A criterion for M\"uger centralizer of a fusion subcategory of a braided non-degenerate fusion category is given. Along the way we extend some identities on the space of class functions of a fusion category introduced by Shimizu in \cite{ scalg}. We also show that in a modular tensor category the product of two conjugacy class sums is a linear combination of conjugacy class sums with rational coefficients.
Categories: math.QA math.CT