Associates of a parabolic subset

PDF p. 5: Definition 2.7

Let $\mathrm{\Gamma }$ be a Coxeter diagram with vertex set ${\mathrm{\Gamma }}_0$ and associated Coxeter group $W$, with simple generators $s_i$ indexed by $i\in {\mathrm{\Gamma }}_0$ as in Coxeter-system, roots, length and weak-order conventions. For $K\subseteq {\mathrm{\Gamma }}_0$, let $W_K=⟨s_k:k\in K⟩$ be its standard parabolic subgroup, as in Standard parabolic subgroups and parabolic roots.

For $J\subseteq {\mathrm{\Gamma }}_0$, an associate of $J$ is a subset $I\subseteq {\mathrm{\Gamma }}_0$ for which $W_I$ and $W_J$ are conjugate in $W$: that is, there exists $w\in W$ such that $W_I=w{W}_J{w}^{-1}=\{wu{w}^{-1}:u\in W_J\}$.

The Coxeter-data definition supplies the group $W$ and its indexed generators. The standard parabolic definition supplies a subgroup $W_K$ for every subset $K\subseteq {\mathrm{\Gamma }}_0$. For each $w\in W$, the displayed conjugate is a well-defined subset of $W$; it is a subgroup because it contains the identity and is closed under products and inverses by the group laws. Thus the existence condition defines a property of each subset $I$ of ${\mathrm{\Gamma }}_0$.