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\usepackage{ytableau}
\usepackage{subfig}
\usepackage{graphicx, caption}
\usepackage{bm}
\usepackage{float}
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\equalenv{corollary}{coro}
\equalenv{definition}{defi}
\equalenv{remark}{rema}
\equalenv{example}{exam}
\equalenv{proposition}{prop}
\title{Stable centres of wreath products}
\author[\initial{C.} Ryba]{\firstname{Christopher} \lastname{Ryba}}
\address{Department of Mathematics, University of California, Berkeley, CA 94720, USA}
\email{ryba@math.berkeley.edu}
\keywords{wreath products, Farahat--Higman algebra, Jucys--Murphy elements}
\subjclass{20C30, 20E22, 16U70}
\newcommand{\bs}{\boldsymbol}
\newcommand{\RGamma}{\mathcal{R}_{\Gamma}}
\newcommand{\LambdaG}{\Lambda(\Gamma_*)}
\newcommand{\FH}{\mathrm{FH}}
\newcommand{\FHG}{\mathrm{FH}_{\Gamma}}
\begin{DefTralics}
\newcommand{\FH}{\mathrm{FH}}
\newcommand{\RGamma}{\mathcal{R}_{\Gamma}}
\newcommand{\LambdaG}{\Lambda(\Gamma_*)}
\newcommand{\FHG}{\mathrm{FH}_{\Gamma}}
\end{DefTralics}
\begin{document}
\begin{abstract}
A result of Farahat and Higman shows that there is a ``universal'' algebra, $\FH$, interpolating the centres of symmetric group algebras, $Z(\mathbb{Z}S_n)$. We explain that this algebra is isomorphic to $\mathcal{R} \otimes \Lambda$, where $\mathcal{R}$ is the ring of integer-valued polynomials and $\Lambda$ is the ring of symmetric functions. Moreover, the isomorphism is via ``evaluation at Jucys--Murphy elements'', which leads to character formulae for symmetric groups. Then, we generalise this result to wreath products $\Gamma \wr S_n$ of a fixed finite group $\Gamma$. This involves constructing wreath-product versions $\RGamma$ and $\LambdaG$ of $\mathcal{R}$ and $\Lambda$, respectively, which are interesting in their own right (for example, both are Hopf algebras). We show that the universal algebra for wreath products, $\FHG$, is isomorphic to $\RGamma \otimes \LambdaG$ and use this to compute the $p$-blocks of wreath products.
\end{abstract}
\maketitle
\section{Introduction}
Let $S_n$ be the $n$-th symmetric group, and $Z(\mathbb{Z}S_n)$ the centre of its group ring over the integers. The centre has a basis consisting of conjugacy class sums. Farahat and Higman \cite{FarahatHigman} showed that the structure constants of the multiplication with respect to this basis are integer-valued polynomials in $n$. For example, multiplying two transposition can give either an element with two cycles of size 2, a 3-cycle, or the identity. If $X_\mu^\prime$ is the sum of all elements of cycle type $\mu$ in $S_n$, so that $X_{(2,1^{n-2})}^\prime$ is the sum of all transpositions, we have
\begin{equation}
(X_{(2,1^{n-2})}^\prime)^2 = 2X_{(2^2,1^{n-4})}^\prime + 3X_{(3,1^{n-3})}^\prime + \binom{n}{2}X_{(1^{n})}^\prime.
\end{equation}
Here, we emphasise that $\binom{n}{2}$ is an integer-valued polynomial. This property of structure constants allowed Farahat and Higman to define an ``interpolating'' algebra, which we denote $\FH$, with coefficients in the ring of integer-valued polynomials, $\mathcal{R}$. See Section \ref{FH_theory_section} for a precise explanation. By construction, there are surjective ``specialisation'' homomorphisms $\FH \to Z(\mathbb{Z}S_n)$ for any $n$. The original motivation for defining the algebra $\FH$ was to give a new proof of Nakayama's Conjecture about $p$-blocks of symmetric groups.
We begin by reviewing the construction of the algebra $\FH$, in particular we explain that it is isomorphic to $\mathcal{R} \otimes \Lambda$, where $\Lambda$ is the ring of symmetric functions. The Jucys--Murphy elements are key in describing this isomorphism, which leads to character formulae for $S_n$. We then fix a finite group $\Gamma$, and generalise the theory to the wreath products $\Gamma \wr S_n$. An algebra analogous to $\FH$ was defined by Wang in \cite{Wang1}, and we denote it $\FHG$. (Wang mostly worked with an associated graded version of $\FHG$ which he used to study Hilbert schemes of points on crepant resolutions of certain plane singularities.) The main result of this paper (Theorem \ref{main_thm}) determines the algebra structure of $\FHG$ via a ``Jucys--Murphy evaluation'' map that implies character formulae for wreath products of the same nature as those for the symmetric~group.
We define two rings, $\RGamma$ and $\LambdaG$, which are, respectively, versions of $\mathcal{R}$ and $\Lambda$ weighted by the conjugacy classes of $\Gamma$. Then we show that $\FHG = \RGamma \otimes \LambdaG$, and mimic the approach of Farahat and Higman to classify the $p$-blocks of $\Gamma \wr S_n$.
In the appendix, we discuss some properties of $\RGamma$ and $\LambdaG$ that are not needed for the main theory, but may be of independent interest. In particular, both $\RGamma$ and $\LambdaG$ are Hopf algebras. The Hopf algebra structure on $\RGamma$ is a consequence of the fact that $\RGamma$ is a certain distribution algebra. We describe homomorphisms from $\RGamma$ to a field; similarly to the case of $\mathcal{R}$, such maps are parametrised by several $p$-adic~numbers.
The paper is organised as follows. In Section~\ref{sec:bg}, we review basic facts about symmetric groups, wreath products, and symmetric functions.
We also briefly summarise the modular representation theory that is necessary for our applications.
In Section~\ref{FH_theory_section}, we review the theory of the Farahat--Higman algebra $\FH$, Jucys--Murphy elements, and explain how $\FH$ is used to prove Nakayama's Conjecture.
In Section~\ref{LambdaG_section}, we define $\LambdaG$, which we call the ring of $\Gamma_*$-weighted symmetric functions.
We discuss the wreath-product Farahat--Higman algebra $\FHG$ and introduce $\RGamma$ in Section~\ref{rgamma_section}.
Then we prove the isomorphism $\FHG = \RGamma \otimes \LambdaG$ in Section~\ref{sec:6}, and use it to study $p$-blocks of wreath products in Section~\ref{applications_section}.
In Appendix~\ref{appendix} we study maps from $\RGamma$ to a field, and show that both $\RGamma$ and $\LambdaG$ are Hopf algebras.
\section{Background}
\label{sec:bg}
\subsection{Symmetric Groups and Wreath Products}
We now introduce the notation and basic properties of symmetric groups and wreath products. Details may be found in \cite[Chapter 1]{Macdonald} (wreath products are discussed in Appendix B).
A \emph{partition}, $\lambda = (\lambda_1, \ldots, \lambda_r)$, is a finite non-increasing sequence of positive integers. The entries $\lambda_i$ are called the \emph{parts} of $\lambda$. It is also common to write $\lambda = (1^{m_1} 2^{m_2} \cdots)$ to mean that $\lambda$ has $m_i$ parts of size $i$ (this information uniquely determines $\lambda$). We will write $m_i(\lambda)$ for the number of parts of $\lambda$ of size $i$. The size of a partition is the sum of its parts: $|\lambda| = \lambda_1 + \cdots + \lambda_r$. If $|\lambda| = n$ it is common to say $\lambda$ is a partition of $n$, and to write $\lambda \vdash n$. The length of a partition is the number of parts: $l(\lambda) = r$. We write $\mathcal{P}$ for the set of partitions of any size.
We depict partitions with \emph{Young diagrams}. The Young diagram of $\lambda$ consists of $l(\lambda)$ rows, where the $i$-th row consists of $\lambda_i$ boxes, left-justified. For example, the Young diagram of the partition $(5,2,1)$ is:
\begin{figure}[H]
\ydiagram{5,2,1}
\end{figure}
The \emph{content} of the box in the $i$-th row from the top and $j$-th column from the left is defined to be $j-i \in \mathbb{Z}$. If the boxes of a Young diagram are labelled (usually with positive integers), we call it a \emph{Young tableau}. If $\lambda$ is a partition of $n$, and a Young diagram is labelled using the numbers $1,\ldots,n$ (each number used once) such that numbers increase in each row from top to bottom, and increase in each column from left to right, then the corresponding Young tableau is called a \emph{standard Young tableau}. Suppose that $\lambda \vdash n$. For a standard Young tableau $T$ of shape $\lambda$, an element $r \in \{1,\ldots,n\}$ labels a unique box of $T$; we write $c_T(r)$ for the content of the box in $T$ labelled $r$. For example, if $T$ is the standard Young tableau of shape $(3,3,1)$ below,
\begin{figure}[H]
\ytableausetup{centertableaux}\begin{ytableau}
1 & 3 & 4 \\
2 & 6 & 7 \\
5
\end{ytableau}
\end{figure}
\noindent
then $c_T(5) = 1-3 = -2$, $c_T(6) = 2-2 = 0$, and $c_T(7) = 3-2 = 1$.
