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Independent spherical ribbon splitting input

Statement

Let $(A, S)$ be a finite-rank spherical Artin system, with associated finite Coxeter system $(W, S)$ . For $K \subseteq S$ , write $W_{K} = ⟨ K ⟩$ and let $w_{K}$ be its unique longest element. In the canonical real reflection representation with simple-root basis $(e_{s})_{s \in S}$ , define the opposition permutation $ι_{K} : K \to K$ by $w_{K} e_{s} = - e_{ι_{K} (s)}$ for $s \in K$ . For $I \subseteq S$ , let $A_{I}$ be the standard parabolic subgroup and $Δ_{I}$ the positive lift of $w_{I}$ , with $Δ_{\emptyset} = 1$ . Give the conjugator groupoid $Conj (A)$ arrows $g : I \to J$ exactly when $g^{- 1} A_{I} g = A_{J}$ , and compose by multiplication from left to right. Let $P$ be the totally disconnected subgroupoid with vertex groups $A_{I}$ , and let $R_{ν}$ be the subgroupoid generated by $r (I, a) = Δ_{I}^{- 1} Δ_{I \cup {a}}$ for $a \notin I$ , directed $I \to ι_{I \cup {a}} (I)$ . Then $Conj (A) = P ⋊ R_{ν}$ . In particular $N_{A} (A_{I}) = A_{I} ⋊ (R_{ν})_{I}$ . The factorization of each conjugator is uniquely $g = p r$ with $p \in A_{I}$ and $r \in R_{ν} (I, J)$ .

Facts & Assumptions

[F1]

The spherical positive monoid is a monoidal Garside structure with group of fractions $A$ . (Independent spherical Artin Garside input)

[F2]

Composition and vertex group conventions are fixed in Set-sized groupoids, functors and composition conversion.

[F3]

The exterior elementary ribbons and their explicit endpoints are supplied by Elementary ribbons have explicitly determined endpoints. Existence of the spherical nu-structure is the separate imported input [F7].

[F4]

Those exterior atoms generate precisely the source's reduced subgroupoid $R^{ν} (A)$ . (The reduced ribbon subgroupoid uses exactly the exterior atoms)

[F5]

Unique conjugator factorization gives the normalizer semidirect decomposition as proved in Unique conjugator factorization restricts to a normalizer splitting.

[F6]

The independent factorization theorem is imported explicitly below (source reference 0).

[F7]

Every spherical Artin monoid with its standard Garside structure has a nu-structure, including both left and right nu-functions. In particular, for $a \notin I$ its left elementary value is $ν_{I} (a) = Δ_{I}^{- 1} Δ_{I \cup {a}}$ . This is Godelle's Example 3.14, with nu-structure defined in Definition 3.12, imported below (source reference 1).

[F8]

Finite parabolics have unique longest elements and opposition permutations characterized by $w_{K} e_{s} = - e_{ι_{K} (s)}$ . These exist also for $K = \emptyset$ , with its empty permutation. (Finite parabolic longest elements and disconnected factors)

External input

Godelle's Example 3.14 states that every spherical Artin monoid has a nu-structure in the sense of Definition 3.12: both its left and right families of positive ribbon atoms satisfy the stated divisibility and lcm conditions. We import this existence result, with the exterior formulas $ν_{I} (a) = Δ_{I}^{- 1} Δ_{I \cup {a}}$ and ${\tilde{ν}}_{I} (a) = Δ_{I \cup {a}} Δ_{I}^{- 1}$ for $a \notin I$ . It applies to the standard spherical Garside structure of [F1]; no local proof of existence or reconstruction of the historical source [15] is claimed.

Godelle's Theorem 4.25(ii) is imported as an established conjugator-factorization theorem. Its objects are standard parabolic subgroups, its arrows satisfy $g^{- 1} A_{I} g = A_{J}$ , and its composition is the product in $A$ . It asserts $Conj (A) = P (A) ⋊ R^{ν} (A)$ ; Definitions 4.23–4.24 define this as unique factorization and transport of vertex factors. The paragraph after the theorem explicitly includes spherical Artin groups. For this adjudication we read the cited statements and definitions in the extracted text; their general proofs were not reconstructed. The endpoint and generator translation uses [F3]–[F4], and the vertex restriction uses [F5].

Proof

Given: The spherical system and the groupoids in the statement.

Goal: Apply the independent splitting theorem with the correct generators, endpoints and product convention.

