Let $(W,S)$ be a finite Coxeter system. Let us consider the following situation:

Let $v_1,v_2,w\in W$ such that $v_1=wv_2w^{-1}$. Let $s_{\beta_r}\ldots s_{\beta_1}$ be a reduced expression of $v_2$. We consider the condition **C** that
$$
1<ws_{\beta_1}w^{-1}<ws_{\beta_2}s_{\beta_1}w^{-1}<\cdots<ws_{\beta_{r-1}}\cdots s_{\beta_1}w^{-1}<ws_{\beta_r}\dots s_{\beta_1}w^{-1}=wv_2w^{-1}=v_1
$$
is a saturated chain in the Bruhat order.

If **C** is satisfied for one reduced expression of $v_2$, it is satisfied for any. Further, we then must have $\ell(v_1)=\ell(v_2)$.

I want to consider solutions for the problem **C**. I know some, e.g., if $w=1$ and $v_1=v_2$ or if $v_1=v_2=w=w_o$. These are rather trivial. It is also possible to find solutions when $v_1,v_2\in S$ (or maybe even in $T$).

Let us consider elements $\pi$ such that $\pi S\pi^{-1}\cap S=\emptyset$.

**Questions.**

- Do permutations $\pi$ as above have a name and where somewhere studied in the literature? Where?
- Can problem
**C**be solved for some non-trivial* $v_1,v_2\neq 1$ and some $w$ of the form $w=v_1\pi$ with $\pi$ as above? Has problem**C**been studied in the literature? Where?

***Footnote:** There is also the trivial solution $v_1=v_2=\pi^{-1}$ which corresponds to $w=1$ listed as trivial above. I am interested if there are any more possible solutions for $w$ of this form.

Partial answers for the symmetric group are also very interesting...