Verma Bases for finite dimensional Representations of the orthosymplectic Lie superalgebra $\mathfrak{spo}(4|1)$

TL;DR

This paper constructs a Verma basis for finite-dimensional irreducible representations of the orthosymplectic Lie superalgebra $rak{spo}(4|1)$ using Kashiwara-Nakashima tableaux, establishing linear independence and basis properties.

Contribution

It introduces a novel Verma basis for $rak{spo}(4|1)$ representations, extending the concept analogous to $rak{sp}_4$ to superalgebras.

Findings

The Verma vector system forms a basis for $L()$.

The basis is constructed via Kashiwara-Nakashima tableaux.

Linear independence of the vector system is proven.

Abstract

We define the Verma vector system for each finite dimensional irreducible representation of the orthosymplectic Lie superalgebra $\mathfrak{spo}(4|1)$ with the highest weight $\lambda,$ via the conditions that making a tableau with shape $\lambda$ to be a Kashiwara-Nakashima tableau. We then show the linearly independence of this vector system. It turns out to be a basis of the finite dimensional irreducible representation $L(\lambda)$ of the orthosymplectic Lie superalgebra $\mathfrak{spo}(4|1)$ with the highest weight $\lambda,$ which analogs to the Verma basis of representations of $\mathfrak{sp}_4,$ called the Verma basis of the finite dimensional irreducible representation of $\mathfrak{spo}(4|1)$.