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Reduced-ribbon lcm presentation including the empty subset

Statement

Let $A^{+}$ be the positive monoid of a finite-rank spherical Artin system on $S$ , and $A$ its Artin group; identify each $s \in S$ with its standard positive generator. For $I \subseteq S$ , let $W_{I}$ be the standard parabolic Coxeter subgroup, $w_{I}$ its longest element, and $Δ_{I}$ its positive reduced-word lift in $A^{+}$ . In particular $Δ_{\emptyset} = 1$ . Write $A_{I}^{+}$ for the submonoid generated by $I$ , including the empty product, and $A_{I}$ for the subgroup generated by $I$ .

Use the ambient ribbon groupoid with objects all subsets $I \subseteq S$ and arrows $(I, g, J)$ for $g \in A$ satisfying $g^{- 1} A_{I}^{+} g = A_{J}^{+}$ . Arrows are equal exactly when their triples are equal. Composition is $(I, g, J) (J, h, K) = (I, g h, K)$ , the identity at $I$ is $(I, 1, I)$ , and the inverse of $(I, g, J)$ is $(J, g^{- 1}, I)$ . Positive arrows have $g \in A^{+}$ .

For $I \subseteq S$ and $a \notin I$ put $ν (I, a) = Δ_{I}^{- 1} Δ_{I \cup {a}}$ . This is positive, and $ν (I, a)^{- 1} I ν (I, a)$ is a subset $J$ of $S$ ; regard it as the arrow $(I, ν (I, a), J)$ . Let $R^{ν} (A)$ be the subgroupoid on the full object set generated by these arrows, their inverses and all object identities. Here a reduced-ribbon path is a finite composable sequence of the positive generators $ν (I, a)$ , and its label is their product in $A^{+}$ . For each $I \subseteq S$ and distinct $a, b \notin I$ , let $d$ be the lcm of $ν (I, a)$ and $ν (I, b)$ for left divisibility in $A^{+}$ . Among reduced-ribbon paths with source $I$ , exactly two have label $d$ , one starting with each of these arrows. These two paths represent the same ribbon arrow with source $I$ and label $d$ . The groupoid $R^{ν} (A)$ has a presentation by the displayed generators and the equalities between these two paths for every such $I, a, b$ . This includes $I = \emptyset$ : then $ν (\emptyset, a) = a$ , and the two paths are the alternating braid words of length $m_{a b}$ .

Statement references

Independent spherical Artin Garside input

Set-sized groupoids, functors and composition conversion

Braid congruence preserves word length and support

Rigidity of two-letter words below braid length

Exactly two words for a two-generator braid element

The two-generator braid element is the lcm

The empty-object ribbon component and its braid relations

Opposition as conjugation by a longest Artin lift

Elementary ribbons have explicitly determined endpoints

The reduced ribbon subgroupoid uses exactly the exterior atoms

Facts & Assumptions

[F1]

The positive monoid, standard generators, Garside structure and both-sided lcms are supplied by Independent spherical Artin Garside input; arrows, inverses and traversal-order composition use Set-sized groupoids, functors and composition conversion.

[F2]

The positive braid congruence preserves support and degree, short two-letter words are rigid, and a full braid word has exactly its two alternating representatives. (Braid congruence preserves word length and support, Rigidity of two-letter words below braid length, Exactly two words for a two-generator braid element)

[F3]

The pairwise generator lcm is that full braid word. (The two-generator braid element is the lcm)

[F4]

The empty ribbon object is an isolated one-object component with exactly this Artin presentation. (The empty-object ribbon component and its braid relations)

[F5]

The parabolic longest elements, their well-defined positive lifts $Δ_{I}$ and the submonoids $A_{I}^{+}$ and subgroups $A_{I}$ are supplied by Opposition as conjugation by a longest Artin lift.

[F6]

Each $ν (I, a)$ with $a \notin I$ is positive and has an endpoint $J \subseteq S$ with $ν (I, a)^{- 1} I ν (I, a) = J$ and $ν (I, a)^{- 1} A_{I}^{+} ν (I, a) = A_{J}^{+}$ . (Elementary ribbons have explicitly determined endpoints)

[F7]

The ambient tagged-arrow ribbon groupoid and its full-object-set reduced subgroupoid are specified by The reduced ribbon subgroupoid uses exactly the exterior atoms; the latter is generated exactly by the exterior elementary arrows used here.

