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Independent spherical Artin Garside input

Statement

For a spherical Artin system, its positive Artin monoid embeds in its Artin group as a Garside monoid. Its simples are the positive lifts of elements of the Coxeter group. Gcds and lcms exist on both sides.

More explicitly, spherical means that the Coxeter group $W$ is finite. The positive monoid $M$ and the Artin group $A$ are given by the same standard braid presentation in the monoid and group categories. Let $w_{0}$ be the longest element of $W$ and $Δ = w_{0}$ its reduced-word lift in $M$ . The canonical map identifies $M$ with the positive submonoid $A^{+} \subseteq A$ and transports the Garside structure with Garside element $σ_{w_{0}}$ . In the categorical convention $x Δ = Δ ϕ (x)$ , simples are the left divisors of $Δ$ ; they are also precisely its right divisors, and are exactly ${w : w \in W}$ . In $A^{+}$ these are ${σ_{w} : w \in W}$ .

Statement references

Categorical Garside structures and simple arrows

Reduced-word lifts and the prefix-order convention

Positive degree and atoms of the presented Artin monoid

Reduced Coxeter lifts in the presented positive monoid

Both-sided divisors of the longest positive lift

Identifying the presented monoid with its positive image

Transport of a Garside structure to the presented monoid

Artin monoid and Artin group presentations, and the canonical monoid-to-group map

The longest element as the opposition of the chamber, and longest elements of finite parabolics

External input

For a spherical Artin system, Godelle's Example 2.3 supplies the monoidal Garside structure on the positive submonoid $A^{+} \subseteq A$ , with $Δ^{+} = σ_{w_{0}}$ ; Definition 2.1 specifies cancellation, both divisibility lattices, noetherianity and simple atoms. The automorphism is taken in the convention $x Δ^{+} = Δ^{+} ϕ^{+} (x)$ . This is an imported structure on the positive image, not a claim that its identification with a presentation was proved there.

Paris's Theorem 1.1 supplies injectivity of the canonical map $j : M \to A$ for the monoid and group defined by the same braid presentation. We use only its finite-rank spherical specialization. These two results are imported established inputs; their proofs have not been reconstructed here.

Facts & Assumptions

[F1]

The categorical Garside convention, simplicity and both divisibility lattices are specified in Categorical Garside structures and simple arrows. Reduced group lifts $σ_{w}$ are specified in Reduced-word lifts and the prefix-order convention.

[F2]

The presented monoid has additive degree, atomicity and bounded nonidentity factorizations. (Positive degree and atoms of the presented Artin monoid)

[F3]

Reduced monoid lifts are well defined and map under $j$ to the reduced group lifts. (Reduced Coxeter lifts in the presented positive monoid)

[F4]

The left and right divisors of $Δ = w_{0}$ are exactly the reduced monoid lifts. (Both-sided divisors of the longest positive lift)

[F5]

An injective canonical map identifies the presented monoid with its positive image. (Identifying the presented monoid with its positive image)

[F6]

A Garside structure, simples and both divisibility lattices transport through that isomorphism. (Transport of a Garside structure to the presented monoid)

[F7]

The positive-image Garside structure is the Godelle input recorded below.

[F8]

The injectivity of the canonical presented-monoid map is the Paris input recorded below.

[F9]

The standard braid presentations of $M$ and $A$ and the canonical map $j : M \to A$ are supplied by Artin monoid and Artin group presentations, and the canonical monoid-to-group map. Since $W$ is finite, its unique longest element $w_{0}$ is supplied by The longest element as the opposition of the chamber, and longest elements of finite parabolics, Statement (1)(iii).

Proof

Given: The objects and hypotheses in the statement.

Goal: Establish all assertions of the statement.

1.1F3F5F8F9

By the Paris input, $j$ is injective. Applying [F5] identifies $M$ with exactly $A^{+}$ , with $j (w) = σ_{w}$ by [F3]. In particular $j (Δ) = σ_{w_{0}}$ . Thus the monoid to which Godelle's positive-submonoid assertion applies is the image of the presented monoid, with the same longest-lift element.

2.1F1F2F6F7step 1.1

Apply the Godelle input to the spherical system. It supplies a cancellative one-object Garside structure on $A^{+}$ with the stated Garside element and automorphism convention. Pull it back by step 1.1 using [F6]. The atomic condition needed by [F1] also follows directly from [F2]. Therefore $M$ is a Garside monoid in the adopted categorical convention, and both divisibility orders are lattices, yielding gcds and lcms on both sides.

3.1F1F4step 1.1step 2.1∎

By [F1] the simples of this structure are precisely the left divisors of $Δ$ . By [F4] these coincide with its right divisors and with all reduced monoid lifts $w$ . Their images are exactly the lifts $σ_{w}$ in [F1] by step 1.1. This proves the simple-element identification and all claims in the statement.

Depends on

Used by · 11

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 · §2.3, Definition 2.1 (printed p. 5), Example 2.3 (printed p. 6), and Definition 2.5 (printed p. 7)

For an Artin–Tits group A of spherical type, the submonoid A+ generated by its standard positive generators, with Delta the reduced-word section image of the longest element of W, is a monoidal Garside structure. The inner automorphism induced by Delta preserves A+, and A is its associated Garside group. Definition 2.1 supplies cancellation, noetherianity, both divisibility lattices and simple atoms; Definition 2.5 defines simple elements as left divisors of Delta.

Hypotheses: A finite-rank Artin–Tits system whose associated Coxeter group W is finite; A+ denotes the positive submonoid of A, rather than an unidentified presented monoid.

Reading limits: For this repair, read the extracted text of Definition 2.1, Example 2.3 (spherical paragraph), and Definition 2.5 on printed/PDF pp. 5–7 of arXiv:0811.0751v1. No new visual check was completed. The established positive-image Garside structure is imported; its proof and historical sources are not reconstructed. Identification with the presented monoid and its simples uses the separate local argument.

Luis Paris, Artin monoids inject in their groups, Commentarii Mathematici Helvetici 77 (2002), 609–637 · §1, Theorem 1.1, printed p. 610; defining presentations on printed pp. 609–610

For the positive Artin monoid presented by the standard braid generators and relations, the canonical homomorphism to the Artin group with the same generators and relations is injective.

Hypotheses: A Coxeter matrix over a finite set S, with off-diagonal entries in {2,3,...,infinity}; the monoid and group are the presented ones specified in §1. Only the spherical specialization is used here.

Reading limits: For this repair, read the extracted presentations and Theorem 1.1 in Section 1, printed pp. 609–610 (PDF pp. 1–2). No new visual check was completed. The injectivity theorem is imported; its proof in later sections is not read or reconstructed.

Sources

Eddy Godelle, Parabolic subgroups of Garside groups II: ribbons, arXiv:0811.0751v1 · §2.3, Definition 2.1 (printed p. 5), Example 2.3 (printed p. 6), and Definition 2.5 (printed p. 7)
Luis Paris, Artin monoids inject in their groups, Commentarii Mathematici Helvetici 77 (2002), 609–637 · §1, Theorem 1.1, printed p. 610; defining presentations on printed pp. 609–610