Let $W=\bigcup\_{n=1}^\infty S\_n$ be the union of all symmetric groups $S\_n$. For an element $w\in W$, denote by $\mathfrak{S}\_w$ the Schubert polynomial associated to $w$, and by $\partial\_w$ the divided difference operator associated to $w$.


**Question:** Let $u,v\in W$. Why is 
$$
\partial\_u(\mathfrak{S}\_u\mathfrak{S}\_v)-\mathfrak{S}\_v
$$
always a positive polynomial, in the sense that all coefficients are non-negative integers?


**Preliminary thoughts:**


* It is clear that $\partial\_u(\mathfrak{S}\_u\mathfrak{S}\_v)$ is positive.
* One may use the twisted Leipniz rule and induction on the length of $u$ to reduce the problem to an apparently simpler one. Indeed, let $s\_i$ be the rightmost factor in a reduced expression of $u$. Then it suffices to show that $\partial\_{us\_i}(s\_i\mathfrak{S}\_u\partial\_i\mathfrak{S}\_v)$ is positive. (Here, $W$ acts on $\mathbb{Z}[x\_1,x\_2,x\_3,\ldots]$ by permuting the variables.)
* I was thinking to realize the polynomials in question as characters of representations, in the spirit of the work of M. Watanabe on KP modules, but had not much success.


Any help would be greatly appreciated. I computed many examples and cannot find a counter example. In fact, I have a stronger "positivity conjecture" which implies this one, and even to this stronger conjecture, I was not able to find neither a counterexample nor a proof.



Positivity of coefficients of a polynomial derived from Schubert polynomials
I found the relevant literature for this question:


<https://arxiv.org/abs/0705.4546>


<https://arxiv.org/abs/1409.7056>


The keyword is "skew divided difference operator". In the first paper, Conjecture 1 states explicitely what I was trying to solve. An affirmative answer to this conjecture would immediately imply the positivity of the polynomial addressed in the original question. 


As to the date of 2007, positivity of skew divided difference operators seems to be an open problem. If someone else knows other results in this direction (the second cited paper goes in the "right" direction), please comment below this answer.


