Deligne Categories and Representations of the Finite General Linear Group, Part 1: Universal Property

We study the Deligne interpolation categories \(\underline{\textrm{Rep}}(GL_{t}({\mathbb F}_q))\) $\underline{\text{Rep}}(G{L}_t({\mathbb{F}}_q))$ for \(t\in \mathbb {C}\) $t\in \mathbb{C}$ , first introduced by F. Knop. These categories interpolate the categories of finite-dimensional complex representations of the finite general linear group \(GL_n(\mathbb {F}_q)\) $G{L}_n({\mathbb{F}}_q)$ . 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 \({\mathbb C}{\mathbb F}_q^n\) $\mathbb{C}{\mathbb{F}}_q^n$ of \(GL_n(\mathbb {F}_q)\) $G{L}_n({\mathbb{F}}_q)$ ) carries the structure of a Frobenius algebra with a compatible \({\mathbb F}_q\) ${\mathbb{F}}_q$ -linear structure; we call such objects \(\mathbb {F}_q\) ${\mathbb{F}}_q$ -linear Frobenius spaces and show that \(\underline{\textrm{Rep}}(GL_{t}({\mathbb F}_q))\) $\underline{\text{Rep}}(G{L}_t({\mathbb{F}}_q))$ is the universal symmetric monoidal category generated by such an \(\mathbb {F}_q\) ${\mathbb{F}}_q$ -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 \(GL_{\infty }(\mathbb {F}_q)\) $G{L}_{\infty }({\mathbb{F}}_q)$ .