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 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.