Nakajima monomials, Young walls and Kashiwara embedding for [formula omitted

Abstract

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 ι λ : B ( λ ) ↪ Z ∞ ⊗ R λ , where ι is some infinite sequence from the index set of simple roots. Finally, we give a U q ( A n ( 1 ) ) -crystal isomorphism between Young wall realization and monomial realization, and so we can understand the image of Kashiwara embedding Ψ ι λ : B ( λ ) ↪ Z ∞ ⊗ R λ using the combinatorics of Young walls.