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

TL;DR

This paper provides explicit combinatorial descriptions of Auslander-Reiten quivers of type A and explores their connections with quantum affine Schur-Weyl duality, linking module categories over quantum affine algebras and quiver Hecke algebras.

Contribution

It offers a detailed combinatorial framework for AR-quivers of type A and investigates their role in the duality between quantum affine algebras and quiver Hecke algebras.

Findings

Explicit combinatorial descriptions of AR-quivers of type A.

Relations between module categories over quantum affine algebras and quiver Hecke algebras.

Enhanced understanding of quantum affine Schur-Weyl duality.

Abstract

The quiver Hecke algebra $R$ can be also understood as a generalization of the affine Hecke algebra of type $A$ in the context of the quantum affine Schur-Weyl duality by the results of Kang, Kashiwara and Kim. On the other hand, it is well-known that the Auslander-Reiten(AR) quivers $\Gamma_Q$ of finite simply-laced types have a deep relation with the positive roots systems of the corresponding types. In this paper, we present explicit combinatorial descriptions for the AR-quivers $\Gamma_Q$ of finite type $A$. Using the combinatorial descriptions, we can investigate relations between finite dimensional module categories over the quantum affine algebra $U'_q(A_n^{(i)})$ $(i=1,2)$ and finite dimensional graded module categories over the quiver Hecke algebra $R_{A_n}$ associated to $A_n$ through the generalized quantum affine Schur-Weyl duality functor.