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\title
[Divisibility of character values of wreath products]
{Almost all wreath product character values are divisible by given primes}
\author[\initial{B.} Dong]{\firstname{Brandon} \lastname{Dong}}
\address[Brandon Dong]
{Carnegie Mellon University\\
Pittsburgh\\
PA}
\email[Brandon Dong]{bjdong@andrew.cmu.edu}
\author[\initial{H.} Graff]{\firstname{Hannah} \lastname{Graff}}
\address[Hannah Graff]
{Creighton University\\
Omaha\\
NE}
\email[Hannah Graff]{hjg46409@creighton.edu}
\author[\initial{J.} Mundinger]{\firstname{Joshua} \lastname{Mundinger}}
\address[Joshua Mundinger]
{University of Chicago\\
Chicago\\
IL}
\email[Joshua Mundinger]{mundinger@uchicago.edu}
\author[\initial{S.} Rothstein]{\firstname{Skye} \lastname{Rothstein}}
\address[Skye Rothstein]
{Bard College\\ Annandale-on-Hudson\\
NY}
\email[Skye Rothstein]{sr5920@bard.edu}
\author[\initial{L.} Vescovo]{\firstname{Lola} \lastname{Vescovo}}
\address[Lola Vescovo]
{Macalester College\\
Saint Paul\\
MN}
\email[Lola Vescovo]{lvescovo@macalester.edu}
%\date{April 19, 2023}
%\keywords{Wreath product, character table}
\subjclass{20C15, 05E10}
\datepublished{2024-01-08}
\begin{document}
\newtheorem{claim}[cdrthm]{Claim}
\begin{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 $S_N$.
\end{abstract}
\maketitle
\section{Introduction}
Let $S_N$ be the symmetric group on $N$ letters.
The complex irreducible characters of $S_N$ were calculated by Frobenius in 1900; in particular, Frobenius showed that the characters are integer-valued \cite{frobenius}.
In 2019, Alex Miller investigated the distribution of the parity of entries of the character table of $S_N$.
He made the remarkable conjecture that for any prime $p$ and exponent $\ell \geq 1$, the proportion of entries of the character table of $S_N$ divisible by $p$ (and later $p^\ell$ for $\ell \geq 1$) tends to 1 as $N \to \infty$ \cite{miller19, miller20}. This conjecture was recently proved by Peluse and Soundararajan in the case $\ell=1$ in \cite{peluse2022almost}.
This leaves the question of investigating the distribution of residues modulo $p$ for more general finite groups with integer-valued character tables. For a fixed group $G$ with integer-valued character table, a natural infinite family of such is the wreath product $G \wr S_N$ as $N \to \infty$. When $G$ has integer-valued character table, it is known that the characters of $G \wr S_N$ are also integer-valued \cite[Corollary 4.4.11]{james2006representation}.
These families of wreath products include the Weyl group of type $B_N$, when $G = \mathbb Z/2\mathbb Z$, and wreath products $S_M \wr S_N$ of symmetric groups.
Our main result is a generalization of Peluse and Soundararajan's theorem:
\begin{thm*}[see Theorem \ref{theorem: main theorem} below]
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 $N \to \infty$.
\end{thm*}
The proof relies on the combinatorics of the representations of $G \wr S_N$. If $G$ has $k$ conjugacy classes, then conjugacy classes and representations of $G \wr S_N$ are both naturally labelled by $k$-multipartitions of $N$. One of the key inputs is characterizing when two elements of $G \wr S_N$ have columns in the character table congruent modulo $p$. In Lemma \ref{lemma: permutation module congruence}, we give a combinatorial characterization directly generalizing the corresponding criterion for $S_N$.
It is known that the character tables of all Weyl groups are integer-valued. The Weyl groups of type $A$ are the symmetric groups, where our question was answered by Peluse and Soundararajan. The Weyl groups of type $B_N$ and $C_N$ are both equal to $\mathbb Z/2\mathbb Z \wr S_N$, handled by our main theorem.
The only remaining infinite family of Weyl groups is that of type $D$. In Section \ref{section: extensions}, we also show that the proportion of character values of the Weyl group of type $D_N$ divisible by a prime $p$ tends to 1 as $N \to \infty$.
\section{Preliminaries}
\subsection{Representation Theory of the Wreath Product}
Let $G$ be a finite group with integer-valued character table and let $S_N$ be the symmetric group on $N$ letters.
\begin{defi}
The \emph{wreath product of $G$ with $S_N$}, denoted $G \wr S_N$, is the group of $N \times N $ permutation matrices with nonzero entries in $G$.
\end{defi}
We begin by recalling the representation theory of $G \wr S_N$. The representation theory of wreath products was first studied in Specht's dissertation \cite{specht32}, anticipated by Young's work on the case $G = \mathbb Z/2\mathbb Z$ \cite{youngfifth}; see also \cite{james2006representation,zelevinsky} for more modern treatments.
If we take the representation theory of $G$ as input data and let $N$ vary, the representation theory has structural similarities to the representation theory of $S_N$, the case when $G = 1$. While representations and conjugacy classes of the symmetric group are labelled by partitions of $N$, representations and conjugacy classes of the wreath product are labelled by multipartitions:
\begin{defi}
A \emph{$k$-multipartition} of an integer $N$ is $\lambda = (\lambda_1, \dots, \lambda_k)$ where $\lambda_i$ is a partition for all $i$ such that $\sum_{i=1}^k \vert \lambda_i \vert = N$.
\end{defi}
We now describe how to label conjugacy classes of $G \wr S_N$.
For an element of $G \wr S_N$, assign to each cycle in its projection to $S_N$ a conjugacy class of $G$, called the \emph{cycle product}, as follows:
if $(i_1 i_2\cdots i_m)$ is a cycle of $\sigma$, then the cycle product of $(g_1,g_2,\ldots, g_N)\sigma \in G \wr S_N$ corresponding to $(i_1 i_2\cdots i_m)$ is defined to be the conjugacy class of $g_{i_m} \cdots g_{i_2}g_{i_1}$ \cite[(4.2.1)]{james2006representation}.
\begin{prop}[{\cite[Theorem 4.2.8]{james2006representation}}]
\label{proposition: cycle products index conj classes}
If $G$ has $k$ conjugacy classes, then the conjugacy classes of $G \wr S_N$ are indexed by $k$-multipartitions of $N$.
Given $x \in G \wr S_N$, the multipartition $\lambda$ corresponding to $x$ is formed as follows: for each cycle in $x$ of length $\ell$, if the corresponding cycle product is the $i$th conjugacy class of $G$, then add a part of size $\ell$ to $\lambda_i$.
