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

TL;DR

This paper introduces a new deflation map for symmetric group characters, generalizes classical combinatorial rules, and applies these results to verify cases of Foulkes' Conjecture.

Contribution

It defines a novel deflation map from S_{mn} to S_n, generalizes the Murnaghan--Nakayama rule, and provides new combinatorial formulas for character multiplicities.

Findings

Derived a combinatorial formula for the deflation map values.

Generalized classical rules to broader contexts.

Verified Foulkes' Conjecture in new cases.

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 S_n. This map is obtained by first restricting a character of S_{mn} to the wreath product S_m \wr S_n, and then taking the sum of the irreducible constituents of the restricted character on which the base group S_m \times ... \times 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.