Representations of General Linear Groups in the Verlinde Category

TL;DR

This paper constructs affine group schemes in the Verlinde category over characteristic p and classifies their irreducible representations, extending classical representation theory to a categorical setting.

Contribution

It introduces a framework for affine group schemes in the Verlinde category and provides a classification of their irreducible representations.

Findings

Representation categories are semisimple for simple objects with categorical dimension i.

Classification of irreducible representations for $GL(nL)$ using Verma modules.

Complete classification of irreducible representations for $GL(X)$ via parabolic induction.

Abstract

In this article, we construct affine group schemes $GL(X)$ where $X$ is any object in the Verlinde category in characteristic $p$ and classify their irreducible representations. We begin by showing that for a simple object $X$ of categorical dimension $i$, this representation category is semisimple and is equivalent to the connected component of the Verlinde category for $SL_{i}$. Subsequently, we use this along with a Verma module construction to classify irreducible representations of $GL(nL)$ for any simple object $L$ and any natural number $n$. Finally, parabolic induction allows us to classify irreducible representations of $GL(X)$ where $X$ is any object in the Verlinde Category.