Crystal bases as tuples of integer sequences

Abstract

We describe a set $\mathcal{R}^{\infty}$ consisting of tuples of integer sequences and provide certain explicit maps on it. We show that this defines a semiregular crystal for $\mathfrak{sl}_{n+1}$ and $\mathfrak{sp}_{2n}$ respectively. Furthermore we define for any dominant integral weight $\lambda$ a connected subcrystal $\mathcal{R}(\lambda)$ in $\mathcal{R}^{\infty}$, such that this crystal is isomorphic to the crystal graph $B(\lambda)$. Finally we provide an explicit description of these connected crystals $\mathcal{R}(\lambda)$.