Let $R$ be a root system (reduced, crystallographic) with Weyl group $W$. Let $u,v\in W$ such that there exists a root $\alpha\in R$ such that $u=vs_\alpha$.
Do we always have that $u$ and $v$ are comparable in the (strong) Bruhat order, i.e. do we have $u\prec v$ or $v\prec u$.
By definition of the Bruhat order, this is equivalent to ask: Do we always have $\ell(u)<\ell(v)$ or $\ell(v)<\ell(u)$ where $\ell$ is the usual length function.
In other words, I am asking: With $u,v,\alpha$ as above, can we have $\ell(u)=\ell(v)$. If yes, I would be very happy to see an example. If not, a reference or an explanation would be very helpful.
Thanks a lot!
NB. I tried Humphreys book on Coxeter groups, but didn't find what I want. Up to now, I only looked at the part of the book available in the google preview. There might be more. As far as I can see, in the definition of the Bruhat order the length constraint is always explicitely mentioned to make the elements $u$ and $v$ comparable in the Bruhat order.
Yes, if $u = v s_{\alpha}$, then $u$ and $v$ comparable. It is not possible that $\ell(u) = \ell(v)$. Fix a Coxeter system $(W,S)$. The relevant facts are:
1) For a simple reflection $s \in S$, if $u = vs$, then $\ell(u) = \ell(v) \pm 1$. This is Proposition 5.2 in Humphreys.
2) Every root $\alpha$ satisfies $\alpha = w(\alpha_s)$ for some simple reflection $s \in S$ and some $w \in W$. Roots are defined this way in 5.4 of Humphreys, but for the early chapters on finite reflection groups, this is Corollary 1.5.
3) For roots $\alpha$ and $\beta$ satisfying $\beta = w(\alpha)$, we have $s_{\beta} = w s_{\alpha} w^{-1}$. This is Lemma 5.7 in Humphreys.
Combining facts 2 and 3, we see that every reflection $s_{\alpha}$ has odd length and thus can be written as a product of an odd number of simple reflections. By fact 1, each multiplication by a simple reflection results in a change in parity of length. An odd number of parity changes results in a parity change. Thus, the lengths differ in parity, which implies $\ell(u) \neq \ell(v)$.