Schur-finiteness and endomorphisms universally of trace zero via certain trace relations

We provide a sufficient condition that ensures the nilpotency of endomorphisms universally of trace zero of Schur-finite objects in a category of homological type, i.e., a Q-linear tensor category with a tensor functor to super vector spaces. This generalizes previous results about finite-dimensional objects, in particular by Kimura in the category of motives. We also present some facts which suggest that this might be the best generalization possible of this line of proof. To get the result we prove an identity of trace relations on super vector spaces which has an independent interest in the field of combinatorics. Our main tool is Berele-Regev’s theory of Hook Schur functions. We use their generalization of the classic Schur-Weyl duality to the “super” case, together with their factorization formula.