External input

Godelle supplies the existence and formula of the spherical nu-structure (Example 3.14), the reduced-ribbon groupoid presentation by lcm relations (Theorem 4.14(3) and Corollary 4.18), and the exactly-two-path theorem for the ribbon with specified nonempty source $I$ (Proposition 4.19). These are imported only under the exact hypotheses recorded in the source reference. Their proofs are not claimed as locally reconstructed. The empty-object extension is a local consequence of the positive braid presentation.

Proof

Given: The hypotheses and notation in the Statement.

Goal: Establish the asserted conclusion.

1.1F1F5F6F7given

The operations in the Statement form a groupoid: for composable triples, $(g h)^{- 1} A_{I}^{+} (g h) = h^{- 1} (g^{- 1} A_{I}^{+} g) h = A_{K}^{+}$ , and the same conjugation equality gives the inverse condition; associativity and the identity laws are inherited from $A$ . F5 defines $Δ_{I}$ for every subset, and F6 ensures that each displayed positive generator has its stated endpoint. By F7 this ambient groupoid and its generated subgroupoid are exactly the ones used in Godelle's reduced-ribbon conventions.

2.1F1F7step 1.1

Apply the external contracts to the spherical Garside monoid of F1 and the identified reduced subgroupoid. Example 3.14 supplies its nu-structure and identifies the generators; Corollary 4.18 gives the reduced subgroupoid presentation, with relation (3) identifying two paths with source $I$ representing the ribbon labelled by the pairwise lcm $d$ . For nonempty $I$ , Proposition 4.19 supplies exactly two reduced-ribbon paths with source $I$ and label $d$ , with the required initial arrows; both represent that ribbon arrow. The source hypotheses are precisely spherical type, nonempty $I$ , and distinct $a, b$ outside $I$ .

3.1F2F3F4step 2.1∎

For empty $I$ , F2 and F3 give the two alternating representatives of its generator lcm $d$ , and F4 identifies these words with exactly the two reduced-ribbon paths with source $\emptyset$ and label $d$ . Both represent the loop $(\emptyset, d, \emptyset)$ . F4 also proves that the empty-object component has precisely the standard braid presentation. Thus this case is supplied independently of the nonempty hypothesis in Proposition 4.19. Together with the general presentation contract this proves the assertion for every subset.

Depends on

Used by · 1

Two ribbon-lcm expressions and presentation

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, arXiv:0811.0751v1 · Example 3.14 (printed p. 15); §4.2 opening notation and Theorem 4.14 relation (3) (printed pp. 20–22); Corollary 4.18 (printed pp. 23–24); Proposition 4.19 (printed p. 24)

Spherical Artin monoids have the standard nu-structure. The reduced ribbon subgroupoid is presented by its reduced nu-generators and the relations equating their lcm decompositions (Corollary 4.18 and Theorem 4.14(3)). For a nonempty parabolic subset $I$ and distinct $a,b\notin I$, the ribbon with source $I$ and label the lcm of $\nu(I,a),\nu(I,b)$ has exactly two representing paths with source $I$, beginning with these generators respectively (Proposition 4.19).

Hypotheses: Finite-rank spherical Artin positive monoid, its standard Garside structure and nu-structure; the two-path clause in Proposition 4.19 requires I nonempty.

Reading limits: For this adjudication, read the extracted PDF statements of Example 3.14, Theorem 4.14 including relation (3), Corollary 4.18 and Proposition 4.19, together with the opening notation of Section 4.2 and the proof of Proposition 4.19, at the cited printed pages of arXiv:0811.0751v1. No visual inspection is claimed for this adjudication. The nu-structure, general presentation and nonempty two-path assertions are imported established results; their proofs and historical sources are not reconstructed or independently verified. The empty-object extension uses the five existing local intermediate items.

Sources

Eddy Godelle, Parabolic subgroups of Garside groups II: ribbons, arXiv:0811.0751v1 · Example 3.14 (printed p. 15); §4.2 opening notation and Theorem 4.14 relation (3) (printed pp. 20–22); Corollary 4.18 (printed pp. 23–24); Proposition 4.19 (printed p. 24)