Segal <i>K</i>-theory of vector spaces with an automorphism

Abstract We describe the Segal $K$-theory of the symmetric monoidal category of finite-dimensional vector spaces over a field $\mathbb{F}$ together with an automorphism, or, equivalently, the group-completion of the $E_\infty $-algebra of maps from $S^{1}$ to the disjoint union of classifying spaces $\mathrm{B}\mathrm{GL}_{d}(\mathbb{F})$, in terms of the $K$-theory of finite field extensions of $\mathbb{F}$. A key ingredient for this is a computation of the Segal $K$-theory of the category of finite-dimensional vector spaces with a nilpotent endomorphism. We also discuss the topological cases of $\mathbb{F}=\mathbb{C},\mathbb{R}$.