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

We study the Deligne interpolation categories (Rep) under bar (GL(t) (F-q)) for t is an element of 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 CFqn 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 (Rep) under bar (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(infinity)(F-q).