Standard parabolic subgroups and parabolic roots

PDF p. 5: §2.3

Let $(W,S)$ be the Coxeter system indexed by $S=\{s_i:i\in {\mathrm{\Gamma }}_0\}$, with geometric action on the simple roots ${\alpha }_i$, root set $\mathrm{\Phi }$, and positive and negative root sets ${\mathrm{\Phi }}^{+}$ and ${\mathrm{\Phi }}^{-}$ as specified in the Coxeter-data definition. The index set may be infinite. Let $A$ be the Artin–Tits group of the same Coxeter matrix: it has generators ${\sigma }_i$ for $i\in {\mathrm{\Gamma }}_0$ and, for each distinct $i,j$ with finite $m(s_i,s_j)$, the relation equating the two alternating words in ${\sigma }_i,{\sigma }_j$ of length $m(s_i,s_j)$, one beginning with each generator. No relation is imposed for a pair with $m(s_i,s_j)=\mathrm{\infty }$.

For $J\subseteq {\mathrm{\Gamma }}_0$, the standard parabolic subgroup of $W$ attached to $J$ is $W_J=⟨s_j:j\in J⟩$, and the standard parabolic subgroup of $A$ attached to $J$ is $A_J=⟨{\sigma }_j:j\in J⟩$. Here brackets denote the smallest subgroup containing the displayed elements. The associated parabolic root set and its positive and negative parts are ${\mathrm{\Phi }}_J=\{w{\alpha }_j:w\in W_J,j\in J\}$ and ${\mathrm{\Phi }}_J^{\pm }={\mathrm{\Phi }}_J\cap {\mathrm{\Phi }}^{\pm }$.

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

Artin–Tits group, pure subgroup and positive lifts

Group presentation by generators and relations

The subgroup ⟨ S ⟩ generated by a subset, the cyclic subgroup ⟨ g ⟩, and cyclic groups

The Coxeter-data definition supplies the group, its action and the root sets. The group-presentation construction applies to the indicated set of generators and braid relations, including an infinite index set. In the finite-rank setting of Artin–Tits group, pure subgroup and positive lifts, the fixed identification of the presented Artin group with the topological Artin–Tits group carries these generators to the elements denoted ${\sigma }_i$ there.

For each displayed generating set, The subgroup ⟨ S ⟩ generated by a subset, the cyclic subgroup ⟨ g ⟩, and cyclic groups gives a unique smallest subgroup; the empty generating set gives the identity subgroup. Since $W_J\subseteq W$ and each ${\alpha }_j$ is a root, every $w{\alpha }_j$ in the displayed set belongs to $\mathrm{\Phi }$ by the definition of $\mathrm{\Phi }$. The intersections with ${\mathrm{\Phi }}^{+}$ and ${\mathrm{\Phi }}^{-}$ therefore define subsets of $\mathrm{\Phi }$. No finiteness of $W_J$, equality with a root-space intersection, or presentation of $A_J$ is asserted here.