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

Abstract

We study the Deligne interpolation categories $\underline{\mathrm{Rep}}(GL_{t}(\mathbb{F}_q))$ 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 $GL_n(\mathbb{F}_q)$. 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 $GL_n(\mathbb{F}_q)$) 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 $\underline{\mathrm{Rep}}(GL_{t}(\mathbb{F}_q))$ 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 $GL_{\infty}(\mathbb{F}_q)$.