%%% 2019-5-31:  Fixing typos
%%% 2019-5-31: Added previously omitted Theorem after Definition 3.3 on page 5
%%% 2019-5-31: removed powers of `t' in formula for H[L^S] on bottom of  page 6; reworded Definition 3.1 on page 4.
%%% 2019-6-1: Corrected mistyped citations in proof of Prop 3.1, page 8.
%%% Theorem 3.10 on page 11; last sentence of Introduction.
%%%
%%%
%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[revision]{FPSAC2019}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

\articlenumber{49}
\addbibresource{49.bib}

%% note that the class file already loads {amsmath, amsthm, amssymb}

\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}{Lemma}[section]
\newtheorem{defn}{Definition}[section]
\newtheorem{cor}{Corollary}[section]
\newtheorem{prop}{Proposition}[section]
\newtheorem{conj}{Conjecture}
\newtheorem{rk}{Remark}[section]

\usepackage{lipsum}

\usepackage{comment}

%% define your title in the usual way
%\title[]{Schur positivity in the character table of $S_n$}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%\author[Optional shorter names]{Longer names me%\thanks{\href{mailto:hello@world.c}{hello@world.c}. Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}\addressmark{1}, \and you\addressmark{2}}

%% then use \addressmark to match authors to institutions here
%\address{\addressmark{1}Department of This, University of That, There \\ \addressmark{2}School of Something, University of Something Else, Somewhere}
\title[Schur positivity of sums of power sums]{On the Schur positivity of sums of power sums}

\author{Sheila Sundaram\thanks{\href{mailto:shsund@comcast.net}{shsund@comcast.net}}}
\address{Pierrepont School, One Sylvan Road North, Westport, CT 06880}
%% put the date of submission here
\received{14 Nov 2018}

%% leave this blank until submitting a revised version
\revised{\today}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{Let $T$ be a nonempty subset of positive integers and $p_n$ the $n$th power sum symmetric function. Consider the multiplicity-free sum of power sums $F_n^T=\sum_{\lambda\vdash n} p_\lambda,$ where the sum ranges over all partitions of $n$ with parts in the set $T.$  We define a new symmetric function $f_T$ and give two descriptions of the (possibly virtual)  symmetric group representation associated to the series $\prod_{n\in T} (1-p_n)^{-1}=\sum_{n\geq 0}F_n^T:$  one in terms of the Lie representation, and another as the  symmetric or exterior power of (again possibly virtual)  modules induced from centralisers of the symmetric group. 

When $T=\{1\},$ the degree $n$ term of $f_T$ reduces to the Frobenius characteristic of the Lie representation $Lie_n.$ At the other extreme, when $T$ is the set of all positive integers, it is  the conjugacy action of $S_n.$  
The function $f_T$ allows us to unify previous results on the Schur positivity of multiplicity-free sums of power sums, as well as investigate new ones.  We also uncover some curious plethystic relationships between the conjugacy action and the Lie representation.

Finally we establish some special cases of an earlier conjecture of this author on the Schur positivity of sums of power sums in the intervals $[(1^n), \mu]$ in reverse lexicographic order.
}
%, namely when $\mu\in [(n-3,3), (n)]$ or when $\mu\in [(1^n), (3,1^{n-3})].$ }

%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
%\keywords{math, maths, mathematics}
\keywords{Schur positivity, plethysm, Lie representation, conjugacy action, symmetric and exterior powers}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations, ie. [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
%\usepackage[backend=bibtex]{biblatex}
%\addbibresource{ExtAbsFormattedfpsac2019FINAL.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

%\bibliography{ExtAbsFormattedfpsac2019FINAL}
%\bibliographystyle{amsplain.bst}
\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

\section{Introduction}

In this paper we investigate the positivity of the row sums in the character table of $S_n.$  
%Our view of the character table has the rows indexed by the irreducible characters of $S_n$, and the columns indexed by its conjugacy classes.  Hence rows and columns are both indexed by integer partitions $\lambda$ of $n.$ 
For each irreducible character $\chi^\lambda$ indexed by a partition $\lambda$ of $n,$ 
and any subset $T$ of the conjugacy classes, one can form the sum $\sum_{\mu\in T} \chi^\lambda(\mu),$ and ask when this sum is nonnegative.   
In the language of symmetric functions, one asks for what subsets $T$ of partitions of $n$ the sum of power sums $\sum_{\mu\in T} p_\mu $ is the Frobenius characteristic of a true representation of $S_n,$ i.e. a symmetric function with nonnegative integer coefficients in the basis of Schur functions.  
A  method for generating such  classes of subsets $T$ was presented in \cite{Su1}.  %This paper  generalises and extends these results with classes indexed by subsets of primes (Theorems 3.6 and 3.9).  The approach in Section 3 is to define certain $S_n$-modules $L_n^S$ induced from the cyclic subgroup  generated by an $n$-cycle, indexed by subsets $S$ of primes, and show that their symmetric powers 
%can be expressed as the product $\prod_{n\in P( S)} (1-p_n)^{-1},$  for an appropriate subset $P(S)$ determined by $S.$  The interesting choices of $S$ other than $Lie^{(2)}$ are recorded in Theorem \ref{OnePrime} and Corollaries \ref{Spositivity1} and \ref{Spositivity2}.

 We present a new approach to the Schur positivity problem for power sums.  Specifically, we give a general formula which expresses the product $\prod_{n\in T} (1-p_n)^{-1}$ as a symmetrised module over a sequence of  possibly virtual representations $f_n^T,$ having the specific property that their characters vanish unless the conjugacy class has all cycles of equal length.  The goal then is to determine for what choices of the set $T$ the $f_n^T$ are true 
$S_n$-modules, thereby establishing the Schur positivity of the 
product $\prod_{n\in T} (1-p_n)^{-1}.$    The module $Lie_n,$ the $S_n$-module afforded by the multilinear component of the free Lie algebra on $n$ generators, plays a prominent role in the construction.
 In particular we describe simple relationships (see \cref{prop6.6} and equations \eqref{prop6.6eqn1} and \eqref{prop6.6eqn4} with $q=2$) between  $Lie_n,$  the conjugacy action, and the variant $Lie_n^{(2)}$. This variant was the subject of \cite{SuFPSAC18}, and was shown to have remarkable properties in  \cite{Su3}.
 
In the course of these calculations many interesting plethystic identities emerge, as well as many  new conjectures on Schur positivity. 

\section{Preliminaries}\label{sec:prelims}

Recall \cite{R} that the $S_n$-module $Lie_n$ is the action of $S_n$ on the multilinear component of the free Lie algebra, and coincides with the  induced representation $\exp(\frac{2i\pi}{n})\uparrow_{C_n}^{S_n},$ where $C_n$ is the cyclic group generated by an $n$-cycle in $S_n$.  

