Verma Bases and Kashiwara-Nakashima Tableaux of $\mathfrak{sp}_4$

Abstract

We construct a one-to-one correspondence between the Verma basis vectors of a finite dimensional irreducible representation $L(\lambda)$ of the symplectic Lie algebra $\mathfrak{sp}_4$ and the Kashiwara-Nakashima tableaux of $\mathfrak{sp}_4$ with shape $\lambda $ naturally. We also give a proof of the linear independence of the Verma vector system directly.