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Two ribbon-lcm expressions and presentation

PDF p. 22: Remark 5.7; (5.8)

Statement

Let $Γ$ be a finite-rank spherical Coxeter diagram with vertex set $S$ , let $A^{+}$ be its positive Artin monoid, and fix $J \subseteq S$ . Use the reduced ribbon groupoid $R (J)$ and elementary arrows $μ_{a, I} = Δ_{I}^{- 1} Δ_{I \cup {a}} : I \to K$ of Reduced ribbon groupoid, where $I$ ranges over the associates of $J$ and $a \in S ∖ I$ . Thus an associate means a subset whose standard parabolic Coxeter subgroup is conjugate to that of $J$ .

For distinct $a, b \in S ∖ I$ , the least common multiple $μ_{a, I} \lor μ_{b, I}$ for left divisibility in $A^{+}$ is represented by a morphism of $R (J)$ with source $I$ . It has exactly two expressions as composable products of elementary arrows: one begins with $μ_{a, I}$ and the other with $μ_{b, I}$ . If their lengths are $m$ and $n$ , respectively, these expressions have the form $μ_{a_{1}, I_{1}} \dots μ_{a_{m}, I_{m}}$ and $μ_{b_{1}, L_{1}} \dots μ_{b_{n}, L_{n}}$ , with $a_{1} = a$ , $b_{1} = b$ , and $I_{1} = L_{1} = I$ .

The elementary arrows, their formal inverses, and the groupoid identity and inverse laws generate $R (J)$ , and the equalities between these two expressions, over all associates $I$ and all distinct pairs $a, b \in S ∖ I$ , give a complete set of defining relations. For $a = b$ , the lcm is $μ_{a, I}$ and has one elementary-arrow expression; the corresponding equality is tautological.

Statement references

Reduced ribbon groupoid

Garside divisibility and lcm assertions for A+

Reduced-ribbon lcm presentation including the empty subset

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

Artin–Tits group, pure subgroup and positive lifts

Set-sized groupoids, functors and composition conversion

Facts & Assumptions

Given: A finite-rank spherical diagram, a subset $J \subseteq S$ , an associate $I$ of $J$ , and, for the two-path assertion, distinct $a, b \in S ∖ I$ .

[F1]

The reduced ribbon groupoid has exactly the associates of $J$ as objects and is generated by the displayed elementary arrows and their inverses. Each elementary arrow has an associate of $J$ as its target. Reduced ribbon groupoid

[F2]

The elementary ribbons belong to $A^{+}$ , and their left-divisibility lcm exists there. Left divisibility means $x$ divides $y$ when $y = x z$ for some $z \in A^{+}$ . Garside divisibility and lcm assertions for A+

[F3]

For the spherical Artin system, the all-subsets reduced ribbon groupoid has the presentation by elementary arrows and pairwise lcm equalities. For distinct $a, b$ outside a subset $I$ , the lcm has exactly two elementary-arrow paths, starting respectively with the two specified arrows. This includes $I = \emptyset$ , where the paths are the two alternating braid words. Reduced-ribbon lcm presentation including the empty subset

[F4]

The Coxeter diagram, its Coxeter system and the spherical convention are supplied by Coxeter-system, roots, length and weak-order conventions. The associated finite-rank Artin–Tits group and its standard generators, identified with the presented Artin group, are supplied by Artin–Tits group, pure subgroup and positive lifts.

[F5]

A groupoid is a category in which every arrow has a two-sided inverse. For composable arrows $u : X \to Y$ and $v : Y \to Z$ , the product $u v = v \circ u$ is read from left to right; associativity, identities and inverse laws use Set-sized groupoids, functors and composition conversion.

External input

Godelle’s positive ribbon category has each elementary ribbon as an atom; an atom has only one-edge representing paths in the atom graph, with equal edges identified. The spherical Artin monoid has the required standard nu-structure.

Proof

technique · direct

Goal: Prove the two-path statement, the presentation on the associate class, and the diagonal qualification.

1.1givenF1F2F3F4

The hypotheses of F3 hold for the given spherical system and distinct $a, b \notin I$ . Its elementary arrow $ν (I, a) = Δ_{I}^{- 1} Δ_{I \cup {a}}$ is exactly $μ_{a, I}$ by F1. F2 identifies the lcm as the left-divisibility lcm in $A^{+}$ . F3 therefore supplies the two and only two elementary-arrow paths with the stated initial arrows. Since each arrow starting at an associate ends at an associate by F1, both paths lie in $R (J)$ and start at $I$ ; writing their successive labels gives the displayed form.

2.1F1F3F5step 1.1

Restrict the presentation in F3 to objects that are associates of $J$ . By F1 every elementary arrow and inverse incident to an associate remains in this object set, so any word in the full generating graph that starts at an associate stays there. Each defining lcm equality has two paths with the same source and target; consequently any application of a defining equality or a groupoid inverse or identity law to such a word also stays in the restricted graph. Thus the restricted relations give all and only the relations among its paths, and F1 identifies the resulting subgroupoid with $R (J)$ .

3.1F2step 1.1∎

If $a = b$ , its lcm with itself is $μ_{a, I}$ . By the independent atomicity input, every positive elementary-ribbon path representing this arrow consists of one edge; equal edges in the atom graph represent the same arrow. Thus its sole elementary-arrow expression is the one-arrow path, and the diagonal equality is tautological.

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

Eddy Godelle, Parabolic subgroups of Garside groups II: ribbons · §2.3, printed p. 4; §4.2 opening, printed p. 20; Example 3.14, printed p. 15

In a spherical Artin positive ribbon category, the elementary nu ribbons are atoms; an atom has only one-edge representing paths in the atom graph.

Hypotheses: A finite-rank spherical Artin–Tits monoid with its standard nu-structure; the positive ribbon category with parabolic submonoids as objects.

Reading limits: Read the category atom definition in §2.3 and the opening of §4.2, including its reference to Example 3.14, and the end of Corollary 4.10. The proof of the nu-structure and the full article are not claimed read or reconstructed.

Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 22, Remark 5.7 and equation (5.8)
Eddy Godelle, Parabolic subgroups of Garside groups II: ribbons · §2.3, printed p. 4; §4.2 opening, printed p. 20; Example 3.14, printed p. 15