Another module that will be of interest is the $S_n$-module $Conj_n$ afforded by the conjugacy action of $S_n$ on the class of $n$-cycles.  Clearly we have $Conj_n\simeq {\mathbf 1}\uparrow_{C_n}^{S_n}.$

We follow \cite{M} and \cite{St4EC2} for notation regarding symmetric functions.  In particular, $h_n,$ $e_n$ and $p_n$ denote respectively the complete homogeneous, elementary and power sum symmetric functions.  If $\mathrm{ch}$ is the Frobenius characteristic map from the representation ring of the symmetric group $S_n$ to the ring of symmetric functions with real coefficients, then 
$h_n=\mathrm{ch}(1_{S_n})$ is the characteristic of the trivial representation, and $e_n=\mathrm{ch}({\rm sgn}_{S_n})$ is the characteristic of the sign representation of $S_n.$   If $\mu$ is a partition of $n$ then 
define $p_\mu =\prod_i p_{\mu_i};$ $h_\mu$ and $e_\mu$ are defined multiplicatively in analogous fashion.  As in \cite{M}, 
the Schur function $s_\mu$  indexed by the partition $\mu$ is the Frobenius characteristic of the $S_n$-irreducible indexed by $\mu.$  
%%%%%%%
  Finally, $\omega$ is the involution on the ring of symmetric functions which takes $h_n$ to $e_n,$ corresponding to tensoring with the sign representation.
  
  By a slight abuse of notation we will also write $Lie_n$ (resp. $Conj_n$) to mean the Frobenius characteristic of the representation $Lie_n$ (resp. $Conj_n$).  Let $\mu(d)$ denote the number-theoretic M\"obius function, and $\phi(d)$ the Euler totient function.  The following facts are well known (see \cite{R}). 
  
\begin{align}\label{LieConj}Lie_n={\rm \, ch\,} \exp(\frac{2i\pi}{n})\large\uparrow_{C_n}^{S_n} &= \frac{1}{n}\sum_{d|n} \mu(d) p_d^{\frac{n}{d}};\\
Conj_n={\rm \, ch\,} \mathbf{1}\large\uparrow_{C_n}^{S_n}&=\frac{1}{n}\sum_{d|n} \phi(d) p_d^{\frac{n}{d}}
\end{align}

If $q$ and $r$ are characteristics of representations of $S_m$ and $S_n$ respectively, they yield a representation of the wreath product $S_m[S_n]$ in a natural way, with the property that when this representation is induced up to $S_{mn},$ its Frobenius characteristic is the plethysm $q[r].$ For more background about this operation, see \cite{M}.  We will make extensive use of the properties of this operation, in particular the fact that plethysm with a symmetric function $r$ is an endomorphism on the ring of symmetric functions \cite[(8.3)]{M}.  See also \cite[Chapter 7, Appendix 2, A2.6]{St4EC2}.
%%%%%%%%
\begin{comment}
Letting $z_\mu$ denote the size of the centraliser in $S_n$ of an element of cycle type $\mu,$ we have that 
$z_\mu^{-1} p_\mu$ is the Frobenius characteristic of the class function whose value is 1 on the conjugacy class of type $\mu$ and zero on all other classes. 
%%%%%%%%%%
In this paper we study representations of $S_n$ induced from centralisers.  The centraliser of an   element with cycle-decomposition corresponding to the partition $\mu$ of $n$ is a direct product of wreath products $\times_i S_{m_i}[C_i],$ where $\mu$ has $m_i$ parts equal to $i$  and $C_i$ is a cyclic subgroup of order $i$ (generated by an $i$-cycle). Note that the centraliser has cardinality $z_\mu=\prod_i i^{m_i} (m_i!).$  As indicated in the preceding paragraph, we will make extensive use  of properties of the  plethysm operation.  
\end{comment}
%%%%%%%%
Define the symmetric functions \begin{align}\label{defHE} %&H(t)=\sum_{i\geq 0}t^i  h_i, \quad E(t) = \sum_{i\geq 0} t^i  e_i; \\
&H=\sum_{i\geq 0}  h_i, \quad E= \sum_{i\geq 0}  e_i; %\quad H^{\pm}=\sum_{r\geq 0} (-1)^r h_r, \quad E^{\pm}=\sum_{r\geq 0} (-1)^r e_r. 
\\
&Lie=\sum_{i\geq 1} Lie_i; \quad Conj=\sum_{i\geq 1}Conj_i.
\end{align} 

We collect some of the key tools used to establish our results.  %The first is the meta-theorem of \cite{Su1}:

Let $\psi(n)$ be any real-valued function defined on the positive integers. 
Define symmetric functions $f_n$ by 
$f_n = \dfrac{1}{n} \sum_{d|n} \psi(d) p_d^{\frac{n}{d}},$
% \begin{equation}\label{definef_n}f_n = \dfrac{1}{n} \sum_{d|n} \psi(d) p_d^{\frac{n}{d}},\end{equation}
%so that \begin{equation}  \omega(f_n) =  \dfrac{1}{n} \sum_{d|n} \psi(d) (-1)^{n-\frac{n}{d}} p_d^{\frac{n}{d}}.\end{equation}
%Note that,  when $\psi(1)$ is a positive integer,  this makes $f_n$ the Frobenius characteristic of a possibly virtual $S_n$-module whose dimension is $(n-1)!\psi(1).$
and the associated polynomial in one variable, $t,$ by 
%
$f_n(t) =\dfrac{1}{n} \sum_{d|n} \psi(d) t^{\frac{n}{d}}.$
%Also define the associated polynomial in one variable, $t,$ by
%\begin{equation}\label{definepolyf_n}
%f_n(t) =\dfrac{1}{n} \sum_{d|n} \psi(d) t^{\frac{n}{d}}.
%\end{equation}

%Theorem~\ref{thm6.4} is proved by invoking the following result.
\begin{thm}\label{metathm} \cite[Theorem 3.2]{Su1} Let $F=\sum_{n\geq 1}  f_n$ where $f_n$ is of the form described above, $H(v)=\sum_{n\geq 0} v^n h_n$ and 
$E(v)=\sum_{n\geq 0}  v^n e_n.$  
We have the following plethystic generating functions:
%\begin{enumerate}
%\item
%
% (Symmetric powers) 
\begin{equation}\label{metaSym}H(v)[F] 
%= \sum_{\lambda\in Par} v^{\ell(\lambda)}  H_\lambda[F]
=\prod_{m\geq 1} (1-p_m)^ {-f_m(v)}\end{equation}
%\item 
%(Exterior powers) 
\begin{equation}\label{metaExt}E(v)[F] 
%=\sum_{\lambda\in Par} v^{\ell(\lambda)} E_\lambda[F] 
=\prod_{m\geq 1} (1- p_m)^{f_m(-v)}\end{equation}
\end{thm}

