Intersection multiplicity one for the Belkale-Kumar product in G/B

Consider the complete flag variety $X$ of a complex semisimple algebraic group $G$. We show that the structure coefficients of the Belkale-Kumar product ${\odot }_0$, on the cohomology ${\mathrm{H}}^{\ast }(X,\mathbf{Z})$, are all either $0$ or $1$. We also derive some consequences. The proof that is mainly geometric also uses new combinatorial results on root systems. Moreover, it is uniform and avoids case by case considerations.