Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras II

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Let be an untwisted affine Kac-Moody algebra of type A⁽¹⁾ₙ (n ≥ 1) or D⁽¹⁾ₙ (n ≥ 4) and let ₀ be the underlying finite-dimensional simple Lie subalgebra of. For each Dynkin quiver Q of type ₀, Hernandez and Leclerc (HL11) introduced a tensor subcategory of the category of finite-dimensional integrable -modules and proved that the Grothendieck ring of is isomorphic to [N], the coordinate ring of the unipotent group N associated with ₀. We apply the generalized quantum affine Schur-Weyl duality introduced in KKK13 to construct an exact functor from the category of finite-dimensional graded R-modules to the category, where R denotes the symmetric quiver Hecke algebra associated to ₀. We prove that the homomorphism induced by the functor coincides with the homomorphism of Hernandez and Leclerc and show that the functor sends the simple modules to the simple modules.