Generalized cohomology for irreducible tensor fields of mixed Young symmetry type

We construct N-complexes of noncompletely antisymmetric irreducible tensor fields on R-D, thereby generalizing the usual complex (N = 2) of differenti al forms. These complexes arise naturally in the description of higher spin gauge fields. Although, for N greater than or equal to 3, the generalized cohomology of these N-complexes is nontrivial, we give a generalization of the Poincare lemma. Several results which have appeared in various contexts are shown to be particular cases of this generalized Poincare lemma.