On the Galois symmetries for the character table of an integral fusion category

In this paper, we show that integral fusion categories with rational structure constants admit a natural group of symmetries given by the Galois group of their character tables. Based on these symmetries, we generalize a well-known result of Burnside from representation theory of finite groups. More precisely, we show that any row corresponding to a non-invertible object in the character table of a weakly integral fusion category contains a zero entry.

Automated summary

In this paper, it is shown that integral fusion categories with rational structure constants have a natural group of symmetries, given by the Galois group of their character tables. These symmetries are then used to generalize a well-known result of Burnside in the representation theory of finite groups. The main result establishes that any row in the character table of a weakly integral fusion category, corresponding to a non-invertible object, contains a zero entry.