Let $(W, S)$ be a Coxeter system, and set $T = \bigcup_{w \in W} wSw^{-1}$ the set of reflections. A subgroup $R \subseteq W$ is a reflection subgroup if it is generated by a subset of $T$, or equivalently $R = \langle R \cap T \rangle$. It is known that $R$ admits a canonical (with respect to $S$) set $R'$ of generators, making $(R, R')$ a Coxeter system.
The usual examples of reflection subgroups are those generated by some subset $I \subseteq S$ of the generating set, or their conjugates (these are collectively called parabolic subgroups). I am interested in the more general theory of reflection subgroups.
Question: What is a standard reference for reflection subgroups, from the perspective of root systems if possible?
I've been able to piece together various results from the work of Deodhar and Dyer around 1987-1990, but I hope that some of this material would have been standardised in a textbook, survey, or course notes. In particular, I want some reference for basic results like:
- The length $l_R(w) \leq l(w)$ for $w \in R$.
- Each coset $Rw$ for $w \in W$ posesses a unique element of minimum length.
- The minimum length coset representatives are the set $\{w \in W \mid w(\alpha) > 0\}$ for all simple roots $\alpha$ of $R$,
and so on.