Deligne categories and representations of the finite general linear group, part 1: universal property

Abstract

We study the Deligne interpolation categories $\underline{\mathrm{Rep}}(G{L}_t({\mathbb{F}}_q))$ for $t\in \mathbb{C}$, first introduced by F. Knop. These categories interpolate the categories of finite dimensional complex representations of the finite general linear group $G{L}_n({\mathbb{F}}_q)$. We describe the morphism spaces in this category via generators and relations. We show that the generating object of this category (an analogue of the representation $\mathbb{C}{\mathbb{F}}_q^n$ of $G{L}_n({\mathbb{F}}_q)$) carries the structure of a Frobenius algebra with a compatible ${\mathbb{F}}_q$-linear structure; we call such objects ${\mathbb{F}}_q$-linear Frobenius spaces, and show that $\underline{\mathrm{Rep}}(G{L}_t({\mathbb{F}}_q))$ is the universal symmetric monoidal category generated by such an ${\mathbb{F}}_q$-linear Frobenius space of categorical dimension $t$. In the second part of the paper, we prove a similar universal property for a category of representations of $G{L}_{\infty }({\mathbb{F}}_q)$.

arXiv commentv2: Minor changes. Renamed the Deligne categories to $\underline{\mathrm{Rep}}(G{L}_t({\mathbb{F}}_q))$ 47 pages; Comments welcome!