LemmaStatement: 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.

Atomic reduced compositions and standard products use the same order

Statement

For a composable sequence of atoms $p_{i} = [I_{i - 1}, T_{i}, I_{i}]$ with $T_{i} = I_{i - 1} \cup {a_{i}}$ finitary, put $g_{i} = w_{I_{i - 1}} w_{T_{i}}$ . The minimum of $p_{i}$ is $g_{i} = v_{a_{i}, I_{i - 1}}$ , and $I_{i} = T_{i} ∖ {w_{T_{i}} a_{i} w_{T_{i}}}$ . The sequence is an atomic reduced expression precisely when $g_{1} \dots g_{m}$ is a standard expression. In that case the composed core coset has minimum $g_{1} \dots g_{m}$ . Products retain their displayed left-to-right order.

Facts & Assumptions

Given: The data and hypotheses in the Statement.

[F1]

The core-minimum dictionary gives the maximum of a core $(I, K)$ -coset as $w_{I} d = d w_{K}$ . Minimal representatives identify core cosets with root-transport arrows

[F2]

The finite wall formula identifies $v_{a, I} = w_{I} w_{I + a}$ and its target, and a standard expression means a composable product whose Coxeter lengths add. Simplified wall crossing for a finite parabolic Standard expressions and their lengths

[F3]

Atoms are the core cosets obtained by deleting one vertex from a finitary parabolic. Core and atomic double cosets

[F4]

Singular composition, evaluation of atomic expressions and sequential reducedness have the finitary conventions imported below from Ko, Definitions 2.1–2.3 and 3.9.

External input

For finitary $I, H, K$ , use Ko's Coxeter monoid $(W, ⋆)$ : the usual Coxeter braid relations and $s ⋆ s = s$ replace $s^{2} = 1$ , identifying underlying elements with $W$ . For an $(I, H)$ -coset $p$ and an $(H, K)$ -coset $q$ with maxima $P, Q$ , define $p ⋆ q = W_{I} (P ⋆ Q) W_{K}$ . It is reduced iff $ℓ ((P w_{H}^{- 1}) Q) = ℓ (P w_{H}^{- 1}) + ℓ (Q)$ ; then $(P w_{H}^{- 1}) Q$ is its maximum. Reduced composition preserves core cosets.

For adjacent nested finitary subsets, $[J_{0}, \dots, J_{n}] = (W_{J_{0}} e W_{J_{1}}) ⋆ \dots ⋆ (W_{J_{n - 1}} e W_{J_{n}})$ . An atomic expression concatenates triples $[I_{i - 1}, T_{i}, I_{i}]$ , each denoting its atom. Evaluate it by $r_{0} = W_{I_{0}}$ , $r_{i} = r_{i - 1} ⋆ p_{i}$ . It is reduced iff each prefix composition $r_{i - 1} ⋆ p_{i}$ is reduced. This is Ko's grouped form of sequential reducedness (Definitions 2.1–2.3 and the following equivalence; Section 1.3; Definition 3.9). The empty expression represents $W_{I_{0}}$ and is reduced (Remark 3.11).

Proof

Given: The data and hypotheses in the Statement.

[F1] The core-minimum dictionary gives the maximum of a core $(I, K)$ -coset as $w_{I} d = d w_{K}$ . Minimal representatives identify core cosets with root-transport arrows

[F2] The finite wall formula identifies $v_{a, I} = w_{I} w_{I + a}$ and its target, and a standard expression means a composable product whose Coxeter lengths add. Simplified wall crossing for a finite parabolic Standard expressions and their lengths

[F3] Atoms are the core cosets obtained by deleting one vertex from a finitary parabolic. Core and atomic double cosets

[F4] Singular composition, evaluation of atomic expressions and sequential reducedness have the finitary conventions imported below from Ko, Definitions 2.1–2.3 and 3.9.

Proof technique: direct.

Goal: Establish the stated dictionary or identification.

1.1F1F2F3

The atom has maximum $w_{T_{i}}$ , so [F1] gives its minimum $g_{i} = w_{I_{i - 1}} w_{T_{i}}$ . Its core condition forces $I_{i} = w_{T_{i}} I_{i - 1} w_{T_{i}}$ , hence the displayed deleted vertex. The longest element on the source changes negative simple roots to positive ones, so this agrees with the root-transport orientation in [F2].

