LemmaStatement: AI-adaptedProof: AI-generated

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AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Normaliser quotient exact sequence

PDF p. 3: (1.6); PDF p. 6: Lemma 2.8; (2.9)

Statement

Let $(W, S)$ be a finite-rank Coxeter system, $A$ its Artin–Tits group, $π : A \to W$ the standard projection and $P = \ker π$ . For a subset $J \subseteq S$ , let $W_{J}$ and $A_{J}$ be the standard parabolic subgroups. Write $N (W, J) = {Norm}_{W} (W_{J}) / W_{J}$ , $N (A, J) = {Norm}_{A} (A_{J}) / A_{J}$ , and let $N (P, J)$ be the image of $P \cap {Norm}_{A} (A_{J})$ in $N (A, J)$ . Then $π$ induces $\overline{π} : N (A, J) \to N (W, J)$ and the sequence $1 \to N (P, J) \to N (A, J) \overset{\overline{π}}{\to} N (W, J) \to 1$ is exact. Here exactness means that the inclusion $N (P, J) ↪ N (A, J)$ is injective, its image equals $\ker \overline{π}$ , and $\overline{π}$ is surjective.

Facts & Assumptions

[F1]

The projection sends each standard Artin generator $σ_{s}$ to $s$ , has kernel $P$ , and maps $A_{J}$ onto $W_{J}$ . The subgroups $A_{J}$ and $W_{J}$ are generated by their respective standard generators indexed by $J$ . Artin–Tits group, pure subgroup and positive lifts Standard parabolic subgroups and parabolic roots

[F2]

A homomorphism $p : G \to Q$ with $p (H) = K$ induces a map $N_{G} (H) / H \to N_{Q} (K) / K$ whose kernel is the image of $\ker p \cap N_{G} (H)$ ; it is surjective exactly when $p (N_{G} (H)) = N_{Q} (K)$ . Normalizer quotients, kernels and conditional surjectivity

[F3]

Every double coset $W_{J} w W_{J}$ contains an element $d$ with $ℓ (s d) > ℓ (d)$ and $ℓ (d s) > ℓ (d)$ for every $s \in J$ ; these inequalities define membership in $^{J} W^{J}$ . Descent reduction, minimum-length elements, and the additive factorization in a double coset Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups

[F4]

If $d \in^{J} W^{J}$ and $s \in J$ with $d^{- 1} s d \in W_{J}$ , then $d^{- 1} s d \in J$ . Also $W_{J} \cap S = J$ . The parabolic intersection W_I∩ dW_Jd⁻¹ for d∈^IW^J Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification

[F5]

For $d \in^{J} W^{J}$ and $s, t \in J$ , the products $s d$ and $d t$ have length $ℓ (d) + 1$ . The positive lift $σ_{x}$ of a Coxeter element $x$ is well defined, projects to $x$ , and satisfies $σ_{x y} = σ_{x} σ_{y}$ whenever $ℓ (x y) = ℓ (x) + ℓ (y)$ . Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups Reduced-word lifts and the prefix-order convention

Proof

Given: The finite-rank Coxeter and Artin–Tits data and the subset $J$ in the Statement.

Proof technique: direct.

Goal: Prove the induced map, its kernel and its surjectivity.

1.1F1F2given

By [F1] the image of $A_{J}$ under $π$ is $W_{J}$ and $\ker π = P$ . Applying [F2] with $G = A$ , $H = A_{J}$ , $Q = W$ and $K = W_{J}$ gives the displayed induced homomorphism and identifies its kernel with the image of $P \cap {Norm}_{A} (A_{J})$ , namely $N (P, J)$ . Its inclusion in $N (A, J)$ is injective by the definition of an image subgroup.

1.2F3F4given

Let $w \in {Norm}_{W} (W_{J})$ . By [F3] choose $d \in^{J} W^{J}$ in $W_{J} w W_{J}$ . Since $w$ normalises $W_{J}$ , this double coset equals $w W_{J}$ . Thus $d = w u$ for some $u \in W_{J}$ , and $d$ also normalises $W_{J}$ . For every $s \in J$ , [F4] gives $d^{- 1} s d \in J$ . Conjugation is injective and $J$ is finite, so it permutes $J$ .

2.1F1F5step 1.2

For $s \in J$ put $t = d^{- 1} s d \in J$ . Then $s d = d t$ . Both sides are length-additive by [F5], so their positive lifts give $σ_{s} σ_{d} = σ_{d} σ_{t}$ . As $s \mapsto t$ permutes $J$ , conjugation by $σ_{d}$ sends the generating set of $A_{J}$ onto itself. Hence $σ_{d} \in {Norm}_{A} (A_{J})$ . Since $u \in W_{J} = π (A_{J})$ , choose $a \in A_{J}$ with $π (a) = u^{- 1}$ . Then $σ_{d} a$ normalises $A_{J}$ and projects to $d u^{- 1} = w$ . Thus every element of ${Norm}_{W} (W_{J})$ lifts to its Artin normaliser.

3.1F1F2step 1.1step 2.1∎

The reverse containment $π ({Norm}_{A} (A_{J})) \subseteq {Norm}_{W} (W_{J})$ follows by applying $π$ to the normalising equality and using $π (A_{J}) = W_{J}$ . Together with step 2.1 this is equality. The surjectivity criterion in [F2], combined with the kernel calculation in step 1.1, proves the asserted short exact sequence.

Depends on

Used by · 3

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF pp. 3 and 6, (1.6) and Lemma 2.8