\end{prop}
One can check the assignment of a conjugacy class to a multipartition is well-defined by checking under conjugation by $S_N$ and by diagonal matrices $G^N \subseteq G \wr S_N$.
Conjugating an element of $G \wr S_N$ by $S_N$ does not change the set of cycle products at all.
If $(g_1,\ldots,g_N) \in G^N$ and $(12\cdots N)$ is an $N$-cycle,
the conjugate of $(12\cdots N)(g_1,g_2,\ldots,g_N)$ by $(g,1,\ldots,1)$ is $(12\cdots N) (g_1g^{-1},g_2,\ldots,gg_N)$; these two elements have conjugate cycle products $g_N\cdots g_2g_1$ and $g(g_N\cdots g_2g_1)g^{-1}$. The general case of conjugation by $G^N$ reduces to the above case.
We will not need to use the specific form of this bijection in this paper; it is used in the proofs of character formulas in Propositions \ref{proposition: wreath-irreducibles} and \ref{proposition: perm_modules}, which we omit.
To find the complex irreducible representations of $G \wr S_N$, we need the complex irreducible representations of $G$ as input; call the irreducible $G$-representations $V_1, \dots, V_k$.
\begin{prop}[{\cite[Theorem 4.4.3]{james2006representation}}]
\label{proposition: wreath-irreducibles}
If $G$ has $k$ conjugacy classes, then the irreducible representations of $G \wr S_N$ are in bijection with $k$-multipartitions of $N$.
For $\lambda=(\lambda_1,\ldots,\lambda_k)$ a $k$-multipartition of $N$, let $a_i = |\lambda_i| $ and $G_a = G \wr S_a$. Then the irreducible representation of $G \wr S_N$ corresponding to $\lambda$ is
\begin{equation*}
V^{\lambda} = \Ind_{G_{a_1}\times \dots \times G_{a_k}}^{G_N}\left(
\boxtimes_{i=1}^k \left(
S^{\lambda_i} \otimes V_i^{\otimes a_i}\right)
\right)
\end{equation*}
where $S^{\lambda_i}$ is the Specht module for $S_N$ corresponding to $\lambda_i$.
\end{prop}
Character values of wreath products can be calculated using a modified version of the Murnaghan--Nakayama rule for the symmetric group.
Let $\chi^\lambda$ be the character of $V^\lambda$ and $\chi^\lambda_\mu$ be the value of $\chi^\lambda$ on the conjugacy class corresponding to $\mu$.
Then $\chi^\lambda_\mu$ is calculated by decomposing the Young diagrams of the partitions $\lambda_i$ for all $i$ using~rimhooks:
\begin{defi}
A \emph{rimhook} of a $k$-multipartition $\lambda=(\lambda_1,\ldots,\lambda_k)$ consists of $k$ adjacent boxes in the Young diagram of a single $\lambda_i$ such that no other boxes are remaining south or east after the rimhook has been removed and no box in the rimhook has a southeast neighbor.
\end{defi}
\begin{figure}[h]
\begin{center}
\begin{tabular}{ccccccccc}
NO & & NO & & NO & & YES \\
\begin{ytableau}
*(white) & *(white) & *(lightgray) \\
*(white) & *(lightgray) \\
\end{ytableau} & & \begin{ytableau}
*(lightgray) & *(lightgray) & *(white) \\
*(white) & *(white) \\
\end{ytableau} & & \begin{ytableau}
*(lightgray) & *(lightgray) & *(lightgray) \\
*(lightgray) & *(lightgray) \\
\end{ytableau} & & \begin{ytableau}
*(white) & *(lightgray) & *(lightgray) \\
*(white) & *(lightgray) \\
\end{ytableau}
\end{tabular}
\end{center}
\caption{
Examples of three invalid and one valid rimhooks in $\lambda = ((3^12^1))$.
}
\label{fig:rimhook example}
\end{figure}
\begin{defi}
For $k$-multipartitions $\lambda$ and $\mu$, a \emph{rimhook decomposition} of $\lambda$ by $\mu$ is obtained by repeatedly removing rimhooks in $\lambda$ according to a fixed ordering of the parts of $\mu$. Further, we define $RHD(\lambda,\mu)$ to consist of all rimhook decompositions of $\lambda$ by $\mu$.
\end{defi}
As in the Murnaghan--Nakayama rule for the usual symmetric group, formulas for irreducible characters involve the height of a rimhook decomposition:
\begin{defi}
The \emph{height} of a rimhook is one less than the number of rows included in that rimhook.
The \emph{height} of a rimhook decomposition $\rho$, denoted $ht(\rho)$, is the sum of the heights of all rimhooks in the decomposition.
\end{defi}
The Murnaghan--Nakayama rule can be modified for wreath products as follows:
\begin{prop}[{\cite[Theorem 4.4.10]{james2006representation}}]
\label{proposition: MN rule}
Let $\lambda$ and $\mu$ be $k$-multipartitions of $N$. Let $\chi^{1}, \chi^{2}, \ldots,\chi^k$ be the irreducible characters of $G$
and $c_1,c_2\ldots, c_k$ the conjugacy classes of $G$.
For $\rho \in RHD(\lambda,\mu)$, let $\psi(\rho)$ be defined by
\begin{equation*}
\psi(\rho) = \prod_{i = 1}^{k}
\prod_{j=1}^k \left( \chi^i(c_j)^{\#\{\text{rimhooks } h\in\rho \text{ from }\mu_j \text{ in } \lambda_i\}}\right).
\end{equation*}
Then
\begin{equation*}
\chi^\lambda_\mu = \sum_{\rho \in \text{RHD}(\lambda, \mu)} (-1)^{\text{ht}(\rho)} \psi(\rho).
\end{equation*}
\end{prop}
The permutation module characters of wreath products form another basis for the space of class functions of $G \wr S_N$ that is easier to work with.
\begin{defi}\label{definition: permutation modules}
Let $\lambda = (\lambda_1,\ldots,\lambda_k)$ be a $k$-multipartition of $N$ and let $a_i = |\lambda_i|$.
For each $\lambda_i$, let $S_{\lambda_i}$ be the Young subgroup of $S_{a_i}$ corresponding to $\lambda_i$ and let $G_{\lambda_i} = G \wr S_{\lambda_i}$.
