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Suppose $(W,S)$ is a Coxeter system and let $<$ denote the (strong) Bruhat Order of $W$; that is $u < b$, there exists some sequence of $t_1,\ldots,t_k \in S^W$ such that $v = ut_1\ldots t_k$ and $l(v) = l(u) + k$ (for the usual length function $l$ with respect to $S$).

Given $w \in W$ and $\{t_1,\ldots,t_k\} \subseteq S^W$ is a set of pairwise commuting elements, is the following true:

$$ w < wt_i \quad \forall i =1,\ldots,k \iff w < wt_1\ldots t_k?$$

I have thought about this for a while and assume the answer should be obvious but I can't see it directly.

Rob Nicolaides
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  • I think only one implication (left-to-right as written) of the if and only if statement is true now but still don't know how to prove it. – Rob Nicolaides Jun 01 '22 at 14:53

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This is a nice question and I'm answering largely because I'm curious whether you've figured it out :).

As you noted in the comments, one direction fails: for example, in $S_4$, let $w = s_2 = 1324 = (23)$, let $t_1 = w$ and let $t_2 = 4231 = (14) = s_1s_2s_3s_2s_1$, so that $t_1t_2 = 4321 = (14)(23) = w_0$. Then $w < wt_1t_2 = 4231$ but obviously $w \not < wt_1$.

It is easy to prove that the other direction holds in $S_n$: $w < wt$ if and only if the two positions permuted by $t$ are inverted, and if a bunch of transpositions commute then they are disjoint so none of them affects whether the entries permuted by the others are inverted. A slightly pickier but similar argument works in finite types B and D using signed permutations, and I think I convinced myself that it's true in dihedral groups. But it certainly feels like it should be true at the level of Coxeter groups rather than finite Coxeter groups. If it's in the book of Bjorner and Brenti, I wasn't able to find it.

JBL
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    Thanks JBL, I could only see the implication is true for the type a Coxeter groups (just as you showed) which is all I needed at the time. I would still be curious to see a general proof :) I wonder if looking at root systems more directly might be the way to go… – Rob Nicolaides Feb 23 '23 at 22:55
  • @RobNicolaides Ok thanks for the update :). Indeed, looking at the general root system seems like a promising idea -- I hope to come back and think about it again at some point in the future. – JBL Mar 26 '23 at 19:32