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Zeros in the character table of the symmetric group

Sarah Peluse, Kannan Soundararajan

Abstract

Computations of Miller and Scheinerman suggest that the vast majority of the zeros appearing in the character table of the symmetric group are of a certain special type. While we cannot prove this, we resolve a conjecture arising in their paper concerning these zeros, and address a related question of Stanley.

Zeros in the character table of the symmetric group

Abstract

Computations of Miller and Scheinerman suggest that the vast majority of the zeros appearing in the character table of the symmetric group are of a certain special type. While we cannot prove this, we resolve a conjecture arising in their paper concerning these zeros, and address a related question of Stanley.

Paper Structure

This paper contains 5 sections, 5 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on the asymptotic for $p(N)$
  3. Distribution of hook lengths
  4. Distribution of parts
  5. Proof of the main theorem

Key Result

Theorem 1

For a natural number $N$, let $Z_I(N)$, $Z_{II}(N)$, and $Z_{III}(N)$ denote the number of pairs $(\lambda,\mu)$ corresponding to zeros of type I, II, and III respectively. Then for integers $N\ge 3$, all three quantities $Z_I(N)$, $Z_{II}(N)$ and $Z_{III}(N)$ are asymptotically where $p(N)$ denotes the number of partitions of $N$. Moreover,

Theorems & Definitions (10)

  • Conjecture 1: Miller and Scheinerman
  • Theorem 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof