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

Abstract We study the Deligne interpolation categories Rep (GLₓ (Fq) ) Rep ̲ (G L t (F q) ) for t C t ∈ C, first introduced by F. Knop. These categories interpolate the categories of finite-dimensional complex representations of the finite general linear group GLₙ (Fq) G L 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 C Fqⁿ C F q n of GLₙ (Fq) G L n (F q) ) carries the structure of a Frobenius algebra with a compatible Fq F q -linear structure; we call such objects Fq F q -linear Frobenius spaces and show that Rep (GLₓ (Fq) ) Rep ̲ (G L t (F q) ) is the universal symmetric monoidal category generated by such an Fq 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 (Fq) G L ∞ (F q).