Normalisers and their quotients

PDF p. 1: (1.2); PDF p. 3: (1.7); PDF p. 6: §2.3

Let $(W,S)$ be a finite-rank Coxeter system indexed by $S=\{s_i:i\in {\mathrm{\Gamma }}_0\}$, let $A$ be its Artin–Tits group, and let $P=\ker (\pi :A\to W)$ be its pure Artin–Tits group. For $J\subseteq {\mathrm{\Gamma }}_0$, let $W_J=⟨s_j:j\in J⟩$ and $A_J=⟨{\sigma }_j:j\in J⟩$ be the standard parabolic subgroups; the brackets denote the subgroups generated by the displayed elements.

For $G=W$ with $G_J=W_J$, or $G=A$ with $G_J=A_J$, define the normaliser of $G_J$ in $G$ by ${\mathrm{Norm}}_G(G_J)=\{g\in G:g{G}_J{g}^{-1}=G_J\}$. Define the normaliser quotient groups $N(W,J)={\mathrm{Norm}}_W(W_J)\mathrm{/}{W}_J$ and $N(A,J)={\mathrm{Norm}}_A(A_J)\mathrm{/}{A}_J$.

Define $N(P,J)=\{\beta A_J\in N(A,J):P\cap \beta A_J\mathrm{\ne }\mathrm{\varnothing }\}$. Equivalently, $N(P,J)$ is the image of $P\cap {\mathrm{Norm}}_A(A_J)$ under the quotient homomorphism ${\mathrm{Norm}}_A(A_J)\to N(A,J)$.

The parabolic-subgroups definition supplies $W_J\le W$ and $A_J\le A$. The Artin–pure-lifts definition supplies $A$, the pure subgroup $P$ and the homomorphism $\pi$. The exact sequence in Classical free action and covering exact sequence, with $P$ identified with its image in $A$ as in Artin–Tits group, pure subgroup and positive lifts, gives $P=\ker \pi$. By Normalizer quotients, kernels and conditional surjectivity, each normaliser is a subgroup and the corresponding parabolic subgroup is normal in it; consequently both displayed quotient groups exist, with elements the indicated cosets.

By The intersection of a nonempty family of subgroups of G is a subgroup of G, applied to the two subgroups of $A$, the intersection $I=P\cap {\mathrm{Norm}}_A(A_J)$ is a subgroup of $A$. Let $q:{\mathrm{Norm}}_A(A_J)\to N(A,J)$ be the quotient homomorphism. Its restriction $q{\mathrm{\mid }}_I:I\to N(A,J)$ is a homomorphism, since for $x,y\in I$ one has $q{\mathrm{\mid }}_I(xy)=q(xy)=q(x)q(y)=q{\mathrm{\mid }}_I(x)q{\mathrm{\mid }}_I(y)$. By The image of a group homomorphism is a subgroup and its kernel is a normal subgroup, its image $q(I)$ is a subgroup of $N(A,J)$. If a coset $\beta A_J$ in $N(A,J)$ meets $P$ at $p$, then $p$ also belongs to ${\mathrm{Norm}}_A(A_J)$ because $\beta$ and $A_J$ do, and $p{A}_J=\beta A_J$. Conversely, the coset of any element of that intersection meets $P$. Thus the displayed set is exactly the stated image.