  The total multiplicity in the decomposition into irreducibles of the tensor
product i x j of two irreducible representations of a simple Lie algebra is
invariant under conjugation of one of them sum_k N_{i j}^{k}= sum_k N_{ibar
j}^{k}. This also applies to the fusion multiplicities of affine algebras in
conformal WZW theories. In that context, the statement is equivalent to a
property of the modular S matrix, Sigma(k)= sum_j S_{j k}=0 if k is a complex
representation. Curiously, this vanishing of Sigma(k) also holds when k is a
quaternionic representation. We provide proofs of all these statements. These
proofs rely on a case-by-case analysis, maybe overlooking some hidden symmetry
principle. We also give various illustrations of these properties in the
contexts of boundary conformal field theories, integrable quantum field
theories and topological field theories.
