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

Alexandra Utiralova

TL;DR

This work extends representation theory in Verlinde categories Ver_p by building a comprehensive highest weight framework for GL(X) across varying Borels. It reduces the problem to GL(L_m⊕L_n) via odd reflections, then leverages a categorical $\widehat{\mathfrak{sl}}_p$-action from translation functors to realize loop-module structures and classify simple objects. It proves that Rep_{Ver_p}(GL(X)) carries a highest weight category structure with generalized Verma modules as standards, and establishes level-rank dualities and a Shapovalov-type criterion governing irreducibility of Kac modules in the super-analytic setting $GL(L_m|L_n)$. The results provide concrete combinatorics via weight diagrams and connect to classical supergroup theory, yielding a robust framework for translating highest weights across Borels and understanding projectives, reciprocity, and lowest weights in Ver_p.

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.

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

TL;DR

This work extends representation theory in Verlinde categories Ver_p by building a comprehensive highest weight framework for GL(X) across varying Borels. It reduces the problem to GL(L_m⊕L_n) via odd reflections, then leverages a categorical -action from translation functors to realize loop-module structures and classify simple objects. It proves that Rep_{Ver_p}(GL(X)) carries a highest weight category structure with generalized Verma modules as standards, and establishes level-rank dualities and a Shapovalov-type criterion governing irreducibility of Kac modules in the super-analytic setting . The results provide concrete combinatorics via weight diagrams and connect to classical supergroup theory, yielding a robust framework for translating highest weights across Borels and understanding projectives, reciprocity, and lowest weights in Ver_p.

Abstract

Following the work of Venkatesh (arXiv:2203.03158), we study further the categories of representations of the general linear groups in the Verlinde category in characteristic . The main question we answer is how to translate between highest weight labelings for different choices of the Borel subgroup . We do this by reducing the general case to the study of representations of the group for using the method of odd reflections. On the category of representations of we introduce the structure of the highest weight category, as well as the categorical action of 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.
Paper Structure (44 sections, 58 theorems, 388 equations, 3 figures)

This paper contains 44 sections, 58 theorems, 388 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Notations and assumptions
  4. Affine group schemes in symmetric tensor categories
  5. The fundamental group of a symmetric tensor category
  6. General linear groups in symmetric tensor categories
  7. Semisimplification of tensor categories
  8. Verlinde category Ver p
  9. Definition and properties
  10. Harish-Chandra pairs for group schemes in Ver p
  11. Triangular decomposition for GL(X) in Ver p
  12. Categories Ver p(G)
  13. Definition and properties
  14. Tensor subcategories
  15. Representations of GL(L n) for simple objects L n
  16. ...and 29 more sections

Key Result

Theorem 2.10

The functor $F$ induces an equivalence of categories

Figures (3)

  • Figure 1: Example of a weight diagram
  • Figure 2:
  • Figure 3: Permutation applied to a weight diagram

Theorems & Definitions (169)

  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Theorem 2.10: deltan, Theorem 8.17
  • ...and 159 more