Then the \emph{permutation module} $M^{\lambda}$ for $G \wr S_N$ is defined by
\begin{equation*}
M^{\lambda} = \Ind_{G_{\lambda_1} \times \cdots \times G_{\lambda_k}}^{G \wr S_N}
\left( \boxtimes_{i=1}^k V_i^{\otimes a_i} \right).
\end{equation*}
\end{defi}
There is a character formula for $M^\lambda$ using row decompositions instead of rimhook decompositions. It is as follows:
\begin{defi}
Let $\lambda$ and $\mu$ be $k$-multipartitions of $N$. A \emph{row decomposition} of $\lambda$ by $\mu$ is
a function $\rho: \{\text{rows of }\mu\} \to \{\text{rows of }\lambda\}$ such that if $r$ is a row of $\lambda$, then the rows in $\rho^{-1}(r)$ have the same total length as $r$.
The set of all row decompositions of $\lambda$ by $\mu$ is denoted $RD(\lambda, \mu)$.
\end{defi}
We will think of row decompositions of $\lambda$ by $\mu$ as a tiling of the Young diagrams of $\lambda$ by rows, where rows of $\mu$ are placed in a fixed ordering.
\begin{figure}[h]
\centering
\ytableausetup{centertableaux, boxsize = 1.5em}
\begin{tabular}{ccc}
$\left( \hspace{1mm} \begin{ytableau}
*(color2)2 & *(color1)1 & *(color1)1 & *(color1)1 \\
*(color4)4
\end{ytableau} \hspace{1mm}, \hspace{1mm} \begin{ytableau}
*(color3)3 & *(color3)3
\end{ytableau}
\right)$ & \hspace{4mm} & $\left( \hspace{1mm} \begin{ytableau}
*(color4)4 & *(color1)1 & *(color1)1 & *(color1)1 \\
*(color2)2
\end{ytableau} \hspace{1mm}, \hspace{1mm} \begin{ytableau}
*(color3)3 & *(color3)3
\end{ytableau}
\right)$
\end{tabular}
\ytableausetup{boxsize=0.4em}
\caption{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.
}
\end{figure}
\begin{prop}\label{proposition: perm_modules}
Let $\lambda$ and $\mu$ be $k$-multipartitions of $N$.
Let $\chi^1,\chi^2,\ldots,\chi^k$ be the irreducible characters of $G$ and $c_1,\ldots,c_k$ the conjugacy classes of $G$.
For $\rho \in RD(\lambda,\mu)$, let $\alpha(\rho)$ be defined by
\begin{equation*}
\alpha(\rho) = \prod_{i = 1}^{k}
\prod_{j=1}^k
\left( \chi^i(c_j)^{\#\{\text{rows from }\mu_j\text{ placed into }\lambda_i\text{ by }\rho\}}\right).
\end{equation*}
Then the character for permutation module $M^\lambda$ at $\mu$ is
\begin{equation*}
M^\lambda_\mu = \sum_{\rho\in \text{RD}(\lambda,\mu)} \alpha(\rho).
\end{equation*}
\end{prop}
The proof follows from the character formula for induced representations.
We now describe the change-of-basis between irreducible and permutation characters.
\begin{defi}
The \emph{dominance order} on partitions is defined by $\lambda \succcurlyeq \eta$ if the row lengths $\lambda^1 \geq \lambda^2 \geq \ldots$ and $\eta^1 \geq \eta^2 \geq \ldots$ of $\lambda$ and $\eta$ satisfy $\sum_{i=1}^j \lambda^i \geq \sum_{i=1}^j \eta^i$ for all $j\geq 1$.
The dominance order on $k$-multipartitions is defined by $\lambda \succcurlyeq \eta$ if and only if $\lambda_i \succcurlyeq \eta_i$ for all $i$.
\end{defi}
\begin{lemma} \label{lemma: unitriangularity}
The matrix of multiplicities $[M^\lambda: V^\eta]$ of the irreducible representations of $G \wr S_N$ in permutation modules is unimodular and lower-triangular with respect to dominance order.
\end{lemma}
\begin{proof}
Recall that the Kostka numbers $K^{\beta,\gamma}$ for $\beta,\gamma$ partitions of $N$ are defined by
\[
M^\beta = \Ind_{S_\beta}^{S_N} 1 = \bigoplus_\gamma \left(V^{\gamma}\right)^{\oplus K^{\gamma,\beta}},
\]
where $S_\beta$ is the Young subgroup corresponding to $\beta$ and $V^\gamma$ is the Specht module corresponding to $\gamma$. Note that our notation for $M^\beta$ and $V^\gamma$ agrees with that of wreath products $G \wr S_N$ when $G=1$.
The Kostka numbers satisfy $K^{\beta,\beta} = 1$ and $K^{\gamma,\beta} > 0$ if and only if $\gamma \succcurlyeq \beta$ in dominance order
\cite[I, (6.5)]{macdonald98}.
We claim that
\begin{equation} \label{equation: permutation-irrep-decomposition}
M^\lambda = \bigoplus_{\eta} \left(V^\eta\right)^{\oplus c(\lambda,\eta)}, \qquad c(\lambda,\eta) = \left(\prod_{i = 1}^k K^{\eta_i, \lambda_i} \right).
\end{equation}
By Definition \ref{definition: permutation modules},
if $a_i = |\lambda_i|$ for all $i$ and $H = G_{a_1} \times G_{a_2} \times \cdots \times G_{a_k}$, then
\[ M^\lambda = \Ind_{G_\lambda}^{G_N} \left(\boxtimes_{i=1}^k V_i^{\otimes a_i} \right) = \Ind_{H}^{G_N} \left( \boxtimes_{i=1}^k M^{\lambda_i} \otimes V_i^{\otimes a_i}\right),\]
where $S_{a_i}$ acts diagonally on the tensor product $M^{\lambda_i} \otimes V_i^{\otimes a_i}$ and $G^{a_i}$ acts naturally on $V_i^{\otimes a_i}$.
Then \eqref{equation: permutation-irrep-decomposition} follows from multilinearity of the tensor product and linearity of~induction.
Now since the matrix of Kostka numbers is unimodular and upper-triangular with respect to dominance order, the matrix $\{c(\lambda,\mu)\}_{\lambda,\mu}$ is unimodular and lower-triangular with respect to dominance order.
\end{proof}
\subsection{Asymptotics of Partitions}
We recall a form of the Hardy--Ramanujan asymptotic for the number of partitions of $N$, denoted $p(N)$.
\begin{prop}[{\cite[(1.36)]{HR}}] \label{proposition: HR asymptotic}
If $\delta > 0$, then
\begin{equation*}
\left(\frac{2 \pi}{\sqrt{6}} - \delta\right)\sqrt{N} \leq \log p(N) \leq \left(\frac{2 \pi}{\sqrt{6}} + \delta\right)\sqrt{N}
\end{equation*}
for sufficiently large $N$.
