ , and $B (e_{s}, e_{t}) = - \cos (π / m (s, t))$ with $m (s, t) = 3$ for adjacent and $2$ for non-adjacent generators of $A_{3}$ . Also $u \leq_{T} v$ means $ℓ_{T} (v) = ℓ_{T} (u) + ℓ_{T} (u^{- 1} v)$ , and $ℓ_{T} (u) = 0$ holds exactly when $u = 1$ , since only the empty product has length zero. Reflection length, the absolute order on a finite Coxeter group, and the moved and fixed spaces of an orthogonal operator The canonical reflection homomorphism, roots, reflections, and the positive cone The real Coxeter form, its radical, reflections, and form-preserving maps Coxeter diagrams: edges, labels, components and finite type

[F7]

Verification

technique · direct

1.1F1F5F6F7given

Put $c : = \cos (π / 3)$ . By [F7], $0 < π / 3 < γ < 2$ gives $c > \cos γ = 0$ , while $2 c^{2} - 1 = \cos (2 π / 3) = - c$ , so $(2 c - 1) (c + 1) = 0$ and $c = \frac{1}{2}$ ; also $\cos (π / 2) = 0$ by [F7]. Put $v_{i} : = \frac{1}{\sqrt{2}} (e_{i} - e_{i + 1}) \in H$ for $i = 1, 2, 3$ . Then $⟨ v_{i}, v_{i} ⟩ = 1$ , $⟨ v_{i}, v_{i + 1} ⟩ = - \frac{1}{2}$ and $⟨ v_{i}, v_{j} ⟩ = 0$ whenever $∣ i - j ∣ \geq 2$ , which by [F6] and [F7] matches $B (e_{s_{i}}, e_{s_{j}}) = - \cos (π / m (s_{i}, s_{j}))$ ; the $v_{i}$ are linearly independent, since the coordinates of $a v_{1} + b v_{2} + c v_{3}$ are $(a, - a + b, - b + c, - c) / \sqrt{2}$ , and every $x \in H$ equals $\sqrt{2} (x_{1} v_{1} + (x_{1} + x_{2}) v_{2} + (x_{1} + x_{2} + x_{3}) v_{3})$ , so they form a basis of the three-dimensional space $H$ and the linear map $Θ : V \to H$ with $Θ (e_{s_{i}}) = v_{i}$ is a linear isometry onto $H$ . For each $i$ the permutation $(i i + 1)$ preserves $H$ , fixes ${v_{i}}^{⊥} \cap H$ pointwise and sends $v_{i}$ to $- v_{i}$ , so it acts on $H$ as an orthogonal involution with moved space $R v_{i}$ ; the image $Θ ρ (s_{i}) Θ^{- 1}$ is an orthogonal involution with the same moved space $R v_{i}$ by [F5], and by the uniqueness in [F5] the two are equal. Since $φ$ is an isomorphism [F1] and $S$ generates $W$ , the two homomorphisms $Θ ρ Θ^{- 1}$ and $σ \mapsto σ ∣_{H}$ from $W$ to the orthogonal group of $H$ agree on $S$ , hence everywhere: $Θ ρ (w) Θ^{- 1} = φ (w) ∣_{H}$ for all $w \in W$ . In particular $Θ Φ = {σ v_{i} : σ \in S_{4}, i \leq 3} = {\pm \frac{1}{\sqrt{2}} (e_{p} - e_{q}) : p \neq q}$ , because $φ$ is onto and the transpositions of $S_{4}$ are the images of the conjugate reflections.

