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Coxeter normaliser semidirect product

PDF p. 17: Corollary 4.8

Statement

Let $(W, S)$ be an arbitrary-rank Coxeter system with $S = {s_{i} : i \in Γ_{0}}$ in its standard real root representation, and let $J \subseteq Γ_{0}$ be arbitrary. Write $W_{J} = ⟨ s_{j} : j \in J ⟩$ , $M = {Norm}_{W} (W_{J})$ and $N (W, J) = M / W_{J}$ . Let $N_{J} = {B H}_{J}$ be the vertex group at $J$ , equivalently the subgroup permuting the simple-root set ${α_{j} : j \in J}$ .

The vertex group $N_{J}$ maps isomorphically onto $N (W, J)$ under the normaliser quotient, and ${N o r m}_{W} (W_{J}) ≅ W_{J} ⋊ N_{J}$ .

Statement references

Root-permuting normaliser complement

Normalizer quotients, kernels and conditional surjectivity

Facts & Assumptions

[F1]

The vertex group $N_{J}$ is the subgroup permuting the simple roots indexed by $J$ and is contained in $M$ . Root-permuting normaliser complement

[F2]

Every coset of $W_{J}$ in $M$ contains exactly one element of $N_{J}$ . Each normaliser coset has exactly one root-permuting representative

[F3]

The subgroup product and intersection are $M = W_{J} N_{J}$ and $W_{J} \cap N_{J} = {1}$ . Every normalising element factors through the root-permuting subgroup The root-permuting subgroup intersects the parabolic trivially

[F4]

The parabolic subgroup is normal in its normaliser, and a complement to the kernel of a quotient map gives the corresponding semidirect-product decomposition. Normalizer quotients, kernels and conditional surjectivity Splitting lemma for groups: a section, a complement, and a semidirect-product decomposition are equivalent

Proof

Given: An arbitrary-rank Coxeter system and arbitrary $J \subseteq Γ_{0}$ . Write $M = {Norm}_{W} (W_{J})$ and $N (W, J) = M / W_{J}$ . No topology, finite-rank or spherical-type assumption is made. Goal: Prove the displayed quotient identification and splitting.

1.1F1F2F4

By [F1] and [F4], the quotient homomorphism $q : M \to M / W_{J}$ restricts to a homomorphism on $N_{J}$ . By [F2], each quotient coset has exactly one representative in that subgroup. Therefore $q ∣_{N_{J}} : N_{J} \to N (W, J)$ is bijective and is a group isomorphism.

2.1F1F3F4step 1.1∎

The product and intersection in [F3] make $N_{J}$ a complement to the normal subgroup $W_{J}$ . Apply the splitting theorem in [F4]: conjugation by $N_{J}$ is the action on $W_{J}$ , and the map $W_{J} ⋊ N_{J} \to M$ , $(u, n) \mapsto u n$ , is an isomorphism. The section of $q$ is the inverse of $q ∣_{N_{J}}$ followed by inclusion into $M$ . This establishes exactly the stated splitting and quotient map.

Source notes

The article proves Corollary 4.8 from its universal covering and attributes the classical result to Lusztig [26, Lemma 5.2] and Howlett [22]. Those works are attribution only here. To retain arbitrary rank without depending on finite-rank geometric suppliers, the five local algebraic helpers instead use Dyer–Gimenez, arXiv:2606.12937v1, §1.4 and Propositions 2.10(b)–(d), 2.11(c). The exact imported statements, hypotheses, locators and reading limits are recorded in the first two helpers. In particular their §1.4 explicitly permits infinite rank. The departure from the article's geometric method is limited to constructing the complement algebraically via minimum coset representatives. No infinite-rank chamber geometry or extension of the covering theorem is claimed, and no original-article defect is asserted.

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF pp. 17; Corollary 4.8