\end{prop}
Let $p_k(N)$ denote the number of $k$-multipartitions of $N$.
\begin{claim}\label{claim: asymptotics of multipartitions}
If $\delta > 0$, then
\begin{equation*}
\left(\frac{2 \pi}{\sqrt{6}} - \delta\right) \sqrt{kN} \leq \log{p_k(N)} \leq \left(\frac{2 \pi}{\sqrt{6}} + \delta\right) \sqrt{kN}
\end{equation*}
for sufficiently large $N$.
\end{claim}
This formula also appears in \cite{murty2013partition}. We provide an elementary inductive proof.
\begin{proof}
We proceed by induction on $k$. The base case $k$ = 1 is Proposition \ref{proposition: HR asymptotic}.
For $\delta > 0$, let $\delta' = \frac{4}{5}\delta$.
By inductive hypothesis, there exists a constant $B$ such that if $C \geq B$, then
\begin{equation*}
\exp\left( \left(\frac{2 \pi}{\sqrt{6}} - \delta'\right) \left( \sqrt{(k-1)C}\right) \right) \le p_{k-1}(C) \\ \le \exp\left( \left(\frac{2 \pi}{\sqrt{6}} + \delta'\right) \left(\sqrt{(k-1)C}\right) \right)
\end{equation*}
and
\begin{equation*}
\exp\left( \left(\frac{2 \pi}{\sqrt{6}} - \delta'\right) \left( \sqrt{C}\right) \right) \le p(C) \\ \le \exp\left( \left(\frac{2 \pi}{\sqrt{6}} + \delta'\right) \left(\sqrt{C}\right) \right).
\end{equation*}
By considering the size of the first partition in a $k$-multipartition, it follows that
\begin{equation*}
p_k(N) = \displaystyle\sum_{a = 0}^N p(a)p_{k-1}(N-a).
\end{equation*}
Now assume that $N \geq 2B - 2$; we can then break up the sum for $p_k(N)$ into the following distinct parts: let
\begin{align*}
D_1 &= \displaystyle\sum_{a=0}^{B-1} p(a)p_{k - 1}(N - a), \\
D_2 &= \displaystyle\sum_{a=B}^{N-B} p(a)p_{k - 1}(N - a), \\
D_3 &= \displaystyle\sum_{a=N-B+1}^N p(a)p_{k - 1}(N - a).
\end{align*}
In $D_2$, for $B \leq a \leq N - B$, we have
\begin{align*}
\exp\left( \left(\frac{2 \pi}{\sqrt{6}} - \delta'\right) \left(\sqrt{a} + \sqrt{(k-1)(N-a)}\right) \right) &\le p(a)p_{k - 1}(N - a),
\end{align*}
and the right hand side in turn satisfies
\begin{align*}
p(a)p_{k - 1}(N - a)\le \exp\left( \left(\frac{2 \pi}{\sqrt{6}} + \delta'\right) \left(\sqrt{a} + \sqrt{(k-1)(N-a)}\right) \right).
\end{align*}
Note that $\sqrt{a} + \sqrt{(k - 1)(N - a)} \leq \sqrt{kN}$,
with equality achieved at $a= \frac{N}{k}$. Summing over $a \in [B,N-B]$, we get
\begin{equation}\label{D_2ref}
\exp\left(\left(\frac{2 \pi}{\sqrt{6}} - \delta\right)\sqrt{kN}\right) \le D_2 \\ \le (N - 2B + 1) \exp\left( \left(\frac{2 \pi}{\sqrt{6}} + \delta'\right) \sqrt{kN} \right).
\end{equation}
We now consider $D_1$ and $D_3$. Note that for $a \in [0, B)$, we have
$p(a)p_{k - 1}(N - a) \leq p(B)p_{k - 1}(N),$ and for $a \in (N - B, N]$, we have $p(a)p_{k - 1}(N - a) \leq p(N)p_{k - 1}(B)$.
Hence
\begin{align}
0 \leq D_1
\leq B p(B)p_{k - 1}(N)
&\le B
\exp\left( \left(\frac{2 \pi}{\sqrt{6}} + \delta'\right) \sqrt{kN} \right) \label{D_1ref}
\end{align}
for sufficiently large $N$. Likewise,
\begin{align}
0 \leq D_3
\leq B p(N)p_{k - 1}(B)
&\le B \exp\left( \left(\frac{2 \pi}{\sqrt{6}} + \delta'\right) \sqrt{kN} \right) \label{D_3ref}.
\end{align}
Combining \eqref{D_2ref}, \eqref{D_1ref}, and \eqref{D_3ref}, we have that
\begin{align}\label{P_k(N)}
\exp \left(
{\left(\frac{2 \pi}{\sqrt{6}} - \delta\right)} \sqrt{kN}\right) \le
p_k(N)
&\le (N + 1)\exp \left( {\left(\frac{2 \pi}{\sqrt{6}} + \delta'\right)} \sqrt{kN}\right) \nonumber \\
&\le \exp \left({\left(\frac{2 \pi}{\sqrt{6}} + \delta\right)} \sqrt{kN}\right) \nonumber
\end{align}
for sufficiently large $N$.
\end{proof}
The above estimate implies $k$-multipartitions concentrate around having close to equal-size parts:
\begin{coro}\label{corollary: concentration of multipartitions}
For all $\delta>0$, the proportion of $k$-multipartitions $\lambda = (\lambda_1, \dots, \lambda_k) \vdash N$ such that
\[
\frac{N}{k}(1-\delta) < \vert \lambda_i \vert < \frac{N}{k}(1 + \delta)
\]
for all $1 \leq i \leq k$ goes to 1 as $N \to \infty$.
\end{coro}
\begin{proof}
Pick $1\le i \le k$.
By partitioning $k$-multipartitions according to the size of $\lambda_i$,
the number of $k$-multipartitions of $N$ where $|\lambda_i| \notin \left(\frac{N}{k}(1-\epsilon),\frac{N}{k}(1+\epsilon)\right)$ is
\begin{equation}\label{equation: badones}
\sum_{a \notin( \frac{N}{k}(1-\varepsilon),\frac{N}{k}(1 + \varepsilon))} p(a) p_{k-1}(N-a).
