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Almost all wreath product character values are divisible by given primes

Brandon Dong, Hannah Graff, Joshua Mundinger, Skye Rothstein, Lola Vescovo

Abstract

For a finite group $G$ with integer-valued character table and a prime $p$, we show that almost every entry in the character table of $G \wr S_N$ is divisible by $p$ as $N \to \infty$. This result generalizes the work of Peluse and Soundararajan on the character table of $S_N$.

Almost all wreath product character values are divisible by given primes

Abstract

For a finite group with integer-valued character table and a prime , we show that almost every entry in the character table of is divisible by as . This result generalizes the work of Peluse and Soundararajan on the character table of .
Paper Structure (8 sections, 16 theorems, 43 equations, 4 figures)

This paper contains 8 sections, 16 theorems, 43 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representation Theory of the Wreath Product
  4. Asymptotics of Partitions
  5. Main Results
  6. Character Table Column Congruences
  7. Proof of Main Theorem
  8. Weyl groups of type D

Key Result

Theorem 1

Let $G$ be a group with integer-valued character table and let $G \wr S_N$ be the wreath product of $G$ with $S_N$. For all primes $p$, the proportion of entries in the character table of $G \wr S_N$ which are divisible by $p$ tends to 1 as $N \to \infty$.

Figures (4)

  • Figure 1: Examples of three invalid and one valid rimhooks in $\lambda = ((3^12^1))$.
  • Figure 2: All valid row decompositions of $\left(\ydiagram{4,1},\ydiagram{2}\right)$ by $({\color{red}{31}}, {\color{blue}{21}})$. The numbers in the boxes indicate the order in which parts of $\mu$ are placed into rows of $\lambda$, with fixed right-to-left placement.
  • Figure 3: Example of three conjugacy classes which are congruent mod 3 in $\mathbb{Z} / 2\mathbb{Z} \wr S_N$ (note that $m=2$ in the first cycle type and $m=1$ in the second).
  • Figure 4: Hook-lengths for $\lambda_i = (4,2,1)$

Theorems & Definitions (36)

  • Theorem : see Theorem \ref{['theorem: main theorem']} below
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: james2006representation, Theorem 4.2.8
  • Proposition 2.4: james2006representation, Theorem 4.4.3
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7: james2006representation, Theorem 4.4.10
  • Definition 2.8
  • Definition 2.9
  • ...and 26 more