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Characters of symmetric groups: sharp bounds on virtual degrees and the Witten zeta function

Lucas Teyssier, Paul Thévenin

Abstract

We prove sharp bounds on the virtual degrees introduced by Larsen and Shalev. This leads to improved bounds on characters of symmetric groups. We then sharpen bounds of Liebeck and Shalev concerning the Witten zeta function. Our main application is a characterization of the fixed-point free conjugacy classes whose associated random walk mixes in 2 steps.

Characters of symmetric groups: sharp bounds on virtual degrees and the Witten zeta function

Abstract

We prove sharp bounds on the virtual degrees introduced by Larsen and Shalev. This leads to improved bounds on characters of symmetric groups. We then sharpen bounds of Liebeck and Shalev concerning the Witten zeta function. Our main application is a characterization of the fixed-point free conjugacy classes whose associated random walk mixes in 2 steps.

Paper Structure

This paper contains 39 sections, 51 theorems, 185 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Context
  3. The Larsen--Shalev character bounds
  4. Main results
  5. Structure of the paper
  6. Preliminaries
  7. Young diagrams, representations and integer partitions
  8. Hook lengths, hook products and the hook-length formula
  9. Diagram notation
  10. Some elementary results
  11. Sliced hook products
  12. Definitions
  13. Bounds on sliced hook products
  14. Virtual degrees and augmented dimensions
  15. Sharp bounds on virtual degrees
  16. ...and 24 more sections

Key Result

Theorem 1.1

For any $n\geq 1$, $\sigma \in \mathfrak{S}_n$ and $\lambda \in \widehat{\mathfrak{S}_n}$, we have the following character bound:

Figures (12)

  • Figure 1: The Young diagram coding the partition $[5,2,1]$ of the integer $8$. It has 5 boxes on the first row, 2 on the second, and 1 on the third.
  • Figure 2: Left: the Young diagram associated to $\lambda = [7,5,4,1]$, with a box $u$ colored in orange. Right: the hook associated to $u$ is in pink. Its length is $H(\lambda, u) = 5$.
  • Figure 3: The Young diagram coding $\lambda = [7,5,4,1]$, with the set $E = \left\{ (1,1), (3,1), (6,1), (1,2),(2,2), (3,3)\right\}$ in orange. Here, $H(\lambda, E) = 10\cdot 7\cdot 2 \cdot 7\cdot 5 \cdot 2=9800$.
  • Figure 4: The diagram $\lambda = [3,2]$ and its $d_\lambda = |\mathop{\mathrm{ST}}\nolimits(\lambda)|= 5$ standard tableaux.
  • Figure 5: Examples of Definition \ref{['def:definitionnotation']}.
  • ...and 7 more figures

Theorems & Definitions (105)

  • Theorem 1.1: LarsenShalev2008
  • Theorem 1.2: LarsenShalev2008, Theorem 2.2
  • Theorem 1.3: LarsenShalev2008, Theorem 1.1 (a)
  • Example 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 95 more