Crystal bases as tuples of integer sequences

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 $λ$ a connected subcrystal $\mathcal{R}(λ)$ in $\mathcal{R}^{\infty}$, such that this crystal is isomorphic to the crystal graph $B(λ)$. Finally we provide an explicit description of these connected crystals $\mathcal{R}(λ)$.

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