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

TL;DR

This paper provides a combinatorial description of the Auslander-Reiten quiver of type D and explores its implications for categories of representations over quantum affine algebras and quiver Hecke algebras, including Dorey's rule.

Contribution

It introduces an explicit combinatorial model for the Auslander-Reiten quiver of type D and applies it to study representation categories via generalized quantum affine Schur-Weyl duality.

Findings

Dorey's rule holds for the category (R_{D_{n+1}})

Identifies differences between multiplicity free and non-free positive roots

Provides a combinatorial description of the Auslander-Reiten quiver of type D

Abstract

We first provide an explicit combinatorial description of the Auslander-Reiten quiver $\Gamma^Q$ of finite type $D$. Then we can investigate the categories of finite dimensional representations over the quantum affine algebra $U_q'(D^{(i)}_{n+1})$ $(i=1,2)$ 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 $\Rep(R_{D_{n+1}})$ and prove an interesting difference between multiplicity free positive roots and multiplicity non-free positive roots.