Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Higher Verlinde categories of reductive groups

Joseph Newton

TL;DR

The paper defines higher Verlinde categories \(\mathsf{Ver}_{p^n}(G)\) for connected reductive groups \(G\) over a field \(\mathbf{k}\) of characteristic \(p>0\), generalizing both the semisimple \(\mathsf{Ver}_p(G)\) and the higher Verlinde categories \(\mathsf{Ver}_{p^n}\) for \(\mathrm{SL}_2\). It builds these categories as abelian envelopes of quotients of tilting module categories, using an expanded construction \(\overline{\mathcal{T}}_n\subseteq \mathsf{Rep}G\) that links to the \(\mathrm{SL}_2\) case via principal morphisms and Frobenius twists; this yields a robust framework with exactness properties and functorial relationships. Key results include a precise description of simple and projective objects, a compatible Frobenius-twist–driven inclusion \(\mathsf{Ver}_{p^n}(G)\hookrightarrow \mathsf{Ver}_{p^{n+1}}(G)\), and a perfection limit \(\mathsf{Ver}_{p^\infty}(G)\) arising from the perfection of \(G\). For \(G=\mathrm{SL}_2\), the authors give alternative abelian descriptions and Serre-quotient realizations that connect \(\mathsf{Ver}_{p^n}(\mathrm{SL}_2)\) to subcategories of \(\mathsf{Rep}\mathrm{SL}_2\) and explicit weight-filtered subcategories. These constructions advance understanding of tensor categories of moderate growth and open directions toward Tannakian interpretations and quantum-group analogues for higher \(n\).

Abstract

We define tensor categories ${\sf Ver}_{p^n}(G)$ in characteristic $p$ for connected reductive groups $G$ and positive integers $n$, generalising the semisimple Verlinde categories ${\sf Ver}_p(G)$ originating from Gelfand-Kazhdan and the higher Verlinde categories ${\sf Ver}_{p^n}$ for ${\rm SL}_2$ defined by Benson-Etingof-Ostrik. The construction is based on the definition of ${\sf Ver}_{p^n}$ as an abelian envelope of a quotient of a category of tilting modules, but we also introduce an expanded construction which refines the ${\rm SL}_2$ case and gives new results. In particular, the union ${\sf Ver}_{p^\infty}(G)$ can be derived from the perfection of $G$; certain exact sequences in ${\sf Rep}G$ map to exact sequences in ${\sf Ver}_{p^n}(G)$; and the underlying abelian category of ${\sf Ver}_{p^n}$ can be expressed as a subcategory of ${\sf Rep}{\rm SL}_2$, or as a Serre quotient of a subcategory of ${\sf Rep}{\rm SL}_2$.

Higher Verlinde categories of reductive groups

TL;DR

The paper defines higher Verlinde categories \(\mathsf{Ver}_{p^n}(G)\) for connected reductive groups over a field of characteristic , generalizing both the semisimple \(\mathsf{Ver}_p(G)\) and the higher Verlinde categories for . It builds these categories as abelian envelopes of quotients of tilting module categories, using an expanded construction that links to the case via principal morphisms and Frobenius twists; this yields a robust framework with exactness properties and functorial relationships. Key results include a precise description of simple and projective objects, a compatible Frobenius-twist–driven inclusion \(\mathsf{Ver}_{p^n}(G)\hookrightarrow \mathsf{Ver}_{p^{n+1}}(G)\), and a perfection limit \(\mathsf{Ver}_{p^\infty}(G)\) arising from the perfection of . For , the authors give alternative abelian descriptions and Serre-quotient realizations that connect \(\mathsf{Ver}_{p^n}(\mathrm{SL}_2)\) to subcategories of and explicit weight-filtered subcategories. These constructions advance understanding of tensor categories of moderate growth and open directions toward Tannakian interpretations and quantum-group analogues for higher .

Abstract

We define tensor categories in characteristic for connected reductive groups and positive integers , generalising the semisimple Verlinde categories originating from Gelfand-Kazhdan and the higher Verlinde categories for defined by Benson-Etingof-Ostrik. The construction is based on the definition of as an abelian envelope of a quotient of a category of tilting modules, but we also introduce an expanded construction which refines the case and gives new results. In particular, the union can be derived from the perfection of ; certain exact sequences in map to exact sequences in ; and the underlying abelian category of can be expressed as a subcategory of , or as a Serre quotient of a subcategory of .
Paper Structure (15 sections, 35 theorems, 53 equations, 1 figure)

This paper contains 15 sections, 35 theorems, 53 equations, 1 figure.

Table of Contents

  1. Preliminaries and assumptions
  2. The cover $\widetilde{G}$ and its weights
  3. Modules of $G$
  4. Principal $\mathrm{SL}_2$ morphisms
  5. Tensor categories and ideals
  6. Tensor ideals in $\mathsf{Tilt}\mathrm{SL}_2$
  7. Abelian envelopes
  8. Existence and construction of abelian envelopes
  9. Definition of $\mathsf{Ver}_{p^n}(G)$
  10. A more comprehensive construction
  11. Properties of $\mathsf{Ver}_{p^n}(G)$
  12. The subcategory $\mathsf{Ver}_{p^n}^+(G)$
  13. Perfection and $\mathsf{Ver}_{p^\infty}(G)$
  14. Alternative descriptions of $\mathsf{Ver}_{p^n}(\mathrm{SL}_2)$
  15. Expressing $\mathcal{C}_n$ and $\mathsf{Ver}_{p^n}$ as a Serre quotient

Key Result

Theorem 1

Let $\mathbf{k}$ be an algebraically closed field with characteristic $p>0$, and let $G$ be a connected reductive linear algebraic group such that the Coxeter number $h$ of $G$ satisfies $p\geq\max(h,2h-4)$. Fix a principal map $\phi:\mathrm{SL}_2\to G$ giving a functor $F:\mathsf{Tilt} G\to\mathsf{

Figures (1)

  • Figure 1: Dominant weights $X(T)^+$ for $\mathrm{SL}_3$ with $p=5$, their corresponding indecomposable tilting modules, and the thick tensor ideals those tilting modules belong to. Each labelled region includes all dominant weights on its lower boundary. The faint lines show translates of $X_1(T)$ shifted by $-\rho$.

Theorems & Definitions (75)

  • Theorem 1
  • Theorem 2
  • Theorem 1.9
  • proof
  • Theorem 1.10
  • Lemma 1
  • proof
  • Example 2.2
  • Lemma 2
  • proof
  • ...and 65 more