The plethystic formulas in this abstract are also consequences of the following propositions:

\begin{prop}\label{Pleth5} Let $F=\sum_{n\geq 1} f_n,$  $G=1+\sum_{n\geq 1} g_n$  be formal series of symmetric functions, as usual with $f_n, g_n$ being of homogeneous degree $n.$
Then
$$H[F]=G\iff E[F]=\dfrac{G}{G[p_2]}=\left(\dfrac{p_1}{p_2}\right)[G].$$
%
%\noindent
Hence the exterior power of the series $F$ may also be obtained as the symmetric power of another series, namely, the plethysm
\begin{center} $H[ \dfrac{F}{F[p_2]}] =E[F]=\dfrac{H[F]}{H[F[p_2]]}$ \end{center}
In particular if $F$ is Schur-positive, then so is $\dfrac{G}{G[p_2]}.$

\end{prop}

\begin{prop}\label{Pleth8} \cite{Su3}  The following pairs are plethystic inverses:
%\begin{equation}\label{Pleth8a} \dfrac{p_1}{1+p_1} {\rm\ and\ } \dfrac{p_1}{1-p_1} ;\end{equation}
%\begin{equation}\label{Pleth8b}\sum_{n\geq 1} g(n) p_n {\rm\ and\ }\sum_{n \geq 1}g(n) \mu(n) p_n, \end{equation}%k=0,\pm 1
%for any function $g(n)$ defined on the positive integers, such that $g(mn)=g(m)g(n);$
\begin{equation}\label{Pleth8c}\sum_{n\text{ odd}} g(n) p_n {\rm\ and\ }\sum_{n\text{ odd}} g(n) \mu(n) p_n, \end{equation}
for any function $g(n)$ defined on the positive integers, such that $g(mn)=g(m)g(n);$
\begin{equation} \label{Pleth8d}
\sum_{n\geq 1} Lie_n  {\rm\ and\ }\dfrac{H-1}{H}=\sum_{n\geq 1} (-1)^{n-1} e_n.
\end{equation}
\begin{equation} \label{Pleth8f}
\sum_{n\geq 1} (-1)^{n-1}\omega(Lie_n ) {\rm\ and\ }H-1
\end{equation}
%\begin{equation} \label{Pleth8e}
%\sum_{n\geq 1} Lie_n^{(2)} {\rm\ and\ } \dfrac{E-1}{E}=\sum_{n\geq 1} (-1)^{n-1} h_n.
%\end{equation}
%and hence $\omega(Lie_n^{\langle-1\rangle})
%=(Lie_n^{(2)})^{\langle-1\rangle}.$
\end{prop}
\section{A formula for $\prod_{n\in T} (1-p_n)^{-1}$} 

In this section we explore, for a fixed subset $T$ of positive integers,   the sum of power sums resulting from the product 
$\prod_{n\in T} (1-p_n)^{-1}.$
(This product is 1 if $T$ is the empty set.)

\begin{defn}\label{def6.1} Fix a nonempty subset $T$ of the positive integers.  Define, on the set of positive integers,  a function $\psi^T$  by $\psi^T(d)=\sum_{m|d,\, m\in T} m\,\mu\left(\frac{d}{m}\right).$
\end{defn}
\begin{defn}\label{def6.2} For each nonempty subset $T$ of positive integers, 
 define a sequence of (possibly virtual) representations  indexed by the subset $T,$ with Frobenius characteristic 
$$
f_n^T=\dfrac{1}{n}\sum_{d|n} \psi^T(d) p_d^{\frac{n}{d}}.$$
Set $F^T=\sum_{n\geq 1} f_n^T.$  Finally let 
 $p^T=\sum_{n\in T} p_n.$
\end{defn}

These definitions imply:
\begin{lem}\label{lem6.3} $f_n^T(1)=1$ if and only if $n\in T,$ and $f_n^T(1)=0$ otherwise.
\end{lem}
\begin{proof} Let us write $\delta(m\in T)$ for the indicator function of the set $T,$ so that $\delta(m\in T)=1$  if $m\in T,$ and is zero otherwise.

By definition of $\psi^T$, we have 
$$\psi^T(n)=\sum_{d|n}\mu(\frac{n}{d})\ d\,\delta(d\in T).$$
Hence M\"obius inversion gives 
$$n\delta(n\in T)=\sum_{d|n}\psi^T(d)
=n\, f_n^T(1),$$
i.e. $f_n^T(1)=\delta(n\in T)$ as claimed.
\end{proof}

With this lemma and \cref{metathm} in \cref{sec:prelims},  we can prove:
\begin{thm}\label{thm6.4} Let $T$ be a nonempty subset of the positive integers.  Then:
\begin{equation}\label{thm6.4eqn1} H[F^T]=\prod_{n\in T} (1-p_n)^{-1}
\end{equation}
%$$\prod_{n\in T} (1-p_n)^{-1}=H[F^T]$$ if and only if 
\begin{equation}\label{thm6.4eqn2} F^T=p^T[Lie]=\sum_{m\in T} Lie[p_m],  \ \mathrm{or\ equivalently\ } 
f_n^T=\sum_{\stackrel{m\in T}{m|n}} Lie_{\frac{n}{m}}[p_m].
\end{equation}
If $G^T=\sum_{k\geq 0}\sum_{m\in T} Lie[p_{m\cdot 2^k}],$ then 
\begin{equation}\label{thm6.4eqn3}
E[G^T]=\prod_{n\in T} (1-p_n)^{-1}=H[F^T].
\end{equation}
\end{thm}
\begin{comment}
\begin{proof} 
Equation (\ref{thm6.4eqn1}) is immediate from \cref{metathm} and \cref{lem6.3}. 
 From \cref{Su1Prop3.1}, we have 
\begin{align*}\exp (F^T)&= \prod_{d\geq 1} (1-p_d)^{-\frac{1}{d}\psi^T(d)} =\prod_{d\geq 1} (1-p_d)^{-\sum_{m|d,\, m\in T} \frac{m}{d}\mu(\frac{d}{m})} \\
&=\prod_{d\geq 1} \prod_{m|d,\, m\in T} (1-p_d)^{- \frac{m}{d}\mu(\frac{d}{m})}
=\prod_{ m\in T}\prod_{r\geq 1}(1-p_{rm})^{- \frac{1}{r}\mu(r)}, \mathrm{\ putting\ }d=rm\\
&=\prod_{ m\in T}\prod_{r\geq 1}(1-p_{r})^{- \frac{1}{r}\mu(r)}[p_m]
=\prod_{ m\in T} \exp(Lie[p_m] )=\exp\sum_{m\in T} Lie[p_m],
\end{align*}
where we have used the first equation in \cref{Su1Prop3.1} for $F=Lie.$  Equation (\ref{thm6.4eqn2}) follows.
Equation (\ref{thm6.4eqn3})  is a consequence of \cref{Pleth3}.
\end{proof}
\end{comment}

