PropositionStatement: AI-generatedProof: AI-generated

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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.

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Reduced-word lifts and the prefix-order convention

Statement

Let $(W, S)$ be a finite-rank Coxeter system, with length $ℓ (w)$ equal to the least length of a word in $S$ representing $w$ . Let $A$ be its Artin group, generated by $σ_{s}$ for $s \in S$ with the braid relations, and let $π : A \to W$ send $σ_{s}$ to $s$ . For a reduced expression $w = s_{1} \dots s_{r}$ , meaning one of length $r = ℓ (w)$ , put $σ_{w} = σ_{s_{1}} \dots σ_{s_{r}}$ , with the empty product equal to the identity. This element is independent of the reduced expression and satisfies $π (σ_{w}) = w$ . If $ℓ (u v) = ℓ (u) + ℓ (v)$ , then $σ_{u v} = σ_{u} σ_{v}$ .

For this interface, a left divisor of $v$ means a Coxeter prefix: an element $u$ such that $v = u x$ and $ℓ (v) = ℓ (u) + ℓ (x)$ for some $x \in W$ . The relation thus defined is the right weak order $u \leq_{R} v$ . The map $w \mapsto σ_{w}$ is a set-theoretic section of $π$ ; it is not asserted to be a homomorphism.

Statement references

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

Artin monoid and Artin group presentations, and the canonical monoid-to-group map

Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares

Matsumoto's theorem: braid connectivity of reduced expressions, with singleton detection in dihedral subgroups

The right and left weak orders, intervals, covers, and meets and joins of subsets

Facts & Assumptions

[F1]

A reduced expression for $w \in W$ is a word in $S$ of length $ℓ (w)$ representing $w$ . (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups)

[F2]

Any two reduced expressions for the same element of $W$ are connected by finitely many braid moves, each replacing one alternating braid word by the other. (Matsumoto's theorem: braid connectivity of reduced expressions, with singleton detection in dihedral subgroups)

[F3]

The corresponding alternating words in the generators $σ_{s}$ are equal in $A$ , and $π : A \to W$ is a homomorphism with $π (σ_{s}) = s$ . (Artin monoid and Artin group presentations, and the canonical monoid-to-group map, Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares)

[F4]

Right weak order is defined by $u \leq_{R} v$ exactly when $v = u x$ and $ℓ (v) = ℓ (u) + ℓ (x)$ for some $x \in W$ . (The right and left weak orders, intervals, covers, and meets and joins of subsets)

Proof

Given: A finite-rank Coxeter system and its Artin group and projection as in the statement.

Goal: Establish the lift, its length-additive multiplication property, and the stated weak-order convention.

1.1F1F2F3

Choose two reduced expressions of $w$ . By [F2] one passes from one to the other by finitely many braid moves. Each move preserves the corresponding product in $A$ by [F3], including when the alternating block occurs inside a longer word. Thus the product $σ_{w}$ is well defined; the empty expression gives the identity. Applying the homomorphism $π$ to any such product gives the product of its letters in $W$ , namely $w$ .

2.1F1F4step 1.1∎

Choose reduced expressions for $u$ and $v$ . Their concatenation represents $u v$ and has length $ℓ (u) + ℓ (v)$ ; under the displayed length equality it is reduced by [F1]. Its Artin product is $σ_{u} σ_{v}$ , so step 1.1 identifies it with $σ_{u v}$ . The definition in [F4] is exactly the stated Coxeter-prefix condition. Finally, $π (σ_{w}) = w$ makes the resulting map a set-theoretic section; the argument asserts multiplication only under the displayed length condition.

Depends on

Used by · 14

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