\end{equation}
Note that $\sqrt{a}+\sqrt{(k-1)(N-a)}$ increases for $a<\frac{N}{k}$ and decreases for $a>\frac{N}{k}$. Then by Claim \ref{claim: asymptotics of multipartitions}, the rate of growth \eqref{equation: badones} is significantly slower than the rate of growth of $p_k(N)$.
\end{proof}
\section{Main Results}
\subsection{Character Table Column Congruences}
Corollary \ref{corollary: character congruence} below, which we call ``the mashing rule,'' gives a criterion for mod $p$ congruence of two columns of the character table of $G \wr S_N$ in terms of $k$-multipartitions.
In this section, we must assume that $G$ has integer-valued character table. By \cite[\S13.1]{serrelinear}, the group $G$ has integer-valued character table if and only if $\sigma\in G$ is conjugate to $\sigma^j$ whenever $j$ is prime to the order of $\sigma$.
\begin{defi}
Let $\sim_p$ be the equivalence relation on $k$-multipartitions generated by the following: $\mu \sim_p \nu$ if there is $j$ such that $\mu_i = \nu_i$ for $i \neq j$, and $\nu_j$ is formed by replacing one part of size $mp$ in $\mu_j$ with $p$ parts of size $m$ in $\nu_j$.
\end{defi}
\begin{figure}[h]
\ytableausetup{boxsize=0.95em}
\centering
\begin{tabular}{ccccc}
$\left(\begin{ytableau}
*(skyeblue) & *(skyeblue) & *(skyeblue) & *(skyeblue) & *(skyeblue) & *(skyeblue)\\
*(white)
\end{ytableau} \> , \> \begin{ytableau}
*(white) & *(white) & *(white) & *(white) \\
*(darkpurp) \\
*(darkpurp) \\
*(darkpurp) \\
\end{ytableau} \right)$
& $\sim_3$
& $\left(\begin{ytableau}
*(skyeblue) & *(skyeblue) \\
*(skyeblue) & *(skyeblue) \\
*(skyeblue) & *(skyeblue) \\
*(white)
\end{ytableau} \> , \> \begin{ytableau}
*(white) & *(white) & *(white) & *(white) \\
*(darkpurp) \\
*(darkpurp) \\
*(darkpurp) \\
\end{ytableau} \right)$
& $\sim_3$
& $\left(\begin{ytableau}
*(skyeblue) & *(skyeblue) \\
*(skyeblue) & *(skyeblue) \\
*(skyeblue) & *(skyeblue) \\
*(white)
\end{ytableau} \> , \> \begin{ytableau}
*(white) & *(white) & *(white) & *(white) \\
*(darkpurp) & *(darkpurp) & *(darkpurp)
\end{ytableau} \right)$ \\
& & & & \\
$(61, 4 1^3)$ & $\sim_{3}$ & $(2^3 1, 4 1^3)$ & $\sim_{3}$ & $(2^3 1, 4 3)$ \\
\end{tabular}
\caption{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).
}
\label{figure: mashing example}
\end{figure}
\begin{lemma}\label{lemma: permutation module congruence}
Let $p$ be a prime and $G$ be a group with integer-valued character table. Let $\mu = (\mu_1,\dots, \mu_k)$ and $\nu = (\nu_1,\dots, \nu_k)$ be $k$-multipartitions of $N$, indexing conjugacy classes of $G \wr S_N$. If $\mu \sim_p \nu$, then $M^\lambda_{ \mu} \equiv M^\lambda_{ \nu} \pmod{p}$ for all $k$-multipartitions $\lambda$ of $N$.
\end{lemma}
\begin{proof}
It suffices to show $M^\lambda_\mu \equiv M^\lambda_\nu \pmod{p}$ if there exists $j$ such that $\mu_i = \nu_i$ for all $i \neq j$, $\mu_j = (\xi, mp)$ for some $\xi$, and $\nu_j = (\xi, m^p)$.
We break $RD(\lambda,\nu)$ into two cases. In case one, we consider the row decompositions of $\nu$ where the $p$ rows not in $\xi$ are tiled into the same row of $\lambda$. In case two we consider the row decompositions where the $p$ rows not in $\xi$ are not tiled in the same row. Recalling our formula for characters of permutation modules in Proposition \ref{proposition: perm_modules}, let
\begin{equation}
\beta = \displaystyle\sum_{\substack{\rho \in RD(\lambda, \nu) \text{ s.t. $m^p$ is tiled} \\ \text{in the same row}}} \alpha(\rho)
\end{equation}
and
\begin{equation}\label{gamma}
\gamma = \displaystyle\sum_{\substack{\rho \in RD(\lambda, \nu) \text{ s.t. $m^p$ is not tiled} \\ \text{in the same row}}} \alpha(\rho),
\end{equation}
so that $M^\lambda_\nu = \beta + \gamma$. In case one, we will show $\beta \equiv M^\lambda_\mu \pmod{p}$ and in case two that $\gamma \equiv 0 \pmod{p}$.
Together, these two congruences imply $M^\lambda_{\mu} \equiv M^\lambda_{\nu} \pmod{p}$.
In both cases, we break into subcases based on the ways to tile $\mu_i$ for $i \neq j$ and $\xi$. In case one, we have compatible tilings for $\mu$ and $\nu$, and in case two, we have additional tilings for $\nu$.
In case one, assume we have tiled all rows of $\mu_i$ for all $i \neq j$ and we have tiled $\xi$. We now have one row remaining. There is only one way to tile the last row for both $\mu$ and $\nu$: put the remaining pieces into the remaining row. Let these row decompositions be denoted $\rho_\mu$ and $\rho_\nu$ respectively.
For $\rho_\mu$, say that we place the final row $r$ of size $mp$ in the partition $\lambda_q$. The associated cycle product is $c_j$ because $mp$ comes from $\mu_j$. Then $mp$ contributes $\chi_q(c_j)$ to the product $\alpha(\rho_\mu)$.
Then for $\rho_\nu$, the $p$ rows of size $m$ are placed into $\lambda_q$. The conjugacy class of $G$ associated with the $p$ rows of size $m$ is again $c_j$, so $m^p$ contributes a factor of $\chi_q(c_j)^p$ to $\alpha(\rho_\nu)$.
By assumption, the character values of $G$ are integral, so by Fermat's little theorem, $\chi_q(c_j) \equiv \chi_q(c_j)^p \pmod{p}$. All other factors in $\alpha(\rho_\mu)$ contributed by $\mu_i$ for $i \neq j$ and $\xi$ are identical to the corresponding factors in $\alpha(\rho_\nu)$.
Hence, $\alpha(\rho_\mu) \equiv \alpha(\rho_\nu) \pmod{p}$.
