Auslander-Reiten quiver of type D and generalized quantum affine Schur-Weyl duality

We first provide an explicit combinatorial description of the Auslander-Reiten quiver ${\mathrm{\Gamma }}^Q$ of finite type $D$. Then we can investigate the categories of finite dimensional representations over the quantum affine algebra $U_q^{\mathrm{\prime }}(D_{n+1}^{(i)})$ $(i=1,2)$ and the quiver Hecke algebra $R_{D_{n+1}}$ associated to $D_{n+1}$ $(n\ge 3)$, by using the combinatorial description and the generalized quantum affine Schur-Weyl duality functor. As applications, we can prove that Dorey’s rule holds for the category $\text{Rep}(R_{D_{n+1}})$ and prove an interesting difference between multiplicity free positive roots and multiplicity non-free positive roots.