We know that every symmetric fusion category (SFC) gives rise to data
$N^{ij}\_k$ that describe the fusion of simple objects:
$i\times j = N^{ij}\_k k$, and the data $\theta\_i =\pm 1$ that describe the twist of simple objects.


My question is what are the conditions on the data $N^{ij}\_k,\theta\_i$ such that the data correspond to a SFC?


Any conditions beyond those for fusion category are welcome. (One may try to classify finite groups via a classification of SFCs.)


One also can ask a related (and more general) question: what are the conditions on the data $N^{ij}\_k,\theta\_i$ such that the data correspond to a pre-modular braided fusion category?



Conditions on the fusion data of symmetric fusion category
The information in the $\theta$'s is very weak, for example if a FC admits a symmetric structure with $\theta\_i=-1$ for at least one $i$, it also admits a symmetric structure with $\theta\equiv 1$. Even, it is possible to have several non-equivalent symmetric structures with $\theta\equiv 1$ over a fixed fusion category. 


On the other hand, since every SFC is equivalent to the representation category of a (super)-group a necessary condition over the fusion algebra $N\_k^{ij}$ is that Frobenius-Perron dimension of every simple object is an integer. 


Let us call a fusion algebra without non-trivial fusion subalgebra a simple fusion algebra. Since quotient groups of a finite group $G$ are in biyective correspondence with fusion subalgebras of $K\_0(Rep(G))$, a condition on the fusion data of symmetric fusion category is that every simple fusion subalgebra have to be realized (in a unique way) as the fusion algebra of the SFC of representation of (a unique) simple group (the fusion data can be read from the character table). Also, you can read from your fusion data if it corresponds to a nilpotent group, see <http://arxiv.org/abs/math/0610726> in this case using group cohomology is possible to find obstructions to the existence of a SFC that realize your fusion data. 