Summing over all the tilings in case one, we find $M^\lambda_\mu \equiv \beta \pmod{p}$.
In case two, assume we have tiled all rows of $\mu_i$ for $i \neq j$ and $\xi$, after which there are $t>1$ remaining unfilled rows of the Young diagrams of $\lambda$. If $T \subseteq RD(\lambda,\nu)$ is the set of row decompositions extending our given tiling by $\mu_i$ for $i\neq j$ and $\xi$, then we will show
\[
\sum_{\rho \in T} \alpha(\rho) \equiv 0 \pmod p.
\]
Then $\gamma$ is the sum over all such $T$ of $\sum_{\rho \in T}\alpha(\rho)$, from which it will follow $\gamma \equiv 0 \mod p$.
Call the lengths of the remaining rows $(m\ell_1 , m\ell_2, \dots, m\ell_t)$.
Since the elements of $T$ are in bijection with choices of placements of $p$ cycles of length $m$ into these rows,
\begin{equation}\label{equation: binomial}
|T| = \binom{p}{\ell_1, \ell_2, \dots, \ell_t}.
\end{equation}
Let $\rho \in T$. Note that all pieces of $m^p$ come from $\mu_j$, and thus have cycle product $c_j$, while all other cycles in $\mu$ are in the same place in $T$. Thus $\alpha(\rho) = \alpha(\rho')$ for all $\rho,\rho' \in T$. Hence $\sum_{\rho \in T}\alpha(\rho)$ is a sum of $|T|$ identical terms. Then $\sum_{\rho \in T}\alpha(\rho) \equiv 0 \pmod{p}$ because $|T|$ is divisible by $p$.
Case one has shown that $M^\lambda_\mu \equiv \beta \pmod{p}$, and case two has shown that $\gamma \equiv 0 \pmod{p}$. Since $M^\lambda_\nu = \beta + \gamma$, we conclude $M^\lambda_{\mu} \equiv M^\lambda_{\nu} \pmod{p}$.
\end{proof}
\begin{coro}[The mashing rule]\label{corollary: character congruence}
Let $G$ have integer-valued character table and $k$ conjugacy classes.
Let $\mu$ and $\nu$ be $k$-multipartitions of $N$.
If $\mu \sim_p \nu$, then $\chi_{ \mu}^{\lambda} \equiv \chi_{ \nu}^{\lambda} \pmod{p}$ for all irreducible characters $\chi^\lambda$ of $G \wr S_N$.
\end{coro}
\begin{proof}
The set of irreducible characters and the set of characters of
permutation modules form bases for the space of class functions on $G \wr S_N$. Since the change of basis matrix between these two bases is unimodular and lower-triangular, as stated in Lemma \ref{lemma: unitriangularity}, $\chi^{\lambda}$ can be expressed as an integral linear combination of $M^{\eta}$ for all $k$-multipartitions $\lambda$. It follows from Lemma $\ref{lemma: permutation module congruence}$ that $\mu \sim_p \nu$ implies $\chi_{ \mu}^{\lambda} \equiv \chi_{ \nu}^{\lambda} \pmod{p}$.
\end{proof}
\begin{rema}
Corollary \ref{corollary: character congruence} may also be proved using the following criterion from modular representation theory: if $G$ has integer-valued character table, $g \in G$, and $g'$ is the $p$-prime part of $g$, then $\chi(g) \equiv \chi(g') \mod p$ for all characters $\chi$ of $G$ \cite[18.1(v)]{serrelinear}. In this situation, it may be seen that the mashing preserves the conjugacy class of the $p$-prime part of a class in $G \wr S_N$. Nonetheless, we have chosen to give a combinatorial proof.
\end{rema}
\subsection{Proof of Main Theorem}
Using Corollary \ref{corollary: character congruence}, the existence of one zero in the character table implies many more entries are divisible by $p$. We proceed, following Peluse and Soundararajan in \cite{peluse2022almost}, by using Proposition \ref{proposition: MN rule} to show sufficiently many entries of the character table are zero.
\begin{defi}
A partition is called a \emph{$t$-core} if none of the hook lengths of its Young diagram are divisible by $t$ where $t \in \mathbb{Z}$. For example, from Figure \ref{t-core} one can see that $(4,2,1)$ is a 5-core.
\end{defi}
\begin{figure}[h]
\centering
\ytableausetup{centertableaux}
\begin{ytableau}
6 & 4 & 2 & 1 \\
3 & 1 \\
1
\end{ytableau}
\caption{Hook-lengths for $\lambda_i = (4,2,1)$}
\label{t-core}
\end{figure}
In the course of proving \cite[Proposition 1]{peluse2022almost}, Peluse and Soundararajan proved the following estimate of the number of $t$-cores when $t$ is slightly larger than the typical longest cycle in a random conjugacy class:
\begin{prop}\label{prop: PSProp1}
Let $L$ be a positive integer, and let $A$ be a real number with $1 \leq A \leq \log L / \log \log L$. Additionally suppose that $t$ is a positive integer with
\begin{equation} \label{eq: t-bound}
t \geq \frac{\sqrt{6}}{2 \pi} \sqrt{L} (\log L) \left(1 + \frac{1}{A}\right).
\end{equation}
Then the number of partitions $\lambda$ of $L$ which are not $t$-cores is at most
\begin{equation*}
O\left(p(L)\frac{\log L}{L^\frac{1}{2A}}\right),
\end{equation*}
independent of $t$ satisfying \eqref{eq: t-bound}.
\end{prop}
Complementing the estimate in Proposition \ref{prop: PSProp1}, Peluse and Soundararajan also estimated how many columns of the character table are congruent to a column corresponding to a partition with a large first part:
\begin{prop}[{\cite[Proposition 2]{peluse2022almost}}]\label{prop: PSProp2}
Let $p \leq\frac{(\log{L})}{(\log{\log{L}})^2}$ be a prime. Starting with a partition $\mu$ of $L$, we repeatedly replace every occurrence of $p$ parts of the same size $m$ by one part of size $mp$ until we arrive at a partition $\tilde{\mu}$ where no part appears more than $p-1$ times. Then the largest part of $\tilde{\mu}$ exceeds
\begin{equation*}
\frac{\sqrt{6}}{2 \pi} \sqrt{L} \left(\log{L}\right)\left(1+\frac{1}{5p}\right),
\end{equation*}
except for at most
\begin{equation*}
O\left(p(L)\exp \left( {-L^{\frac{1}{15p}}} \right) \right)
\end{equation*}
partitions $\mu$.
