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

TL;DR

This paper classifies indecomposable objects in the monoidal center of Deligne's category $Rep(S_t)$, showing semisimplicity conditions and relating its Grothendieck ring to symmetric group centers.

Contribution

It provides a classification of indecomposables in the monoidal center of $Rep(S_t)$ and characterizes when this center is semisimple, connecting it to symmetric group representation theory.

Findings

Center is semisimple iff $t$ is not a non-negative integer.

Classified indecomposable objects in the monoidal center of $Rep(S_t)$.

Identified the graded Grothendieck ring with that of symmetric group centers.

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.