Generalized cohomology for irreducible tensor fields of mixed Young symmetry type

Abstract

We construct N-complexes of non completely antisymmetric irreducible tensor fields on $\mathbb R^D$ generalizing thereby the usual complex (N=2) of differential forms. These complexes arise naturally in the description of higher spin gauge fields. Although, for $N\geq 3$, the generalized cohomology of these N-complexes is non trivial, we prove a generalization of the Poincar\'e lemma. Several results which appeared in various contexts are shown to be particular cases of this generalized Poincar\'e lemma.