1.1F1F6F7

By [F1] the positive monoid and $A$ have the monoidal Garside/group-of-fractions relationship required by Theorem 4.25. By the independent spherical existence input [F7], Example 3.14 supplies the required nu-structure for this same positive monoid. Its positive parabolic objects and the corresponding standard parabolic subgroups are those used here. Thus the theorem's hypotheses are met.

1.2F2F3F4F6F8

By [F4] the source's $R^{ν} (A)$ equals exactly $R_{ν}$ as defined here. Every $W_{K}$ is finite because it is a subgroup of finite $W$ , so [F8] supplies the opposition permutation used in those endpoints. Its elementary arrows therefore have exactly the endpoints proved in [F3]. The source's morphisms satisfy $g^{- 1} A_{I} g = A_{J}$ and its composition is group multiplication, precisely [F2] and the convention in our statement; no inversion or product reversal is being made.

2.1F5F6step 1.1step 1.2∎

Apply Theorem 4.25(ii), interpreted through Definitions 4.23–4.24. It gives unique factorization of every arrow as $g = p r$ with the indicated source-parabolic factor and reduced-ribbon factor. This proves the groupoid splitting. Applying [F5] at each object yields $N_{A} (A_{I}) = A_{I} ⋊ (R_{ν})_{I}$ , including $I = \emptyset$ and $I = S$ . No assertion about arbitrary Artin type or the article's universal wall cover is used or obtained.

Depends on

Used by · 1

Spherical Artin normaliser splitting

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 · Definitions 4.23–4.24 (printed p. 27); Theorem 4.25(ii) and following explanatory paragraph (printed p. 28); standard parabolics in Example 3.3 (printed p. 12)

For a monoidal Garside structure (A^+,Delta) with a nu-structure and fraction group A, the groupoid Conj(A), with standard parabolic objects and arrows g:A_I to A_J satisfying g^{-1}A_I g=A_J and product composition, equals P(A) semidirect R^nu(A). P(A) is totally disconnected with vertex groups A_I. The internal semidirect notation entails unique factorization g=pr and transport A_I r=r A_J. Standard parabolic subgroups of an Artin–Tits group are generated by subsets of its standard atoms. Its vertex conjugator groups are normalizers.

Hypotheses: A monoidal Garside structure with a nu-structure and its group of fractions; applied only to the spherical Artin specialization explicitly included after Theorem 4.25.

Reading limits: For this adjudication, read Example 3.3 on printed/PDF p. 12 and Definitions 4.23–4.24 and Theorem 4.25(ii), including its following specialization paragraph, on printed/PDF pp. 27–28 in the frozen extracted text of arXiv:0811.0751v1. No new visual PDF check was performed. The factorization theorem is imported; its proof and the cited historical sources were not reconstructed. No arbitrary-type assertion is imported.

Eddy Godelle, Parabolic subgroups of Garside groups II: ribbons, arXiv:0811.0751v1 · §3.3, Definition 3.12 (printed/PDF p. 14), and Example 3.14 (printed/PDF p. 15)

Every spherical Artin–Tits monoid has a nu-structure, meaning both the left nu-functions and right tilde-nu-functions of Definition 3.12, whose values are positive ribbon atoms and satisfy its divisibility and lcm conditions. For I contained in S and a outside I, nu_I(a)=Delta_I^{-1}Delta_{I union {a}} and tilde-nu_I(a)=Delta_{I union {a}}Delta_I^{-1}; for s in I, both values equal Delta_{I_s}, where I_s is the connected component containing s in the induced Coxeter diagram on I.

Hypotheses: The positive monoid of a finite-rank Artin–Tits system of spherical type, equipped with its standard monoidal Garside structure; I ranges over subsets of its standard atom set S and Delta_I is the standard parabolic longest positive lift.

Reading limits: For this adjudication, read §3.3, Definition 3.12 and Example 3.14 on printed/PDF pp. 14–15 in the frozen extracted text of arXiv:0811.0751v1; also located Example 3.14 in the online PDF text. No new visual PDF check was performed. Import only the stated spherical existence result and formulas; the historical source [15] and its proof were not read or reconstructed. No general Artin-type existence is asserted.

Sources

Eddy Godelle, Parabolic subgroups of Garside groups II: ribbons, arXiv:0811.0751v1 · Definitions 4.23–4.24 (printed p. 27); Theorem 4.25(ii) and following explanatory paragraph (printed p. 28); standard parabolics in Example 3.3 (printed p. 12)
Eddy Godelle, Parabolic subgroups of Garside groups II: ribbons, arXiv:0811.0751v1 · §3.3, Definition 3.12 (printed/PDF p. 14), and Example 3.14 (printed/PDF p. 15)