In my endless fiddling with formulas I discovered one that fills in the blanks in a generic formula I saw in a paper, but I'm wondering if maybe it's already known and the paper was just mentioning the form of it casually. The formula I saw, which has undetermined constants, expresses a Schubert polynomial in $n$ variables as a sum of products of Schubert polynomials in a smaller number of variables with nonnegative coefficients. It looks like
$$S\_w(x\_1,x\_2,\ldots,x\_n)=\sum\_{u,v}{d\_{u,v}^wS\_u(x\_1,x\_2,\ldots,x\_k)S\_v(x\_{k+1},x\_{k+2},\ldots,x\_n)}$$
where $d\_{u,v}^w$ is mentioned to be nonnegative. I discovered that if you let $w\_0$ be the longest element of $S\_{n+1}$ and let $w\_0'$ be the longest element of $S\_{n+1-k}$ (identified with the parabolic subgroup of $S\_{n+1}$ corresponding to the elements in the first $n+1-k$ positions) then in fact
$$S\_w(x\_1,x\_2,\ldots,x\_n)=\sum\_{a,b}{c\_{a,b}^{ww\_0}S\_{aw\_0'w\_0}(x\_1,x\_2,\ldots,x\_k)S\_{bw\_0'}(x\_{k+1},x\_{k+2},\ldots,x\_n)}$$
where $\ell(aw\_0'w\_0)+\ell(a)=\ell(w\_0'w\_0)$, $\ell(bw\_0')+\ell(b)=\ell(w\_0')$, and $c\_{a,b}^{ww\_0}$ is the corresponding structure constant for multiplying Schubert polynomials, which of course is known to be nonnegative. Is the formula known to this degree of specificity?



Reduction formula for Schubert polynomials
This is essentially Theorem 4.6.1 in (<http://arxiv.org/pdf/alg-geom/9703001v1.pdf>) Schubert polynomials, the Bruhat order, and the geometry of flag manifolds by Bergeron and Sottile.


