Double Koszul complex and construction of irreducible representations of $\mathfrak{gl}(3\vert 1)$

Let $V$ be a super vector space with super dimension $(m\vert n)$. Manin introduced the Koszul complex associated to $V$, which is denoted $K$. There is another Koszul complex, denoted $L$. Our observation is that these two Koszul complexes can be combined into a double complex, which we call the double Koszul complex. By using the differential of this complex, we give a way to describe all irreducible representations of $\frak {gl}(V)$ when $V$ has super dimension $(3\vert 1)$.