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

Abstract

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