On the monoidal center of Deligne's category Re̲p(St)

Abstract

We explicitly compute a monoidal subcategory of the monoidal center of Deligne's interpolation category normalRe̲pfalse(Stfalse), 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 Sn which is essentially surjective and full. Hence the new ribbon categories interpolate 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.

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

“…The categories Z(Rep S t ), for t generic, and Z(Rep ab S d ), for d ∈ Z ≥0 , are infinite analogues of modular categories. This interpretation follows from [FL21,Theorem 3.27] where Z(Rep S t ) was shown to be a ribbon category and Section 3.8 where we prove that Z(Rep S t ) and Z(Rep ab S d ) are non-degenerate braided tensor categories. We note that these categories are also factorizable braided tensor categories by [EGNO15,Proposition 8.6.3].…”

“…In [FL21], the first and third listed authors started the investigation of Z(Rep S t ), showed that the latter is a ribbon category, and obtained invariants of framed links as an application. It was shown that the braided categories Z(Rep S t ) interpolate the braided categories Z(Rep S d ) in the sense that Z(Rep S d ) is the semisimplification of Z(Rep S d ) for d ∈ Z ≥0 .…”

“…⊕ X ⊗m k , with m i ≤ m. With respect to this filtration, we show that Z(Rep S t ) ≤k,m is equivalent to the ultraproduct of the categories Z(Rep Fp i S n i ) ≤k,m with partially defined monoidal and additive structures, see Proposition 3.1. In particular, this enables us to solve the question of semisimplicity of Z(Rep S t ) raised in [FL21,Question 3.31 for every n ≥ 0 and t ∈ C. This functor Ind is a separable Frobenius monoidal functor compatible with the braidings, see Proposition 3.7. It enables us to classify the indecomposable objects in Z(Rep S t ).…”

“…The question of classifying the indecomposable objects in Z(Rep S t ) naturally emerged from the paper [FL21] but was independently raised by P. Etingof in his research statement. Moreover, in Corollary 3.14 we describe the blocks of the category Z(Rep S t ) as…”

“…We note that the first paper [FL21] on Z(Rep S t ) already contained a general construction of objects via explicit idempotents and half-braidings using the combinatorial description of Rep S t by partitions. We identify the objects constructed in [FL21] in the image of Ind in Section 3.6 and prove that these objects generate Z(Rep S t ) as a Karoubian tensor category but are, in general, not indecomposable.…”