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.

lexicographically smallestreduced expression of $u$, you get many summands. One of them is $\mathfrak{S}_v$. I expect each of the summands to be a positive polynomial, but I am not completely sure about this, because I have only checked it heuristically for $u=s_\alpha$ - the case I am mostly interested in. $\endgroup$