The indecomposable objects in the center of Deligne's category $Rep(S_t)$

Abstract

We classify the indecomposable objects in the monoidal center of Deligne's interpolation category $Rep(S_t)$ by viewing $Rep(S_t)$ as a model-theoretic limit in rank and characteristic. We further prove that the center of $Rep(S_t)$ is semisimple if and only if $t$ is not a non-negative integer. In addition, we identify the associated graded Grothendieck ring of this monoidal center with that of the graded sum of the centers of representation categories of finite symmetric groups with an induction product. We prove analogous statements for the abelian envelope.

arXiv commentv2: Final accepted manuscript after minor edits, to appear in Proceedings of the LMS

References in corpus (8)

• On blocks of Deligne's category Rep(S_t)
• Consistent systems of correlators in non-semisimple conformal field theory
• On Deligne's category $\uRep^{ab}(S_d)$
• Deligne categories and the limit of categories $Rep(GL(m|n))$
• Categorification and Heisenberg doubles arising from towers of algebras
• Note on Frobenius monoidal functors
• Modular Categories and TQFTs Beyond Semisimplicity
• On the Monoidal Center of Deligne's Category Rep(S_t)

Cited by in corpus (3)

• Constructing non-semisimple modular categories with local modules
• Frobenius monoidal functors from ambiadjunctions and their lifts to Drinfeld centers
• Frobenius Monoidal Functors of Dijkgraaf-Witten Categories and Rigid Frobenius Algebras