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Let $(W,S)$ be a Coxeter group with Bruhat order $\leq$ and length function $\ell(w)$.

Definition: a subset $X \subset W$ has the chain property if whenever $x,y \in X$ with $x < y$, there exists $x_0, x_1, \ldots, x_k \in X$ such that $x=x_0 < x_1 < \ldots < x_k = y$, with $\ell(x_i) = \ell(u) + i$.

Remark: clearly, for this property to hold, it is sufficient the existence of a $x_1 \in X$ such that $x < x_1 \leq y$ and $\ell(x_1) = \ell(x) + 1$.

Let $J \subset S$ and consider $W_J$ the parabolic subgroup spanned by $J$. Then the right quotient $W/W_J$ has a system of representatives given by $W^J = \{ w :\, \ell(ws) > \ell(w),\, \forall s \in J \}$ where each $w \in W^J$ is the minimal element of the right coset $w W_J$.

It is well known that the system of representatives $W^J$ of the right coset has the chain property (Theorem 2.5.5 p.45 of Bjorner-Brenti's Combinatorics of Coxeter Groups).

Now let $I,J \subset S$. Then the double quotient $W_I \backslash W / W_J$ has a system of representatives given by $^I W^J = (W^I)^{-1} \cap W^J$ where each $w \in$$^I W^J$ is the minimal element of the double coset $W_I w W_J$ (Proposition 2.7.3 p.64 of Carter's Finite Groups of Lie Type).

My question is then: does the system of representatives $^IW^J$ of the double coset has the chain property?

I am actually interested in this result only in the case of Weyl groups, but it seemed best to ask it in the above generality. This looks like a textbook question but I could not find a reference for this nor did I succeed in adapting the cited Bjorner-Brenti's Theorem to this case.

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By part (b) of Remark 1.6 on page 7 of http://www.math.lsa.umich.edu/~jrs/papers/affine.ps.gz, the answer seems to be: "Not always"!

A shorter counterexample is $W=S_4$ the symmetric group on 4 letters and $I=J=\{ (12), (34) \}$.

Then $\# W_I=4$ so that $W^I = \left\{ w \in S_4:\, w(1) < w(2),\, w(3) < w(4) \right\}$ has $\# W^I= \# W/W_I = 3! = 6$ which are easy to find by construction. This reduces the work to pin down $^I W^I = (W^I)^{-1} W^I$ which can be computed to be the three permutations $1$, $(23)$, $(13)(24)$.

They have length 0,1 and 4, respectively, which shows that $^I W^I$ does not have the chain property at all!

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  • $\begingroup$ I saw your question and was on my way to find Stembridge's counterexample, but you were too fast for me. :) Stembridge actually says this in part (b) of Remark 1.6 of the same paper. Good question, and good answer. $\endgroup$ – Nathan Reading Jul 24 '14 at 13:10
  • $\begingroup$ Thanks @NathanReading! I corrected the reference to Stembridge and gave a shorter counterexample :) Actually, I had googled on the subject before, but I only found Stembridge reference a few minutes after I posted the question here: it is a very nice reference since not all references on Bruhat order deal with the double coset. $\endgroup$ – Lucas Seco Jul 25 '14 at 3:32

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