\end{prop}
We now extend Peluse and Soundarajan's estimate in Proposition \ref{prop: PSProp2} to $k$-multipartitions.
\begin{prop}\label{proposition: mash to the max}
Let $p\ll N$ be a prime. Given a $k$-multipartition $\mu = (\mu_1, \dots \mu_k)$ of $N$, for all $\mu_i$ with $1 \leq i \leq k$, we repeatedly replace every occurrence of $p$ parts of the same size $m$ by one part of size $mp$ until we arrive at a $k$-multipartition $\tilde{\mu}$ where no part in any $\tilde{\mu}_i$ appears more than $p - 1$ times.
Then the largest part of $\tilde{\mu}$ is of size at least
\begin{equation}\label{equation: t lower bound}
\frac{\sqrt{6}}{2 \pi} \sqrt{\frac{N}{k}} \left(\log{\frac{N}{k}}\right)\left(1+\frac{1}{5p}\right)
\end{equation}
except for a number of multipartitions $\mu$ which is at most
\begin{equation*}
O\left(\exp\left(-\left(\frac{N}{k}\right)^\frac{1}{15p}\right) p_k(N)\right).
\end{equation*}
\end{prop}
\begin{proof}
For a $k$-multipartition $\mu = (\mu_1, \mu_2, \dots \mu_k)$ of $N$, let $\tilde{\mu}$ be as above. We will bound above the number of $k$-multipartitions $\mu$ such that $\tilde{\mu}$ has largest part less than \eqref{equation: t lower bound}.
For any $\mu$, we know that for some $1 \leq i \leq k$, $\vert \mu_i \vert \geq \frac{N}{k}$.
Fix $i$ such that $\mu_i$ has size $\vert \mu_i \vert = a \geq \frac{N}{k}$. Then Proposition \ref{prop: PSProp2} tells us that the largest part of ${\tilde{\mu}}_i$ exceeds
\begin{equation*}
\frac{\sqrt{6}}{2 \pi} \sqrt{a} \left(\log{a}\right)\left(1+\frac{1}{5p}\right) \geq \frac{\sqrt{6}}{2 \pi} \sqrt{\frac{N}{k}} \left(\log{\frac{N}{k}}\right)\left(1+\frac{1}{5p}\right)
\end{equation*}
except for at most
\begin{equation*}
O\left(p(a) \exp\left(-a^\frac{1}{15p}\right) \right)
\end{equation*}
partitions $\mu_i$ of size $a$ and therefore at most
\begin{equation*}
O\left(p(a) \exp\left(-a^\frac{1}{15p}\right) p_{k-1}(N - a)\right)
\end{equation*}
total $k$-multipartitions $\mu$ with $\vert \mu_i \vert = a$. Furthermore, since $a \geq \frac{N}{k}$,
\begin{equation*}
\exp\left(-a^\frac{1}{15p}\right) \leq \exp\left(-\left(\frac{N}{k}\right)^\frac{1}{15p}\right),
\end{equation*}
and therefore summing over all $ a\geq \frac{N}{k}$ we have that the number of multipartitions $\mu$ such that $\vert \mu_i \vert \geq \frac{N}{k}$ with no part in ${\tilde{\mu}}_i$ exceeding \eqref{equation: t lower bound} is at most an absolute constant times
\begin{align*}
\sum_{a = \frac{N}{k}}^N \exp\left(-a^\frac{1}{15p}\right) p(a)p_{k - 1}(N - a) &\ll \exp\left(-\left(\frac{N}{k}\right)^\frac{1}{15p}\right) \sum_{a = \frac{N}{k}}^N p(a)p_{k - 1}(N - a) \\
&\ll \exp\left(-\left(\frac{N}{k}\right)^\frac{1}{15p}\right) \sum_{a = 0}^N p(a)p_{k - 1}(N - a) \\
&= \exp\left(-\left(\frac{N}{k}\right)^\frac{1}{15p}\right) p_k(N).
\end{align*}
Since this bound is identical for each $i$, the number of $k$-multipartitions $\mu$ such that $\tilde{\mu}$ does not have a part of size greater than \eqref{equation: t lower bound} is at most a factor of $k$ greater than the bound above, and therefore also at most
\begin{equation*}
O\left(\exp\left(-\left(\frac{N}{k}\right)^\frac{1}{15p}\right) p_k(N)\right).\qedhere
\end{equation*}
\end{proof}
\begin{theorem}\label{theorem: main theorem}
Let $G$ be a group with integer-valued character table, and let $G \wr S_N$ be the wreath product of $G$ with the symmetric group $S_N$. For all primes $p$, the proportion of entries in the character table of $G \wr S_N$ divisible by $p$ tends to 1 as $N \to \infty$.
\end{theorem}
\begin{proof}
Let $k$ be the number of conjugacy classes of $G$.
Given a $k$-multipartition $\mu$, let ${\tilde{\mu}}$ be the multipartition obtained by repeatedly replacing $p$ parts of $\mu_i$ of size $m$ with one part of size $mp$ until no $\mu_i$ has a part appearing more than $p-1$ times.
For $A=5p$, Proposition \ref{proposition: mash to the max} implies that the largest part of ${\tilde{\mu}}$ has size
\begin{equation}
\label{equation: final-proof-t-condition}
t \geq \frac{\sqrt{6}}{2\pi} \sqrt{\frac{N}{k}}\left(\log\frac{N}{k}\right)\left(1 + \frac{1}{A}\right)
\end{equation}
for a proportion of $\mu$ tending to 1 as $N \to \infty$.
Now pick $A' \geq 1$ and $\delta > 0$ such that
\[ \left(\log\frac{N}{k}\right)\left(1 + \frac{1}{A}\right) \geq \sqrt{1+\delta} \left(\log\left(\frac{N}{k}(1+\delta)\right)\right) \left(1 + \frac{1}{A'}\right).\]
By Corollary \ref{corollary: concentration of multipartitions}, the proportion of $k$-multipartitions $\lambda = (\lambda_1,\ldots,\lambda_k)\vdash N$ such that $|\lambda_i| \in \left(\frac{N}{k}(1-\delta),\frac{N}{k}(1+\delta)\right)$ for all $i$ tends to 1 as $N \to \infty$.
Thus, consider only $(\lambda,\mu)$ satisfying the above conditions.
