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Let $(W, S)$ be a Coxeter system. Let $J \subseteq S$ and recall that $W_J = \langle s: s \in J\rangle \subseteq W$.

Define $W^J = \{w \in W : \ell(ws) > \ell(w)\ \text{for all } s \in J\}$.

Proposition. For each $w \in W$ there is a unique $u \in W^J$ and $v \in W_J$ such that $w = uv$. Moreover, it holds for these elements that $\ell(w) = \ell(u) + \ell(v)$. Also, $u$ is the unique element of smallest length in the coset $wW_J = \{wx : x \in W_J\}$.

Question: Does $W^J\subseteq \langle t\in T: t\not\in W_J\rangle$, where $T$ is the set of reflections?

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    $\begingroup$ I must be missing something. This seems obviously true and like it has nothing to do with Coxeter group theory. The $W^J$ are coset representatives for the subgroup $W_J$ so none of them other than the identity belong to $W_J$. $\endgroup$ – Sam Hopkins Oct 30 '18 at 2:58
  • $\begingroup$ What is the quantification of t in the question? If it is $\langle t\in W:t\not\in W_J\rangle$, then as Sam says, the answer is obviously yes. If it is $\langle t\in S:t\not\in W_J\rangle$, then the answer is no. Or is it $\langle t\in T:t\not\in W_J\rangle$, where $T$ is the set of reflections, as the use of the letter $t$ might suggest to some? $\endgroup$ – Nathan Reading Oct 30 '18 at 11:58
  • $\begingroup$ The qualification of $t$ is $\langle t\in T:t\not\in W_J\rangle$, where $T$ is the set of reflections. $\endgroup$ – James Cheung Oct 30 '18 at 12:47
  • $\begingroup$ You should edit the question to make that clear. $\endgroup$ – Derek Holt Oct 31 '18 at 10:03
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In fact the minimal length coset representatives are exactly the elements $w$ such that for all reflections $t\in W_J$ we have $\ell(wt) \gt \ell(w) $. So the answer to your question is yes. Specifically, suppose $w\in W^J$ has the reduced word $s_1\cdots s_k$. Define $$t_i=s_ks_{k-1}\cdots s_i\cdots s_{k-1}s_k$$ Then none of the $t_i$ are in $W_J$ and $$t_kt_{k-1}\cdots t_2t_1=w$$

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