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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Simultaneous elementary prefixes force the relative longest prefix

Statement

Let q be the morphism of Finite relative longest element without spherical ambient rank. If an article morphism w:I→K′ has both va,I and vb,I as length-additive left divisors, then q⪯w and q−1w is an article morphism from the target of q to K′.

Facts & Assumptions

[F1]

The definitions and results used in this argument are supplied, under their stated hypotheses, by Finite relative longest element without spherical ambient rank; Elementary inversions detect the two initial prefixes; Standard expressions and their lengths.

Proof

Given: The stated Coxeter and groupoid data and the explicit independent inputs cited below.

Goal: Establish the assertion in the Statement.

1.1given

By Elementary inversions detect the two initial prefixes and §1.5, αa,αb∈N(w). The inverse label u=w−1 is an ordinary root-action arrow out of I (the article condition is wΠK′=ΠI). Thus {αa,αb}⊆Π∩Φu−1. Apply Dyer–Gimenez Lemma 3.12(a): ν({a,b},I)−1⪯u−1=w. By Finite relative longest element without spherical ambient rank its left side is q.

2.1step 1.1∎

If q:I→K, then q−1wΠK′=q−1ΠI=ΠK. This is the required article morphism K→K′, and the prefix conclusion gives ℓ(w)=ℓ(q)+ℓ(q−1w).

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 · Lemma 3.12(a), PDF pp. 42–43; §1.5, PDF p. 4

For a root-action arrow u:I→K, L⊆Π∩Φ_(u^-1) implies ν(L,I)^-1≤u^-1. Weak order is length-additive prefix order.

Hypotheses: Arbitrary-rank Coxeter system with standard based root system; a Brink–Howlett component. For ν(M,I), M is disjoint from I and the relative root difference is finite. No spherical ambient-system hypothesis.

Reading limits: Read extracted definitions and statements in §§1.4–1.5, 1.8–1.9, 1.19–1.27, 3.1–3.4, Proposition 3.11, Lemma 3.12 (including its proof), Proposition 3.13 and Proposition 3.14. Visually inspected PDF pp. 10, 11 and 43. No complete reading of this paper or reconstruction of its rank-two presentation proof is claimed. Brink–Howlett (1999) was not accessible and was not read. Section 1.27 prints ν(α,K),ν(β,K) although α,β belong to K, contrary to Remark 1.26; only its unambiguous exactly-two minimum-word assertion is imported, and initial factors are derived independently.

Sources