Generalized Young walls and crystal bases for quantum affine algebra of type $A$

We give a new realization of the crystal $B(\infty)$ of $U_q^{-}(A_n^{(1)})$ using generalized Young walls, a modified notion of Young walls of type $A_n^{(1)}$. Moreover, by the fact that the irreducible highest weight crystal $B(\lambda)$ lies in the crystal $B(\infty)\otimes R_{\lambda}$, we construct the crystal $B(\lambda)$ using generalized Young walls.