Computations in symmetric fusion categories in characteristic p

Etingof, Pavel; Ostrik, Victor; Venkatesh, Siddharth

doi:10.48550/arxiv.1512.02309

2015

DOI: 10.48550/arxiv.1512.02309

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Computations in symmetric fusion categories in characteristic p

Pavel Etingof¹,

Victor Ostrik²,

Siddharth Venkatesh³

Abstract: 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 an explicit formula for the Verlinde fiber functor F (X) of X (defined by the second author) in terms of the usual and super Frobenius-Perron dimensions of X. We also compute the decomposition of symmetric powers of objects of the Verlinde category, generalizing a classical formula of Cayley and Sylvester for invariant… Show more

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“…There is one small change required: the existence of the right adjoint functor I : Verp → C as in[EOV, 3.2] follows from the exactness of F : C → Verp and finiteness of C.…”

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confidence: 99%

On the Frobenius functor for symmetric tensor categories in positive characteristic

Etingof¹,

Ostrik²

2019

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We develop a theory of Frobenius functors for symmetric tensor categories (STC) C over a field 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 ⊠ Verp, where Verp is the Verlinde category (the semisimplification of Rep k (Z/p)); a similar construction of the underlying additive functor appeared independently in [Co]. This generalizes the usual Frobenius twist functor in modular representation theory and also the one defined in [O], where it is used to show that if C is finite and semisimple then it admits a fiber functor to Verp. The main new feature is that when C is not semisimple, F need not be left or right exact, and in fact this lack of exactness is the main obstruction to the existence of a fiber functor C → Verp. We show, however, that there is a 6-periodic long exact sequence which is a replacement for the exactness of F , and use it to show that for categories with finitely many simple objects F does not increase the Frobenius-Perron dimension. We also define the notion of a Frobenius exact category, which is a STC on which F is exact, and define the canonical maximal Frobenius exact subcategory Cex inside any STC C with finitely many simple objects. Namely, this is the subcategory of all objects whose Frobenius-Perron dimension is preserved by F . One of our main results is that a finite STC is Frobenius exact if and only if it admits a (necessarily unique) fiber functor to Verp. This is the strongest currently available characteristic p version of Deligne's theorem (stating that a STC of moderate growth in characteristic zero is the representation category of a supergroup). We also show that a sufficiently large power of F lands in Cex. Also, in characteristic 2 we introduce a slightly weaker notion of an almost Frobenius exact category (namely, one having a fiber functor into the category of representations of the triangular Hopf algebra k[d]/d 2 with d primitive and R-matrix R = 1 ⊗ 1 + d ⊗ d), and show that a STC with Chevalley property is (almost) Frobenius exact. Finally, as a by-product, we resolve Question 2.15 of [EG1].

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“…There is one small change required: the existence of the right adjoint functor I : Verp → C as in[EOV, 3.2] follows from the exactness of F : C → Verp and finiteness of C.…”

mentioning

confidence: 99%

On the Frobenius functor for symmetric tensor categories in positive characteristic

Etingof¹,

Ostrik²

2019

Preprint

Self Cite

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show abstract

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Computations in symmetric fusion categories in characteristic p

Cited by 1 publication

References 3 publications

On the Frobenius functor for symmetric tensor categories in positive characteristic

On the Frobenius functor for symmetric tensor categories in positive characteristic

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