by Markus Rost (Notes, February 2019, 9 pages)
The text grew out of an attempt to formalize the matrix calculus for the exterior algebra used in the text Notes on Schur functors.
It considers functors from the category of free abelian groups to a monoidal category. Closely related is the question for generators and relations for integral matrices of any size with respect to direct sums and products. There are only some rudiments.
Meanwhile (2021-2022) I have understood things better, see the first remarks in the text.
Full text (version of June 11, 2022): [pdf]
by Markus Rost (Notes, February 2019, 8 pages)
The main results are alternative descriptions of the module generated by the quadratic "exchange" relations for Schur modules. A first step is a reformulation of the relations in terms of the Hopf algebra structure. The proofs are written in a pretty compact form using a (rather obvious) matrix calculus for the exterior algebra. Other related topics are briefly mentioned.
Full text (version of Feb 28, 2019): [pdf]
by Markus Rost (Notes, February 2017, 10 pages)
For an endomorphism f of an R-module M we define certain tensors Ak(f) of degree k. If M is locally free of rank k+1, then Ak(f) is the adjunct of f. For each k there is a generalization of the Cayley-Hamilton theorem which reduces to the classical Cayley-Hamilton theorem for f when M is locally free of rank k+1. Various other tensors and relations among them are considered. Everything follows from formal properties of the exterior algebra.
In a future version one should perhaps use the matrix notation as described in Notes on strict bicommutative Hopf algebras.
Full text (version of Feb 28, 2017): [pdf]