The highest weight theory for Representations of General Linear groups in the Verlinde categories in positive characteristic

TL;DR

This paper explores the highest weight theory for representations of general linear groups within Verlinde categories over fields of positive characteristic, establishing structural and categorical properties.

Contribution

It introduces a highest weight category structure and categorical actions for representations of GL(X) in Verlinde categories, extending understanding of their module theory.

Findings

Established highest weight category structure for GL(X) representations

Defined categorical actions of affine sl_p on these categories

Analyzed projective, injective objects, and dualities in the category

Abstract

Following the work of Venkatesh (arXiv:2203.03158), we study further the categories of representations of the general linear groups $GL(X)$ in the Verlinde category $Ver_p$ in characteristic $p$. The main question we answer is how to translate between highest weight labelings for different choices of the Borel subgroup $B(X)\subset GL(X)$. We do this by reducing the general case to the study of representations of the group $GL(X)$ for $X=L_m\oplus L_{n}$ using the method of odd reflections. On the category of representations of $GL(L_m\oplus L_{n})$ we introduce the structure of the highest weight category, as well as the categorical action of $\widehat{\mathfrak{sl}}_p$ through translation functors. It allows us to understand projective and injective objects, BGG reciprocity, duality and lowest weights for simple modules, and standard filtration multiplicities for projective objects.