Thoughts on the paper ``Generic representation theory of finite fields in nondescribing characteristic'' by N.J. Kuhn

Ross Street – 13 July 2022

Let bbF be a finite field, let scrF be the category of finite dimensional vector spaces and linear functions over bbF, and let scrG be the subcategory of scrF with all the objects and only the bijective linear functions. Let scrV be the category of vector spaces over the complex numbers (say). André Joyal and I studied a monoidal structure on the functor category [scrG,scrV] as recorded in our publication [49]. So I was quite interested and surprised when Nicholas Kuhn told Steve Lack and me about his equivalence of categories [scrF,scrV] \simeq [scrG,scrV] when we were discussing Dold-Kan-type theorems. The proof depends on a remarkable piece of linear algebra by our late ANU colleague Laci Kov\'aks. Given Kov\'aks' result (1992), I am studying the rest of the argument. It is at a fairly early stage but that has not in the past stopped me from giving a premature talk!