The indecomposable objects in the center of Deligne's category Rep St

Flake, Johannes; Harman, Nate; Laugwitz, Robert

Abstract

We classify the indecomposable objects in the monoidal center of Deligne's interpolation category Rep St by viewing Rep St as a model‐theoretic limit in rank and characteristic. We further prove that the center of Rep St 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.

• Journal Article Type: Article
• Acceptance Date: Nov 23, 2022
• Online Publication Date: Jan 11, 2023
• Publication Date: 2023-04
• Deposit Date: Apr 5, 2023
• Publicly Available Date: Apr 5, 2023
• Journal: Proceedings of the London Mathematical Society
• Print ISSN: 0024-6115
• Electronic ISSN: 1460-244X
• Publisher: Wiley
• Peer Reviewed: Peer Reviewed
• Volume: 126
• Issue: 4
• Pages: 1134-1181
• DOI: https://doi.org/10.1112/plms.12509
• Public URL: https://nottingham-repository.worktribe.com/output/16216867
• Publisher URL: https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/plms.12509
• Additional Information: Received: 2021-06-10; Accepted: 2022-11-23; Published: 2023-01-11