Character deflations and a generalization of the Murnaghan--Nakayama rule

Given natural numbers m and n, we define a deflation map from the characters of the symmetric group S-mn to the characters of S-n. This map is obtained by first restricting a character of S-mn to the wreath product S-m (sic) S-n, and then taking the sum of the irreducible constituents of the restricted character on which the base group S-m x...x S-m acts trivially. We prove a combinatorial formula which gives the values of the images of the irreducible characters of S-mn under this map. We also prove an analogous result for more general deflation maps in which the base group is not required to act trivially. These results generalize the Murnaghan-Nakayama rule and special cases of the Littlewood-Richardson rule. As a corollary we obtain a new combinatorial formula for the character multiplicities that are the subject of the long-standing Foulkes' Conjecture. Using this formula we verify Foulkes' Conjecture in some new cases.