A \emph{border strip} $R$ of $\lambda$ is a subset of the boxes of the Young diagram of $\lambda$ satisfying the following conditions. Firstly, $\lambda \backslash R$ (i.e. the diagram of $\lambda$ with the border strip $R$ removed) should be the Young diagram of a partition. Secondly, the subset $R$ should be contiguous (boxes sharing an edge are considered adjacent, but those sharing only a vertex are not). Finally, $R$ should not contain any $2\times2$ square of boxes. Below are some examples of border strips $R$ for the partition $(4,2,2)$, where in each case $R$ is indicated by the shaded squares:
\begin{figure}[H]
\centering
\ytableausetup{nosmalltableaux}
\begin{ytableau}
*(white) &*(white) &*(gray) &*(gray) \\
*(white) &*(white) \\
*(white) &*(white)
\end{ytableau}
\hspace{10mm}
\begin{ytableau}
*(white) &*(white) &*(white) &*(white) \\
*(white) &*(gray) \\
*(gray) &*(gray)
\end{ytableau}
\hspace{10mm}
\begin{ytableau}
*(white) &*(gray) &*(gray) &*(gray) \\
*(white) &*(gray) \\
*(white) &*(gray)
\end{ytableau}
\end{figure}
If $p$ is a prime number, the \emph{$p$-core} of $\lambda$ is the partition obtained by successively removing border strips of size $p$ from the diagram of $\lambda$ until it is no longer possible to do so. It turns out that the resulting partition is independent of the choice of how to remove border strips of size $p$.
\begin{example} \label{2-core_example}
The $2$-cores of the partitions $(3)$, $(2,1)$, and $(1,1,1)$ are $(1)$, $(2,1)$, and $(1)$ respectively. The diagrams below illustrate this (grey boxes indicate a border strip of size 2 to be removed):
\begin{figure}[H]
\centering
\ytableausetup{nosmalltableaux}
\begin{ytableau}
*(white) &*(gray) &*(gray)
\end{ytableau}
\hspace{10mm}
\begin{ytableau}
*(white) &*(white) \\
*(white)
\end{ytableau}
\hspace{10mm}
\begin{ytableau}
*(white) \\
*(gray) \\
*(gray)
\end{ytableau}
\end{figure}
\end{example}
Consider a border strip $R$ of size $p$, viewed as a sequence of adjacent boxes, starting at the bottom-left-most square, with each subsequent box either above or to the right of the previous one. Then the content of each subsequent box is $1$ larger than the content of the preceding box as either the column index $j$ increases or the row index $i$ decreases. This means that the content of the boxes in $R$ attain each congruence class in $\mathbb{Z}/p\mathbb{Z}$ exactly once. It follows that two partitions of the same size with the same $p$-core have the same multiset of content modulo $p$. In fact, two partitions of the same size have the same multiset of content modulo $p$ if and only if the two partitions have the same $p$-core (e.g. in Example \ref{2-core_example} above). Details can be found in \cite[Examples 8 and 11, Section 1.1]{Macdonald}.
The \emph{$n$-th symmetric group}, $S_n$, is the set of bijections from the set $\{1, \ldots, n\}$ to itself, which is a group with the operation of function composition. The \emph{cycle type} of an element $\sigma \in S_n$ is a partition $\lambda$ such that $m_i(\lambda)$ is the number of cycles of size $i$ in $\sigma$. Two elements of $S_n$ are conjugate if and only if they have the same cycle type, so the conjugacy classes of $S_n$ are in bijection with partitions of size $n$.
Let $\Gamma$ be a finite group. We let $\Gamma_* = \{c_1, \ldots, c_l\}$ be the set of conjugacy classes of $\Gamma$. Sometimes we will write $1$ to denote the identity conjugacy class. We will need to consider the set $\Gamma^*$ of irreducible representations of $\Gamma$ over $\mathbb{C}$ (we could work with any characteristic-zero splitting field of $\Gamma$, but we stick to the complex numbers for concreteness). By abuse of notation, we write $\chi \in \Gamma^*$ to mean both the irreducible representation and its character, for example we may write $\chi(1) = \dim(\chi)$. The centre of the group algebra $\mathbb{Z}\Gamma$ is a free $\mathbb{Z}$-module with a basis consisting of conjugacy-class sums. By further abuse of notation, we use the same symbol to denote a conjugacy class and the sum of its elements. This allows us to write $Z(\mathbb{Z}\Gamma) = \mathbb{Z}\Gamma_*$. In addition, we define the structure constants $A_{i,j}^k \in \mathbb{Z}$ via the following equations in $\mathbb{Z}\Gamma_*$:
\begin{equation} \label{gp_alg_centre_str_consts}
c_i c_j = \sum_k A_{i,j}^k c_k.
\end{equation}
By the Artin--Wedderburn theorem, $\mathbb{C}\Gamma$ is a product of matrix algebras indexed by $\Gamma^*$. Since the centre of a matrix algebra is one dimensional, we have an isomorphism of algebras $\mathbb{C}\Gamma_* = \mathbb{C}^{\Gamma^*}$, where the implied basis of the latter space consists of orthogonal central idempotents in $\mathbb{C}\Gamma_*$.
The $n$-fold product of $\Gamma$ with itself has an action of the symmetric group by permutation of factors. Concretely, if $(g_1, \ldots, g_n) \in \Gamma^n$ and $\sigma \in S_n$, then
\begin{equation} \label{semidirect_mult}
\sigma (g_1, \ldots, g_n) = (g_{\sigma^{-1}(1)}, \ldots, g_{\sigma^{-1}(n)}).
\end{equation}
This defines an automorphism of $\Gamma^n$, and in fact we obtain a homomorphism $S_n \to \mathrm{Aut}(\Gamma^n)$. The corresponding semidirect product $\Gamma^n \rtimes S_n$ is called the \emph{wreath product} of $\Gamma$ with $S_n$, and is denoted $\Gamma \wr S_n$. As a set, $\Gamma \wr S_n$ is equal to $\Gamma^n \times S_n$. We write an element $(g_1, \ldots, g_n, \sigma)$ as $(\mathbf{g}, \sigma)$ where $\mathbf{g} \in \Gamma^n$. Then the group operation is
\[
(\mathbf{g}, \sigma) (\mathbf{h}, \rho) = (\mathbf{g} \sigma(\mathbf{h}), \sigma \rho),
\]
where $\sigma(\mathbf{h})$ has the meaning in Equation \eqref{semidirect_mult}. Both $\Gamma^n$ and $S_n$ are subgroups of $\Gamma \wr S_n$ in the obvious way. If $g \in \Gamma$, we write $g^{(i)} \in \Gamma^n$ for the element whose $i$-th component is $g$, and all other components are the identity. This notation extends linearly to give us an embedding of $\mathbb{Z}\Gamma$ into $\mathbb{Z}\Gamma^n$, so for example we may write $c^{(i)} = \sum_{g \in c} g^{(i)}$ for the conjugacy-class sum $c$ embedded in the $i$-th component of $\mathbb{Z}\Gamma^n$. We also let $\mathbf{g}_i$ be the $i$-th entry in $\mathbf{g} \in \Gamma^n$, so we have the tautological equality
\[
\mathbf{g} = \prod_{i=1}^n \mathbf{g}_i^{(i)}.
\]
This also makes it simpler to write the $S_n$ action:
\[
\sigma(\mathbf{g}) = \prod_{i=1}^n \mathbf{g}_{i}^{(\sigma(i))}.
\]
\begin{example}
An important special case is $\Gamma = C_2$, the cyclic group of order 2. In that case, $C_2 \wr S_n$ is called \emph{the $n$-th hyperoctahedral group}. It arises in Lie theory as the Weyl group of types $B_n$ and $C_n$. We will use the case $\Gamma = C_2$ for examples throughout the paper.
\end{example}
We now describe the conjugacy classes in $\Gamma \wr S_n$. Suppose that $(\mathbf{g}, \sigma) \in \Gamma \wr S_n$, and that $(i_1, \ldots, i_r)$ is a cycle of $\sigma \in S_n$. We also say that it is a cycle of $(\mathbf{g}, \sigma)$, and we define its \emph{type} to be the conjugacy class of $g_{i_r} \cdots g_{i_1}$. Note that $g_{i_1}(g_{i_r} \cdots g_{i_1})g_{i_1}^{-1} = g_{i_1} g_{i_r}\cdots g_{i_2}$, so that $g_{i_r} \cdots g_{i_1}$ and $g_{i_1} g_{i_r} \cdots g_{i_2}$ are conjugate. This means that the two (equal) cycles $(i_1, \ldots, i_r)$ and $(i_2, \ldots, i_r, i_1)$ have the same type, and iterating though all cyclic permutations we see that the type of a cycle is well defined. We record the sizes and types of an element of $\Gamma \wr S_n$ in a multipartition.
\begin{definition}
If $D$ is a set, we say that a \emph{multipartition} indexed by $D$ is a function from $D$ to the set of partitions. We denote the set of multipartitions indexed by $D$ by $\mathcal{P}(D)$. If $D$ is not specified, we take $D = \Gamma_*$. The \emph{size} of a multipartition $\bs \lambda$ is the sum of the sizes of its constituent partitions:
\[
|\bs \lambda| = \sum_{x \in D} |\bs\lambda(x)|.
\]
Similarly, the \emph{length} of a multipartition is the sum of the lengths of its consituent partitions:
\[
l(\bs \lambda) = \sum_{x \in D} l(\bs\lambda(x)).
\]
To express a multipartition we write the juxtaposition of its constituent partitions with the corresponding element of $D$ as a subscript \textup{(}omitting empty partitions for brevity\textup{)}. If $\bs\lambda(y)$ is the empty partition for $y \neq x$, we say that $\bs\lambda$ is \emph{concentrated in type $x$}.
\end{definition}
For example, $(2,1)_{c_1} (1,1,1)_{c_2}$ is the multipartition in $\mathcal{P}(\Gamma_*)$ taking the value $(2,1)$ at $c_1 \in \Gamma_*$ and taking the value $(1,1,1)$ at $c_2 \in \Gamma_*$, while $(3,1)_{c_3}$ is concentrated in type $c_3$.
\begin{definition}
The \emph{cycle type} of an element $(\mathbf{g}, \sigma)$ of $\Gamma \wr S_n$ is the $\Gamma_*$-indexed multipartition $\bs \mu$ such that for each $i \in \mathbb{Z}_{>0}$ and $c \in \Gamma_*$, $m_i(\bs\mu(c))$ is equal to the number of cycles of type $c$ and size $i$ in $(\mathbf{g}, \sigma)$.
\end{definition}
By construction, the size of the cycle type of an element of $\Gamma \wr S_n$ is $n$. Two elements of $\Gamma \wr S_n$ are conjugate if and only if they have the same cycle type. So the conjugacy classes of $\Gamma \wr S_n$ correspond to $\Gamma_*$-indexed multipartitions of $n$.
\begin{lemma} \label{cycle_label_compatibility}
Let $\sigma = (i_1, \ldots, i_r)$ be an $r$-cycle in $S_n$. Let $X_\sigma(c) \in \mathbb{Z}\Gamma \wr S_n$ be the sum of all elements of the form $(\mathbf{g}, \sigma) \in \Gamma \wr S_n$ where the type of the cycle $\sigma$ is $c \in \Gamma_*$, and $\mathbf{g}_i = 1$ for all $i$ not in the cycle. Then for any $i_j$ in the cycle, we have
\[
X_\sigma(c) = c^{(i_j)}X_\sigma(1) = X_\sigma(1) c^{(i_j)},
\]
where $c^{(i)} = \sum_{g \in c} g^{(i)}$ is the conjugacy class sum $c$ embedded in the $i$-th component of $\mathbb{Z}\Gamma^n$.
