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 Smn to the characters of Sn. This map is defined by first restricting a character of Smn to the wreath product Sm oSn, and then taking the sum of the irreducible constituents of the restricted character on which the base group Sm × · · · × Sm acts trivially. We prove a combinatorial formula which gives the values of the images of the irreducible characters of Smn under this map. This formula is shown to generalize the Murnaghan–Nakayama rule. We also prove some analogous results for more general deflation maps in which the base group in the wreath product is not required to act trivially.