# co.combinatorics – Product of two reflections lying in a parabolic subgroup of a Coxeter group

Let $$(W,S)$$ be a Coxeter group, $$Isubseteq S$$ a subset of simple reflections, and $$W_I subseteq W$$ the corresponding parabolic subgroup (we could also assume $$|W_I|, if needed).

Let also $$t_1,t_2in W$$ be two reflections (i.e. elements in $$W$$ conjugated to some $$s_1,s_2 in S$$ respectively) such that $$t_1t_2in W_I$$ but $$t_inotin W_I$$, $$i=1,2$$.

Is it then true that the only possibility is $$t_1t_2=e$$ (i.e. $$t_1=t_2$$)?

If I look at the geometric picture with alcoves and reflection hyperplanes this seems to me to be true, but I don’t know how to prove it in full generality via geometric arguments. I rather tried with some combinatoric method (e.g. using the strong exchange condition or some result about parabolic double cosets) but I have succeeded only in the case $$t_i in S$$ for at least one $$i$$.

Thanks a lot for any comment about that!