This question is related to Pivotal functors of that are substantially different from finite group homomorphisms.

A tensor functor F:C→D is called

dominant(sometimes called “surjective”) if for any Y:D, there is an X:C such that Y is a subobject of FX.It is known (“On fusion categories” by Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik, proposition 8.8), that for any dominant tensor functor of fusion categories F:C→D we have:

FRC=FPDim(C)FPDim(D)RD

Here, FPDim is the Frobenius-Perron-dimension and RC=⨁FPDim(X)X is the regular representation with respect to the Frobenius-Perron-dimensions. (The sum ranges over all simple objects.)Now, if C and D are spherical, and F is pivotal, we can ask the same for categorical dimensions. Denoting the categorical dimension by dim and defining ΩC:=⨁dim(X)X, we can ask:

For which pivotal, dominant tensor functors does the following equation hold?

FΩC=dim(ΩC)dim(ΩD)ΩD

One instance where I know that it holds, is when Frobenius-Perron-dimensions and categorical dimensions coincide, i.e. in “pseudo-unitary” fusion categories, so for example (quantum) group representations. Does this hold in general? Is there a counterexample?P.S.: Unitary fusion categories are fusion categories with a compatible C-* structure (dagger structure with Hilbert space enrichment). Pseudo-unitary fusion categories are fusion categories where FPdim(C)=dim(ΩC). Unitary fusion categories are always pseudo-unitary.

P.P.S.: Here is a sub-question, that may make the original question easier to answer: What are examples for non-unitary spherical categories?

**Answer**

**Attribution***Source : Link , Question Author : Manuel Bärenz , Answer Author : Community*