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Coulembier, Kevin; Etingof, Pavel; Ostrik, Victor

On Frobenius exact symmetric tensor categories. (English) Zbl 1537.18019

Ann. Math. (2) 197, No. 3, 1235-1279 (2023).
Let \(\mathbf{k}\) be an algebraically closed field and \(\mathcal{C}\) a pre-Tannakian category over \(\mathbf{k}\), which means a \(\mathbf{k}\)-linear abelian symmetric rigid monoidal category with bilinear tensor product and \(\text{End}(\mathbf{1}) = \mathbf{k}\), in which objects have finite length. Assume that \(\mathcal{C}\) has moderate growth; that is, lengths of \(V^{\otimes n}\) for \(V \in \mathcal{C}\) grow at most exponentially with \(n\). A fundamental theorem of Deligne states that a pre-Tannakian category over an algebraically closed field of characteristic zero admits a fiber functor to the category of supervector spaces if and only if it has moderate growth. The authors prove a characteristic \(p\) version of this theorem. They show that a pre-Tannakian category over an algebraically closed field of characteristic \(p > 0\) admits a fiber functor into the Verlinde category \(\text{Ver}_p\) if and only if it has moderate growth and is Frobenius exact. This implies that Frobenius exact pre-Tannakian categories of moderate growth admit a well behaved notion of Frobenius-Perron dimension. It follows that any semisimple pre-Tannakian category of moderate growth has a fiber functor to \(\text{Ver}_p\), so Deligne’s theorem holds for semisimple pre-Tannakian categories in characteristics \(2\), \(3\), settling a conjecture of the third author from 2015.
This result applies to semisimplifications of categories of modular representations of finite group schemes, which gives new applications to classical modular representation theory. In one particular instance, it allows one to characterize, for a modular representation \(V\), the possible growth rates of the number of indecomposable summands in \(V^{\otimes n}\) of dimension prime to \(p\).
Reviewer: Mee Seong Im (Annapolis)

Cited in 2 Reviews
Cited in 3 Documents

MSC:

18M05 Monoidal categories, symmetric monoidal categories
14L15 Group schemes
16T05 Hopf algebras and their applications
20C20 Modular representations and characters

Keywords:

modular representations; Frobenius functor; tensor categories; Verlinde category
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References:

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