Fundamental weights

Let $\mathrm{\Phi }\subseteq E$ be a reduced crystallographic root system with simple roots $\mathrm{\Delta }=\{{\alpha }_1,\dots ,{\alpha }_r\}$ (Positive systems and simple roots) and lattices $Q,Q^{\vee },P,P^{\vee }$ (Root, coroot, weight, and coweight lattices). The fundamental weights ${\omega }_1,\dots ,{\omega }_r$ are the vectors of $E$ dual to the simple coroots: $({\omega }_i,{\alpha }_j^{\vee })={\delta }_{ij}(1\le i,j\le r)\mathrm{.}$ They are well defined and unique because the ${\alpha }_j^{\vee }$ form a basis of $E$, the inner product is nondegenerate, and the linear functionals ${\alpha }_j^{\vee }\mapsto {\delta }_{ij}$ extend uniquely. The fundamental weights form a basis of $P$, called the fundamental weight basis: indeed $({\omega }_i,{\alpha }_j^{\vee })={\delta }_{ij}$ shows ${\omega }_i\in P$, and for $\lambda \in P$ the coefficients $c_i=(\lambda ,{\alpha }_i^{\vee })\in \mathbb{Z}$ in $\lambda ={\sum }_i{c}_i{\omega }_i$ are integers, so $P={⨁}_i\mathbb{Z}{\omega }_i$. Symmetrically, the fundamental coweights ${\omega }_i^{\vee }$, dual to the simple roots, form a basis of $P^{\vee }$.