A Higher Frobenius–Schur Indicator Formula for Group-Theoretical Fusion Categories

Group-theoretical fusion categories are defined by data concerning finite groups and their cohomology: a finite group G endowed with a three-cocycle ω, and a subgroup H⊂G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H\subset G}$$\end{document} endowed with a two-cochain whose coboundary is the restriction of ω. The objects of the category are G-graded vector spaces with suitably twisted H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}$$\end{document}-actions; the associativity of tensor products is controlled by ω. Simple objects are parametrized in terms of projective representations of finite groups, namely of the stabilizers in H of right H-cosets in G, with respect to two-cocycles defined by the initial data. We derive and study general formulas that express the higher Frobenius–Schur indicators of simple objects in a group-theoretical fusion category in terms of the group-theoretical and cohomological data defining the category and describing its simples.