Let $W$ be a Coxeter Group generated by simple reflections $S$. If $I,J\subseteq S$ and $W_I = \langle s | s \in I \rangle$ when is it true that $W_IW_J = W$?
I am secretly hoping that the answer might be that this holds if and only if $I$ or $J$ = $S$. So a better follow-up question might be: for which Coxeter Groups is it true that $W_IW_J = W \iff \{I,J\} \cap \{S\} \ne \emptyset$.
I think I recall that this might be equivalent to a notion of flatness for some Coxeter Groups but I can't remember the source exactly. Any sources or comments that might be helpful would be really appreciated.
Thanks!