2.1step 1.1F1F2

Under the identification of step 1.1, $F (w) = Θ^{- 1} {x \in H : φ (w) x = x}$ and $ℓ_{T} (w) = \dim V - \dim F (w) = 3 - \dim {x \in H : φ (w) x = x}$ by [F2]. For $γ = φ^{- 1} (1 2 3 4)$ the fixed space in $R^{4}$ of the $4$ -cycle $(1 2 3 4)$ is $R (e_{1} + e_{2} + e_{3} + e_{4})$ , which meets $H$ in $0$ , so $ℓ_{T} (γ) = 3$ . For $α = φ^{- 1} ((1 2) (3 4))$ the fixed space in $R^{4}$ is ${x : x_{1} = x_{2}, x_{3} = x_{4}}$ , whose intersection with $H$ is $R (e_{1} + e_{2} - e_{3} - e_{4})$ , of dimension $1$ ; hence $ℓ_{T} (α) = 2$ , and the same computation with $x_{1} = x_{4}$ , $x_{2} = x_{3}$ gives fixed space $R (e_{1} - e_{2} - e_{3} + e_{4}) \cap H$ of dimension $1$ and $ℓ_{T} (β) = 2$ . Finally $α γ = φ^{- 1} (2 4)$ and $β γ = φ^{- 1} (1 3)$ by the multiplication convention $(σ τ) (x) = σ (τ (x))$ applied in $S_{4}$ : for instance $α γ$ sends $1 \mapsto 1$ , $2 \mapsto 4$ , $3 \mapsto 3$ , $4 \mapsto 2$ , so it is the transposition $(2 4)$ ; and for a transposition $(p q)$ the fixed space in $R^{4}$ has dimension $3$ (inside $H$ its coordinates satisfy $x_{p} = x_{q} = a$ and $2 a + x_{r} + x_{s} = 0$ for the remaining indices $r, s$ , leaving two free parameters), so $ℓ_{T} (φ^{- 1} (p q)) = 3 - 2 = 1$ .

2.2step 1.1F2

Since $F (α)$ is the line $R (e_{1} + e_{2} - e_{3} - e_{4})$ in the model of step 1.1 and $Θ$ is an isometry, $M (α) = F (α)^{⊥}$ is, inside $H$ , the orthogonal complement of $e_{1} + e_{2} - e_{3} - e_{4}$ , namely ${x \in H : x_{1} + x_{2} - x_{3} - x_{4} = 0} = {x \in H : x_{2} = - x_{1}, x_{4} = - x_{3}}$ ; likewise $F (β) = R (e_{1} - e_{2} - e_{3} + e_{4})$ gives $M (β) = {x \in H : x_{1} - x_{2} - x_{3} + x_{4} = 0} = {x \in H : x_{2} = - x_{3}, x_{4} = - x_{1}}$ . Intersecting the two sets gives $x_{2} = - x_{1}$ , $x_{3} = - x_{2} = x_{1}$ , $x_{4} = - x_{1}$ (the sum condition is then automatic), so $M (α) \cap M (β) = R (e_{1} - e_{2} + e_{3} - e_{4})$ is a line.

3.1step 2.1F6

Since $α^{- 1} γ = α γ = φ^{- 1} (2 4)$ because $α$ is an involution, step 2.1 gives $ℓ_{T} (γ) = 3 = 2 + 1 = ℓ_{T} (α) + ℓ_{T} (α^{- 1} γ)$ , so $α \leq_{T} γ$ by the definition of $\leq_{T}$ in [F6], and the analogous computation with $β γ = φ^{- 1} (1 3)$ gives $β \leq_{T} γ$ .

3.2step 1.1step 2.2F3

By step 1.1 the roots of $A_{3}$ in the model are the twelve vectors $\pm \frac{1}{\sqrt{2}} (e_{p} - e_{q})$ , $p \neq q$ , and none of these is a real multiple of $e_{1} - e_{2} + e_{3} - e_{4}$ , so the line $R (e_{1} - e_{2} + e_{3} - e_{4})$ of step 2.2 contains no root; hence no $w \in W$ has $M (w) = R (e_{1} - e_{2} + e_{3} - e_{4})$ , because a nonzero moved space contains a root by [F3].

4.1step 2.2step 3.1step 3.2F2F4F6∎

Let $τ \in W$ be a common lower bound of $α$ and $β$ , so $M (τ) \subseteq M (α) \cap M (β) = R (e_{1} - e_{2} + e_{3} - e_{4})$ by [F4] and step 2.2. If $M (τ) \neq 0$ , then $M (τ)$ equals that line and $τ$ is an element with moved space the line, contradicting step 3.2; hence $M (τ) = 0$ and $τ = 1$ by [F2]; since $1 \leq_{T} α, β$ , the greatest common lower bound of $α$ and $β$ is $1$ , and $M (1) = 0$ is strictly smaller than the line $M (α) \cap M (β)$ . This verifies (i), (ii) and (iii).

Finite Reflection Length and Orthogonal Moved Spaces — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Simple and reflection lengths of a long transposition in S5S5​

Example

Facts & Assumptions

Verification

The Wall form and line restrictions of a plane rotation, and the necessity of a common upper bound

Example

Facts & Assumptions

Verification

A moved-space intersection in A3A3​ that is not the meet

Example

Facts & Assumptions

Verification

Sources

Simple and reflection lengths of a long transposition in $S_{5}$

A moved-space intersection in $A_{3}$ that is not the meet