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

TL;DR

This paper establishes a natural bijection between Verma basis vectors and Kashiwara-Nakashima tableaux for $rak{sp}_4$, providing a direct proof of their linear independence.

Contribution

It constructs a natural one-to-one correspondence between Verma basis vectors and Kashiwara-Nakashima tableaux for $rak{sp}_4$, and proves their linear independence directly.

Findings

Established a bijection between Verma basis vectors and Kashiwara-Nakashima tableaux.

Provided a direct proof of the linear independence of Verma vectors.

Enhanced understanding of the structure of $rak{sp}_4$ representations.

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.