THE K-RING OF E_6/Spin(10)

Let ${\mathrm{E}}_6$ denote the simply-connected compact exceptional Lie group of rank 6. The Lie group $\mathrm{S}pin(10)$ naturally embeds in ${\mathrm{E}}_6$, corresponding to the inclusion of the Dynkin diagrams. We determine the K-ring of the coset space ${\mathrm{E}}_6\mathrm{/}\mathrm{S}pin(10)$. We identify the class of the tangent bundle of ${\mathrm{E}}_6\mathrm{/}\mathrm{S}pin(10)$ in $KO({\mathrm{E}}_6\mathrm{/}\mathrm{S}pin(10))$. As an application we show that ${\mathrm{E}}_6\mathrm{/}\mathrm{S}pin(10)$ can be immersed in the Euclidean space ${\mathbb{R}}^{53}$.