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.

Minimum representatives of normaliser cosets preserve the simple roots

Statement

Let $(W, S)$ be an arbitrary-rank Coxeter system in its standard real root representation, with simple roots $Π = {α_{i} : i \in Γ_{0}}$ and length $ℓ$ . Fix $J \subseteq Γ_{0}$ , put $A_{J} = {α_{j} : j \in J}$ and $W_{J} = ⟨ s_{j} : j \in J ⟩$ , and write $M = {Norm}_{W} (W_{J})$ and $N_{J} = {n \in W : n A_{J} = A_{J}}$ . These are the conventions of Root-permuting normaliser complement; $N_{J}$ is a subgroup of $M$ .

For every $g \in M$ , the coset $g W_{J}$ has a unique minimum-length element $d$ , and $d A_{J} = A_{J}$ . In particular $d \in N_{J}$ .

External input

Dyer–Gimenez, Proposition 2.10(b),(d), states that for a reflection subgroup $U$ and $x \in W$ , the coset $x U$ has a unique minimum-length element $d$ and the canonical simple system satisfies $Π_{x U x^{- 1}} = d Π_{U}$ . Proposition 2.11(c) states that for the standard parabolic subgroup $W_{J}$ , its canonical simple system is $Π_{W_{J}} = A_{J}$ . Here the canonical simple system is the distinguished positive simple-root system of the reflection subgroup in the ambient based root system.

Facts & Assumptions

[F99]

Proof

Given: The data in the statement. Goal: Prove its conclusion.

[F1] The declared suppliers provide the subgroup and previously established assertions:

Root-permuting normaliser complement

Arbitrary-rank Coxeter presentations, length, reflections, and Bruhat order

Given: The reflection subgroup and normalising element in the statement.

1.1F1givenF99

Apply the external input to the reflection subgroup $U = W_{J}$ and $x = g$ . It gives a unique minimum $d$ in $g W_{J}$ and $Π_{g W_{J} g^{- 1}} = d Π_{W_{J}}$ . Since $g$ normalises $W_{J}$ , the left side is $Π_{W_{J}}$ . The standard-parabolic part of the same input identifies this canonical simple system with $A_{J}$ . Thus $A_{J} = d A_{J}$ .

2.1F1step 1.1∎

The equality says exactly that $d \in N_{J}$ , by [F1]. No part of the argument requires $J$ or $S$ finite.

Source notes

The displayed external input is imported from the exact independent source locations recorded above. The argument applies those assertions to a standard parabolic subgroup; it does not reconstruct the source proofs or use the article under study as a prerequisite.

Depends on

Used by · 2

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 · §1.4, PDF p.4; Proposition 2.10(b),(d) and Proposition 2.11(c), PDF p.23

Dyer–Gimenez, Proposition 2.10(b),(d), states that for a reflection subgroup $U$ and $x\in W$, the coset $xU$ has a unique minimum-length element $d$ and the canonical simple system satisfies $\Pi_{xUx^{-1}}=d\Pi_U$. Proposition 2.11(c) states that for the standard parabolic subgroup $W_J$, its canonical simple system is $\Pi_{W_J}=A_J$. Here the canonical simple system is the distinguished positive simple-root system of the reflection subgroup in the ambient based root system.

Hypotheses: Arbitrary-rank Coxeter system with standard based root system; arbitrary subset of simple roots; W_J is a reflection subgroup. No sphericity or finite-rank assumption.

Reading limits: Checked the retrieved v1 PDF text in §§1.3–1.4, §1.7 and §2.8, Proposition 2.10(b),(d) and Proposition 2.11(c), PDF pp.3–5 and 22–23, including the displayed proof/reference paragraphs relevant to the imported assertions. Only the exact stated inputs and their application are checked; the older works cited there were not read, and no reconstruction of their proofs or visual PDF inspection is claimed. Infinite rank is explicitly permitted in §1.4.

Sources

Matthew Dyer and Harrison Gimenez, Root systems, Tits cones and imaginary cones of Brink–Howlett groupoids · §1.4, PDF p.4; Proposition 2.10(b),(d) and Proposition 2.11(c), PDF p.23