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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 condition is 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?

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?

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 condition is welcome.

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?

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 condition is 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?

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 condition is welcome.

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 condition is welcome.

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?