Schur times Schubert via the Fomin-Kirillov algebra

Understand

We study multiplication of any Schubert polynomial $\mathfrak{S}_w$ by a Schur polynomial $s_\lambda$ (the Schubert polynomial of a Grassmannian permutation) and the expansion of this product in the ring of Schubert polynomials.

• We derive explicit nonnegative combinatorial expressions for the expansion coefficients for certain special partitions $\lambda$, including hooks and the 2x2 box.
• We also prove combinatorially the existence of such nonnegative expansion when the Young diagram of $\lambda$ is a hook plus a box at the (2,2) corner.
• We achieve this by evaluating Schubert polynomials at the Dunkl elements of the Fomin-Kirillov algebra and proving special cases of the nonnegativity conjecture of Fomin and Kirillov.