# On the Frobenius functor for symmetric tensor categories in positive characteristic

@article{Etingof2019OnTF, title={On the Frobenius functor for symmetric tensor categories in positive characteristic}, author={Pavel Etingof and Victor Ostrik}, journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)}, year={2019}, volume={2021}, pages={165 - 198} }

Abstract We develop a theory of Frobenius functors for symmetric tensor categories (STC) 𝒞 {\mathcal{C}} over a field 𝒌 {\boldsymbol{k}} of characteristic p, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor F : 𝒞 → 𝒞 ⊠ Ver p {F:\mathcal{C}\to\mathcal{C}\boxtimes{\rm Ver}_{p}} , where Ver p {{\rm Ver}_{p}} is the Verlinde category (the semisimplification of Rep 𝐤 ( ℤ / p ) {\mathop{\mathrm{Rep}}\nolimits_{\mathbf{k…

## 12 Citations

### New incompressible symmetric tensor categories in positive characteristic

- MathematicsDuke Mathematical Journal
- 2023

We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field $\bf k$. If ${\rm char}({\bf k})=p>0$, we use this method to…

### Monoidal abelian envelopes with a quotient property

- MathematicsJournal für die reine und angewandte Mathematik (Crelles Journal)
- 2022

Abstract We study abelian envelopes for pseudo-tensor categories with the property that every object in the envelope is a quotient of an object in the pseudo-tensor category. We establish an…

### Monoidal abelian envelopes

- MathematicsCompositio Mathematica
- 2021

We prove a constructive existence theorem for abelian envelopes of non-abelian monoidal categories. This establishes a new tool for the construction of tensor categories. As an example we obtain new…

### The indecomposable objects in the center of Deligne's category Re̲pSt$\protect\underline{{\rm Re}}\!\operatorname{p}S_t$

- MathematicsProceedings of the London Mathematical Society
- 2023

### The Delannoy category

- Mathematics
- 2022

. Let G be the group of all order-preserving self-maps of the real line. In previous work, the ﬁrst two authors constructed a pre-Tannakian category…

### Exact factorizations and extensions of finite tensor categories

- Mathematics
- 2022

We extend [G1] to the nonsemisimple case. We define and study exact factorizations B = A • C of a finite tensor category B into a product of two tensor subcategories A ,C ⊂ B, and relate exact…

### Minimal extensions of Tannakian categories in positive characteristic

- MathematicsJournal of Algebra
- 2022

### Monoidal abelian envelopes and a conjecture of Benson and Etingof

- MathematicsAlgebra & Number Theory
- 2022

We give several criteria to decide whether a given tensor category is the abelian envelope of a fixed symmetric monoidal category. Benson and Etingof conjectured that a certain limit of finite…

### On Frobenius exact symmetric tensor categories

- Mathematics
- 2021

. A fundamental theorem of P. Deligne (2002) states that a pre- Tannakian category over an algebraically closed ﬁeld of characteristic zero admits a ﬁber functor to the category of supervector spaces…

### Lectures on Symmetric Tensor Categories

- Mathematics
- 2021

This is an expanded version of the notes by the second author of the lectures on symmetric tensor categories given by the first author at Ohio State University in March 2019 and later at ICRA-2020 in…

## References

SHOWING 1-10 OF 34 REFERENCES

### Finite Symmetric Integral Tensor Categories with the Chevalley Property with an Appendix by Kevin Coulembier and Pavel Etingof

- MathematicsInternational Mathematics Research Notices
- 2019

We prove that every finite symmetric integral tensor category $\mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $p>2$ admits a symmetric fiber…

### Finite symmetric tensor categories with the Chevalley property in characteristic 2

- MathematicsJournal of Algebra and Its Applications
- 2020

We prove an analog of Deligne’s theorem for finite symmetric tensor categories [Formula: see text] with the Chevalley property over an algebraically closed field [Formula: see text] of characteristic…

### On semisimplification of tensor categories

- Mathematics
- 2018

We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. In particular, we compute the semisimplification of the category of representations of a finite…

### Hilbert Basis Theorem and Finite Generation of Invariants in Symmetric Fusion Categories in Positive Characteristic

- Mathematics
- 2015

In this paper, we conjecture an extension of the Hilbert basis theorem and the finite generation of invariants to commutative algebras in symmetric finite tensor categories over fields of positive…

### Koszul duality and the PBW theorem in symmetric tensor categories in positive characteristic

- Mathematics
- 2016

### Finite tensor categories

- Mathematics
- 2003

We start the general structure theory of not necessarily semisimple finite tensor categories, generalizing the results in the semisimple case (i.e. for fusion categories), obtained recently in our…

### Computations in symmetric fusion categories in characteristic p

- Mathematics
- 2015

We study properties of symmetric fusion categories in characteristic $p$. In particular, we introduce the notion of a super Frobenius-Perron dimension of an object $X$ of such a category, and derive…

### Tensor structures arising from affine Lie algebras. III

- Mathematics
- 1993

This paper is a part of the series [KL]; however, it can be read independently of the first two parts. In [D3], Drinfeld proved the existence of an equivalence between a tensor category of…

### Representations of algebraic groups

- Mathematics
- 1987

Part I. General theory: Schemes Group schemes and representations Induction and injective modules Cohomology Quotients and associated sheaves Factor groups Algebras of distributions Representations…