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The classification of thick representations of the symmetric group

Kazunori Nakamoto, Shingo Okuyama, Yasuhiro Omoda

Abstract

We give the classification of thick representations and dense representations of the symmetric group over a field of characteristic zero.

The classification of thick representations of the symmetric group

Abstract

We give the classification of thick representations and dense representations of the symmetric group over a field of characteristic zero.
Paper Structure (19 sections, 55 theorems, 125 equations)

This paper contains 19 sections, 55 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Review of Specht modules
  4. Dense representations of the symmetric group
  5. Thick representations of the symmetric group
  6. Case: $(a,b) = (1,2)$
  7. Subcase: $c=3$
  8. Subcase: $c = 4$
  9. Subcase: $c \ge 5$
  10. Case: $(a,b) = (1,3)$
  11. Case: $a = 1, b\ge 4$
  12. Case: $a = 2$
  13. Case: $a \ge 3$
  14. The case $\lambda = (m_1, m_2)$ with $m_1\ge m_2 \ge 2$
  15. The case $\lambda = (m_1, m_2, m_3)$ with $m_1 \ge m_2 \ge m_3 \ge 1$
  16. ...and 4 more sections

Key Result

Theorem 1.1

For a finite-dimensional irreducible representation $\rho$ of $S_n$ over a field of characteristic zero, $\rho$ is thick if and only if it is dense.

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: NO1 and NO2
  • Proposition 2.2: NO1 and NO2
  • Proposition 2.3: NO1 and NO2
  • Corollary 2.4: NO1 and NO2
  • Definition 2.6: NO1 and NO2
  • Proposition 2.7: NO1 and NO2
  • Definition 2.8: NO4
  • Proposition 2.9: NO4
  • ...and 74 more