Simple coroot (${\alpha }_i^{\vee }$)

For chosen simple roots, the corresponding coroots are the simple coroots. They form the simple system of the dual root system. The fundamental weights are defined by $⟨{\omega }_j,{\alpha }_i^{\vee }⟩={\delta }_{ij}$, so pairings with simple coroots record coordinates in the fundamental-weight basis.