Nakajima monomials, Young walls and Kashiwara embedding for U-q(A(n)((1)))

Abstract

In this paper, we realize the crystal basis B(A) of the irreducible highest weight module V(A) of level 1 for U(A) using Nakajima monomials satisfying some conditions. Also, from this monomial realization, we obtain the image of Kashiwara embedding Psi(lambda)(l) : B(lambda) -> Z(infinity) circle times R-lambda, where l 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 Psi(lambda)(l) : B(lambda) -> Z(infinity) circle times R-lambda using the combinatorics of Young walls.