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\title{On a variant of $Lie_n$} 

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%\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}}
\author{Sheila Sundaram\thanks{\href{mailto:shsund@comcast.net}{shsund@comcast.net}}\addressmark{1}}

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\address{\addressmark{1}Pierrepont School, One Sylvan Road North, Westport, CT 06880}

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\received{14 November 2017}
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\abstract{We introduce a new $S_n$-module $Lie_n^{(2)}$ which 
  interpolates between the representation $Lie_n$ of the symmetric group $S_n$ afforded by the free Lie algebra, and the module $Conj_n$ of the conjugacy action of $S_n$ on $n$-cycles.
%
 Using plethystic identities from our previous work, we establish a decomposition of the regular representation as a sum of {\it exterior} powers of the modules $Lie_n^{(2)}.$    By contrast,  the 
classical result of Thrall decomposes the regular representation into a sum of symmetric powers of the representation $Lie_n.$  We show that nearly every known property of $Lie_n$ in the literature appears to have a counterpart for $Lie_n^{(2)},$  suggesting connections to the cohomology of configuration spaces and other areas.  
%
The construction of $Lie_n^{(2)}$ can be generalised to a module  $Lie_n^{S}$ indexed by subsets $S$ of distinct primes.
This in turn yields new Schur-positivity results for multiplicity-free sums of power sums, extending our previous results. }

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\keywords{plethysm, Schur positivity, symmetric powers, exterior powers, power-sum symmetric functions, configuration space, braid arrangement}

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\begin{document}
\maketitle

\section{Introduction}
The original motivation for this paper was the investigation begun in ~\cite{Su1} on the positivity of the row sums in the character table of $S_n.$  %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}.    Section 2 reviews the key theorems of \cite{Su1}.  We describe some of our new  Schur-positivity results in Section 3.

Our main focus here, however, is  the unexpected discovery of a curious
variant of $Lie_n,$ the representation of $S_n$ on the multilinear
component of the free Lie algebra with $n$ generators.  The theorem of
Poincar\'e--Birkhoff--Witt  states that the universal enveloping algebra
of the free Lie algebra is the full tensor algebra. By Schur--Weyl
duality, this is equivalent to Thrall's decomposition of the regular
representation into a sum of symmetric powers of the representations
$Lie_n.$ By contrast, here we obtain a decomposition of the regular
representation as a sum of exterior powers of modules (see Theorem
\ref{T8}).  The key ingredient is our variant of $Lie_n,$ an $S_n$-module that we denote by $Lie_n^{(2)},$ which turns out to possess remarkable properties similar to those of  $Lie_n.$ We show that $Lie_n^{(2)}$ admits a filtration  analogous to the one arising from the derived series of the free Lie algebra. Together with the plethystic identities derived in Section 2, they indicate the possibility of an 
underlying algebra structure for $Lie_n^{(2)}$ involving an acyclic complex.  The striking similarity of our results with the Whitney homology of the partition lattice also suggests a cohomological context similar to  the configuration space of the braid arrangement.  There is an interesting  action on derangements  arising from $Lie_n^{(2)}$ as well; we show that $Lie_n^{(2)}$ gives rise to a new decomposition of the homology of the complex of injective words studied by Reiner and Webb \cite{RW}, one that is different from the Hodge decomposition of Hanlon and Hersh \cite{HH}. These results are collected in Section 4. %,  showing that for every well-known property of  $Lie_n,$  the representation $Lie_n^{(2)}$  offers an interesting counterpart.

\section{Preliminaries}
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.

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}.  
%%%%%%%%

%%%%%%%%
%Define \begin{align} &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. 
%\end{align} 
%
Define \begin{equation} H(t)=\sum_{i\geq 0}t^i  h_i, \quad E(t) = \sum_{i\geq 0} t^i  e_i; \qquad\qquad
H=\sum_{i\geq 0}  h_i, \quad E= \sum_{i\geq 0}  e_i.
\end{equation}
Now let $\{q_i\}_{i\geq 1}$ be a sequence of symmetric functions, each $q_i$ homogeneous of degree $i.$  %For example, each $q_i$ may be the Frobenius characteristic of a possibly virtual representation of $S_i.$  
Let $Q=\sum_{i\geq 1}q_i,$  $Q(t)=\sum_{n\geq 1} t^n q_n.$
%
For each partition $\lambda$ of $n\geq 1$  with $m_i(\lambda)=m_i$ parts equal to $i,$ let $|\lambda|=n = \sum_i i m_i$ be the size of $\lambda,$ and 
$\ell(\lambda)=\sum_i m_i(\lambda)=\sum_i m_i$ be the length (total number of parts) of $\lambda.$ 
%%%%%

