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.

Finite relative longest element without spherical ambient rank

Statement

Let $(W, S)$ be a Coxeter system with $S = {s_{i} : i \in Γ_{0}}$ , where $Γ_{0}$ may be infinite, and use its standard geometric root system $Φ = Φ^{+} ⊔ Φ^{-}$ . For $L \subseteq Γ_{0}$ , write $W_{L}$ and $Φ_{L}$ for its standard parabolic subgroup and roots, and put $Π_{L} = {α_{i} : i \in L}$ . Let $I \subseteq Γ_{0}$ , let $a, b \in Γ_{0} ∖ I$ be distinct, and put $T = I \cup {a, b}$ . Suppose $∣ Φ_{T} ∖ Φ_{I} ∣ < \infty$ . Let $D$ be the union of the connected components of the diagram induced on $T$ that meet ${a, b}$ . Then $D$ is spherical, meaning $W_{D}$ is finite. With $w_{L}$ denoting the longest element whenever $W_{L}$ is finite, $r = w_{D} w_{D \cap I}$ maps $Π_{I}$ onto $Π_{K}$ for an associate $K \subseteq T$ of $I$ , and $q = r^{- 1}$ is an article Brink–Howlett morphism $I \to K$ in the groupoid of associates of $I$ . Writing $N (x) = Φ^{+} \cap x Φ^{-}$ , one has $N (q) = Φ_{T}^{+} ∖ Φ_{I}^{+}$ .

Statement references

Coxeter-system, roots, length and weak-order conventions

Standard parabolic subgroups and parabolic roots

Longest parabolic elements and opposition permutation

Associates of a parabolic subset

Brink–Howlett groupoid

Opposite-category convention for Brink–Howlett

Facts & Assumptions

[F1]

The ambient diagram and geometric action, parabolic root sets, longest elements, associates and article morphisms use Coxeter-system, roots, length and weak-order conventions; Standard parabolic subgroups and parabolic roots; Longest parabolic elements and opposition permutation; Associates of a parabolic subset; Brink–Howlett groupoid; Opposite-category convention for Brink–Howlett.

[F2]

The root-sign decomposition and parabolic root–reflection correspondence in the External input section are the independent inputs from Dyer–Gimenez §§1.2–1.4 and 1.28.

[F3]

The finite-relative-longest result in the External input section is Dyer–Gimenez Lemma 1.25(a),(c), with the component convention preceding it.

External input

Dyer–Gimenez §§1.2–1.4 and 1.28 give $Φ = Φ^{+} ⊔ (- Φ^{+})$ , the bijection $α \mapsto s_{α}$ from $Φ^{+}$ to $Ref (W)$ , and $Φ_{L} = {α \in Φ : s_{α} \in W_{L}}$ for every standard parabolic $W_{L}$ . Thus this bijection restricts to $Φ_{L}^{+} \leftrightarrow W_{L} \cap Ref (W)$ . Here $Ref (W) = {w s_{i} w^{- 1} : w \in W, i \in Γ_{0}}$ . The reflection formula and the $W$ -invariance of the bilinear form give $s_{w α} = w s_{α} w^{- 1}$ . These inputs apply in arbitrary rank. Root-system and parabolic-subsystem source.

Dyer–Gimenez Lemma 1.25 says that for simple-root subsets $J_{0} \subseteq H_{0}$ , finiteness of $(W_{H_{0}} ∖ W_{J_{0}}) \cap Ref (W)$ makes the union $D_{0}$ of components of $H_{0}$ meeting $H_{0} ∖ J_{0}$ spherical. For $r_{0} = w_{D_{0}} w_{D_{0} \cap J_{0}}$ , it gives $r_{0} J_{0} \subseteq H_{0}$ as a simple-root subset and $Φ^{+} \cap r_{0}^{- 1} Φ^{-} = Φ_{H_{0}}^{+} ∖ Φ_{J_{0}}^{+}$ . The notation here treats $J_{0}, H_{0}, D_{0}$ as sets of simple roots, as in that source. Relative-longest source, Lemma 1.25(a),(c).

Proof

Given: The Coxeter system, subsets and finite relative-root difference in the Statement, the definitions in [F1], and the explicitly imported premises [F2]–[F3].

Goal: Establish the stated relative-longest construction and inversion set.

