The (projective) irreducible representations of the Lorentz group which I denote by $\rho_{m,n}$ are classified by two nonnegative half-integers. I would like to decompose the grades $$ \wedge^k\rho_{m,n} $$ of the exterior algebra into irreducibles. I have found out how to compute the tensor products of these representations, but I don't know how to identify the wedge product inside this decomposition.
My motivation for asking this is that in QFT fermion fields should not be directly observable due to causality, only the even grades of the exterior algebra they generate should be observable, and I'm interested in the symmetries of these objects.