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?

## 1 Answer

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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.

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