%%%%%

Define % for each partition $\lambda$ of $n\geq 1$  with $m_i(\lambda)=m_i$ parts equal to $i,$
 $H_\lambda[Q]=\prod_{i:m_i(\lambda)\geq 1} h_{m_i}[q_i]$ and $E_\lambda[Q]=\prod_{i:m_i(\lambda)\geq 1} e_{m_i}[q_i].$  
%%Also define \begin{center}$H^{\pm}_\lambda[Q]=\prod_{i:m_i(\lambda)\geq 1}(-1)^{m_i}h_{m_i}[q_i]$ and $E^{\pm}_\lambda[Q]=\prod_{i:m_i(\lambda)\geq 1} (-1)^{m_i}e_{m_i}[q_i].$  \end{center}

For the empty partition (of zero) we define $H_\emptyset [Q]=1=
E_\emptyset[Q]=H^{\pm}_\emptyset [Q]=
E^{\pm}_\emptyset[Q].$

 %%We have, by definition of $H^{\pm}$ (respectively $E^{\pm}$), and since $\sum_i m_i=\ell(\lambda),$ 
%
%%\begin{center}
%%$H^{\pm}_\lambda[Q]=(-1)^{\ell(\lambda)} H_\lambda[Q]$ 
%%and $E^{\pm}_\lambda[Q]=(-1)^{\ell(\lambda)} E_\lambda[Q].$ 
%%\end{center}

 Consider the generating functions 
$H[Q](t)$ and $E[Q](t).$ 
With the convention that $Par$, the set of all partitions of nonnegative integers,
 includes the unique empty partition of zero,  by the preceding observations and standard properties of plethysm \cite{M} we have 

$$H[Q](t)=\sum_{\lambda\in Par} t^{|\lambda|} H_\lambda[Q], \qquad \text{and } E[Q](t)=\sum_{\lambda\in Par} t^{|\lambda|} E_\lambda[Q];$$  

%%%

Also write $Q^{alt}(t)$ for the alternating sum $\sum_{n\geq 1} (-1)^{i-1} t^i q_i = t q_1-t^2 q_2+t^3 q_3-\ldots.$  
%%%Note the difference with the definition of $H^{\pm}$ and $E^{\pm},$ namely, here $Q^{alt}(t)$ is not the same as $Q(-t),$ (which we would denote by $Q^{\pm}$) but rather equals $-Q(-t).$

Let $\psi(n)$ be any real-valued function defined on the positive integers. 
Define symmetric functions $f_n$ by 
 \begin{equation}f_n = \dfrac{1}{n} \sum_{d|n} \psi(d) p_d^{\frac{n}{d}}, \qquad\text{ so that }\qquad  \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 $(n-1)!\psi(1).$

Also define the associated polynomial in one variable, $t,$ by
\begin{equation}
f_n(t) =\dfrac{1}{n} \sum_{d|n} \psi(d) t^{\frac{n}{d}}.
\end{equation}

The following theorem regarding the sequence of symmetric functions $f_n$ is a key element in our work.  For the purposes of this abstract we have taken $v=1$ in the original statement of \cite[Theorem 3.2]{Su1}.

\begin{thm}[{\cite[Theorem 3.2]{Su1}}]\label{T1}
Let $F=\sum_{n\geq 1}  f_n,$ $H=\sum_{n\geq 0}  h_n$ and 
$E=\sum_{n\geq 0}  e_n.$  
We have the following plethystic generating functions:
%\begin{enumerate}
%\item
\begin{equation}
 \text{(Symmetric powers)     }\  
H[F] (t)= \sum_{\lambda\in Par} t^{|\lambda|}  H_\lambda[F]=\prod_{m\geq 1} (1-t^m p_m)^ {-f_m(1)}\end{equation}
%\item 
\begin{equation}
\text{(Exterior powers)     }\ 
E[F](t)=\sum_{\lambda\in Par} t^{|\lambda|} E_\lambda[F] =\prod_{m\geq 1} (1-t^m p_m)^{f_m(-1)}\end{equation}
%\item 
\begin{center}
 \text{(Alternating exterior powers)    }
$\sum_{\lambda\in Par} t^{|\lambda|} (-1)^{|\lambda|-\ell(\lambda)} \omega(E_\lambda[F])$\end{center}
\begin{equation}=\sum_{\lambda \in Par} t^{|\lambda|}H_\lambda[\omega(F)^{alt}]
=H[\omega(F)^{alt}](t)
= \prod_{m\geq 1} (1+t^m p_m)^{f_m(1)}\end{equation}
%\item 
\begin{center} \text{(Alternating symmetric powers)    } 
$\sum_{\lambda\in Par} t^{|\lambda|} (-1)^{|\lambda|-\ell(\lambda)} \omega(H_\lambda[F])$\end{center}
\begin{equation}=\sum_{\lambda \in Par} t^{|\lambda|}E_\lambda[\omega(F)^{alt}]
=E[\omega(F)^{alt}](t)
= \prod_{m\geq 1} (1+t^m p_m)^{-f_m(-1)}\end{equation}
%\end{enumerate}
\end{thm}

