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

TL;DR

This paper establishes a sufficient condition for the nilpotency of trace-zero endomorphisms in Schur-finite objects within tensor categories, extending previous results and introducing new trace identities involving super vector spaces.

Contribution

It generalizes Kimura's results on finite-dimensional objects to Schur-finite objects in categories of homological type, using super vector space trace relations and Schur-Weyl duality.

Findings

Proves a trace relation identity on super vector spaces of independent interest.

Provides a sufficient condition for nilpotency of trace-zero endomorphisms in a broad categorical setting.

Suggests the generalization might be optimal based on the presented facts.

Abstract

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.