K-theory of real Grassmann manifolds

Let $G_{n,k}$ denote the real Grassmann manifold of $k$-dimensional vector subspaces of ${\mathbb{R}}^n$. Using the Hodgkin spectral sequence, we compute the complex $K$-ring of $G_{n,k}$, up to a small indeterminacy, for all values of $n,k$ where $2\le k\le n-2$. When $n\equiv 0\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{m}\mathrm{o}\mathrm{d}4,k\equiv 1\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{m}\mathrm{o}\mathrm{d}2$ our result is complete.