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

Evseev, Anton, Paget, Rowena E., Wildon, Mark (2014) Character deflations and a generalization of the Murnaghan--Nakayama rule. Journal of Group Theory, 17 (6). pp. 1035-1070. (doi:10.1515/jgth-2014-0023)

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## 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 ?S_n, and then taking the sum of the irreducible constituents of the restricted character on which the base group S_m ×?×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.

Item Type: Article 10.1515/jgth-2014-0023 Q Science > QA Mathematics (inc Computing science) > QA165 Combinatorics Q Science > QA Mathematics (inc Computing science) > QA171 Representation theory Faculties > Sciences > School of Mathematics Statistics and Actuarial ScienceFaculties > Sciences > School of Mathematics Statistics and Actuarial Science > Pure Mathematics Rowena E Paget 25 Apr 2012 09:42 UTC 29 May 2019 08:57 UTC https://kar.kent.ac.uk/id/eprint/29323 (The current URI for this page, for reference purposes)