Katriel's operators for products of conjugacy classes of S _n

We define a family of differential operators indexed with fixed point free partitions. When these differential operators act on normalized power sum symmetric functions qλ(x), the coefficients in the decomposition of this action in the basis qλ(x) are precisely those of the decomposition of products of corresponding conjugacy classes of the symmetric group S _n. The existence of such operators provides a rigorous definition of Katriel's elementary operator representation of conjugacy classes and allows to prove the conjectures he made on their properties.