Nakajima monomials and crystals for special linear Lie algebras

Abstract

Nakajima introduced a certain set of monomials realizing the irreducible highest weight crystals $\mathcal{B}(\lambda)$. The monomial set can be extended so that it contains crystal $\mathcal{B}(\infty)$ in addition to $\mathcal{B}(\lambda)$. We present explicit descriptions of the crystals $\mathcal{B}(\infty)$ and $\mathcal{B}(\lambda)$ over special linear Lie algebras in the language of extended Nakajima monomials. There is a natural correspondence between the monomial description and Young tableau realization, which is another realization of crystals $\mathcal{B}(\infty)$ and $\mathcal{B}(\lambda)$.