\end{lemma}
\begin{proof}
The second equality follows from the first because
\[
(\mathbf{g}, \sigma) c^{(i_j)} = (\mathbf{g}c^{(i_{j+1})}, \sigma) = c^{(i_{j+1})}(\mathbf{g}, \sigma),
\]
where $i_{r+1}$ is taken to mean $i_1$ when $j=r$. In the second step we use the fact that $c$ is a central element of $\mathbb{Z}\Gamma^n$. We write
\[
X_\sigma(c) = \sum_{\mathbf{g}_{i_r} \cdots \mathbf{g}_{i_1} \in c} (\mathbf{g}, \sigma)
\]
where $\mathbf{g}_i = 1$ for $i \neq i_1, \ldots, i_r$. The condition $\mathbf{g}_{i_r} \cdots \mathbf{g}_{i_1} \in c$ may be written as
\[
\mathbf{g}_{i_j} \in \mathbf{g}_{i_{j+1}}^{-1} \cdots \mathbf{g}_{i_{r}}^{-1}c\mathbf{g}_{i_1}^{-1} \cdots \mathbf{g}_{i_{j-1}}^{-1} = c (\mathbf{g}_{i_{j+1}}^{-1} \cdots \mathbf{g}_{i_{r}}^{-1}\mathbf{g}_{i_1}^{-1} \cdots \mathbf{g}_{i_{j-1}}^{-1}),
\]
where we have used the fact that $c$ is conjugation invariant to move it to the front of the product. In the case where $c=1$, choosing all the elements other than $\mathbf{g}_{i_j}$ arbitrarily uniquely determines $\mathbf{g}_{i_j}$. In the case of general $c$, we again may choose the elements other than $\mathbf{g}_{i_j}$ arbitrarily, and then $\mathbf{g}_{i_j}$ may be any element of $c$ multiplied by the value of $\mathbf{g}_{i_j}$ from the $c=1$ case.
\end{proof}
\begin{lemma} \label{cycle_label_product}
Let $r~~0}$ is $m_i(\lambda)$.
Let $\Lambda_n$ be the $S_n$-invariants of $\mathbb{Z}[x_1, \ldots, x_n]$, and $\Lambda_n^{k}$ be the degree $k$ component of $\Lambda_n$. For $m > n$ we have a map $\rho_{m,n}: \Lambda_m^k \to \Lambda_n^k$ defined by setting the variables $x_{n+1}, \ldots, x_m$ to zero. For $m > k > n$, we have $\rho_{m,k} \circ \rho_{k,n} = \rho_{m,n}$, and so these $\rho_{m,n}$ define an inverse system on the $\mathbb{Z}$-modules $\Lambda_n^k$ (for fixed $k$ and $n \in \mathbb{Z}_{\geq 0}$), and we take
\[
\Lambda^k = \varprojlim \Lambda_n^{k}.
\]
In explicit terms, $\Lambda^k$ consists of sequences $(q_1, q_2, \ldots)$, where $q_n \in \Lambda_n^k$, that satisfy the condition $\rho_{m,n}(q_m) = q_n$. It is well known that $\rho_{m,n}$ is an isomorphism when $m > n\geq k$, and in that case $\Lambda_n^k$ has a basis of monomial symmetric functions $m_\lambda$ indexed by partitions $\lambda$ of size $k$. Moreover, the monomial symmetric functions with different numbers of variables are stable with respect to the maps $\rho_{m,n}$, i.e.
\[
m_\lambda(x_1, \ldots, x_n) = \rho_{m,n}(m_\lambda(x_1, \ldots, x_m)) = m_{\lambda}(x_1, \ldots, x_n, 0, \ldots, 0).
\]
Therefore the monomial symmetric polynomials define elements of the inverse limit $\Lambda^k$, which are called \emph{monomial symmetric functions} and denoted $m_\lambda$. From this it follows that $\Lambda^k$ is a free $\mathbb{Z}$-module with basis $m_\lambda$ indexed by partitions $\lambda$ of size $k$. Finally, the ring of symmetric functions is
\[
\Lambda = \bigoplus_{k=0}^\infty \Lambda^k,
\]
which is free as a $\mathbb{Z}$-module with basis consisting of the monomial symmetric functions. In fact, $\Lambda$ is a graded ring because the operation of setting a variable $x_m$ to zero respects degree and multiplication. By general properties of inverse limits, we have a canonical ring homomorphism $\Lambda \to \Lambda_n$ by sending $(q_1, q_2, \ldots)$ to $q_n$.
The \emph{elementary symmetric functions} $e_r$ are elements of $\Lambda$ given by $e_r = m_{(1^r)}$. When evaluated in $n$ variables they are
\[
e_r(x_1, \ldots, x_n) = \sum_{i_1 < i_2 < \cdots < i_r} x_{i_1} x_{i_2} \cdots x_{i_r},
\]
namely, the sum of all products of $r$ distinct variables. Often it is convenient to work with generating functions, in which case we have
\[
\sum_{r \geq 0} e_r(x_1, \ldots, x_n) t^r = \prod_{i=1}^n (1 + x_i t).
\]
A key fact is that the elementary symmetric functions generate $\Lambda$ as a free polynomial algebra: $\Lambda = \mathbb{Z}[e_1, e_2, \ldots ]$.
\subsection{Central Characters and Blocks} \label{mod_rep_thy_subsect}
In this subsection, we review the modular representation theory needed for our applications and for the appendix.
The notion of central character is a slight variation of a character.
\begin{definition}
Suppose that $\Gamma$ is a finite group. The \emph{central character} $\omega_c^\chi$ of a conjugacy class $c \in \Gamma_*$ on an irreducible representation $\chi \in \Gamma^*$ is the scalar by which $c \in \mathbb{Z}\Gamma$ acts on $\chi$ \textup{(}central elements always act by scalar multiplication on irreducible representations\textup{)}.
\end{definition}
There is a formula for the central characters in terms of the usual characters:
\[
\omega_c^\chi = \frac{\sum_{g \in c} \chi(g)}{\chi(1)}.
\]
This formula is obtained by taking the trace of $c$, viewed as a linear operator on the irreducible representation, and dividing by the dimension (which is the trace of the identity). Since characters are constant on conjugacy classes,
\[
\omega_c^\chi = \frac{|c|\chi(g)}{\chi(1)},
\]
where $|c|$ is the size of the conjugacy class $c$, and $g$ is any element of $c$.
Central characters satisfy a variety of properties, for example they are always algebraic integers (see \cite[Proposition 5.3.2]{EtingofEtAl}). We will use the fact that the central characters control the blocks of the modular representations of $\Gamma$. We briefly review some of the key definitions and properties of blocks.
For the rest of this section, $\mathbb{F}$ is an algebraically closed field of any characteristic. We still have $Z(\mathbb{F}\Gamma) = \mathbb{F}\Gamma_*$. Recall that the centre $\mathbb{F}\Gamma_*$ acts on a simple $\mathbb{F}\Gamma$-module $M$ by multiplication by scalars; multiplication by an element of the centre commutes with the module action, and hence defines an element of $\mathrm{End}(M)$, which equals $\mathbb{F}$ by Schur's Lemma. This gives a homomorphism $\mathbb{F}\Gamma_* \to \mathbb{F}$, which is also called a central character. When $\mathbb{F} = \mathbb{C}$ and $M = \chi \in \Gamma^*$, $\omega_c^\chi$ is the value of this homomorphism on the conjugacy-class sum $c$.
\begin{definition}
The \emph{blocks} of the group algebra $\mathbb{F}\Gamma$ are the minimal indecomposable two-sided ideals $B_i$ of $\mathbb{F}\Gamma$.
\end{definition}
It is well known that $\mathbb{F}\Gamma$ is the direct sum of its constituent blocks:
\begin{equation} \label{block_sum}
\mathbb{F}\Gamma = \bigoplus_i B_i.
\end{equation}
In particular, the intersection of distinct blocks is zero. Because the blocks are ideals, $B_i B_j \subseteq B_i \cap B_j = 0$ for $i \neq j$. This means that Equation \eqref{block_sum} is a decomposition of $\mathbb{F}$-algebras. So if we express the identity element of $\mathbb{F}\Gamma$ as $1 = \sum_i e_i$ with $e_i \in B_i$, it follows that $e_i$ is the identity element of $B_i$, and moreover $B_i = e_i \mathbb{F}\Gamma e_i$. The upshot of this is that if $M$ is any $\mathbb{F}\Gamma$-module, then
\[
M = \bigoplus_i e_i M.
\]
So if $M$ is indecomposable, $e_i M$ is nonzero for exactly one value of $i$, and for that particular $i$, $e_i M = M$. This makes $M$ into a module for some $B_i$. It is common to say that $M$ belongs to the block $B_i$. As a result, we have a decomposition of module categories
\[
\mathbb{F}\Gamma - \mathrm{mod} = \bigoplus_i \left(B_i -\mathrm{mod}\right),
\]
because any module $M$ splits as a direct sum of modules $e_iM$ belonging to each block $B_i$. We can determine when two simple modules belong to the same block in terms of central characters.
\begin{proposition} \label{central_char_blocks}
Two simple modules for $\mathbb{F}\Gamma$ belong to the same block if and only if every element $Z(\mathbb{F}\Gamma)$ acts on each of them by the same scalar \textup{(}i.e. they have the same central character\textup{)}.
\end{proposition}
\begin{proof}
Suppose that $B_i$ is a block of $\mathbb{F}\Gamma$. \cite[Lemma 4.1]{modrepfingrp} shows that $Z(B_i)/J(Z(B_i)) = \mathbb{F}$, where $J$ is the Jacobson radical. According to \cite{modrepfingrp}, a central character of a block $B_i$ is defined as the composite homomorphism
\[
Z(\mathbb{F}\Gamma) \to Z(B_i) \to Z(B_i)/J(Z(B_i)) = \mathbb{F}.
\]
By definition, the Jacobson radical acts by zero on any simple module, so the action of $Z(\mathbb{F}\Gamma)$ on a simple module factors through the above homomorphism. This shows that this definition of central character is consistent with ours. Moreover, two simple modules in the same block have the same central character. Simple modules in different blocks must have different central characters, because the central idempotents $e_i$ act differently ($e_i$ acts by the identity or zero depending on whether the module belongs to $B_i$ or not).
