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 $\mathcal{R}(\lambda )$.

References in corpus (1)

• Quiver varieties and tensor products