# Representations of General Linear Groups in the Verlinde Category

**Authors:** Siddharth Venkatesh  
**arXiv:** [2203.03158](https://arxiv.org/abs/2203.03158)  
**Date:** 2022-03-07  
**Summary page:** https://www.summarizepaper.com/en/paper/2203.03158v1/

## Key points

- Focus on constructing affine group schemes $GL(X)$ for objects in the Verlinde category in characteristic $p
- Classification of irreducible representations of these affine group schemes
- Demonstration that representation category is semisimple for simple object X with categorical dimension i
- Equivalence to connected component of Verlinde category for $SL_i$
- Utilization of Verma module construction to classify irreducible representations of $GL(nL)$ for any simple object L and natural number n
- Expansion of study's scope to encompass broader possibilities within general linear groups using this approach
- Categorization of irreducible representations of $GL(X)$ through parabolic induction techniques for any object X within the Verlinde Category

## Summary

In this article by Siddharth Venkatesh, the focus is on constructing affine group schemes $GL(X)$ for any object in the Verlinde category in characteristic $p$. The study aims to classify their irreducible representations and begins by demonstrating that for a simple object X with categorical dimension i, the representation category is semisimple. It is also equivalent to the connected component of the Verlinde category for $SL_{i}$, providing a foundational understanding for further exploration. The research then delves into utilizing a Verma module construction to classify irreducible representations of $GL(nL)$ for any simple object L and any natural number n. By leveraging this approach, the study expands its scope to encompass a broader range of possibilities within the realm of general linear groups. Furthermore, through the application of parabolic induction techniques, it successfully categorizes irreducible representations of $GL(X)$ where X represents any object within the Verlinde Category. This comprehensive analysis not only sheds light on the intricate structures within affine group schemes but also offers valuable insights into the classification of irreducible representations across varying contexts. Overall, Siddharth Venkatesh's work provides a detailed and systematic exploration of representations of general linear groups in the Verlinde category. It offers a significant contribution to mathematical research in this area.