\end{proof}
There is a procedure for taking a representation of $\Gamma$ in characteristic zero, and producing a representation in positive characteristic. The details are technical, so we only sketch the main idea, directing the interested reader to \cite{modrepfingrp}. We begin with a \emph{$p$-modular system} (here $p$ is a prime), which is a triple $(K, \mathcal{O}, k)$ defined as follows. Firstly, $\mathcal{O}$ is a complete discrete valuation ring. Secondly, $k$ is the residue field of $\mathcal{O}$ which is required to be of characteristic $p$. Thirdly, $K$ is the fraction field of $\mathcal{O}$, which is required to be of characteristic zero. We recall several facts:
\begin{itemize}
\item We may find a $p$-modular system such that every simple representation of $\Gamma$ over either $K$ or $k$ is absolutely irreducible. For example, we may take this $K$ to be a finite extension of the $p$-adic numbers, $\mathbb{Q}_p$, in which the polynomial $x^{|\Gamma|}-1$ splits. Then we may also take $\mathcal{O}$ to be the integral closure of $\mathbb{Z}$ in $K$, which makes $k$ a finite field. (See \cite[Section 3.3]{modrepfingrp}.)
\item The representation theory of $\Gamma$ over $\mathbb{C}$ and $K$ is essentially the same. In either case, irreducible representations are defined over $\mathbb{Q}[\zeta]$, where $\zeta$ is a primitive $|\Gamma|$-th root of unity. Then an identification of $\mathbb{Q}[\zeta]$ as a subring of $\mathbb{C}$ with $\mathbb{Q}[\zeta]$ as a subring of $K$ gives a correspondence of irreducible representations that respects central characters. (This follows from \cite[Theorem 2.7A]{modrepfingrp}.)
\item Given a $K\Gamma$-module $M$, we may find a $\Gamma$-stable free $\mathcal{O}$-submodule $M_{\mathcal{O}}$ such that $KM_{\mathcal{O}} = M$ (i.e. $M_{\mathcal{O}}$ spans $M$ over $K$). Taking the quotient by the maximal ideal of $\mathcal{O}$ gives a $k\Gamma$-module, $M_k =M_{\mathcal{O}} \otimes_{\mathcal{O}} k$. While the isomorphism class of $M_k$ depends on the choice of integral form $M_{\mathcal{O}}$, the composition factors (with multiplicity) of $M_k$ do not. (This is \cite[Theorem 3.6]{modrepfingrp}.)
\end{itemize}
As a result, the operation $M \to M_k$ is well-defined on the level of Grothendieck groups:
\[
d: K_0(K\Gamma-\mathrm{mod}) \to K_0(k\Gamma-\mathrm{mod}).
\]
The map $d$ is called the \emph{decomposition matrix}, and its entries (with respect to the bases coming from simple modules) are called \emph{decomposition numbers}. These definitions turn out to be independent of the choice of $p$-modular system. Decomposition numbers are poorly understood outside of a few special cases, and are an active area of research (one recent breakthrough was the disproof of the \emph{James conjecture} \cite{Williamson} about decomposition numbers for the symmetric groups).
An easy observation from the form of the map $d$, is that since the centre $K\Gamma_*$ acts by scalars on a simple $K\Gamma$-module $M$, any conjugacy class sum $c$ acts by scalars on $M_{\mathcal{O}}$ and also $M_{k}$. This implies that every composition factor of $M_k$ has the same central character. So if $M$ is a simple $K\Gamma$-module, $M_{k}$ belongs to a single block $B_i$. The (necessarily disjoint) subsets of the simple (characteristic zero) modules mapping to a given (characteristic $p$) block are called \emph{$p$-blocks}. The $p$-blocks determine a block-diagonal structure of the decomposition matrix. We will determine the $p$-blocks of $\Gamma \wr S_n$ in terms of the $p$-blocks of $\Gamma$ in Theorem \ref{wreath_nakayama_thm}.
This will use the following tool.
\begin{proposition}[{\cite[Theorem 4.2B]{modrepfingrp}}] \label{block_prop}
Two irreducible representations $\chi_1, \chi_2$ of $K\Gamma$ are in the same $p$-block if and only if
\[
\omega_c^{\chi_1} \equiv \omega_c^{\chi_2} \hspace{5mm} (\mbox{$\mathrm{mod}$ $\pi$}),
\]
for all $c \in \Gamma_*$, where $\pi$ is a uniformiser for $\mathcal{O}$.
\end{proposition}
\begin{proof}
The $p$-block of an irreducible representation $\chi$ is determined by the central character of any composition factor of $(\chi)_k$ by Proposition \ref{central_char_blocks}. But the central character can be computed by taking the central character of $\chi$ and passing to $k = \mathcal{O}/(\pi)$. Since $\Gamma_*$ is a basis for the group algebra of $\Gamma$, the equality of central characters is equivalent to the stated equations.
\end{proof}
Our classification will use the following well-known result.
\begin{proposition} \label{brauer_nesbitt_prop}
Suppose that $|\Gamma| = p^r m$, where $p \nmid m$ and $M$ is a simple $K\Gamma$-module. Then the block $B_i$ to which $M_k$ belongs is semisimple (as a $k$-algebra) if and only if $p^r | \dim_K(M)$, in which case $M_k$ is simple.
\end{proposition}
\begin{proof}
The ``if'' direction is known as the Brauer--Nesbitt theorem (although there are also other results with that name). On the other hand, if $B_i$ is semisimple, then the $B_i$-module $M_k$ is projective for $B_i$ and therefore for $k\Gamma$. If $P$ is a $p$-Sylow subgroup of $\Gamma$, then $M_k$ is a projective $kP$-module. But since $p$-groups only have one irreducible representation in characteristic $p$ (the trivial representation), $kP$ is a (not necessarily commutative) basic local algebra. This implies that a projective $kP$-module is free. In particular, $\dim_k(M_k)$ is a multiple of $\dim_k(kP) = p^r$.
\end{proof}
\section{The Farahat--Higman algebra} \label{FH_theory_section}
Let $X_{\mu}^\prime$ be the sum of all elements of cycle type $\mu$ in $S_n$. These conjugacy-class sums define a $\mathbb{Z}$-basis of $Z(\mathbb{Z}S_n)$ indexed by partitions of $n$. For example
\[
X_{(2,1^{n-2})}^\prime = \sum_{i0}$.
\end{theorem}
Recall that $\Lambda = \mathbb{Z}[e_1, e_2, \ldots]$. It is graded:
\[
\Lambda = \bigoplus_{k \geq 0} \Lambda^k
\]
where $e_r$ has degree $r$. In particular, if $\nu = (\nu_1, \nu_2, \ldots, \nu_l)$ is a partition, then $e_\nu = e_{\nu_1} e_{\nu_2} \cdots e_{\nu_l}$ is in degree $|\nu|$, and a basis of $\Lambda^n$ as a $\mathbb{Z}$-module is given by the set of $e_\nu$ as $\nu$ varies across partitions of $n$. The grading on $\Lambda$ induces a filtration
\[
\Lambda = \bigcup_{i \geq 0} \Lambda^{\leq i},
\]
where
\[
\Lambda^{\leq i} = \bigoplus_{k \leq i} \Lambda^k,
\]
and $\Lambda^{\leq i}$ has $\mathbb{Z}$-basis consisting of $e_\nu$ where $|\nu| \leq i$.
In fact $\FH$ is also a filtered $\mathcal{R}$-algebra,
\[
\FH = \bigcup_{i \geq 0} \mathcal{F}^i
\]
where $\mathcal{F}^i$ is the $\mathcal{R}$-submodule of $\FH$ spanned by $K_\mu$ with $|\mu| \leq i$ (see \cite[Lemma 3.9]{FarahatHigman}). This filtration may be interpreted as follows. A $k$-cycle in a symmetric group may be written as a product of $k-1$ transpositions (and no fewer), for example:
\[
(i_1, i_2, i_3, \ldots, i_k) = (i_1, i_2)(i_2, i_3)\cdots(i_{k-1},i_k).
\]
For an arbitrary permutation of cycle type $\nu = (\nu_1, \nu_2, \ldots, \nu_l)$, the number of transpositions needed is
\[
\sum_i (\nu_i - 1),
\]
which is precisely the size of the corresponding reduced cycle type $(\nu_1-1, \nu_2-1, \ldots, \nu_k-1)$. So $\mathcal{F}^i$ may be seen as filtering permutations according to how many transpositions are needed to construct them.
\begin{theorem} \label{thm:FH_lambda_iso}
There is an isomorphism $\Psi: \mathcal{R} \otimes_{\mathbb{Z}} \Lambda \to \FH$ of filtered $\mathcal{R}$-algebras defined by
\[
\Psi(e_n) = g_n = \sum_{\mu \vdash n} K_\mu.
\]
\end{theorem}
We give two proofs. The first proof is intended to be in the spirit of the original work of Farahat and Higman. The second proof relies on the later work of Jucys, and it is this second proof that will generalise to the wreath-product setting.
\begin{proof}[First Proof of Theorem \ref{thm:FH_lambda_iso}]
The homomorphism $\Psi$ is well defined because $\Lambda$ is a free polynomial algebra, so there are no relations that need to be checked. Since $\Psi(e_i) = g_i \in \mathcal{F}^i$, it is immediate that $\Psi$ respects the filtrations on both spaces. Additionally, Theorem \ref{thm:FH_generation_thm} shows $\Psi$ is surjective. However, the proof in \cite{FarahatHigman} proceeds by showing that \[
\mathcal{F}^i = \mathcal{R}g_i + \sum_{\substack{j+k = i \\ j,k \geq 1}}\mathcal{F}^j \cdot \mathcal{F}^k.
\]
From this, it follows by induction on $i$ that the restriction of $\Psi$ to a map from the $i$-th filtered component of $\Lambda$ to the $i$-th filtered component of $\FH$ is a surjection:
\begin{eqnarray*}
\mathcal{F}^i &=& \mathcal{R}g_i + \sum_{\substack{j+k = i \\ j,k \geq 1}}\mathcal{F}^j \cdot \mathcal{F}^k \\
&=& \mathcal{R}\Psi(e_i) + \sum_{\substack{j+k = i \\ j,k \geq 1}}\Psi(\Lambda^{\leq j}) \cdot \Psi(\Lambda^{\leq k}) \\
&\subseteq& \Psi(\Lambda^{\leq i}).
