The 2-categorical S-matrix of a braided fusion 1-category is a character table
Alea Hofstetter, Christoph Schweigert
TL;DR
The paper analyzes the 2-categorical S-matrix associated to the Drinfeld center of the module 2-category $\mathcal{Z}(\mathrm{Mod}(\mathcal{C}))$ for a braided fusion 1-category $\mathcal{C}$ and proves that, when viewed as an invariant of $\mathcal{C}$, this matrix coincides with the character table of the Müger center $\mathcal{Z}_2(\mathcal{C})$ (or the corresponding group $G$ in the symmetric cases). It develops two equivalent categorical models for $\mathcal{Z}(\mathrm{Mod}(\mathcal{C}))$ and shows how the 2-categorical $S$-matrix can be computed from the 1-categorical $S$-matrix of $\mathcal{Z}(\mathcal{C})$ via the inclusion of the Müger center into the Drinfeld center. The main result is made precise for pivotal braided fusion categories with $\mathcal{Z}_2(\mathcal{C})\simeq \mathrm{Rep}(G)$ or $\mathrm{Rep}(G,z)$, yielding the character table of $G$ (up to relabeling) as the 2-categorical $S$-matrix. The pointed example $\mathcal{C}=\mathrm{vect}_G^{(\Psi,\Omega)}$ demonstrates that $S^{\mathcal{Z}(\mathrm{Mod}(\mathcal{C}))}$ reduces to the character table of $Z_2(G)$, with entries $S_{(\mathcal{C},\chi),U_g}=\chi(g)$. This links higher-categorical invariants to classical representation-theoretic data and clarifies the role of Schur equivalence in classifying braided module categories.
Abstract
The semisimple module categories over a braided fusion category $\mathcal{C}$ form a connected fusion 2-category $\text{Mod}(\mathcal{C})$. Its Drinfeld center $\mathcal{Z}(\text{Mod}(\mathcal{C}))$ is a braided fusion 2-category. To any braided fusion 2-category, Johnson-Freyd and Reutter arXiv:2105.15167v3 [math.QA] have associated a matrix-valued invariant, the 2-categorical $S$-matrix. In this short note we investigate this matrix of $\mathcal{Z}(\text{Mod}(\mathcal{C}))$ as an invariant for the braided fusion 1-category $\mathcal{C}$ and show that it reduces to the character table of the Müger center of $\mathcal{C}$.