Character deflations and a generalization of the Murnaghan–Nakayama rule

Abstract

Abstract Given natural numbers m and n, we define a deflation map from the characters of the symmetric group S mn to the characters of Sn . This map is obtained by first restricting a character of S mn to the wreath product Sm ≀ Sn , 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 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.