$S$-Matrix
$S$-matrix of a fusion ring
On the pages of fusion rings we use the following definition for the $S$-matrix.
Definition ($S$-matrix of a fusion ring) An $S$-matrix associated to a fusion ring $R$ is a square, symmetric, unitary matrix that diagonalises the set of matrices $[N_i]_{i=1}^r$, and satisfies
- $ \left[ S^2\right]^i_j = N_{ij}^1 $,
- $ \left[ S \right]^1_i = \frac{\fpdim(N_i)}{\sqrt{\fpdim(R)}},\quad i = 1,\ldots,r $.
Here $\fpdim$ stands for the quantum dimension.
Not every fusion ring has an associated $S$-matrix and the existence of an $S$-matrix for a fusion ring does not guarantee that the ring is categorifiable into a modular fusion category. In constrast to the $S$-matrix of a category a fusion ring can have multiple $S$-matrices.
$S$-matrix of a braided spherical category
In the EGNO one finds the following definition.
Definition ($S$-matrix of a braided spherical category) Let $\mathcal{C}$ be a braided spherical category with braiding $c_{X,Y}: X \otimes Y \rightarrow Y \otimes X $ then the $\mathcal{S}$-matrix of $\mathcal{C}$ is defined as \[ \mathcal{S}{X,Y} := \tr (c{Y,X}c_{X,Y}). \]
From the definition it follows that the $\mathcal{S}$-matrix is a square, symmetric matrix with $\mathcal{S}_{\mathbf{1},Y} = \fpdim (Y)$. The definition from the EGNO and the one used on this site differs in a factor $\frac{1}{\fpdim{\mathcal{C}}} = \frac{1}{\fpdim{R}}$ which we have added to make the matrix unitary.
In contrast to the definition for an $S$-matrix of a fusion ring above the $S$-matrix of a category need not be invertible. If it is then the category $\mathcal{C}$ is called a modular category.
Note: There is also a more general definition of an $S$-matrix, defined in 8.19 of the EGNO