PropositionStatement: AI-generatedProof: AI-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.

Root signs, inversions and descent translation

Statement

Let $S$ be finite and let $(W, S)$ , $V$ , its basis $(e_{s})_{s \in S}$ , the Coxeter form $B$ , the canonical representation, the root set $Φ$ , the dual action, and the open chamber $C^{\circ}$ be as in the canonical real Coxeter interface. Let $ℓ (w)$ be the minimum length of a word in $S$ representing $w$ . Put $V_{+} = {\sum_{s \in S} a_{s} e_{s} : a_{s} \geq 0}$ and $Φ_{+} = Φ \cap V_{+}$ . A root in $- Φ_{+}$ is called negative.

Every root has exactly one sign: $Φ = Φ_{+} ⊔ (- Φ_{+})$ . For $N (w) = {α \in Φ_{+} : w α \in - Φ_{+}}$ one has $∣ N (w) ∣ = ℓ (w) = ℓ (w^{- 1})$ . For $s \in S$ , the chamber $w C^{\circ}$ lies in ${f : f (e_{s}) < 0}$ exactly when $ℓ (s w) < ℓ (w)$ , equivalently when $w^{- 1} e_{s}$ is negative.

For a root $α$ , write $H_{α} = {f \in V^{*} : f (α) = 0}$ . Distinct positive roots have distinct full hyperplanes $H_{α}$ . The map $α \mapsto H_{α} \cap U$ need not be injective when restricted to a real linear subspace $U \subseteq V^{*}$ .

Statement references

Real Coxeter representation and contragredient notation

Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups

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

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

Descent of the reflection representation, unit root norms, and conjugation of reflections

Linear subspace of a vector space

The dual action, the faces, and the rank-two chamber tiling

The vector space F^X of all functions X → F with pointwise operations, and Fⁿ as the case X = n = {0, 1, …, n-1}

Linear functionals and the algebraic dual V^*=L(V,F)

Facts & Assumptions

[F1]

Roots split disjointly into $Φ_{+}$ and $- Φ_{+}$ ; a positive root is evaluated positively, and a negative root negatively, by every member of $C^{\circ}$ . Root sign coherence and the action of simple reflections on positive roots

[F2]

For every $w \in W$ , $∣ N (w) ∣ = ℓ (w)$ . The inversion formula |N(w)|=ℓ(w), the root-reflection dictionary and strong exchange

[F3]

For $x \in W$ and $s \in S$ , $x e_{s}$ is negative if and only if $ℓ (x s) < ℓ (x)$ . The root-length criterion and faithfulness of the canonical reflection representation

[F4]

Every root $α$ satisfies $B (α, α) = 1$ . Descent of the reflection representation, unit root norms, and conjugation of reflections

[F5]

Each $s \in S$ is an involution, and $ℓ (x)$ is the least length of a representing word. Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups

[F6]

The dual action satisfies $(w f) (v) = f (w^{- 1} v)$ , and $e_{s}$ are basis vectors and roots. Real Coxeter representation and contragredient notation

[F7]

The chamber $C^{\circ}$ is nonempty, by statement (2) of The dual action, the faces, and the rank-two chamber tiling, applied to this finite $S$ and its canonical real representation.

[F8]

A real linear subspace contains zero and is closed under addition and real scalar multiplication; in particular ${0}$ is a real linear subspace of $V^{*}$ . Linear subspace of a vector space

[F9]

In the function space $V = R^{S}$ , operations are pointwise and vectors are equal exactly when all their values agree. The full algebraic dual $V^{*}$ consists of all real linear functionals on $V$ , with pointwise operations. The vector space F^X of all functions X → F with pointwise operations, and Fⁿ as the case X = n = {0, 1, …, n-1} Linear functionals and the algebraic dual V^*=L(V,F)

Proof

Given: the finite Coxeter system and the representation and notation in the Statement.

Proof technique: direct.

Goal: establish the sign, inversion, side and hyperplane assertions in the Statement.

1.1F1F2F5given

Signs and inversion counts. The sign decomposition is [F1], and the first equality $∣ N (w) ∣ = ℓ (w)$ is [F2]. Reversing any word in the involutive generators represents the inverse element and has the same length. Applying this observation to reduced words for $w$ and $w^{- 1}$ gives both inequalities and hence $ℓ (w) = ℓ (w^{- 1})$ .

1.2F6F9givenconstruct

Coordinate separation. For $s \in S$ define $ε_{s} (v) = v (s)$ on the supplied function space $V = R^{S}$ . Pointwise operations give $ε_{s} (a v + b z) = a ε_{s} (v) + b ε_{s} (z)$ , so $ε_{s} \in V^{*}$ . If $v \neq 0$ , some $s$ satisfies $v (s) \neq 0$ , and then $f_{0} = ε_{s} / v (s)$ satisfies $f_{0} (v) = 1$ . Moreover, if $f (v) = f (z)$ for every $f \in V^{*}$ , applying each $ε_{s}$ gives $v (s) = z (s)$ for every $s$ , hence $v = z$ . Thus the explicitly constructed coordinate functionals separate vectors; no basis extension or choice principle is used.

2.1F4F6F9step 1.2givenalgebra

Full hyperplanes. Suppose $H_{α} = H_{β}$ for positive roots. By [F4] both roots are nonzero. Step 1.2 gives $f_{0} \in V^{*}$ with $f_{0} (α) = 1$ . Put $c = f_{0} (β)$ . For any $f \in V^{*}$ the functional $g = f - f (α) f_{0}$ belongs to $V^{*}$ by [F9] and satisfies $g (α) = 0$ . Equality of the hyperplanes implies $g (β) = 0$ , so $f (β) = c f (α) = f (c α)$ for every $f \in V^{*}$ . Coordinate separation from 1.2 gives $β = c α$ , and $c \neq 0$ because $β \neq 0$ . A positive root $α$ has some strictly positive coordinate, and $β = c α$ has only nonnegative coordinates; hence $c > 0$ . By [F4], $1 = B (β, β) = c^{2} B (α, α) = c^{2}$ , so $c = 1$ and $α = β$ .

2.2F1F3F6F7step 1.1

Translate the side condition. By [F6], for $f \in C^{\circ}$ one has $(w f) (e_{s}) = f (w^{- 1} e_{s})$ . If $w^{- 1} e_{s}$ is negative, [F1] makes every such value negative. Conversely, if all these values are negative, the root cannot be positive because [F7] supplies a member of $C^{\circ}$ and positive roots evaluate positively there; the sign decomposition makes it negative. By [F3], applied to $x = w^{- 1}$ , this is equivalent to $ℓ (w^{- 1} s) < ℓ (w^{- 1})$ . The length equality from 1.1, applied also to $w^{- 1} s$ , converts this to $ℓ (s w) < ℓ (w)$ .

3.1F1F6F8step 1.1step 2.1step 2.2∎

Restriction can identify hyperplanes. Take $S = {s, t}$ with $m (s, t) = 2$ . The distinct basis roots $e_{s}, e_{t}$ are positive. For the real linear subspace $U = {0} \subseteq V^{*}$ supplied by [F8], both $H_{e_{s}} \cap U$ and $H_{e_{t}} \cap U$ equal ${0}$ . Thus restriction need not preserve injectivity. Together with steps 1.1, 2.1 and 2.2 this proves all assertions.

Depends on

Used by · 10

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