Katriel's operations for products of conjugacy classes of Sn
Résumé
We define a family of differential operators indexed with fixed point free partitions. When these differential operators act on normalized power sums symmetric functions $q_\lambda(x)$, the coefficients in the decomposition of this action in the basis $q_\lambda(x)$ 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.