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A note on irreducible representations of symmetric groups and Sergeev superalgebras

Minjia Chen, Jinkui Wan, Hongbo Zhao

Abstract

We provide an explicit construction and a closed dimension formula in terms of hook lengths for the irreducible representations for the symmetric groups $\mathfrak{S}_p$ and the Sergeev superalgebras $\mathcal{Y}_p$ over an algebraically closed field $\mathbb{F}$ of characteristic $p>0$.

A note on irreducible representations of symmetric groups and Sergeev superalgebras

Abstract

We provide an explicit construction and a closed dimension formula in terms of hook lengths for the irreducible representations for the symmetric groups and the Sergeev superalgebras over an algebraically closed field of characteristic .

Paper Structure

This paper contains 8 sections, 20 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Irreducible representations of symmetric group $\mathfrak{S}_p$
  3. Basics on partitions and Specht modules
  4. Irreducible $\mathbb{F}\mathfrak{S}_n$-modules in the case $n=p$
  5. Irreducible representations of Sergeev superalgebra $\mathcal{Y}_p$
  6. Basics on affine Sergeev superalgebras
  7. Irreducible completely splittable representations of $\mathcal{Y}_n$
  8. Irreducible representations of $\mathcal{Y}_n$ in the case $n=p$.

Key Result

Theorem 2.1

Ja For $\lambda\in\mathcal{P}(n)$, $D^\lambda\neq 0$ if and only if $\lambda\in\mathcal{P}_p(n)$. Moreover, the set $\{D^\lambda\mid \lambda\in\mathcal{P}_p(n)\}$ is a complete set of non-isomorphic irreducible $\mathbb{F}\mathfrak{S}_n$-modules.

Theorems & Definitions (40)

  • Theorem 2.1
  • Example 2.2
  • Example 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • proof
  • Theorem 2.8
  • ...and 30 more