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.
Highlights
Deligne introduced a symmetric monoidal category Rep(St) for any complex number t in [Del07] which interpolates the representation categories of symmetric groups Rep(Sn) for n ≥ 1
For n ≥ 0, let μ be a partition of n, and eρ be an idempotent in Z ⊗ Mk(k) obtained, for example, from a simple representation ρ of Z = Z(σ), the centralizer of an element σ of cycle type μ in Sn
The object Wσ,ρ gives an object of the monoidal center of Rep(St)
Summary
Deligne introduced a symmetric monoidal category Rep(St) for any complex number t in [Del07] which interpolates the representation categories of symmetric groups Rep(Sn) for n ≥ 1. The category Rep(Sn) of finite-dimensional Sn-modules is the idempotent completion (or Karoubi envelope, Cauchy completion, [BD86]) of a monoidal category generated by the single object Vn. The morphism spaces Hom(Vn⊗k, Vn⊗l) are given by the Sn-invariants of Vn⊗(k+l) and can be described combinatorially using the partition algebras Pk(n), see e.g. We recall basic facts about the monoidal center of the category Rep(Sn) of finite-dimensional Sn-modules over the field k. On the Monoidal Center of Deligne’s Category Rep(St) To illustrate the notation from the proof with an example, take t1 = 4, t2 = 3, π4,3 = We note that this set of generating morphisms can alternatively be derived from the universal property of Rep(St) proved in [Del, Proposition 8.3], which exhibits Rep(St) as the universal symmetric monoidal category with a Frobenius algebra object. The morphisms in Hom, Hom are matrices of morphisms generated by M under composition, tensor product, and k-linear combination
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