\begin{cor}\label{cor6.5} If either $F^T$ or $G^T$ is Schur-positive, then so is 
$$\prod_{n\in T} (1-p_n)^{-1}=\sum_{\stackrel{\lambda\in Par}{\lambda_i\in T}}p_\lambda.$$
\end{cor}

Recall from \cite{SuFPSAC18} the following definitions:

\begin{defn} Let $S=\{q_1, \ldots, q_k,\ldots\}$ be a set of distinct primes.  Every positive integer $n$ factors uniquely into $n=Q_n\ell_n$ where $Q_n= \prod_{q\in S} q^{a_q(n)} $ for nonnegative integers $a_q(n),$ and $(\ell_n, q)=1$ for all $q\in S. $  We associate to the set $S$ two symmetric functions, defined as follows.  For each $n\geq 1:$

\begin{equation}\label{LnS} 
L_n^S=\frac{1}{n}\sum_{d|n} \psi(d) p_d^{\frac{n}{d}} \quad 
\text{ with } \psi(d)=\phi(Q_d)\mu(\ell_d), \text{ and}
\end{equation}
\begin{equation} \label{LnbarS}
L_n^{\bar S}=\frac{1}{n}\sum_{d|n} \bar\psi(d) p_d^{\frac{n}{d}} \quad 
\text{ with } \bar\psi(d)=\phi(\ell_d)\mu(Q_d).
\end{equation}
Note that $L_n^{\bar S}=L_n^T$ where $T$ is the set of primes not in $S.$
\end{defn}

\begin{thm}\label{thm3.1defn3.2}\cite[Theorem~3.2]{SuFPSAC18}, \cite[Theorem~3.1, Definition~3.2]{Su3} The symmetric functions $L_n^S$ and $L_n^{\bar S}$ are Frobenius characteristics of true $S_n$-modules, and are thus Schur positive.
\end{thm}

When $S$ consists of a single prime $\{q\},$  we write $L_n^S=Lie_n^{(q)}.$  The particular case $q=2$ was investigated extensively in \cite{SuFPSAC18} and \cite{Su3}. In that case, the $S_n$-module $Lie_n^{(2)}$ was shown to have remarkable properties closely parallelling those of the Lie representation.

\vskip .1in

We illustrate \cref{thm6.4} and the functions $f_n^T$ by examining  the following special cases:
\begin{enumerate}
\item If $T=\{1\},$ then $\psi^T(d)=\mu(d)$ and $f_n^T$ corresponds to the representation $Lie_n.$ In this case \eqref{thm6.4eqn1} of \cref{thm6.4} gives the classical result of Thrall and Poincar\'e-Birkhoff-Witt \cite{R}: 
\begin{equation*} H[Lie]= (1-p_1)^{-1}, 
\end{equation*}
and \eqref{thm6.4eqn3} gives the surprising exterior power analogue for the variant $S_n$-modules $Lie_n^{(2)}$ studied in \cite{SuFPSAC18} and \cite{Su3}:
\begin{equation*} E[Lie^{(2)}]= (1-p_1)^{-1}. 
\end{equation*}

\item If $T$ is the set of all positive integers, then $\psi^T(d)=\phi(d)$ by M\"obius inversion of the well-known identity $m=\sum_{d|m}\phi(d).$  Thus $f_n^T$ is the characteristic of the conjugacy action on the class of $n$-cycles 
$\textbf{ 1}\uparrow_{C_n}^{S_n},$ i.e. $f_n^T=Conj_n.$
Now \eqref{thm6.4eqn1} of \cref{thm6.4} reduces to a theorem of Solomon \cite{So}: 
\begin{equation*} H[Conj]=\prod_{n\geq 1} (1-p_n)^{-1} =\sum_{\lambda\vdash n, n\geq 0} p_\lambda.
\end{equation*}
Also \eqref{thm6.4eqn3} reduces to a  result of this author, first proved in \cite{Su1}, implying Schur positivity of the right-hand side below:
\begin{equation*} E[Conj]=\prod_{n\geq 1, n \text{ odd}} (1-p_n)^{-1} =\sum_{\lambda\vdash n, n\geq 0, \lambda_i \text{ odd}} p_\lambda.
\end{equation*}
\item Fix a set $S$ of primes.  Let $T$ be the set of all integers whose prime factors are all in $S.$  Then clearly if $d\in T,$ $\psi^T(d)=\phi(d)$ by the identity used above. Otherwise $d=Q_d\ell_d$ with $Q_d\in T$ and $\ell_d$ relatively prime to $Q_d$ and also relatively prime to all integers in $T.$ Hence, since  $\mu$ is multiplicative,
\cref{def6.1} gives 
$$\psi^T(d)=\sum_{m\in T} m \mu(Q_d/m) \mu(\ell_d)
=\mu(\ell_d)\cdot \sum_{m\in T} m \mu(Q_d/m)=\mu(\ell_d)\psi^T(Q_d),$$ 
 Since $Q_d\in T,$  we obtain  $\psi^T(d)=\mu(\ell_d)\phi(Q_d),$ which is precisely the formula given by \eqref{LnS}. Thus $f_n^T=L_n^S.$  Setting $L^S=\sum_{n\geq 1}L_n^S, $ and defining $P(S)$ to be the set of positive integers whose prime factors are a subset of $S,$ \eqref{thm6.4eqn1} becomes the result of  \cite[Theorem 3.5]{Su3}, again  establishing  Schur positivity of the sum on the right. (Note that $1\in P(S).$) 
 \begin{equation*}    \label{SymLS}
 H[L^S] = \prod_{n\in P(S)}  (1- p_n)^{-1}
=\sum_{\lambda\in Par:\lambda_i\in P(S)}  p_\lambda.
 \end{equation*}
 %
%\item Fix a set $S$ of primes. Now let $T$ be the set of all integers relatively prime to every element of $S$ (so $T$ is the set of integers none of whose prime factors is in $S$). Exactly as above, we see that Definition \ref{def6.1} reduces to Eqn.~\eqref{LnbarS}, and hence $f_n^T$ is the characteristic $L_n^{\bar S}.$
\end{enumerate}


