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Unisingular Specht Modules

John Cullinan

Abstract

Let $G$ be a finite group and $ρ:G \to \GL(V)$ a finite dimensional representation of $G$. We say that $ρ$ is unisingular if $\det(1-ρ(g)) = 0$ for all $g \in G$. Building on previous work in \cite{cullinan}, we consider the symmetric groups $S_n$ and prove that certain families of Specht modules are always unisingular as well as raise new questions for future study.

Unisingular Specht Modules

Abstract

Let be a finite group and a finite dimensional representation of . We say that is unisingular if for all . Building on previous work in \cite{cullinan}, we consider the symmetric groups and prove that certain families of Specht modules are always unisingular as well as raise new questions for future study.

Paper Structure

This paper contains 11 sections, 19 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Notation and Background
  3. Partitions
  4. Specht Modules
  5. Linear Algebra
  6. Gamma-Shaped Specht Modules
  7. The module $\mathcal{S}^{(2^2,1^{n-4})}$
  8. Generalities
  9. Tabloids
  10. Proof of Theorem \ref{['(2^2,1^n-4)']}
  11. Observations and Future Work

Key Result

Theorem 1.2

Let $n\geq 5$ be a positive integer. Then

Theorems & Definitions (41)

  • Example 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Proposition 2.9
  • proof
  • Corollary 2.11
  • proof
  • Proposition 2.12
  • proof
  • ...and 31 more