1.1F1F2given

By [F2], $α \mapsto s_{α}$ identifies $Φ_{T}^{+} ∖ Φ_{I}^{+}$ with $(W_{T} ∖ W_{I}) \cap Ref (W)$ : its restrictions to the two nested parabolics are bijections. Also $Φ_{T} ∖ Φ_{I}$ is the disjoint union of $Φ_{T}^{+} ∖ Φ_{I}^{+}$ and its negative, since both parabolic root sets are closed under negation. Thus finiteness of the full root difference is equivalent to finiteness of that reflection difference.

2.1F1F3step 1.1

Apply [F3] to the simple-root subsets $J_{0} = Π_{I} \subseteq H_{0} = Π_{T}$ . Under the index-to-simple-root identification, $D_{0} = Π_{D}$ . Hence $W_{D}$ is finite, as is its subgroup $W_{D \cap I}$ , so the longest elements in $r = w_{D} w_{D \cap I}$ are defined. The imported conclusions give $r Π_{I} = Π_{K}$ for some $K \subseteq T$ , and $N (r^{- 1}) = Φ_{T}^{+} ∖ Φ_{I}^{+}$ . This is the source element $ν ({a, b}, I)$ with its arguments written in index notation.

3.1F1F2step 2.1∎

By the reflection equivariance in [F2], $r Π_{I} = Π_{K}$ gives $r W_{I} r^{- 1} = W_{K}$ , so $K$ is an associate of $I$ . Inverting the root-set equality gives $q Π_{K} = Π_{I}$ for $q = r^{- 1}$ . By Brink–Howlett groupoid, this is precisely an article morphism $I \to K$ ; Opposite-category convention for Brink–Howlett identifies the same arrow convention. Step 2.1 supplies the required $N (q)$ . No finiteness condition has been placed on components of $I$ outside $D$ .

Depends on

Used by · 8

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

Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · §1.24 (component convention) and Lemma 1.25, especially (a),(c), PDF/printed p. 10; arbitrary-rank convention §1.4, p. 4

For simple-root subsets J⊆H, finiteness of (W_H\W_J)∩Ref(W) is equivalent to finitely many components of H meeting H\J, all of finite type. Their union D is spherical. For r=ν(H\J,J)=w_D w_(D∩J), Lemma 1.25(a) gives Φ_H^+\Φ_J^+=Φ_D^+\Φ_(D∩J)^+=Φ^+∩r^(-1)Φ^-, and (c) gives rJ=(J\D)∪r(J∩D)⊆H.

Hypotheses: A Coxeter system with a based root system, including the standard geometric root system; J⊆H⊆Π and (W_H\W_J)∩Ref(W) finite. Arbitrary rank is allowed; neither W nor W_J is assumed finite.

Reading limits: For this repair, read the frozen extracted §1.24 and Lemma 1.25 and visually checked Lemma 1.25 on PDF p. 10; read the arbitrary-rank convention in §1.4. Only the component convention and the stated relative-longest result are imported. No reconstruction of its proof or reading of Brink–Howlett (1999) is claimed.

Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · §§1.2–1.4, PDF/printed pp. 3–4 (reflections, root signs and positive-root bijection); §1.28, PDF/printed p. 11 (reflection-subgroup root subsystem and standard parabolics)

The standard based root system has the disjoint decomposition Φ=Φ^+⊔Φ^- with Φ^-=-Φ^+. The map α↦s_α is a bijection Φ^+→Ref(W). For a reflection subgroup U, its root subsystem is Φ_U={α∈Φ:s_α∈U}; for U=W_L its canonical simple-root set is Π_L. Consequently α↦s_α restricts to Φ_L^+↔W_L∩Ref(W). The reflection formula in §1.2 is equivariant under W⊆O(V,B): s_(wα)=w s_α w^(-1).

Hypotheses: A Coxeter system with its based root system in (V,B), in particular the standard geometric root system; L is any subset of its simple indices. The ambient system and W_L may have infinite rank or be infinite.

Reading limits: For this repair, read the frozen extracted §§1.2–1.4 and §1.28 through its reflection-subgroup conventions, and visually checked the root-subsystem formulas on PDF p. 11. Import only these statements and definitions, not their background proofs or any corank theorem. The equivariance identity follows directly by substituting the B-preserving W-action into the displayed reflection formula. No reading of the cited background references is claimed.

Sources

Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · §1.24 (component convention) and Lemma 1.25, especially (a),(c), PDF/printed p. 10; arbitrary-rank convention §1.4, p. 4
Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · §§1.2–1.4, PDF/printed pp. 3–4 (reflections, root signs and positive-root bijection); §1.28, PDF/printed p. 11 (reflection-subgroup root subsystem and standard parabolics)