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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.

  1. Do permutations $\pi$ as above have a name and where somewhere studied in the literature? Where?
  2. 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...

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  • $\begingroup$ I think for the symmetric group at least, we have $\pi S \pi^{-1}=\varnothing$ if and only if the one line notation of $\pi$ has no two adjacent letters $i$ and $j$ with $|i-j|=1$. E.g., $\pi=13524$ satisfies this. $\endgroup$
    – Sam Hopkins
    Commented Apr 19, 2019 at 16:54
  • $\begingroup$ Yes, this is true. I studied those for a longer time now, and if the number of letters grows, there are plenty. That's why I ask if there is something about it in the literature. $\endgroup$
    – Christoph Mark
    Commented Apr 19, 2019 at 16:58
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    $\begingroup$ okay, yes, this is just an obvious reformulation of your description. I haven't seen these permutations studied specifically. It's an interesting question. $\endgroup$
    – Sam Hopkins
    Commented Apr 19, 2019 at 16:59
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    $\begingroup$ See oeis.org/A002464. $\endgroup$
    – Richard Stanley
    Commented Apr 21, 2019 at 1:03

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