\end{eqnarray*}
We may pass to the fraction field of $\mathcal{R}$, namely the field of rational functions $\mathbb{Q}(t)$. Now $\Psi$ restricts to a surjection of vector spaces:
\[
\left.\Psi\right|_{\Lambda^{\leq i}}:
\mathbb{Q}(t) \otimes_{\mathbb{Z}} \Lambda^{\leq i} \to \mathbb{Q}(t) \otimes_{\mathcal{R}} \mathcal{F}^i.
\]
Since both spaces have the same dimension (each has a basis indexed by partitions of size at most $i$), standard linear algebra shows that since the linear map $\left.\Psi\right|_{\Lambda^{\leq_i}}$ is surjective, it must also be injective. This in turn implies that $\Psi$ is injective.
\end{proof}
Theorem \ref{thm:FH_lambda_iso} is essentially a strengthening of \cite[Theorem 3.1]{CorteelGoupilSchaeffer}, the difference being that they work rationally (i.e. over $\mathbb{Q} \otimes_{\mathbb{Z}} \mathcal{R} = \mathbb{Q}[t]$), while we work integrally (i.e. over $\mathcal{R}$). This will be essential when we consider modular representation theory in Section~\ref{applications_section}.
At this point it may be unclear why we choose to work with the ring of symmetric functions when the only property we have used is that it is a free polynomial algebra with one generator in each positive degree. In the next subsection, we will summarise the theory of Jucys--Murphy elements, which will allow us to interpret the isomorphism $\Psi$ as ``Jucys--Murphy evaluation'' of symmetric functions. This will give us formulae for central characters. To that end, we will need the following definition.
\begin{definition} \label{char_sym_fun_def}
For a partition $\mu$, the \emph{character symmetric function} is $f_\mu = \Psi^{-1}(K_\mu)$ \textup{(}it is an element of $\mathcal{R} \otimes_{\mathbb{Z}} \Lambda$\textup{)}.
\end{definition}
\begin{example} \label{char_sym_fn_example}
By the definition of $\Psi$, $\Psi(e_1) = K_{(1)}$ and $\Psi(e_2) = K_{(1,1)} + K_{(2)}$. Additionally, Equation \eqref{precise_FH_alg_example} shows that
\[
\Psi(e_1^2) = 2 K_{(1,1)} + 3K_{(2)} + \binom{n}{2}.
\]
From this we conclude that $f_{(1)} = e_1$, $f_{(2)} = e_1^2 - 2e_2 - \binom{n}{2}$, and $f_{(1,1)} = 3e_2 - e_1^2 + \binom{n}{2}$.
\end{example}
One may consult \cite[Section 5.4]{CST} for an alternative exposition of character symmetric functions. Note however, that our indexing variable $\mu$ is a partition corresponding to a reduced cycle type, while some of the literature uses partitions corresponding to cycle types, but with parts of size $1$ removed.
\subsection{Jucys--Murphy Elements} The Jucys--Murphy (``JM'') elements are a key part of what has come to be known as the Okounkov--Vershik approach to the representations of symmetric groups (\cite{OkounkovVershik} and \cite{Kleshchev} are both excellent references), some parts of which we briefly review before explaining the connection to the Farahat--Higman algebra.
The JM elements $L_1, \ldots, L_n \in \mathbb{Z}S_n$ are sums of certain transpositions:
\[
L_m = \sum_{1\leq i < m} (i,m).
\]
For example $L_1 = 0$, $L_2 = (1,2)$, and $L_3 = (1,3)+(2,3)$. It is well known that the JM elements pairwise commute, so if $P(x_1, \ldots, x_n) \in \mathbb{Z}[x_1, \ldots, x_n]$, the expression $P(L_1, \ldots, L_n) \in \mathbb{Z}S_n$ is unambiguous. Furthermore, if $P(x_1, \ldots, x_n)$ is symmetric in the $x_i$, then $P(L_1, \ldots, L_n)$ is known to be a central element of $\mathbb{Z}S_n$.
\begin{proposition}[{\cite[Theorem 1.9]{Murphy}}] \label{murphy_prop}
If $m_\mu$ is the monomial symmetric function, then $m_\mu(L_1, \ldots, L_n)$ is equal to $X_\mu$ plus a linear combination of $X_\nu$ such that either $|\nu| < |\mu|$ or $|\nu| = |\mu|$ and $l(\nu) < l(\mu)$.
\end{proposition}
\begin{proof}
Suppose that $g \in S_n$ has reduced cycle type $\mu$. Then multiplying $g$ by $(i,j)$ either merges two (possibly trivial) cycles if $i$ and $j$ are in distinct cycles of $g$, or splits an individual cycle into two cycles if $i$ and $j$ are in the same cycle of $g$. The merging of two cycles has the effect of increasing the size of the reduced cycle type by 1, while splitting subtracts 1 from the size. Since each JM element is a sum of transpositions, $m_\mu(L_1, \ldots, L_n)$ is a sum of products of $|\mu|$ transpositions. So to compute the leading order term, we only consider products that merge cycles at each step.
We consider $L_j^r$. This is a sum of products
\[
(i_1, j)\cdots (i_r,j)
\]
of $r$ transpositions $(i_k,j)$ where the $i_k$ ($k=1,\ldots,r$) may be any numbers less than $j$. In order for the cycle to grow with each multiplication, it is necessary and sufficient that values of $i_k$ must be distinct. The result of such a product will be the $(r+1)$-cycle
\[
(i_r, i_{r-1}, \ldots, i_1, j).
\]
So we get every $(r+1)$ cycle with largest element $j$ exactly once because the elements of the cycle determine the transpositions involved in the product, and their order in the cycle determines the order of the transpositions in the product.
Finally, we note that $m_\mu(L_1, \ldots, L_n)$ is a sum of products of $L_j^r$ where the exponents $r$ are the parts of $\mu$ in some order. By the previous paragraph, up to leading order, we get a product of $(r+1)$-cycles, and the length of the reduced cycle type is maximised when they do not intersect. In that case we get an element of reduced cycle type $\mu$, and moreover each such element arises exactly once because the length of the cycle whose largest element is $j$ must have been the exponent of $L_j$ in the monomial that gave rise to the cycle in consideration.
\end{proof}
In the case where $\mu = (1^r)$ so that $m_\mu = e_r$, the above argument simplifies; it never happens that a transposition splits a cycle. This makes it possible to keep track of all the resulting permutations; we get the sum of all elements in $S_n$ with $n-r$ cycles.
\begin{theorem}[{\cite[Section 3]{Jucys}}] \label{thm:elem_JM}
If $e_r$ is the $r$-th elementary symmetric function, then
\[
e_r(L_1, \ldots, L_n) = \Phi_n(g_r) = \sum_{\mu \vdash r} X_\mu,
\]
where $X_\mu$ is the sum of elements of reduced cycle type $\mu$ in $S_n$.
\end{theorem}
\begin{remark}
The formulation of Theorem \ref{thm:elem_JM} appearing in \cite[Section 3]{Jucys} states that
\[
e_r(L_1, \ldots, L_n) = \sum_{\substack{\mu \vdash n, \\ l(\mu) = n-r}} X_\mu^\prime.
\]
The partitions $\mu$ correspond to reduced cycle types $\nu$ of size $r$ and length at most $n-r$. So while the sum in Theorem \ref{thm:elem_JM},
\[
\sum_{\nu \vdash r} X_{\nu},
\]
features more summands than the first sum, the summands with $l(\nu) > n-r$ have $|\nu| + l(\nu) > n$ and therefore do not correspond to any permutations in $S_n$ (so the corresponding terms $X_{\nu}$ are zero).
\end{remark}
Now we can give the other proof of Theorem \ref{thm:FH_lambda_iso}.
\begin{proof}[Second Proof of Theorem \ref{thm:FH_lambda_iso}]
By Theorem \ref{thm:elem_JM}, $\Phi_n(\Psi(e_r)) \in Z(\mathbb{Z}S_n)$ may be interpreted as the evaluation of the elementary symmetric function $e_r$ at the Jucys--Murphy elements $L_1, \ldots, L_n$. By Proposition \ref{murphy_prop}, applying the same operation to the monomial symmetric functions $m_\mu$, we get $X_\mu$ plus terms lower in a certain partial order. Since $\Phi_n(\Psi(m_\mu))$ is equal to $X_\mu$ plus lower-order terms, $\Psi(m_\mu)$ is equal to $K_\mu$ plus lower-order terms. Since the $m_\mu$ and $K_\mu$ are $\mathcal{R}$-bases of the respective spaces, this shows that $\Psi$ is an isomorphism.
\end{proof}
In view of Theorems \ref{thm:FH_spec_hom} and \ref{thm:FH_lambda_iso}, we have the following commutative diagram for each $n$,
\begin{equation}
\begin{tikzcd}
\mathcal{R} \otimes_{\mathbb{Z}} \Lambda \arrow[r, "\Psi"] \arrow[dr, "ev_n"] & \FH \arrow[d, "\Phi_n"]\\
& Z(\mathbb{Z}S_n)
\end{tikzcd}
\end{equation}
where $ev_n: \mathcal{R} \otimes_{\mathbb{Z}} \Lambda \to Z(\mathbb{Z}S_n)$ evaluates an integer-valued polynomial at $n$ and evaluates a symmetric function at $L_1, \ldots, L_n$. To see that the diagram commutes, it suffices to check the generating elements $e_r$ of $\Lambda$, which is the statement of Theorem \ref{thm:elem_JM}. This suggests that $\FH$ can be thought of as being ``the ring of symmetric functions evaluated at JM elements''.
Recall that the irreducible representations of $\mathbb{C}S_n$ are the Specht modules $S^\lambda$, indexed by partitions $\lambda$ of size $n$. The Specht module $S^\lambda$ possesses a Gelfand-Zetlin (``GZ'') basis $v_T$ indexed by standard Young tableaux $T$ of shape $\lambda$. A very special property of the GZ basis is that it is diagonal for the action of the JM elements, and the eigenvalues are given by the content of boxes in $T$:
\[
L_i v_T = c_T(i) v_T.
\]
Now suppose that $P(x_1, \ldots, x_n)$ is a symmetric polynomial. Evaluating at the JM elements and applying it to $v_T$, we obtain
\[
P(L_1, \ldots, L_n)v_T = P(c_T(1), \ldots, c_T(n))v_T.
