The Fusion Algebra of Bimodule Categories

We establish an algebra-isomorphism between the complexified Grothendieck ring \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{F}$\end{document} of certain bimodule categories over a modular tensor category and the endomorphism algebra of appropriate morphism spaces of those bimodule categories. This provides a purely categorical proof of a conjecture by Ostrik concerning the structure of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{F}$\end{document}. As a by-product we obtain a concrete expression for the structure constants of the Grothendieck ring of the bimodule category in terms of endomorphisms of the tensor unit of the underlying modular tensor category.