A proof of the Murnaghan--Nakayama rule using Specht modules and tableau combinatorics

2018/05/01 by Kochhar, Jasdeep, Wildon, Mark
#05E10 #05E18 (Secondary) #20C30 (Primary) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1805.00255

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.

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