\]
Since $P$ is symmetric, $P(c_T(1), \ldots, c_T(n))$ does not depend on the order of the content of $T$. But for any $T$ of shape $\lambda$, the multiset of $c_T(i)$ is precisely the content of the boxes in $\lambda$, in particular, it is the same for every $T$ of shape $\lambda$. This implies that $P(L_1, \ldots, L_n)$ acts by the same scalar on any $v_T$ of fixed shape $\lambda$. This implies that $P(L_1, \ldots, L_n) \in \mathbb{Z}S_n$ acts by scalar multiplication on each irreducible representation of the symmetric group, and is therefore a central element of $\mathbb{Z}S_n$ as stated earlier.
\begin{theorem} \label{sym_gp_eval_thm}
The central character for $S_n$, $\omega_\mu^\lambda$ \textup{(}$\mu$ defines a conjugacy class via reduced cycle type\textup{)}, is equal to the character symmetric function $f_\mu$ evaluated at the content of the partition $\lambda$, with the integer-valued polynomial variable $t$ evaluated at~$n$.
\end{theorem}
\begin{proof}
By definition $\omega_\mu^\lambda$ is the scalar by which $X_\mu$ acts on $S^\lambda$. To calculate the scalar, we may choose an arbitrary GZ basis vector $v_T$ and act on it by $ev_n(f_\mu) = X_\mu$. But $ev_n$ evaluates elements of $\mathcal{R}$ at $t=n$ and evaluates the symmetric function variables at JM elements, which act on $v_T$ by the contents of $\lambda$.
\end{proof}
\begin{example}
We know from Example \ref{char_sym_fn_example} that $f_{(1)} = e_1$. As $(1)$ is the reduced cycle type of a transposition, we get that sum of all transpositions acts on the Specht module $S^\lambda$ as multiplication by the sum of the contents of the boxes in the Young diagram $\lambda$. For a box $\square$ in the diagram of $\lambda$, let $row(\square)$ and $col(\square)$ be the number of the row and column containing $\square$ respectively. Then the sum of the contents of $\lambda$ is
\[
\sum_{\square \in \lambda} col(\square) - row(\square) = \sum_{\square \in \lambda} col(\square) - \sum_{\square \in \lambda} row(\square) = \sum_i \binom{\lambda_i +1}{2} - \sum_{j} \binom{\lambda_j^\prime +1}{2},
\]
where $\lambda_j^\prime$ are the lengths of the columns in the diagram of $\lambda$ (equivalently, the parts of the partition dual to $\lambda$). Here we have used the identity $1 + 2 + \cdots + m = \binom{m +1}{2}$ to sum each row/column. We have recovered the celebrated Frobenius formula \cite{Frobenius}, see also \cite[Example 7, Section 1.7]{Macdonald}.
\end{example}
We are now able to give a proof of Nakayama's Conjecture, which is a characterisation of the $p$-blocks of $S_n$. Although it was first proved by Brauer and Robinson \cite{BrauerRobinson}, we take the simpler approach of Farahat and Higman.
\begin{theorem}[{\cite{FarahatHigman}}] \label{nakayama_conjecture}
Two irreducible representations of $\mathbb{C}S_n$ are in the same $p$-block if and only if they are labelled by partitions with the same $p$-core.
\end{theorem}
\begin{proof}
The definition of $p$-block makes reference to a $p$-modular system $(K, \mathcal{O}, k)$. Proposition \ref{block_prop} provides a way to determine the blocks in terms of central characters. Note that since $\mathcal{O}/(\pi) = k$ has characteristic $p$, $p \in \mathcal{O}$ is contained in the ideal $(\pi)$. Suppose that $a, b \in \mathbb{Z} \subseteq \mathcal{O}$ are integers viewed as elements of $\mathcal{O}$. Then if $p | a-b$, certainly $\pi | a-b$. However, if $\pi | a-b$, then $a-b$ is an integer divisible by $\pi$. But if $\pi$ divides an integer $m$ coprime to $p$, then by B\'{e}zout's identity, $\pi$ divides $\gcd(m,p)=1$, which is a contradiction as $\pi$ is not a unit. Thus $\pi | a-b$ implies $p | a-b$, and we conclude that the $a$ and $b$ are congruent modulo $\pi$ if and only if the are congruent modulo $p$.
Theorem \ref{sym_gp_eval_thm} shows that the central characters of $S_n$ are integers because the value of an element of $\mathcal{R}$ at $n$ is by definition an integer, and evaluating symmetric polynomials with integer coefficients at integers (contents of $\lambda$) will give integers. So we are left to determine them modulo $\pi$, or equivalently, modulo $p$. The centre of the group algebra of $S_n$ is $ev_n(\mathcal{R} \otimes \Lambda)$. Because $\mathcal{R} \otimes \Lambda$ is generated by $\mathcal{R}$ and the elementary symmetric functions $e_i$, it is enough to consider the action of $ev_n(\mathcal{R})$ and $ev_n(e_i)$ on Specht modules $S^\lambda$. The action of $ev_n(\mathcal{R})$ is independent of the partition $\lambda$. To understand the action of the elementary symmetric polynomials, we assemble them into a generating function:
\[
\sum_i ev_n(e_i) t^i =
\sum_i e_i(L_1, \ldots, L_n)t^i = \prod_{i=1}^n (1 + L_i t),
\]
which acts on a GZ basis vector $v_T$ ($T$ is a standard Young tableau of shape $\lambda$) by the scalar
\[
\prod_{i=1}^n (1 + c_T(i) t).
\]
By the unique factorisation of polynomials in $k[t]$, this generating function determines, and is determined by, the content of $\lambda$ viewed as elements of $k$, i.e. taken modulo $\mathrm{char}(k)=p$. Two irreducibles are in the same $p$-block if and only if they have the same central characters, which holds if and only if they are labelled by partitions with the same content modulo $p$, which holds if and only if the partitions have the same~$p$-core.
\end{proof}
\begin{remark}
Some authors (e.g. \cite{Wang1} and \cite[Example 25, Section 1.7]{Macdonald}) consider the associated graded algebra of $\FH$ (with respect to the filtration defined immediately before Theorem \ref{thm:FH_lambda_iso}). One advantage of this is that the structure constants in the associated graded algebra are integers, rather than arbitrary elements of $\mathcal{R}$, so one may avoid working over $\mathcal{R}$ altogether. One disadvantage is that the maps $ev_n: \FH \to Z(\mathbb{Z}S_n)$ become maps to an associated graded version of the centre of the group algebra (where $X_\mu$ is in degree $|\mu|$). Although this obstructs applications to modular representation theory, it turns out to be the right thing to do in the setting of Hilbert schemes. In fact, this associated graded version of $Z(\mathbb{Z}S_n)$ is isomorphic to $H^*(\mathrm{Hilb}^n(\mathbb{A}_{\mathbb{C}}^2), \mathbb{Z})$, the cohomology ring (with $\mathbb{Z}$ coefficients) of the Hilbert scheme of $n$ points in the plane. This was shown in \cite{LehnSorger}. We will not discuss Hilbert schemes any further in this paper.
\end{remark}
\section{\texorpdfstring{$\Gamma_*$}{Gamma}-Weighted Symmetric Functions} \label{LambdaG_section}
We now generalise the construction of the ring of symmetric functions in a way that incorporates $\Gamma_*$, which will define a ring $\LambdaG$ that will play a central role in this paper. Let $Q$ be the subring of $\mathbb{Z}\Gamma_*[x]$ consisting of polynomials whose constant term is a multiple of the identity (rather than an arbitrary element of $\mathbb{Z}\Gamma_*$). Consider the $n$-th tensor power (over $\mathbb{Z}$) of $Q$, which has an action of $S_n$ by permutation of tensor factors. As shorthand, for $c \in \mathbb{Z}\Gamma_*$ we write
\[
x_i^r(c) = 1^{\otimes (i-1)} \otimes cx^r \otimes 1^{\otimes (n-i)}.
\]
This means that $x_i^r(c) x_i^s(c^\prime) = x_i^{r+s}(cc^\prime)$, and that $x_i^r(c) + x_i^r(c^\prime) = x_i^r(c+c^\prime)$. Since $Q$ has a grading (inherited from $\mathbb{Z}\Gamma_*[x]$), there is a grading on the ring of $S_n$ invariants of $Q^{\otimes n}$. We write $\Lambda_n(\Gamma_*)$ for the $S_n$-invariants of $Q^{\otimes n}$, and $\Lambda_n^k(\Gamma_*)$ for the degree $k$ component of $\Lambda_n(\Gamma_*)$. As before, for $m>n$ there are homomorphisms $\rho_{m,n}: \Lambda_{m}^k(\Gamma_*) \to \Lambda_{n}^k(\Gamma_*)$ which evaluate the polynomial variables of all tensor factors past the $n$-th at zero. These define an inverse system, and we write
\[
\Lambda^k(\Gamma_*) = \varprojlim \Lambda_n^k(\Gamma_*).
\]
\begin{remark}
It may seem unnatural to work with the ring $Q$ rather than the full polynomial ring $\mathbb{Z}\Gamma_*[x]$. The reason we do this is that evaluating elements of $Q$ at zero yields an element of $\mathbb{Z}$ rather than $\mathbb{Z}\Gamma_*$, so the codomain of $\rho_{m,n}$ is
\[
\Lambda_n^k(\Gamma_*) \otimes \mathbb{Z}^{\otimes (m-n)} = \Lambda_n^k(\Gamma_*),
\]
rather than
\[
\Lambda_n^k(\Gamma_*) \otimes (\mathbb{Z}\Gamma_*)^{\otimes (m-n)},
\]
which, being different from $\Lambda_n^k(\Gamma_*)$, would not allow us to construct an inverse system. Later we will account for these ``missing'' constant terms (see Theorem \ref{r_gamma_hom_thm}), using the ring $\RGamma$ which is defined in Section \ref{rgamma_section}.
\end{remark}
The ring $Q$ has a basis consisting of elements of the form $c x^j$ where $c \in \Gamma_*$ and $j \in \mathbb{Z}_{\geq 0}$ (we require $c = 1$ if $j=0$). Thus the ring $Q^{\otimes n}$ has a basis consisting of pure tensors in this basis of $Q$. We refer to such a pure tensor as a \emph{$\Gamma_*$-weighted monomial} and note that the $S_n$ action sends $\Gamma_*$-weighted monomials to other $\Gamma_*$-weighted monomials.
\begin{definition}
Let $\bs \lambda$ be a multipartition indexed by $\Gamma_*$. The \emph{$\Gamma_*$-weighted monomial symmetric polynomial}, $m_{\bs \lambda}(x_1, \ldots, x_n) \in Q^{\otimes n}$ is the sum of all $\Gamma_*$-weighted monomials in $Q^{\otimes n}$ that contain $c x^j$ with $j \geq 1$ as a tensor factor exactly $m_j(\bs \lambda(c))$ times. There is no restriction on the number of times $1$ may appear as a tensor factor.