From \cref{thm6.4} and the preceding observations, we have the following decompositions of the representations $Conj_n,$ $Lie_n^{(q)}:$
\begin{thm}\label{prop6.6}
 \begin{equation}\label{prop6.6eqn1}\sum_{m\geq 1} p_m[Lie]=\sum_{n\geq 1} Conj_n;
\end{equation}
\begin{equation}\label{prop6.6eqn2}\sum_{m\geq 1} p_m=\sum_{n\geq 1}Conj_n[\sum_{r\geq 1}(-1)^{r-1} e_r]
\end{equation}
The plethystic inverse of $ Conj$ is 
\begin{equation}\label{prop6.6eqn3} (\sum_{n\geq 1} Conj_n)^{\langle -1\rangle}
= \sum_{r\geq 1}(-1)^{r-1} e_r[\sum_{n\geq 1} \mu(n) p_n].
\end{equation}
Let $q$ be prime, and let $n=\ell q^k $ 
where $(\ell, q)=1.$ Then 
\begin{equation}\label{prop6.6eqn4} Lie_n^{(q)} =\sum_{r=0}^k Lie_{\ell q^{k-r}}[p_{q^r}].\end{equation}
The plethystic inverse of $Lie^{(q)}$ is 
\begin{equation}\label{prop6.6eqn5}\sum_{n\geq 1}(Lie_n^{(q)})^{\langle -1\rangle}
=Lie^{\langle -1\rangle}[p_1-p_q]=(\sum_{r\geq 1} (-1)^{r-1} e_r)[p_1-p_q]. 
\end{equation} 


\end{thm}

Computations for $q=2 , 3, 5$ (verified for $n\leq 32,$ $n\leq 27$ and $n\leq 25$ respectively) support a curious conjecture on the partial sums in (\ref{prop6.6eqn4}):

\begin{conj} Fix a prime $q$ and $n=\ell q^k$ where $(q,\ell)=1.$ 
Define $W_i=\sum_{r=0}^i Lie_{\ell q^{k-r}}[p_{q^r}].$ Then $W_i$ is Schur-positive for all $i=0, \ldots, k.$   Note that the $W_i$ are all modules of dimension $(n-1)!$, and $W_0=Lie_n,$ while $W_k=Lie_n^{(q)}.$
\end{conj}

The construct of \cref{def6.2} allows us to remove the restriction that $q$ be prime, as follows.  Let $k\geq 2$ be any positive integer, and take $T$ to be the set of all nonnegative powers of $k.$ In this case \cref{thm6.4} gives
\begin{equation}\label{eqn6.9} H[\sum_{n\geq 1} f_n^T]=\prod_{r\geq 0} (1-p_{k^r})^{-1}, \qquad \sum_{n\geq 1} f_n^T=\sum_{r\geq 0} p_{k^r}[Lie].
\end{equation}
By inverting this equation plethystically, we obtain the recurrence 
\begin{equation}\label{eqn6.10} \text{For } k\geq 2, \qquad f_n^T=\begin{cases}  Lie_n+f^T_{\frac{n}{k}}[p_k], & k|n;\\
                          Lie_n, &\mathrm{otherwise},\\
\end{cases}
\end{equation}

However computations show that for $k=4,$ $f_n^T$ is not Schur-positive when $n=4, 16,$ and the degree 16 term in the product 
$\prod_{r\geq 0} (1-p_{4^r})^{-1}$ is not Schur-positive.  In both cases it is the sign representation that appears with coefficient $(-1).$

\begin{conj} For any ODD positive integer $k,$ $f_n^T$ as defined above is Schur-positive.  %, and  so is the product $\prod_{r\geq 0} (1-p_{k^r})^{-1}.$  
%In fact we conjecture that, if  $t$ is the highest power of $k$ dividing $n,$ for $i=0$ or $2\leq i\leq t, $ all the partial sums in the right-hand side of (\ref{eqn6.9}), namely, $\sum_{r=0}^i Lie_{\frac{n}{k^{r}}}[p_{k^{r}}],$  are themselves Schur-positive.  
\end{conj}

\begin{conj} The product $\prod_{r\geq 0} (1-p_{k^r})^{-1}$ is Schur-positive for any ODD positive integer $k.$
\end{conj}

Fix $k\geq 2$ and consider the subset $T=\{1,k\}.$ It was shown in \cite[Theorem 4.23]{Su1} that the symmetric function 
$$W_{n,k}=\sum_{\mu\vdash n, \mu_i=1\, or \, k} p_\mu$$ is Schur-positive.  Define $W_{0,k}=1.$ Then
$$\sum_{n\geq 0} W_{n,k}=\prod_{n\in T}(1-p_n)^{-1}=(1-p_1)^{-1} (1-p_k)^{-1}.$$
For $k=1$ we set $W_{n,1}=p_1^n$ for all $n\geq 0,$ so that the  preceding equation reduces, as expected, to
\begin{center} 
$\sum_{n\geq 0} W_{n,1}=\prod_{n\in T}(1-p_n)^{-1}=(1-p_1)^{-1}.$\end{center}


\begin{prop}\label{prop6.7} If $T=\{1,k\}$ and $k\geq 2,$ then 
\begin{equation}\label{eqn6.11}f_n^T
=\begin{cases}  Lie_n+Lie_{\frac{n}{k}}[p_k], & k|n;\\
                          Lie_n, &\mathrm{otherwise},\\
\end{cases}
\end{equation}
and hence $\sum_{n\geq 0} W_{n,k}=H[\sum_{n\geq 0} f_n^T].$

