# On the Monoidal Center of Deligne's Category Rep(S_t)

## TL;DR

This paper computes a ribbon subcategory within the monoidal center of Deligne's Rep(S_t), providing a framework that interpolates symmetric group invariants and yields polynomial invariants of framed ribbon links.

## Contribution

It explicitly constructs a ribbon subcategory of the monoidal center of Deligne's Rep(S_t) for arbitrary t, extending the connection to symmetric group modules and link invariants.

## Key findings

- Constructed a ribbon category interpolating Rep(S_n)
- Established a functor to modules over the Drinfeld double of S_n
- Produced polynomial invariants of framed ribbon links

## Abstract

We explicitly compute a monoidal subcategory of the monoidal center of Deligne's interpolation category Rep(S_t), for t not necessarily a natural number, and we show that this subcategory is a ribbon category. For t=n, a natural number, there exists a functor onto the braided monoidal category of modules over the Drinfeld double of S_n which is essentially surjective and full. Hence the new ribbon category interpolates the categories of crossed modules over the symmetric groups. As an application, we obtain invariants of framed ribbon links which are polynomials in the interpolating variable t. These polynomials interpolate untwisted Dijkgraaf-Witten invariants of the symmetric groups.