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Spherical Artin normaliser splitting

PDF p. 22: Theorem 5.6

Statement

Let $Γ$ be a finite-rank spherical Coxeter diagram, meaning that its Coxeter group $W$ is finite, and let $J$ be a subset of its vertex set. Let $A$ be the associated Artin–Tits group and $A_{J}$ its standard parabolic subgroup. Let $R (J)$ be the reduced ribbon groupoid, and write $R_{J}$ for its vertex group at $J$ , consisting of its arrows from $J$ to $J$ under composition.

The normaliser ${Norm}_{A} (A_{J})$ is the internal semidirect product $A_{J} ⋊ R_{J}$ : each normaliser element has a unique expression $p r$ with $p \in A_{J}$ and $r \in R_{J}$ . Consequently the quotient homomorphism restricts to an isomorphism $R_{J} ≅ N (A, J)$ , where $N (A, J) = {Norm}_{A} (A_{J}) / A_{J}$ .

Statement references

Reduced ribbon groupoid

Normalisers and their quotients

Standard parabolic subgroups and parabolic roots

Facts & Assumptions

[F1]

The independent spherical ribbon splitting gives a groupoid on all subsets whose elementary arrows are $Δ_{I}^{- 1} Δ_{I \cup {a}} : I \to ι_{I \cup {a}} (I)$ for $a \notin I$ . Its arrows conjugate the source standard parabolic subgroup to the target, using $g^{- 1} A_{I} g = A_{K}$ for an arrow $g : I \to K$ . Its vertex group $R_{J}^{a l l}$ satisfies ${Norm}_{A} (A_{J}) = A_{J} ⋊ R_{J}^{a l l}$ , with unique factorisation $p r$ . Independent spherical ribbon splitting input

[F2]

The objects of $R (J)$ are the associates of $J$ , meaning the subsets $I$ for which $W_{I}$ is conjugate to $W_{J}$ in $W$ . Its arrows are finite composable products of the same elementary ribbons and their inverses, with left-to-right composition. The target of an elementary ribbon is $ι_{I \cup {a}} (I)$ . Reduced ribbon groupoid

[F3]

The homomorphism $π : A \to W$ sends each standard Artin generator $σ_{i}$ to $s_{i}$ ; the subgroups $A_{I}$ and $W_{I}$ are generated by these respective generators. Thus $π (A_{I}) = W_{I}$ . Artin–Tits group, pure subgroup and positive lifts Standard parabolic subgroups and parabolic roots

[F4]

$A_{J}$ is normal in its normaliser, and $N (A, J)$ is the corresponding quotient group. Normalisers and their quotients

Proof

Given: The spherical diagram, subset $J$ , Artin group, and reduced ribbon groupoid in the Statement.

Proof technique: direct.

Goal: Identify the vertex group in the independent splitting with $R_{J}$ and prove the quotient assertion.

1.1F1F2

Every generating arrow of $R (J)$ is an elementary arrow in the all-subsets groupoid of [F1], with the same element, source, and target; the targets agree because the opposition permutation carries $I$ onto $(I \cup {a}) ∖ {ι_{I \cup {a}} (a)}$ . The same holds for inverses and products. Hence $R_{J} \subseteq R_{J}^{a l l}$ .

1.2F1F2F3

Conversely, express an arrow in $R_{J}^{a l l}$ as a finite composable product of elementary arrows and their inverses. For any intermediate object $I$ , the initial segment $g : J \to I$ satisfies $g^{- 1} A_{J} g = A_{I}$ by [F1]. Applying $π$ and [F3] gives $π (g)^{- 1} W_{J} π (g) = W_{I}$ , so $I$ is an associate of $J$ . Every factor of the product therefore occurs between objects of $R (J)$ and is one of its generators or an inverse. Thus $R_{J}^{a l l} \subseteq R_{J}$ .

2.1F1F4step 1.1step 1.2∎

By 1.1 and 1.2 the two vertex groups agree. Substituting this equality into [F1] gives the asserted internal semidirect product and unique factorisation. The quotient map in [F4] is surjective on $R_{J}$ : the coset of $p r$ is the coset of $r$ . Its kernel on $R_{J}$ is $A_{J} \cap R_{J} = {1}$ by uniqueness of the factorisation. It is therefore an isomorphism onto $N (A, J)$ .

Source notes

The cited article attributes its splitting statement to its reference [14, Proposition 4.4]. The proof above uses the independently supplied spherical ribbon splitting and the explicit comparison of groupoids.

Depends on

Used by · 3

Dependency tree · 2 levels

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 22, Theorem 5.6