# About generator of minimal length coset representatives

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?

• 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 $W_J$. – Sam Hopkins Oct 30 '18 at 2:58
• 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? – Nathan Reading Oct 30 '18 at 11:58
• The qualification of $t$ is $\langle t\in T:t\not\in W_J\rangle$, where $T$ is the set of reflections. – James Cheung Oct 30 '18 at 12:47
• You should edit the question to make that clear. – Derek Holt Oct 31 '18 at 10:03

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 $$t_kt_{k-1}\cdots t_2t_1=w$$