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

TL;DR

This paper provides a new combinatorial proof of the Murnaghan--Nakayama rule for symmetric group characters using Specht modules and tableau combinatorics, also offering simplified proofs of Pieri's and Young's rules.

Contribution

It introduces a novel combinatorial proof of the Murnaghan--Nakayama rule via explicit trace calculations in Specht modules, extending straightening lemmas to skew-tableaux.

Findings

New combinatorial proof of Murnaghan--Nakayama rule

Short proofs of Pieri's and Young's rules

Extension of straightening lemma to skew-tableaux

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.