A modular category is a braided category with some additional algebraic
features. The interest of this concept is that it provides a Topological Quantum Field
Theory in dimension 3. The Verlinde formulas associated with a modular category are
the dimensions of the TQFT modules. We discuss reductions and refinements of these
formulas for modular categories related withSU (N ). Our main result is a splitting of the
Verlinde formula, corresponding to a brick decomposition of the TQFT modules whose
summands are indexed by spin structures modulo an even integer. We introduce here the
notion of a spin modular category, and give the proof of the decomposition theorem in
this general context.