  We exhibit a vertex operator which implements multiplication by power-sums of
Jucys-Murphy elements in the centers of the group algebras of all symmetric
groups simultaneously. The coefficients of this operator generate a
representation of ${\cal W}_{1+\infty}$, to which operators multiplying by
normalized conjugacy classes are also shown to belong. A new derivation of such
operators based on matrix integrals is proposed, and our vertex operator is
used to give an alternative approach to the polynomial functions on Young
diagrams introduced by Kerov and Olshanski.
