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.

Invariant choices of facet points for finite linear symmetry groups

Statement

Let $A$ be a finite arrangement of proper linear hyperplanes in a finite-dimensional real vector space $E$ , and let a finite group $G$ act linearly, permuting $A$ . Set $I = E$ in the facet and induced-action conventions of Sign facets of a cone arrangement, Facet incidence and localized chamber signs and Linear symmetries act on the Salvetti model. There are choices $x (F) \in F$ for every facet satisfying $x (g F) = g x (F)$ for all $g \in G$ .

Statement references

Sign facets of a cone arrangement

Facet incidence and localized chamber signs

Linear symmetries act on the Salvetti model

Facts & Assumptions

Given: The finite arrangement and group in the Statement, with $I = E$ . Goal: Make equivariant facet choices.

[F1]

With $I = E$ , the finite arrangement is locally finite in a nonempty open convex cone. Facets are nonempty relatively open convex sign cones with the conventions of Sign facets of a cone arrangement and Facet incidence and localized chamber signs.

[F2]

Every $g \in G$ preserves $I = E$ and permutes $A$ , so Linear symmetries act on the Salvetti model applies and gives an action on the facets and their incidence.

Proof

technique · direct

1.1F1F2given

There are finitely many facets, since each is specified by a sign in ${-, 0, +}$ for each member of the finite arrangement. For one facet $F$ in each $G$ -orbit choose $v_{F} \in F$ , and let $G_{F} = {g : g F = F}$ . Put $x (F) = ∣ G_{F} ∣^{- 1} \sum_{h \in G_{F}} h v_{F}$ . Every summand lies in the convex set $F$ , so $x (F) \in F$ . Reindexing the finite sum shows that $G_{F}$ fixes $x (F)$ .

2.1step 1.1∎

Set $x (g F) = g x (F)$ on that orbit. If $g F = g^{'} F$ , then $g^{- 1} g^{'} \in G_{F}$ , so the assigned points coincide. This proves well-definedness and equivariance. The construction also covers the facet containing the origin and the empty arrangement; only finite choices were used.

Depends on

Used by · 2

Dependency tree · 2 levels

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