# Non-Bruhat inversions and reduced expressions in finite Coxeter groups

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...

• Nice question! I take it you're assuming that $s_{i_l}$ is a non-Bruhat inversion? – darij grinberg Jul 15 at 15:06
• 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().) – darij grinberg Jul 15 at 15:40
• I am still curious as to whether this holds for type A. – darij grinberg Jul 15 at 15:44
• @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. – David E Speyer Jul 15 at 16:00
• @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. – David E Speyer Jul 15 at 16:06