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Let $W$ be a finite Coxeter group (edit: I should assume $W$ is a Weyl group of type ADE, see Edit below) and take $w \in W$. I am interested in the relation between non-Bruhat inversions of $w$ (inversions which reduce length of $w$ by more than one) and reduced expressions of $w$.

Some Definitions. A reflection $t \in W$ is an inversion of $w$ if $l(t w) < l(w)$ holds. Let us call a inversion $t$ is a Bruhat inversion if $l(tw) = l(w)-1$ holds. This corresponds to a covering relation $tw \lessdot w$ in the Bruhat order on $W$.

From now on, fix a reduced expression $w = s_{i_1} s_{i_2} \cdots s_{i_l} \cdots s_{i_n}$. Then for $l=1,\cdots, n$, we have inversions $t_l = s_{i_1} \cdots s_{i_l} \cdots s_{i_1}$ of $w$ which gives $t_l w = s_{i_1} s_{i_2} \cdots \hat s_{i_l} \cdots s_{i_n}$, and these are precisely all the inversions of $w$. Suppose that a inversion $t_l$ of $w$ as above is not a Bruhat inversion of $w$. This means that $t_l w= s_{i_1} s_{i_2} \cdots \hat s_{i_l} \cdots s_{i_n}$ is not a reduced expression by $l(t w) < l(w) - 1$.

A typical example: take $W=S_3$ and $w = s_1 s_2 s_1$. Then $t_2$ is not Bruhat, and actually $t_2 w = s_1 \hat s_2 s_1 = s_1 s_1$ is not a reduced expression.

My NAIVE question is: does such $t_l$ arise as in this example? To be precise, let's mark the letter as $s_{i_1} s_{i_2} \cdots \underline{s_{i_l}} \cdots s_{i_n}$ and write this expression as $s_L \underline{s_{i_l}} s_R$, and consider the following two moves of reduced expressions of $w$ (intuitively, this is a braid move with the letter $\underline{s_{i_l}}$ untouched):

1. Change reduced expressions of $s_L$ and $s_R$ (of course, with $\underline{s_{i_l}}$ untouched). This can be done with repeating braid moves on the left and right parts by Tits' word theorem.

2. If we have expressions like $\cdots s_a \underline{s_{i_l}} \cdots$ such that $s_a$ and $s_{i_l}$ commutes, we may replace this with $\cdots \underline{s_{i_l}} s_a \cdots$ (and vice versa).

Question: By repeating these two moves, can one always obtain a reduced expression of $w$ like: $\cdots s_b \underline{s_{i_l}} s_b \cdots$? [Edit: I assume that $W$ is a Weyl group for a simply laced root system.]

Example: In $W=S_4$, consider $w = s_1 s_2 s_3 s_1 s_2$. Then $t_3$ is not Bruhat because $t_3 w = s_1 s_2 \hat s_3 s_1 s_2$ is not reduced. But one can move as $s_1 s_2 \underline{s_3} s_1 s_2 = s_1 s_2 s_1 \underline{s_3} s_2 = s_2 s_1 s_2 \underline{s_3} s_2$, and this is what we want.

Counter Example in Infinite case: In $W=\tilde A_2$ with label $1,2,3$, consider $w = s_1 s_2 s_3 s_1 s_2$. As above, $t_3$ is not Bruhat, but this is the only reduced expression of $w$, so the answer is no.

Probably, the answer is yes for type A by this question: A question about set of inversion

Edit: Couneter-example in type $B_3$ exists by darij grinberg's comment. But I expect that the answer is yes for type ADE...

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  • $\begingroup$ Nice question! I take it you're assuming that $s_{i_l}$ is a non-Bruhat inversion? $\endgroup$ – darij grinberg Jul 15 at 15:06
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    $\begingroup$ What about this: Let $W = B_3$ with generators $a, b, c$ such that $a \overset{4}{-} b - c$ is the Coxeter diagram. Consider $w = abacba$. Then, removing the $c$ from $w$ is a non-Bruhat inversion, since $ababa = babaa = bab$. But the only reduced expressions of $w$ are $abacba$ and $abcaba$, according to SageMath; none of them has $c$ bordered by two equal letters. (SageMath code: W = WeylGroup(["B", 3]); c, b, a = W.simple_reflections(); w = a * b * a * c * b * a; w.reduced_words().) $\endgroup$ – darij grinberg Jul 15 at 15:40
  • $\begingroup$ I am still curious as to whether this holds for type A. $\endgroup$ – darij grinberg Jul 15 at 15:44
  • $\begingroup$ @darijgrinberg It does hold in type $A$. If $(i k)$ is a non-Bruhat inversion of $w$, then there is $j$ with $i < j < k$ such that $(i j)$ and $(j k)$ are also inversions of $w$. As explained in the question the OP links, that means $w$ has one reduced word where the inversions are added in order $(i j)$, $(i k)$, $(j k)$, and another where they are added in the reverse order. Take a sequence of braid moves changing one such word into the other. There must be a braid which reverses the order of those 3 inversions. That braid has a $s_1 s_2 s_1$ string as required. $\endgroup$ – David E Speyer Jul 15 at 16:00
  • $\begingroup$ @darijgrinberg I confirm your example, but the OP asked for ADE. In type ADE, the two notions of "non-Bruhat reflection" and "in the middle of a complete rank 2 parabolic" coincide, and I'm pretty sure the second one is the right notion if we wish to go outside ADE. $\endgroup$ – David E Speyer Jul 15 at 16:06

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