\end{definition}
It is immediate that the degree of $m_{\bs\lambda}(x_1, \ldots, x_n)$ is $|\bs\lambda|$. Since the $\Gamma_*$-weighted monomial symmetric polynomials are orbit sums for the $S_n$ action on our basis of $Q^{\otimes n}$, it follows that $m_{\bs\lambda}(x_1, \ldots, x_n)$ with $|\bs\lambda|=k$ span $\Lambda_{n}^k(\Gamma_*)$, and the nonzero ones form a basis of this space. Moreover $m_{\bs\lambda}(x_1, \ldots, x_n)$ is nonzero as soon as there are enough tensor factors to accommodate all the basis vectors prescribed by $\bs \lambda$, i.e. as soon as $n \geq l(\bs\lambda)$. Finally, we observe that
\[
\rho_{m,n}(m_{\bs\lambda}(x_1,\ldots, x_m)) = m_{\bs\lambda}(x_1,\ldots, x_n),
\]
so these elements define an element of the inverse limit $\Lambda^k(\Gamma_*)$.
\begin{definition}
The \emph{$\Gamma_*$-weighted monomial symmetric function} $m_{\bs\lambda}$ is the element of $\Lambda^k(\Gamma_*)$ defined by the sequence of elements $m_{\bs\lambda}(x_1, \dots, x_n)$ as $n$ varies \textup{(}here $k = |\bs\lambda|$\textup{)}.
\end{definition}
It now follows that $\Lambda^k(\Gamma_*)$ is free as a $\mathbb{Z}$-module with basis $m_{\bs\lambda}$ indexed by all multipartitions $\bs\lambda$ of size $k$.
\begin{definition}
The ring of $\Gamma_*$-weighted symmetric functions is
\[
\LambdaG = \bigoplus_{k=0}^{\infty} \Lambda^k(\Gamma_*).
\]
\end{definition}
Similarly to the case of ordinary symmetric functions, the fact that evaluating a polynomial variable at zero is a ring homomorphism implies that $\LambdaG$ inherits the structure of a graded ring. In fact, if $\Gamma$ is the trivial group, then $\LambdaG = \Lambda$. Just as for $\Lambda$, formal properties of inverse limits automatically give us ring homomorphisms
\[
\LambdaG \to \Lambda_n(\Gamma_*).
\]
\begin{example} \label{degree_2_example}
We demonstrate how to multiply two degree 1 elements of $\LambdaG$. Suppose that $c_r \in \Gamma_*$ is a conjugacy class. Then $m_{(1)_{c_r}}$ is the element of $\Lambda^1(\Gamma)$ defined by the sequence of elements $\sum_{i=1}^n x_i(c_r) \in \Lambda_n^1(\Gamma_*)$. So we must express products of such elements in terms of $\Gamma_*$-weighted monomial symmetric polynomials $m_{\bs\lambda}(x_1, \ldots, x_n)$. Recall that we place subscripts on partitions to indicate multipartitions. For any number of variables, we have
\begin{eqnarray*}
m_{(1)_{c_r}}^2 &=& \left( \sum_i x_i(c_r) \right)^2 \\
&=& \sum_i x_i^2(c_r^2) + \sum_{i \neq j} x_i(c_r) x_j(c_r) \\
&=& \sum_i \sum_{c_s \in \Gamma_*} A_{r,r}^s x_i^2(c_s) + 2 \sum_{i0}$ and $c \in \Gamma_*$. The counit $\varepsilon$ obeys $\varepsilon(m_{\bs\mu}) = 0$ when $|\bs\mu| > 0$ \textup{(}and $\varepsilon(1) = 1$\textup{)}.
\end{theorem}
\begin{proof}
We have that $(\Delta \otimes 1) \circ \Delta (m_{\bs\lambda}) = (1 \otimes \Delta) \circ \Delta (m_{\bs\lambda})$, both being equal to
\[
\sum_{\bs\mu, \bs\nu,\bs\rho} m_{\bs\mu} \otimes m_{\bs\nu} \otimes m_{\bs\rho},
\]
where the sum is over all multipartitions obeying $m_i(\bs\mu(c)) + m_i(\bs\nu(c)) + m_i(\bs\rho(c)) = m_i(\bs\lambda(c))$. This verifies that $\Delta$ is coassociative. To see that the $\varepsilon$ defines the counit of a bialgebra, we note that
\[
(\varepsilon \otimes 1) \circ \Delta (m_{\bs\lambda}) = \sum_{\bs\mu, \bs\nu} \varepsilon(m_{\bs\mu}) \otimes m_{\bs\nu},
\]
and $\varepsilon(m_{\bs\mu})$ is nonzero only if $m_i(\bs\mu(c)) = 0$ for all $i$ and $c$, in which case $m_{\bs\nu} = m_{\bs\lambda}$. The case of $(1 \otimes \varepsilon) \circ \Delta$ is identical. Finally, we must show there is an antipode $S$. We do not give an explicit formula, but we prove it exists by induction on $|\bs\lambda|$. We take $S(1) = 1$, which serves as the base case $|\bs\lambda| = 0$. For $|\bs\lambda| \geq 1$, we write
\[
\Delta(m_{\bs\lambda}) = m_{\bs\lambda} \otimes 1 + \sum_{\bs\mu,\bs\nu} m_{\bs\mu} \otimes m_{\bs\nu},
\]
where we have written the term where $\bs\nu$ is the empty partition (and $\bs\mu = \bs\lambda$) separately. The antipode axiom becomes
\[
S(m_{\bs\lambda}) + \sum_{\bs\mu, \bs\nu} S(m_{\bs\mu}) m_{\bs\nu} = 0.
\]
But every term in the sum has $|\bs\mu| < |\bs\lambda|$, so this serves to define $S(m_{\bs\lambda})$ inductively. By construction, the resulting map $S$ obeys the antipode axiom.
\end{proof}
\begin{remark}
The Hopf algebra structure on $\LambdaG$ agrees with the usual Hopf algebra structure on $\Lambda$ (see \cite[Section 1.5, Example 25]{Macdonald}) when we set $\Gamma$ to be the trivial group. In fact, it can be shown that the isomorphism
\[
\mathbb{C} \otimes \LambdaG = \mathbb{C} \otimes \Lambda^{\otimes |\Gamma^*|}
\]
from Proposition \ref{complex_bas_change_prop} is an isomorphism of Hopf algebras, where $\Lambda^{\otimes |\Gamma^*|}$ is a tensor product of Hopf algebras, and is therefore a Hopf algebra.
\end{remark}
\begin{remark}
One of the most important tools in the theory of symmetric functions is the Cauchy identity:
\[
\prod_{i,j} \frac{1}{1-x_iy_j} = \sum_{\lambda} s_\lambda(x) s_\lambda(y).
\]
It would be desirable to have a similar identity for $\LambdaG$. However, the most natural formulation of the Cauchy identity is as a formula for the series
\[
\sum_i b_i \otimes b_i^*,
\]
where $b_i$ is a basis of $\Lambda$ and $b_i$ is the dual basis with respect to the Hall inner product on $\Lambda$. The equation above takes both $b_i$ and $b_i^*$ to be the Schur functions $s_\lambda$, which are self dual. For this notion to make sense in $\LambdaG$, we need an inner product on $\LambdaG$. The natural choice, $\langle -,- \rangle$, comes from the isomorphism
\[
\mathbb{C} \otimes \LambdaG = \mathbb{C} \otimes \Lambda^{\otimes |\Gamma^*|},
\]
where each tensor factor of $\Lambda$ is given the Hall inner product. The reason for this choice is that the Hall inner product defines a Hopf pairing on $\Lambda$, and since the Hopf algebra structure on $\LambdaG$ comes from the tensor product of Hopf algebra strutures on $\Lambda$, the tensor product of Hall inner products defines a Hopf pairing on $\mathbb{C} \otimes \LambdaG$. Now we encounter a problem: $\LambdaG$ is not self dual with respect to this inner product: if $b_i$ is a basis of $\LambdaG$, then $b_i^*$ will be contained in
\[
\LambdaG^* = \{f \in \mathbb{C} \otimes \Lambda^{\Gamma_*} \mid \langle f, g \rangle \in \mathbb{Z} \mbox{ for all $g \in \LambdaG$}\},
\]
which is not a subset of $\LambdaG$; roughly speaking it is the ``dual'' of $\LambdaG$, so $\LambdaG^*$ inherits the structure of a Hopf algebra. It would be interesting to give an independent description of $\LambdaG^*$.
\end{remark}
It would also be interesting to have a presentation of $\LambdaG$ by generators and relations. The argument from \cite[Corollary 2.3]{Vaccarino} (filtration/induction on $l(\bs\lambda)$) shows that $\LambdaG$ is generated by $m_{(r^n)_c}$ for $r,n \in \mathbb{Z}_{>0}$ and $c \in \Gamma_*$. However the argument from \cite[Proposition 2.5]{Vaccarino} can be adapted to show that there are redundancies; the generator $m_{(r^n)_c}$ may be omitted if $c$ is an $n$-th power in the ring $\mathbb{Z}\Gamma_*$.
Now we turn our attention to $\RGamma$. Let $R$ be an arbitrary commutative ring, and let us view $R\Gamma_* = \hom_{alg}(A, R)$ as the affine scheme represented by $A = \mathbb{Z}[t_{c_1}, \ldots, t_{c_l}]$, where the polynomial variables are indexed by elements of $\Gamma_*$. Now, $R\Gamma_*$ is a ring whose addition and multiplication maps are algebraic, i.e. the corresponding maps
\[
R\Gamma_* \times R\Gamma_* \to R\Gamma_*
\]
are morphisms of affine schemes. This amounts to saying that $R\Gamma_*$ is a (commutative) ring object in the category of affine schemes. On the level of coordinate rings, we get homomorphisms $\Delta^{(+)}$, $\Delta^{(\times)}$ from $A$ to $A \otimes A$. Explicitly, they are defined by
\[
\Delta^{(+)}(t_c) = t_c \otimes 1 + 1 \otimes t_c
\]
and
\[
\Delta^{(\times)}(t_c) = \sum_{d,e \in \Gamma_*} A_{d,e}^c t_d \otimes t_e.
