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Limit Values of Character Sums in Frobenius Formula of Three Permutations

Dun Liang, Bin Xu, Wenyan Yang

Abstract

We study the asymptotic behavior of the character sums appearing in the Frobenius formula for three conjugacy classes of symmetric groups. We show that if all three conjugacy classes contain no cycles of lengths $1$, $2$, or $3$, then the character sum converges to $2$. On the other hand, if two of the conjugacy classes contain $H\sqrt{n}$ fixed points while all other cycle lengths in all three conjugacy classes are large, then the character sum converges to $2e^{-H^2}$. As consequences of these results, we obtain several corollaries and propose conjectures related to the Hurwitz problem with three branching points on the sphere.

Limit Values of Character Sums in Frobenius Formula of Three Permutations

Abstract

We study the asymptotic behavior of the character sums appearing in the Frobenius formula for three conjugacy classes of symmetric groups. We show that if all three conjugacy classes contain no cycles of lengths , , or , then the character sum converges to . On the other hand, if two of the conjugacy classes contain fixed points while all other cycle lengths in all three conjugacy classes are large, then the character sum converges to . As consequences of these results, we obtain several corollaries and propose conjectures related to the Hurwitz problem with three branching points on the sphere.
Paper Structure (14 sections, 26 theorems, 173 equations)

This paper contains 14 sections, 26 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Main Results and Hurwitz Problem
  3. Sketch of the Proof
  4. Organization of the Paper
  5. Preliminaries and Regroups of the Character Sums
  6. Computations on Character Values
  7. Regroups of the Character Sums
  8. The Limit Two Theorem
  9. Proof of Proposition \ref{['Zlim2']}
  10. Proof of Proposition \ref{['Xlim2']}
  11. The Semi-Gaussian Law
  12. Proof of Proposition \ref{['Zsemigauss']}
  13. Proof of Proposition \ref{['Xsemigauss']}
  14. Comments and Conjecture on Transitive Subgroups

Key Result

Theorem 1.1

For all $\varepsilon > 0$, there exists $N > 0$, such that for all $n > N$, for all triples of conjugacy classes $C_1,C_2,C_3$ satisfy Condition conditiona, define ${\bf Y}_n(C_1,C_2,C_3)$ as in (Yn), then

Theorems & Definitions (32)

  • Theorem 1.1: The Limit 2 Theorem
  • Theorem 1.2: The Semi-Gaussian Law
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 2.1: Hardy-Ramanujan HardyRamanujan1918
  • Theorem 2.2: Alternative Version of the Murnaghan--Nakayama Rule, see Corollary 4.10.6 in GTM203
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • ...and 22 more