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

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 $\mathfrak{spo}(4|1)$.