The Weyl vector

Let $\mathrm{\Phi }\subseteq E$ be a reduced crystallographic root system in the real inner product space $E$ (Reduced crystallographic Euclidean root system) with a chosen positive system ${\mathrm{\Phi }}^{+}$ (Positive systems and simple roots). The Weyl vector of this choice is $\rho =\frac{1}{2}{\sum }_{\alpha \in {\mathrm{\Phi }}^{+}}\alpha \in E\mathrm{.}$ The sum is finite because a root system is finite, and it is taken in the real vector space $E$, so the factor $\frac{1}{2}$ is the real scalar $\frac{1}{2}$; no integrality of $\rho$ is asserted. Since every positive root lies in the root lattice $Q$ (Root, coroot, weight, and coweight lattices), one has $2\rho ={\sum }_{\alpha \in {\mathrm{\Phi }}^{+}}\alpha \in Q$, so $\rho$ lies in $\frac{1}{2}Q$.

Replacing the positive system by its opposite replaces $\rho$ by $\frac{1}{2}{\sum }_{\alpha \in {\mathrm{\Phi }}^{+}}(-\alpha )=-\rho ,$ so the Weyl vector depends on the choice of positive system and is not an invariant of the root system alone.