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Vanishing pairs of conjugacy classes for the symmetric group

Velmurugan S

TL;DR

This paper classifies pairs of conjugacy classes in the symmetric group $S_n$ for which every non-linear irreducible character vanishes on at least one member of the pair, establishing that for $n>6$ the only such pair is $\\{(n),(n-1,1)\\}$. The authors deploy the Murnaghan-Nakayama rule, Schur's second orthogonality relations, and Navarro's product-of-conjugacy-classes lemma to derive necessary constraints, followed by a comprehensive case analysis on cycle-type partitions to rule out all alternatives. The result complements existing bounds on the number of vanishing classes (k(S_n)=2) and parallels related theorems about common zeros of irreducible characters. The findings hold for large $n$ with small cases amenable to direct check, highlighting precise vanishing behavior in $S_n$ tied to specific cycle structures.

Abstract

In this short note, we classify pairs of conjugacy classes of the symmetric group such that any non-linear irreducible character of the symmetric group vanishes on at least one of them.

Vanishing pairs of conjugacy classes for the symmetric group

TL;DR

This paper classifies pairs of conjugacy classes in the symmetric group for which every non-linear irreducible character vanishes on at least one member of the pair, establishing that for the only such pair is . The authors deploy the Murnaghan-Nakayama rule, Schur's second orthogonality relations, and Navarro's product-of-conjugacy-classes lemma to derive necessary constraints, followed by a comprehensive case analysis on cycle-type partitions to rule out all alternatives. The result complements existing bounds on the number of vanishing classes (k(S_n)=2) and parallels related theorems about common zeros of irreducible characters. The findings hold for large with small cases amenable to direct check, highlighting precise vanishing behavior in tied to specific cycle structures.

Abstract

In this short note, we classify pairs of conjugacy classes of the symmetric group such that any non-linear irreducible character of the symmetric group vanishes on at least one of them.
Paper Structure (2 sections, 5 theorems, 18 equations)

This paper contains 2 sections, 5 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Proof of the theorem

Key Result

Theorem 1.1

Hung Let $n>8$ be a positive integer. Then for non-linear characters $\chi_\lambda$ and $\chi_\nu$ of $S_n$, they have a common zero if and only if $\{\chi_\lambda(1),\chi_\mu(1)\}\neq\{n(n-3)/2, (n-1)(n-2)/2\}$.

Theorems & Definitions (7)

  • Conjecture 1
  • Theorem 1.1
  • Lemma 2.1: Murnaghan-Nakayama rule
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof