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

We study the Deligne interpolation categories Rep̲(GL_t(F_q)) for t∈C, first introduced by F. Knop. These categories interpolate the categories of finite-dimensional complex representations of the finite general linear group GL_n(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 CF_qⁿ of GL_n(F_q)) carries the structure of a Frobenius algebra with a compatible F_q-linear structure; we call such objects F_q-linear Frobenius spaces and show that Rep̲(GL_t(F_q)) is the universal symmetric monoidal category generated by such an 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_∞(F_q).