If $k$ is  prime, then 
$f_n^{\{1,k\}}=\mathrm{ch} (\exp\frac{2ki\pi}{n})\big\uparrow_{C_n}^{S_n}=\ell_n^{(k)},$ 
and hence the symmetric function defined by (\ref{eqn6.11}) is  
Schur-positive. \end{prop}
\begin{proof} Equation (\ref{eqn6.11}) is immediate from \cref{thm6.4}.  Now let $k$ be prime. It was shown in \cite[Lemma 5.5, Theorem 5.6]{Su1}   (see also \cref{Su1Thm5.6} below)  that the following identity holds:
$$\sum_{\lambda\vdash n: \lambda_i=1, k} p_\lambda
=H[\sum_{m\geq 1} \text{ch}\, (\exp(\frac{2\pi i k}{n})\uparrow_{C_n}^{S_n})]\vert_{{\rm\, deg\,} n},$$ and thus the left-hand side 
is precisely $p^T$ for $k$ prime and $T=\{1,k\}.$ But $H-1$ is invertible with respect to plethysm (see \cref{Pleth8}), so  $H[F]=H[G]$ if and only if $F=G.$ Hence $f_n^T$ must coincide with $\ell_n^{(k)}=\mathrm{ch} (\exp\frac{2ki\pi}{n})\big\uparrow_{C_n}^{S_n}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{comment}
Since  $f_n=\frac{1}{n}\sum_{d|n}\psi_k(d) p_d^{\frac{n}{d}},$ $\psi_k(n)$ satisfies
$\sum_{d|n}\psi_k(d)=n\delta(n=1, k)$ and 
M\"obius inversion immediately gives 
$\psi_k(n)= \sum_{d|n, d=1, k}\mu(n/d) d$ 

=$\begin{cases} \mu(n)+k\mu(n/k), &\text{if } k|n;\\
                        \mu(n), &\text{ otherwise}.\\
\end{cases}$
\end{comment}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{proof}

Computations  indicate  that 
\begin{conj}  $f_n^{\{1,k\}}$ is Schur-positive for $k=2$  and for all odd $k\geq 3.$  (This is trivially true if $k=1.$)
\end{conj}

When $k$ is even and not equal to 2, this fails.  For instance, if $n=k=4m,$ it is easy to see that $Lie_{4m}+p_{4m}$ contains the sign representation with coefficient $(-1).$ However 
we have $H[F^{\{1,k\}}]=(1-p_1)^{-1}(1-p_k)^{-1},$ which we know to be Schur-positive from \cite[Proposition 4.23]{Su1}.  This example shows that it is not always possible to write a Schur positive sum of power sums as a symmetrised module over a sequence of true $S_n$-modules, since $Lie_k+p_k$ fails to be Schur-positive when $k$ is even.

\begin{comment}
\begin{prop}\label{prop6.8} Let $k\geq 2.$ Then $\omega (E[F^{\{1,k\}}])=(1-p_1)^{-1}(1-(-1)^{k-1}p_k)^{-1} (1+p_2)(1+p_{2k}).$  If $k$ is prime, this is Schur-positive.
\end{prop}
\begin{proof}  We calculate $E[F^{\{1,k\}}]$ using \cref{Pleth5}.  Since the series $F^{\{1,k\}}$ is Schur-positive when $k$ is prime, the claim follows.\end{proof}
\end{comment}

%The next three propositions are also consequences of Theorem \ref{thm6.4}.

\begin{thm}\label{prop6.9}  Let $k\geq 2$ and $T=\{n:n\leq k\}.$ 
Then \begin{equation}\label{eqn6.12} f_n^T=\sum_{\stackrel{m=1}{m|n}}^k Lie_{\frac{n}{m}}[p_m]%\end{equation} 
\quad and \quad
\prod_{n=1}^k (1-p_n)^{-1}=H[\sum_{n} f_n^T]. \end{equation}
%
\end{thm}


\begin{cor}\label{cor6.10}   Let $T=\{n:n\leq k\}, k\geq 2.$  If $n$ is prime, or $n=k,$ or $n>k$ and $n$ is such that its greatest proper divisor is at most $k,$ then $f_n^T$ is Schur-positive.
\end{cor}
%%%%%%%%%%%
\begin{comment}
\begin{proof} If $n$ is prime it is easy to see that 
$f_n^T=\begin{cases} Lie_n, & n>k\\
                                 Lie_n+p_n, & \mathrm{otherwise.}
\end{cases}$
Since in this case, $\mu(n)=-1$ and $\phi(n)=n-1,$ and thus $Lie_n=\frac{1}{n}(p_1^n-p_n),$ we conclude that 
$Lie_n+p_n=\frac{1}{n}(p_1^n +\phi(n)p_n) =Conj_n.$

If $n=k$ then the sum in (\ref{eqn6.12}) ranges over all divisors of $n$ and hence by (\ref{prop6.6eqn1}) we have $f_n^T=Conj_n.$ 

If $n>k$ and  the largest proper divisor of $n$ is at most $k,$ we have $f_n^T=Conj_n-p_n,$ again by (\ref{prop6.6eqn1}).  But it is well known that 
$p_n=\sum_{r\geq 0} (-1)^r s_{(n-r, 1^r)}.$  Also by a  result of 
\cite{Sw}, $Conj_n$ contains all hooks except for the following: $(n-1,1)$ for all $n\geq 2,$ 
$(2, 1^{n-2})$ for odd $n\geq 3$, and $(1^n)$ for even $n.$ In all three cases, the corresponding Schur function appears with coefficient $-1$ in $p_n,$ hence with coefficient  
$+1$ in $Conj_n-p_n.$ This finishes the proof. \end{proof}
\end{comment}

\begin{conj} (See also \cite[Conjecture 1]{Su1}.) $f_n^{\{1,\ldots,k\}}$ is Schur positive for all $n$ and $k,$ and hence so is $\prod_{n=1}^k (1-p_n)^{-1}.$
\end{conj}

\begin{thm}\label{prop6.11} Let $k\geq 2$ and $T=\{n: n|k\}.$ Then  %\begin{equation}\label{eqn6.14} 
\[f_n^T=\sum_{m|(k,n)} Lie_{\frac{n}{m}}[p_m]
%\end{equation} 
\quad and \quad 
%\begin{equation}\label{eqn6.13} 
\prod_{n|k} (1-p_n)^{-1} = H[\sum_n f_n^T].\]
%\end{equation} 
\end{thm}
%
For $k\geq 2$ recall from \cref{prop6.7} that  $\ell_n^{(k)}$ denotes the characteristic of the Foulkes character ${\rm ch}\, (\exp(\frac{2\pi i k}{n})\uparrow_{C_n}^{S_n}).$ %of the Foulkes character (see \cite[Theorem 3.13]{Su3}).   
%
\begin{thm}\label{Su1Thm5.6}\cite[Lemma 5.5, Theorem 5.6]{Su1} %If $k$ is any fixed positive integer, then $\ell_n^{(k)}(1)$ is nonzero if and only if $n$ divides $k,$ in which case it equals 1.  Hence one has the Schur positive sum 
The following sum is Schur-positive:
\begin{equation*}\sum_{\lambda\vdash n: \lambda_i|k} p_\lambda
=H[\sum_{m\geq 1} \ell_m^{(k)} ]\vert_{{\rm\, deg\,} n}.
\end{equation*}
\end{thm}
 In particular %from Theorem \ref{Su1Thm5.6} 
 we immediately have 
\begin{cor}\label{cor6.12} $$\ell_n^{(k)} ={\rm ch } \exp(2i\pi\cdot k/n)\uparrow_{C_n}^{S_n}= \sum_{m|(k,n)} Lie_{\frac{n}{m}}[p_m].$$
\end{cor}
Hence we have the following curious decomposition of the regular representation into virtual representations:
\begin{cor}\label{cor6.13} 
\begin{equation}\label{eqn6.15} p_1^n=\sum_{k=1}^n \sum_{m|(k,n)} Lie_{\frac{n}{m}}[p_m]=\sum_{d|n}d\, Lie_d[p_{\frac{n}{d}}].
\end{equation}
\end{cor}
\begin{proof} The first sum can be rewritten as  $$\sum_{m|n} \sum_{\stackrel{r=1}{k=rm\leq n}}^{ \frac{m}{n}} Lie_{\frac{m}{n}}[p_m]=\sum_{m|n}\frac{m}{n}\ Lie_{\frac{m}{n}}[p_m].$$
The result now follows from the well-known decomposition \cite[Theorem 8.8]{R} 
$ p_1^n=\sum_{k=1}^n \ell_n^{(k)}$ of the regular representation, 
and the preceding corollary.
\end{proof}

