Nakajima monomials, Young walls and Kashiwara embedding for Uq(An(1))

In this paper, we realize the crystal basis B(λ) of the irreducible highest weight module V(λ) of level 1 for U_q(A_n⁽¹⁾) using Nakajima monomials satisfying some conditions. Also, from this monomial realization, we obtain the image of Kashiwara embedding ψ^λ_ι:B(λ){right arrow, hooked}Z^∞⊗R_λ, where ι is some infinite sequence from the index set of simple roots. Finally, we give a Uq(An(1))-crystal isomorphism between Young wall realization and monomial realization, and so we can understand the image of Kashiwara embedding ψ^λ_ι:B(λ){right arrow, hooked}Z^∞⊗R_λ using the combinatorics of Young walls.