  We introduce a commutative associative graded algebra structure on the direct
sum Z of the centers of the Hecke algebras associated to the symmetric groups
in n letters for all n. As a natural deformation of the classical construction
of Frobenius, we establish an algebra isomorphism from the algebra Z to the
ring of symmetric functions. This isomorphism provides an identification
between several distinguished bases for the centers (introduced by
Geck-Rouquier, Jones, Lascoux) and explicit bases of symmetric functions.
