S-Matrix

$S$-matrix of a fusion ring

On the pages of fusion rings we use the following definition for the $S$-matrix.

Not every fusion ring has associated $S$-matrices and the existence of an $S$-matrix for a fusion ring does not guarantee that the ring is categorifiable into a modular fusion category. If the fusion ring is categorifiable into a modular fusion category $\mathcal{C}$ then the $S$ matrix of $\mathcal{C}$ must be one of the fusion ring's up to scaling (see definition of $S$-matrix of a category below). This does not imply that all $S$-matrices of a fusion ring appear as modular data of fusion categories.

Note: If the fusion ring is categorifiable to a ribbon category $\mathcal{C}$ that is not modular then the $S$-matrix of $\mathcal{C}$ is not one of the ones on the fusion ring page. This is because we demand that an $S$-matrix belonging to a fusion ring is unitary, and thus invertible.

$S$-matrix of a braided spherical category

In the EGNO one finds the following definition.

From the definition it follows that the $\mathcal{S}$-matrix is a square, symmetric matrix with $\mathcal{S}_{\mathbf{1},Y} = \mathrm{FPdim}(Y)$.

In contrast to the $S$-matrices of a fusion ring the $S$-matrix from $\mathcal{C}$ is not necessarily invertible. If it is invertible, it is not unitary but can be made unitary by scaling with a factor $\frac{1}{\mathrm{FPdim}(\mathcal{C})} = \frac{1}{\mathrm{FPdim}(R)}$. Only after scaling it corresponding to one of the $S$-matrices of its fusion ring.

If the $S$-matrix of $\mathcal{C}$ is invertible, $\mathcal{C}$ is called a modular category.

Note: There is also a more general definition of an $S$-matrix, defined in 8.19 of the EGNO but we don't use that definition anywhere on the site at the moment.