1.2F1F4

Let two core cosets have minima $x, y$ and intermediate subset $H$ . By [F1], their maxima are $P = w_{I} x = x w_{H}$ and $Q = w_{H} y$ . Thus [F4] tests whether $x (w_{H} y)$ is length additive. The product $x y$ is a root-transport arrow, so [F1] gives $ℓ (w_{I} x y) = ℓ (w_{I}) + ℓ (x y)$ and $ℓ (w_{H} y) = ℓ (w_{H}) + ℓ (y)$ . Conjugation by $x$ bijects the parabolic simple generating sets $H$ and $I$ , preserving their word lengths in both directions; hence $ℓ (w_{H}) = ℓ (w_{I})$ . Since $x w_{H} = w_{I} x$ , the criterion is equivalent to $ℓ (x y) = ℓ (x) + ℓ (y)$ . When it holds, [F4] gives maximum $w_{I} x y$ and a core composite, whose minimum is $x y$ by [F1].

2.1F1F2F4step 1.1step 1.2∎

Evaluate the atomic expression by the prefixes $r_{i}$ of [F4]. The initial coset $r_{0} = W_{I_{0}}$ is core with minimum $e$ . Whenever the prefix is reduced, step 1.2 and step 1.1 show that its minimum is $g_{1} \dots g_{i}$ and that adjoining $p_{i + 1}$ is reduced exactly when $ℓ (g_{1} \dots g_{i} g_{i + 1}) = ℓ (g_{1} \dots g_{i}) + ℓ (g_{i + 1})$ . Thus [F4] makes atomic reducedness equivalent to all these prefix length equalities. They telescope to the total length equality; conversely, a shorter word for any prefix, followed by shortest words for the remaining factors, would contradict the total equality. By [F2] this is exactly a standard expression. Step 1.2 gives the asserted minimum of its composite, with no reversal of factors. The empty case has minimum $e$ and length sum zero on both sides.

Depends on

Used by · 3

Dependency tree · 2 levels

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External dependencies

Hankyung Ko, An atomic Coxeter presentation, arXiv:2312.16666v3 · Section 2.2 and Definition 2.1, PDF p. 9, equation (2.4); Definitions 2.2–2.3 and following equivalence, p. 10; Section 1.3, p. 4; Lemma 3.4, p. 12; Definition 3.9 and Remark 3.11, p. 13; Definition 1.4 and equation (1.4), p. 3

For finitary I,H,K and cosets p:(I,H), q:(H,K), singular composition is p*q=W_I(P*Q)W_K, with P,Q the maxima and * the Coxeter monoid product. It is reduced exactly when (P w_H^{-1})Q is length additive, then has that maximum, and preserves core cosets. Atomic expressions concatenate atomic triples; reducedness tests successive prefix compositions. The empty expression at I represents W_I.

Hypotheses: An arbitrary Coxeter system (W,S); every subset indexing a coset or expression is finitary. Each atomic triple has a finitary larger subset and is core. No finite ambient Coxeter group is assumed.

Reading limits: For this repair, read the extracted v3 PDF text at Section 2.2, Definitions 2.1–2.3, equation (2.4), the equivalence immediately after Definition 2.3, Section 1.3, Lemma 3.4, Definition 3.9, Remark 3.11 and Definition 1.4, with their immediate context. PDF screenshot retrieval failed; no successful visual inspection is claimed. The displayed composition and reducedness definitions and the stated core-closure result are imported, not reproved. The papers cited by Ko for these results were not independently read or reconstructed.

Sources

Hankyung Ko, An atomic Coxeter presentation, arXiv:2312.16666v3 · Section 2.2 and Definition 2.1, PDF p. 9, equation (2.4); Definitions 2.2–2.3 and following equivalence, p. 10; Section 1.3, p. 4; Lemma 3.4, p. 12; Definition 3.9 and Remark 3.11, p. 13; Definition 1.4 and equation (1.4), p. 3