Explicit construction of harmonic two-spheres into the complex Grassmannian

We present an explicit description of all harmonic maps of finite uniton number from a Riemann surface into a complex Grassmannian. Namely, starting from a constant map $Q$ and a collection of meromorphic functions and their derivatives, we show how to algebraically construct all harmonic maps from the two-sphere into a given Grassmannian $G_p({\mathbb{C}}^n)$. In this setting the uniton number depends on $Q$ and $p$ and we obtain a sharp estimate for it.