%One can also arrive at the decomposition of equation (\ref{eqn6.15}) directly by using the expansion of $Lie_n$ into power sums.
%Looks like M\''obius inversion but not quite.

\begin{thm}\label{prop6.14} Let $T=\{n:n\equiv 1\,\mathrm{mod}\, k\}.$
Then  
%
\[f_n^T=\sum_{\stackrel{m\equiv 1\,\mathrm{mod}\, k}{ m|n}} Lie_{\frac{n}{m}}[p_m]  \quad    \text{  and }
\prod_{n\equiv 1\,\mathrm{mod}\, k} (1-p_n)^{-1}=H[\sum_{n} f_n^T].\]
\end{thm}

After seeing an early version  of \cite{Su1}, Richard Stanley made the following conjecture,  verifying it for $n\leq 24$ and $k\leq 6$.
\begin{conj}\label{RPS1} (R. Stanley, 2015)  $\prod_{n\equiv 1\,\mathrm{mod}\, k} (1-p_n)^{-1}$ is Schur-positive for all $k.$ 
\end{conj}

Note that \cref{RPS1} holds  for $k=2.$   We have two  different ways of identifying the associated $S_n$-module.
\begin{thm}\cite{Su1}, \cite{SuFPSAC18} and \cite[Corollary 3.11]{Su3}
$$ \prod_{n\equiv 1\,\mathrm{mod}\, 2} (1-p_n)^{-1} =E[Conj]=H[L^{\overline{{(2)}}}].$$ \end{thm}
The equivalence of the two plethystic expressions is a consequence of 
\cref{Pleth5} in \cref{sec:prelims}.
%
In this case, writing $p^{\mathrm{ odd}}$ for 
$\sum_{n\, \mathrm{odd}} p_n, $ we have the identity
$ p^{\mathrm {odd}}[Lie]=L^{\overline{{(2)}}},$ and hence:

\begin{thm}\label{prop6.15}  $\sum_{\stackrel{m\,\mathrm{odd}}{ m|n}} Lie_{\frac{n}{m}}[p_m]=p^{\mathrm {odd}}[Lie]\vert_{{\rm deg\ } n}$ is  Schur-positive; it is the Frobenius characteristic  $L_n^{\overline{{(2)}}}$ of the representation $\exp(2i\pi\ell/n)\uparrow_{C_n}^{S_n},$ where 
$n=2^k\cdot\ell$ and $\ell$ is odd.
\end{thm}
\begin{proof} This is clear by using \eqref{Pleth8d} of \cref{Pleth8} since the symmetric powers of the two modules coincide, both being equal to  $\prod_{n\, \mathrm{ odd}}(1-p_n)^{-1}.$
\end{proof}

The module $Lie_n^{(2)}$ whose many intriguing properties are described in \cite{SuFPSAC18} and \cite{Su3}, makes an appearance in the decomposition of the module $Conj_n$ of the conjugacy action on the class of $n$-cycles as well.  In fact 
\cref{prop6.6} leads to several different decompositions of  $Conj_n$, some of which we collect in the following:

\begin{thm}\label{thm6.16} 
%\begin{equation}\label{eqn6.16}\sum_n Conj_n=p^{\mathrm {odd}}[Lie^{(2)}],\end{equation} 
%and hence $p^{\mathrm {odd}}[Lie^{(2)}]$ is Schur-positive.%
 For any prime $q,$ we have 
\begin{equation}\label{eqn6.17} \sum_n Conj_n=\sum_{\stackrel{n}{ q \text{ does not divide } n}} p_n[Lie^{(q)}]=\sum_np_n[Lie],
\end{equation}
and hence the two sums on the right are Schur-positive.

In fact for any positive integer $q$ we have 
\begin{equation}\label{eqn6.18} \sum_n Conj_n=\sum_{\stackrel{n}{q \text{ does not divide } n}} p_n[\sum_{k\geq 0} Lie[p_{q^k}]].
\end{equation}

\end{thm}
%\begin{comment}
%\begin{proof} We start with equation (\ref{prop6.6eqn1}) of Proposition \ref{prop6.6}, and use the plethystic inverse formula established in Section 6,  Proposition \ref{Pleth1}.
%We have, by associativity of plethysm,
%\begin{align*} \sum_{n\geq 1} Conj_n&= \sum_{n\geq 1} p_n[Lie]\\
%&=\sum_{n\geq 1} (p_n[p_1-p_q] )[\sum_{k\geq 0} p_{q^k}[Lie]]\\
%&=\sum_{n\geq 1} (p_n-p_{nq} )[\sum_{k\geq 0} p_{q^k}[Lie]]\\
%&=(\sum_{n\geq 1} p_n -\sum_{n\geq 1} p_{nq})[\sum_{k\geq 0} Lie[p_{q^k}]];
%\end{align*}
%invoking Theorem \ref{thm6.4}, this establishes equation (\ref{eqn6.18}).\end{proof}
%\end{comment}



\section{Reverse Lexicographic Order}
The  previous sections focused on sums of power sums for partitions with restricted parts.  In our original paper \cite{Su1}, however, other families of sums were considered, and shown to be Schur positive by identifying the sum as the characteristic of actual $S_n$-modules.  In this section we describe partial progress on a conjecture of \cite{Su1}.  More details and a numerical analysis of the number of such Schur positive families can be found in \cite{Su4}.

