LemmaStatement: AI-adaptedProof: 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.

Positive gallery category, cancellation and length

Statement

Let $V$ be a finite-dimensional real vector space and $A$ a finite set of linear hyperplanes in $V$ (a finite central arrangement). Chambers are the connected components of $V ∖ ⋃_{H \in A} H$ . Assume every chamber is an open simplicial cone in $V$ , meaning the set of positive linear combinations of some basis of $V$ . A wall face of a chamber $C$ is an intersection $\overline{C} \cap H$ of dimension $\dim V - 1$ with $H \in A$ ; its supporting wall hyperplane is $H$ , distinct from that face. Distinct chambers are adjacent when they share a wall face.

A gallery $(C_{0}, \dots, C_{n})$ has adjacent consecutive chambers and length $n$ ; a length-zero gallery consists of one chamber. It is minimal if its length is least among galleries with the same endpoints. Form the category $P$ with chambers as objects and galleries as arrows, modulo the path congruence generated by replacing any minimal subgallery by another with the same endpoints. Products follow traversal order, and empty paths give identities.

For every pair $A, B$ , a minimal gallery exists and its class $u (A, B)$ is unique. Then $P$ is small, cancellative and atomic, with additive nonnegative integer length, length-zero arrows exactly identities, and atoms exactly one-step gallery classes. Here an atom is a nonidentity arrow admitting no factorization into two nonidentity arrows; atomicity means bounded lengths of factorizations of each arrow into nonidentity arrows. Reversal $(C_{0}, \dots, C_{n}) \mapsto (C_{n}, \dots, C_{0})$ induces an involutive anti-isomorphism $g \mapsto g^{*}$ . The antipodal map $(C_{0}, \dots, C_{n}) \mapsto (- C_{0}, \dots, - C_{n})$ induces an involutive automorphism $g \mapsto - g$ .

Statement references

Path presentations, congruences and formal inverses

Locally finite arrangements and relative face terminology

Divisibility and atomicity in a category

External input

[E1] Deligne (1.3) supplies existence and the separating-hyperplane length formula; (1.10), (1.12)–(1.13) supply the rank-two congruence and minimal class; (1.19)(i),(ii) supply two-sided cancellation. These are imported results under the Statement's hypotheses.

Facts & Assumptions

[F1]

Use the path-category and quotient conventions of Path presentations, congruences and formal inverses.

[F2]

Use Locally finite arrangements and relative face terminology with $T = V$ for chambers, wall faces, adjacency, galleries and open simplicial cones; retain the Statement's assumption that the chambers themselves are open simplicial cones in $V$ .

[F3]

Use Divisibility and atomicity in a category for cancellation, atoms and atomicity.

[F4]

The results stated in External input apply under the finite central simplicial hypotheses; their substantial proofs are imported, not reconstructed.

Proof

Given: The data and hypotheses in the Statement.

Goal: Establish the Statement.

1.1F1F2

The chamber-adjacency quiver has one directed edge $C \to D$ for each ordered pair of adjacent chambers. Choose for each $H \in A$ a nonzero linear functional with kernel $H$ . Each nonempty region with prescribed strict signs is convex, hence connected, and each connected component of the complement has constant signs. Thus chambers are exactly these regions, so there are finitely many. Finite paths on this finite quiver form a set. The endpoint-preserving quotient construction therefore gives a small category $P$ .

2.1F1F2F3step 1.1

Two minimal galleries with the same endpoints have the same length by the definition of minimality, so every defining replacement preserves length. Hence gallery length descends to an additive nonnegative integer length on $P$ . A zero-length arrow is represented by the empty path and is an identity. A one-step class cannot factor into two nonidentity arrows, since both factors would have positive length; every longer path factors after its first edge into two positive-length paths. These are exactly the atoms. For an arrow of length $n$ , every factorization into nonidentity arrows has at most $n$ factors, proving atomicity in the stated sense.

2.2F1F2F3F4step 1.1

Deligne (1.3) gives a minimal gallery between every pair of chambers. His rank-two replacements (1.10) are minimal replacements in our presentation; conversely (1.12) equates every pair of minimal galleries using his replacements. The congruences therefore coincide. In particular $u (A, B)$ is well-defined, and (1.19)(i),(ii) give both cancellations for $P$ .

3.1F1F2step 1.1∎

Reversal preserves adjacency and path length, and sends a least-length path to a least-length path: a shorter reversed path would reverse to a shorter original path. Since every hyperplane is linear, negation permutes the hyperplanes and chambers, preserves adjacency, and likewise preserves minimality. Both operations therefore preserve the generating relations and descend to the quotient. On paths they square to the identity and satisfy $(g h)^{*} = h^{*} g^{*}$ and $- (g h) = (- g) (- h)$ , so they give the required involutive anti-isomorphism and automorphism.

Depends on

Used by · 5

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Pierre Deligne, Les immeubles des groupes de tresses généralisés, Inventiones mathematicae 17 (1972), 273–302 · §1, (1.1)–(1.3), (1.8), (1.10), (1.12)–(1.13), (1.19)(i),(ii); printed pp. 274–275, 277–278, 280

Every pair of chambers has a minimal gallery of length equal to the number of separating hyperplanes. The rank-two replacements of (1.10) equate all minimal galleries with the same endpoints, and their gallery congruence admits cancellation on both sides.

Hypotheses: A finite set of homogeneous real hyperplanes in a finite-dimensional real vector space, with every chamber an open simplicial cone in that space; composition follows traversal order.

Reading limits: Checked the PDF text at the listed definitions, hypotheses and result statements, including the argument of (1.3) and the presentation comparison using (1.12). Cancellation is imported from (1.19)(i),(ii), without reconstructing its proof. No whole-paper reading or rendered-formula inspection is claimed.

Sources

Pierre Deligne, Les immeubles des groupes de tresses généralisés, Inventiones mathematicae 17 (1972), 273–302 · §1, (1.1)–(1.3), (1.8), (1.10), (1.12)–(1.13), (1.19)(i),(ii); printed pp. 274–275, 277–278, 280