Abstract
The quiver Hecke algebra R R can be also understood as a generalization of the affine Hecke algebra of type A 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 Γ Q \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 Γ Q \Gamma _Q of finite type A A . Using the combinatorial descriptions, we can investigate relations between finite dimensional module categories over the quantum affine algebra U q ′ ( A n ( i ) ) U’_q(A_n^{(i)}) ( i = 1 , 2 ) (i=1,2) and finite dimensional graded module categories over the quiver Hecke algebra R A n R_{A_n} associated to A n A_n through the generalized quantum affine Schur-Weyl duality functor.
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