Root-permuting normaliser complement

PDF p. 17: paragraph before Corollary 4.8

Let $(W,S)$ be a Coxeter system with simple generators $S=\{s_i:i\in {\mathrm{\Gamma }}_0\}$ and geometric representation on the real vector space with basis $\{{\alpha }_i:i\in {\mathrm{\Gamma }}_0\}$. The index set may be infinite. For $J\subseteq {\mathrm{\Gamma }}_0$, write $W_J=⟨s_j:j\in J⟩$ and ${\mathrm{Norm}}_W(W_J)=\{w\in W:w{W}_J{w}^{-1}=W_J\}$.

The root-permuting normaliser subgroup $N_J$, also denoted ${\mathcal{B}\mathcal{H}}_J$, is

$N_J={\mathcal{B}\mathcal{H}}_J=\{x\in W:(\mathrm{\forall }j\in J)(\mathrm{\exists }{j}^{\mathrm{\prime }}\in J)x{\alpha }_j={\alpha }_{j^{\mathrm{\prime }}}\}\mathrm{.}$

It is the vertex group at $J$ of the Brink–Howlett groupoid and is a subgroup of ${\mathrm{Norm}}_W(W_J)$.

Brink–Howlett groupoid

Connected small groupoids and vertex groups

Coxeter-system, roots, length and weak-order conventions

Standard parabolic subgroups and parabolic roots

The normalizer N_G(H)={g∈ G:gHg⁻¹=H} of a subgroup

Put $A_J=\{{\alpha }_j:j\in J\}$. By the Coxeter-data definition, each $x\in W$ has a word with finite index support $F$, and each simple reflection changes any vector by a multiple of its own simple root. Consequently $x$ preserves the span of the roots indexed by $F$, and $x{\alpha }_j-{\alpha }_j$ lies in that span for every $j$.

Suppose $x{A}_J\subseteq A_J$. If $j\in J\setminus F$, comparison of basis coefficients shows that $x{\alpha }_j={\alpha }_j$. If $j\in J\cap F$, the image lies in the span of the roots indexed by $F$, so $x{\alpha }_j$ belongs to $\{{\alpha }_k:k\in J\cap F\}$. Since $x$ acts invertibly, its restriction to this finite set is injective and hence bijective. Thus $x{A}_J=A_J$, including when $J$ is infinite. The converse is immediate. The Brink–Howlett definition identifies the elements satisfying this equality with the labels of endomorphisms of the object $J$. By Connected small groupoids and vertex groups, these endomorphisms, with multiplication and inverses inherited from $W$, form the vertex group ${\mathcal{B}\mathcal{H}}_J$.

It remains to check inclusion in the normaliser. The reflection formula in the Coxeter-data definition shows that each simple reflection preserves the Coxeter form $B$; hence so does $x$. If $x{\alpha }_j={\alpha }_k$, conjugating the reflection formula gives $\rho (x{s}_j{x}^{-1})=\rho (s_k)$ on the geometric representation. Choose a finite set of indices containing $j$, $k$ and the support of a word for $x$. The corresponding finite-generator Coxeter group embeds in $W$ by Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank, and its geometric representation is faithful by The root-length criterion and faithfulness of the canonical reflection representation. Restricting the displayed equality to its root span therefore gives $x{s}_j{x}^{-1}=s_k$ in $W$. Since $x$ permutes $A_J$, it permutes the generators $\{s_j:j\in J\}$ under conjugation. It follows from the definition of the generated subgroup that $x{W}_J{x}^{-1}=W_J$, as required.