Schubert puzzles and integrability III: separated descents

Abstract: In paper I of this series we gave positive formulae for expanding the product $\mathfrak S^\pi \mathfrak S^\rho$ of two Schubert polynomials, in the case that both $\pi,\rho$ had shared descent set of size $\leq 3$. Here we introduce and give positive formulae for two new classes of Schubert product problems: separated descent in which $\pi$'s last descent occurs at (or before) $\rho$'s first, and almost separated descent in which $\pi$'s last two descents occur at (or before) $\rho$'s first two respectively. In both cases our puzzle formulae extend to $K$-theory (multiplying Grothendieck polynomials), and in the separated descent case, to equivariant $K$-theory. The two formulae arise (via quantum integrability) from fusion of minuscule quantized loop algebra representations in types $A$, $D$ respectively.