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

Abstract

Let be a super vector space with super dimension . Manin introduced the Koszul complex associated to , which is denoted . There is another Koszul complex, denoted . 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 when has super dimension .