Let $R$ be an irreducible root system with a basis $\Pi$. We obtain the Dynkin diagram $D$ and the extended Dynkin diagram ${\widetilde{D}}$ of $R$ with respect to $\Pi$. Let $Q^\vee\subset P^\vee$ denote the coroot lattice and the coweight lattice, respectively. It is known that the finite abelian group $P^\vee/Q^\vee$ (which is isomorphic to the center of the corresponding simply connected compact Lie group) acts on ${\widetilde{D}}$.

I am looking for references where the action of $P^\vee/Q^\vee$ on ${\widetilde{D}}$ is described in detail (preferably with examples, say for a root system of type $D_n$).