Our choice of $\delta$ and $A'$ imply that
\[ \frac{\sqrt{6}}{2\pi} \sqrt{\frac{N}{k}}\left(\log\frac{N}{k}\right)\left(1 + \frac{1}{A}\right)
\geq
\frac{\sqrt{6}}{2\pi}\sqrt{\frac{N(1+\delta)}{k}}\left(\log\left(\frac{N}{k}(1+\delta)\right)\right)\left(1 + \frac{1}{A'}\right)
.\]
If $(N_1,\ldots,N_k)$ is a partition of sufficiently large $N$ with $N_i \in \left(\frac{N}{k}(1-\delta),\frac{N}{k}(1+\delta)\right)$,
we can assume
$A' \leq \frac{\log N_i}{\log \log N_i}$ for each $i$, as $N_i \geq \frac{N}{k(1 - \delta)} \to \infty$.
Then by Proposition \ref{prop: PSProp1}, the proportion of $k$-multipartitions $\lambda$ with $|\lambda_i| = N_i$ such that some $\lambda_i$ is not a $t$-core is
\[ \sum_{i=1}^k O\left(\frac{\log N_i}{N_i^{\frac{1}{2A'}}}\right) \]
for all $t$ satisfying \eqref{equation: final-proof-t-condition}, independent of $t$.
Hence, over all $k$-multipartitions $\lambda$ satisfying $|\lambda_i| \in \left(\frac{N}{k}(1-\delta),\frac{N}{k}(1+\delta)\right)$, the proportion of $\lambda$ such that some $\lambda_i$ is not a $t$-core is
\[ O\left( \frac{\log\left(\frac{N}{k}(1+\delta)\right)}{\left(\frac{N}{k}(1-\delta)\right)^{\frac{1}{2A'}}}\right).\]
It follows that most $(\lambda,\mu)$ satisfy that $\lambda_i$ is a $t$-core for $t$ the largest part of ${\tilde{\mu}}$.
Thus, for a proportion of $(\lambda,\mu)$ tending to 1 as $N \to \infty$, we have $\chi^\lambda_{{\tilde{\mu}}} = 0$ by Proposition \ref{proposition: MN rule} and therefore $\chi^\lambda_\mu \equiv 0\mod p$ by Corollary \ref{corollary: character congruence}.
\end{proof}
\section{Weyl groups of type D}\label{section: extensions}
\begin{defi}
The \emph{Weyl group of type $D_N$} is the group of $N\times N$ signed permutation matrices with an even number of entries equal to $-1$.
\end{defi}
We will denote this group by $D_N$ also (note that it is distinct from the dihedral group).
The study of representations of $D_N$ was taken up by Young in \cite[ยง7-10]{youngfifth}; see \cite{geckpfeiffer00} for a more modern treatment.
Let $B_N = \{\pm 1\} \wr S_N$; then $D_N$ is a subgroup of $B_N$ of index two.
Recall that conjugacy classes of $B_N$ are labelled by $2$-multipartitions $(\eta,\nu)$ of $N$, where we take $\nu$ to be those cycles with nontrivial cycle product.
\begin{prop}[{\cite[Proposition 3.4.12]{geckpfeiffer00}}]
\label{prop: conjugacy-of-DN}
The conjugacy classes $B_N$ which meet $D_N$ correspond to 2-multipartitions $(\eta,\nu)$ of $N$ where $\nu$ has an even number of parts.
The conjugacy classes of $B_N$ which split in $D_N$ are exactly those $(\eta,\nu)$ where $\nu = \varnothing$ and $\eta$ has only even parts.
\end{prop}
Since $D_N$ is a subgroup of $B_N$ of index two, Clifford theory determines its representations:
\begin{prop}[{\cite[\S5.6.1]{geckpfeiffer00}}]
The irreducible representations of the Weyl group of type $D_N$ are as follows:
\begin{enumerate}
\item if $(\lambda,\mu)$ is a 2-multipartition of $N$ such that $\lambda \neq \mu$, then
\[ \Res^{B_N}_{D_N} V^{\lambda,\mu} = \Res^{B_N}_{D_N} V^{\mu,\lambda}\]
is an irreducible representation of $D_N$;
\item if $(\lambda,\lambda)$ is a 2-multipartition of $N$ with equal parts, then \[ Res^{B_N}_{D_N} V^{\lambda,\lambda}\]
is the sum of two irreducible representations of $D_N$.
\item Each irreducible representation of $D_N$ appears exactly once in (1) or (2).
\end{enumerate}
\end{prop}
\begin{coro}\label{corollary: D_n}
For all primes $p$, the proportion of entries in the character table of $D_N$ which are divisible by $p$ tends to 1 as $N \to \infty$.
\end{coro}
\begin{proof}
The number of irreducible representations of $D_N$ of the form $\Res_{D_N}^{B_N} V^{\lambda,\mu}$ for $\lambda \neq \mu$ equals
$\frac{1}{2}\left( p_2(N) - p(N/2)\right)$
when $N$ is even, and $\frac{1}{2}p_2(N)$ when $N$ is odd.
The number of irreducible representations appearing as a summand of $\Res^{B_N}_{D_N} V^{\lambda,\lambda}$ is $2p(N/2)$ when $N$ is even and $0$ when $N$ is odd.
By Claim \ref{claim: asymptotics of multipartitions}, we have $p_2(N) \gg p(N/2)$ for large enough $N$, so the proportion of irreducibles of the form $\Res_{D_N}^{B_N} V^{\lambda,\mu}$ goes to 1 as $N \to \infty$.
We must also analyze the splitting of conjugacy classes from $B_N$ in $D_N$.
By Proposition \ref{prop: conjugacy-of-DN},
the number of conjugacy classes of $B_N$ which split in $D_N$ is
$p(N/2)$ when $N$ is even and $0$ when $N$ is odd.
We conclude just as above that the proportion of non-split conjugacy classes tends to 1 as $N \to \infty$.
We have shown that almost every entry of the character table of $D_N$ is of the form $V^{\lambda,\mu}_{\eta,\nu}$ where $\lambda \neq \mu$ and $(\eta,\nu)$ is a non-split conjugacy class in $D_N$.
Such entries occupy at least a constant fraction of the character table of $B_N$ as $N \to \infty$.
By Theorem \ref{theorem: main theorem},
almost all of such entries are divisible by $p$ as $N \to \infty$.
Hence, the proportion of entries of the character table of $D_N$ which are divisible by $p$ goes to 1 as $N \to\infty$.
\end{proof}
\longthanks{
This work was supported by NSF Grant DMS-2149647, and conducted at the MathILy-EST 2022 REU under the direction of Nate Harman. We thank Sarah Peluse for comments on an earlier version of this paper.
}
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\end{document}