A Proof of the Murnaghan–Nakayama Rule Using Specht Modules and Tableau Combinatorics

Jasdeep Kochhar, Mark Wildon

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Abstract

The Murnaghan–Nakayama rule is a combinatorial rule for the character values of symmetric groups. We give a new combinatorial proof by explicitly finding the trace of the representing matrices in the standard basis of Specht modules. This gives an essentially bijective proof of the rule. A key lemma is an extension of a straightening result proved by the second author to skew tableaux. Our module theoretic methods also give short proofs of Pieri's rule and Young's rule.
Original languageEnglish
Pages (from-to)1-22
Number of pages22
JournalAnnals of combinatorics
DOIs
Publication statusPublished - 20 Feb 2020

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