Invertible Bimodule Categories and Generalized Schur Orthogonality

The Schur orthogonality relations are a cornerstone in the representation theory of groups. We utilize a generalization to weak Hopf algebras to provide a new, readily verifiable condition on the skeletal data for deciding whether a given bimodule category is invertible and therefore defines a Morita equivalence. Ultimately, the condition arises from Schur orthogonality relations on the characters of the annular algebra associated to a module category. As a first application, we provide an algorithm for the construction of the full skeletal data of the invertible bimodule category associated to a given module category, which is obtained in a unitary gauge when the underlying categories are unitary. As a second application, we show that our condition for invertibility is equivalent to the notion of MPO-injectivity, thereby closing an open question concerning tensor network representations of string-net models exhibiting topological order. We discuss applications to generalized symmetries, including a generalized Wigner-Eckart theorem.

Automated summary

The study utilizes a generalization to weak Hopf algebras to provide a new and verifiable condition on skeletal data for determining the invertibility of a given bimodule category, which defines a Morita equivalence. The condition is derived from Schur orthogonality relations on the characters of the annular algebra associated with a module category. As applications, an algorithm is provided for constructing the complete skeletal data of the invertible bimodule category related to a given module category, and the condition for invertibility is shown to be equivalent to MPO-injectivity in tensor network representations of string-net models with topological order. The study also discusses applications to generalized symmetries, including a generalized Wigner-Eckart theorem.