Nakajima monomials and crystals for special linear Lie algebras

We present explicit descriptions of the crystals $\mathcal{B}(\mathrm{\infty })$ and $\mathcal{B}(\lambda )$ over special linear Lie algebras in the language of \emph{extended Nakajima monomials}. There is a natural correspondence between the monomial description and Young tableau realization, which is another realization of crystals $\mathcal{B}(\mathrm{\infty })$ and $\mathcal{B}(\lambda )$.