\]
We call these two maps the \emph{coaddition} and \emph{comultiplication}, respectively. They satisfy relations coming from the associative and distributive laws in a ring. Because of these two maps, a commutative ring object in the category of affine schemes is sometimes called a \emph{biring}, although this is not the same thing as a bialgebra.
Now we let $I$ be the ideal of $A$ vanishing at the multiplicative identity; $I$ is generated by $t_1-1$ and $t_c$ for $c \neq 1$. We have $\Delta^{(\times)}(I) \subseteq I \otimes A + A \otimes I$ (this follows from the fact that $I$ is the ideal corresponding to the identity element of $R\Gamma_*$, but can also be checked directly). We use $I$ to define the distribution algebra of $R\Gamma_*$ (see \cite[Chapter 7 of Part I]{Jantzen_alggp}).
\begin{definition}
The \emph{distribution algebra} of $R\Gamma_*$ is
\[
D(R\Gamma_*) = \bigcup_{n \geq 0} (A/I^n)^*,
\]
where $(A/I^n)^*$ is viewed as a subspace of $A^* = \hom_{\mathbb{Z}}(A, \mathbb{Z})$.
\end{definition}
So $D(R\Gamma_*)$ is the set of linear functionals on $A$ that vanish on some power of $I$. We obtain a multiplication on $D(R\Gamma_*)$ by dualising $\Delta^{(\times)}$. If $f_1, f_2 \in D(R\Gamma_*)$, then we take
\[
(f_1f_2)(a) = (f_1 \otimes f_2)(\Delta^{(\times)}(a))
\]
for $a \in A$. The product $f_1f_2$ vanishes on some power of $I$ because we have
\[
\Delta^{(\times)}(I^n) \subseteq \sum_{m=0}^n I^m \otimes I^{n-m},
\]
and taking $n$ sufficiently large we guarantee that each summand is annihilated by $f_1 \otimes f_2$. Finally, we see that the coassociativity of $\Delta^{(\times)}$ implies the associativity of the multiplication, and the canonical map $A \to A/I = \mathbb{Z}$ is the multiplicative identity. Thus $D(R\Gamma_*)$ is an associative ring.
\begin{theorem} \label{dist_thm}
We have that $D(R\Gamma_*) = \RGamma$.
\end{theorem}
\begin{proof}
Define the variables $z_1 = t_1 - 1$ and $z_c = t_c$ for $c \neq 1$. The coordinate ring $A$ has a basis consisting of monomials $\prod_{c \in \Gamma_*} z_c^{\mathbf{N}(c)}$, indexed by $\mathbf{N} \in \mathbb{Z}_{\geq 0}^{\Gamma_*}$. Such a monomial is contained in $I^{|\mathbf{N}|}$. Hence $D(R\Gamma_*)$ has $\mathbb{Z}$-basis consisting of the dual basis, which we denote $\beta_{\mathbf{N}}$. The structure constants of the multiplication with respect to the basis $\beta_{\mathbf{N}}$ are precisely the structure constants of $\Delta^{(\times)}$ with respect to the monomials $\prod_c z_c^{\mathbf{N}_c}$. To compute the multiplication, note that for any $c \in \Gamma_*$,
\[
\Delta^{(\times)} (z_c) = z_c \otimes 1 + 1 \otimes z_c + \sum_{d,e}A_{d,e}^c z_d \otimes z_e.
\]
We recognise this as the change of variables appearing in Proposition \ref{r_gamma_is_an_algebra_prop} (subject to the relabelling $z_i \otimes 1 = x_i$ and $1 \otimes z_i = y_i$) that describes the structure constants of the multiplication in $\RGamma$. This implies that the linear map $\RGamma \to D(R\Gamma_*)$ given by $B_{\mathbf{N}} \mapsto \beta_{\mathbf{N}}$ is a ring isomorphism.
\end{proof}
It is well known that over a field of characteristic zero, the distribution algebra of an algebraic group $G$ coincides with the universal enveloping algebra of the Lie algebra of $G$. Proposition \ref{r_gamma_is_an_algebra_prop} witnesses this fact; the proof explains that $\mathbb{Q} \otimes \RGamma = \mathcal{U}(\mathbb{Q}\Gamma_*)$. Although $R\Gamma_*$ is a monoid rather than a group, we may consider the group of units, which has same distribution algebra. Since distribution algebras of algebraic groups are Hopf algebras, the same is true of $\RGamma$.
\begin{proposition}
We have that $\RGamma \subseteq \mathcal{U}(\mathbb{Q}\Gamma_*)$ is a Hopf subalgebra.
\end{proposition}
Now we turn our attention to classifying maps into a field. For context and background, we direct the reader to \cite{HarmanHopkins}. We write $\mathbb{Z}_p$ for the set of $p$-adic integers. The motivating example is the following.
\begin{example}[{\cite[Section 8]{HarmanHopkins}}] \label{hom_classification_example}
Homomorphisms from the ring $\mathcal{R}$ of integer-valued polynomials to a field $\mathbb{F}$ are given by the following. If $\mathbb{F}$ has characteristic zero, they are given by evaluating the polynomial variable $t$ at an arbitrary element of $\mathbb{F}$. If $\mathbb{F}$ has characteristic $p>0$, then they are given by evaluating the polynomial variable $t$ at a $p$-adic integer. Evaluation of an integer-valued polynomial at a $p$-adic integer yields another $p$-adic integer, which may be viewed as an element of $\mathbb{F}$ via the canonical map
\[
\mathbb{Z}_p \to \mathbb{Z}_p/(p) = \mathbb{Z}/p\mathbb{Z} \subseteq \mathbb{F}.
\]
In fact, if $t \in \mathbb{Z}_p$, $\binom{t}{p^r}$ is the $r$-th $p$-adic digit of $t$ (viewed as an element of $\mathbb{F}$).
\end{example}
First we get the characteristic-zero case out of the way.
\begin{proposition}
Suppose that $\mathbb{F}$ is a field of characteristic zero. Homomorphisms $\RGamma \to \mathbb{F}$ are given by evaluating each $T(c)$ at an arbitrary element of $\mathbb{F}$ \textup{(}and are hence parametrised by $\mathbb{F}^{\Gamma_*} = \mathbb{F}\Gamma_*$\textup{)}.
\end{proposition}
\begin{proof}
We have $\mathbb{Q} \subseteq \mathbb{F}$, and $\mathbb{Q} \otimes \RGamma = \mathcal{U}(\mathbb{Q}\Gamma_*)$ is the free polynomial algebra in variables $T(c)$ for $c \in \Gamma_*$. Homomorphisms from a polynomial algebra to $\mathbb{F}$ amount to evaluating each variable at an element of $\mathbb{F}$.
\end{proof}
Now we move on to the positive characteristic case. Fix a prime $p$, and let $\mathbb{F}$ be a field of characteristic $p$. Since any field embeds in its algebraic closure, we may as well take $\mathbb{F}$ to be algebraically closed.
\begin{definition}
Let $q \in \mathbb{F}\Gamma_*$ be $q = \sum_{c \in \Gamma_*} m_c c$. We define
\[
B_{q,r} = \sum_{|\mathbf{M}| = p^r} \prod_{c \in \Gamma_*} m_c^{\mathbf{M}(c)} B_{\mathbf{M}}.
\]
\end{definition}
For example, if $q=c \in \Gamma_*$, we have
\[
B_{c,r} = B_{\mathbf{M}}
\]
where $\mathbf{M}(c) = p^r$ and $\mathbf{M}(c^\prime) = 0$ for $c^\prime \neq c$. Also, $B_{0,r} = 0$ for all $r$.
\begin{lemma} \label{rgamma_modular_generation_lemma}
The algebra $\mathbb{F} \otimes \RGamma$ is generated by the elements $B_{q,r}$, where $q$ varies across a basis of $\mathbb{F}\Gamma_*$ and $r \in \mathbb{Z}_{\geq 0}$,
\end{lemma}
\begin{proof}
Lemma \ref{leading_term_lemma} asserts that $B_{\mathbf{M}}$ has leading order term
\[
\prod_{c \in \Gamma_*} \frac{T(c)^{\mathbf{M}(c)}}{\mathbf{M}(c)!}.
\]
Writing $q = \sum_{c \in \Gamma_*} m_c c$ and using the multinomial theorem, we see that the leading order term of $B_{q,r}$ is
\[
\sum_{|\mathbf{M}| = p^r} \prod_{c \in \Gamma_*} m_c^{\mathbf{M}(c)} \frac{T(c)^{\mathbf{M}(c)}}{\mathbf{M}(c)!} = \frac{T(q)^{p^r}}{p^r!}.
\]
If the $p$-adic decomposition of $n$ is $\sum_r n_r p^r$, then over $\mathbb{F}$ we have
\[
\prod_r \left(\frac{T(q)^{p^r}}{p^r!}\right)^{n_r} = \frac{n!}{\prod_{r} (p^r!)^{n_r}} \cdot \frac{T(q)^n}{n!}.
\]
Here, the multinomial coefficient
\[
\frac{n!}{\prod_{r} (p^r!)^{n_r}}
\]
is invertible in $\mathbb{F}$. This implies that the leading order term of any $B_{\mathbf{M}}$ is a (scalar multiple of a) product of leading order terms of the $B_{c,r}$, where we are using the particular basis of conjugacy class sums, proving the lemma for that particular basis. To prove the lemma for an arbitrary basis $q_i$, we express $B_{c,r}$ in terms of the $B_{q_i,r}$. Let $c = \sum_i k_i q_i$, and observe that to leading order, $B_{c,r}$ may be written
\[
\frac{T(\sum_i k_i q_i)^{p^r}}{p^r!} = \sum_{|\mathbf{W}| = p^r} \prod_{i} k_i^{\mathbf{W}(q_i)} \frac{T(q_i)^{\mathbf{W}(q_i)}}{\mathbf{W}(q_i)!},
\]
where the sum is over functions $\mathbf{W}$ from the basis $q_i$ to $\mathbb{Z}_{\geq 0}$, such that $\sum_i \mathbf{W}(q_i) = p^r$. We observe that this is equal to
\[
\sum_i k_i^{p^r} \frac{T(q_i)^{p^r}}{p^r!}
\]
plus ``mixed'' terms whose exponents (values of $\mathbf{W}(q)$) are less than $p^r$ and may therefore be written in terms of $B_{q,s}$ for $s ~~