 Recall that the reverse lexicographic order on partitions is defined as follows \cite[p. 6]{M}.  For partitions $\lambda, \mu$ of the same integer $n,$ we say a partition $\lambda$ is preceded by a partition $\mu$ in reverse lexicographic order  if $\lambda_1>\mu_1$ or there is an index $j\geq 2$ such that $\lambda_i=\mu_i$ for $i<j$ and $\lambda_j>\mu_j.$  Thus for $n=4$ we have the total order 
$(1^4) < (2,1^2)< (2^2) <(3,1) <(4).$ In particular our convention  is that the minimal and maximal elements in this total order are $(1^n)$ and $(n)$   respectively.  

\begin{conj}\cite[Conjecture 1]{Su1, Su4} Let $L_n$ denote the reverse lexicographic ordering on the set of partitions of $n.$ Then the sum of power sum symmetric functions $\sum p_\lambda,$ taken over any initial segment of the total order $L_n,$
i.e. any interval of the form $[ (1^n), \mu]$ for fixed $\mu,$  (and thus  necessarily including  the partition $(1^n)$), is Schur-positive. 
\end{conj}

We are able to prove the following special cases of this conjecture.  
%\label{lexorderconj}\cite{Su4}
\begin{thm}\label{lexorderconj}\cite[Theorem 18, Theorem 23, Proposition 36]{Su4}  The symmetric function 
$\psi_\mu=\sum_{(1^n)\leq \lambda\leq \mu} p_\lambda$ is Schur-positive if  $\mu\leq (3, 1^{n-3})$ or $\mu\geq (n-4,1^4)$  in reverse lexicographic order, 
 and also if $\mu=(3, 2^k, 1^r)$ for $k\geq 1$ and $0\leq r\leq 2.$  
\end{thm}

By a theorem of Solomon \cite{So}, $\psi_{(n)}=\psi_n$  is the Frobenius characteristic of $S_n$ acting on itself by conjugation. The proof for the Schur positivity of $\psi_\mu$  when $\mu$ lies in the interval 
$[(1^n), (3, 1^{n-3})]$ relies on some interesting symmetric function identities that we believe are new.
%
The other half of this theorem is established by using bounds on the multiplicity of the irreducibles occurring in  $\psi_n.$   Specifically, we use 
\begin{thm}\cite[Theorem 5.1]{Su2} Let $n\neq 4, 8.$ Then the conjugacy class indexed by a partition $\lambda$ contains all irreducibles %i.e. it is a global class, 
if and  only if $\lambda$ has at least two parts, and all its parts are distinct and odd. If $n=8,$ the conjugacy class indexed by $(7,1)$ 
contains all irreducibles,
%is global, 
while the class of the partition $(5,3)$ contains all irreducibles except those indexed by $(4^2)$ and $(2^4).$ \end{thm}

This in turn allows us to establish the following bounds:

\begin{lem}\cite[Lemma~2.6]{Su2} Let $n\geq 5.$ Let $do_n$ denote the number of partitions of $n$ with at least two parts and with all parts odd and distinct.  In  the conjugacy representation 
$\psi_n,$ every irreducible except possibly the sign occurs with multiplicity at least 
$\begin{cases} 4+do_n, & n \text{ odd};\\
                      3+do_n, & n \text{ even}.
\end{cases}$
 This number is at least 5 for odd $n\geq 7,$ and at least 4 for even $n\geq 6.$
 \end{lem}
 
 The proof of \cref{lexorderconj} then proceeds by a careful (and tedious) analysis showing that the multiplicity of any irreducible in $\psi_\mu$ cannot exceed the above bounds in absolute value.
In contrast to the previous proofs of Schur positivity, we are unable to identify a representation-theoretic context for this symmetric function. It would of course be of interest to find such a context.

Finally, we remark (see \cite{Su4})  that it is easy to find counterexamples showing that the analogous conjecture is false for dominance order.  

\printbibliography
\end{document}


%\bibliographystyle{amsplain.bst}
%\begin{thebibliography}{<99>}


%\bibitem{M}  I.~G.~Macdonald, \textit{Symmetric functions and 
%Hall polynomials}, Second Edition, Oxford University Press (1995).

%\bibitem{R} C.~Reutenauer, \textit{Free Lie Algebras}, London Mathematical Society Monographs, Oxford Science Publications (1993).

%\bibitem{So}  L.~Solomon, \textit{On the sum of the elements in the character table of a finite group}, Proc. Amer. Math. Soc. \textbf{12} No. 6 (1961), 962--963.

%\bibitem{St4EC2} R.~P.~Stanley, {Enumerative 
%Combinatorics, Vol.2}, Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, Cambridge, 1999.

%\bibitem{Su1} S.~Sundaram, \textit{The conjugacy action of $S_n$ and modules induced from centralisers},  J. Algebraic Combin. 48 (2018), no. 2, 179--225. DOI: 10.1007/s10801-017-0796-9
%(arXiv:160305898v2).

%\bibitem{Su2} S.~Sundaram, \textit{On conjugacy classes of $S_n$ containing all irreducibles},  Isr. J. Math. {\bf 225(1)} (2018), 321--342. DOI: 10.1007/s11856-018-1659-3. %(arXiv.math.GR/1607.08581v3)

%\bibitem{SuFPSAC18} S.~Sundaram, \textit{On a variant of $Lie_n$,} {Proceedings of the 30th International Conference on Formal Power Series and Algebraic Combinatorics, Hanover, USA,} Issue 80B (2018).

%\bibitem{Su3} S.~Sundaram, \textit{Variations on the $S_n$-module $Lie_n$}. (arXiv.math.RT/1803.09368v2).


%\end{thebibliography}


In this paper we investigate the Schur positivity of multiplicity-free sums of power sums.  This is equivalent to examining the character table of the symmetric group $S_n$, selecting a subset of the conjugacy classes, and asking whether the row sum corresponding to each irreducible character  is nonnegative.

In previous work, many such subsets have been identified, and the positivity question has been settled by  constructing explicitly a representation whose Frobenius characteristic is that sum of power sums.  Here we define a new symmetric function $f_T$ for each nonempty subset $T$ of positive integers, and describe how to realise the sum of power sums over partitions of $n$ with parts in $T$ as the characteristic of a possibly virtual representation.  We then investigate when the representation is a true $S_n$-module.  The function $f_T$ unifies many of our previous results, and leads to some new ones.  

We discover many curious and unexpected plethystic identities, notably one relating the conjugacy action and the Lie representation, as well as relationships between the latter 
and the remarkable variant $Lie_n^{2)}$ described in last year's FPSAC submission.
