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

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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.