We also have, with  $F,$ $H,$ and $E$ as defined above, the following generating functions for the symmetric and exterior powers on conjugacy classes of derangements.  

\begin{thm}[{\cite[Proposition 2.3, Corollary 2.4]{Su1}}]\label{T2}
Assume that $\psi(1)=1$ in the definition (2.3) of $f_n,$ i.e. $f_1=p_1.$ 

\begin{center}
$H[\sum_{n\geq 2} f_n] (t)
= \left(\sum_{n\geq 0}t^n (-1)^n e_n \right)\prod_{m\geq 1} (1-t^m p_m)^ {-f_m(1)}$
\end{center}

\begin{center}
$E[\sum_{n\geq 2} f_n] (t)
= \left(\sum_{n\geq 0}t^n (-1)^n h_n \right)\prod_{m\geq 1} (1-t^m p_m)^ {f_m(-1)}$
\end{center}
\end{thm}

\section{A class of symmetric functions indexed by subsets of primes}

Define $Lie_n$ to be $f_n$ with the choice $\psi(d)=\mu(d)$ where $\mu$ is the number-theoretic M\"obius function, and $Conj_n$ to be $f_n$ with the choice $\psi(d)=\phi(d)$  where $\phi$ is Euler's phi-function. It is well known that $Lie_n$ is the Frobenius characteristic of the action of $S_n$ on the multilinear component of the free Lie algebra, and also of the induced representation $\exp(\frac{2i\pi}{n})\big\uparrow_{C_n}^{S_n},$ where $C_n$ is the cyclic group generated by an $n$-cycle in $S_n$ (see \cite{R} for background and history.) Likewise $Conj_n$ is the Frobenius characteristic of the 
induced representation $1\big\uparrow_{C_n}^{S_n},$ i.e. the conjugacy action of $S_n$ on the class of $n$-cycles.

%Finally let $\pi_n=\omega(Lie_n).$ 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%

\begin{defn}\label{D3}
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 representations, defined via their Frobenius characteristics, as follows.  For each $n\geq 1:$

\begin{equation} 
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} 
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}
\end{defn}

Let $\mathcal{P}$ denote the set of all primes. Clearly 
$L_n^\emptyset=Lie_n=L_n^{\bar {\mathcal{P}} }$ and $L_n^{\bar{\emptyset}}=Conj_n=L_n^{\mathcal{P}}.$  More generally, we have:

\begin{thm}\label{T4}
$L_n^S$ and $L_n^{\bar S}$ are Schur-positive symmetric functions.     If  $S=\{q\}$ where  $q$ is prime, and $k$ is the largest power of q which divides $n,$ and $C_n$ is the cyclic subgroup of $S_n$ generated by an $n$-cycle, 
 then  $L_n^S=Lie^{(q)}_n$ is the Frobenius characteristic of the induced representation 

\begin{center}$\exp(\frac{2i\pi}{n}\cdot q^k)\big\uparrow_{C_n}^{S_n}.$\end{center}
%where $k$ is the largest power of q which divides $n.$
\end{thm}

%The important point here is that the functions $\psi(d)$ and $\bar\psi(d)$ depend only on the fixed set $S,$ and not on $n.$ Hence these expressions match the form of the functions $f_n$ defined in (2.3).  This means that we can apply Theorem 1.

Let $P(S)$ denote the set of positive integers whose  prime divisors constitute a subset of the set of primes $S;$  note that $1\in P(S).$ 
Thus $P(S)=\{n\geq 1: q|n \text{ and } q  \text{ prime } \Longrightarrow  q\in S \}.$ Similarly let $P(\bar{S})$ be the set of positive integers whose set of prime divisors is disjoint from  $S;$ note that $P(\bar{S})$ is precisely the set of integers that are relatively prime to every prime in $S:$  
$P(\bar{S})=\{n\geq 1: q \text{ is a prime factor of }n \Longrightarrow q\notin S\}.$ Also $P(S)\cap P(\bar{S})=\{1\}.$  

Using a technical calculation of the values $L_n^S(1)$ and $L_n^{\bar
S}(1),$ we now obtain,  from equations (2.4) and (2.6)  of Theorem
\ref{T1} applied to
the sequences of symmetric functions $f_n=L_n^S$ and $f_n=L_n^{\bar S}:$ 

\begin{thm}\label{T5}
Let $L^S=\sum_{n\geq 1} L_n^S, $   
$L^{\bar{S}}=\sum_{n\geq 1} L_n^{\bar{S}}.$   Let $D\!Par$ denote the subset of all partitions $Par$ consisting of partitions with distinct parts, and the empty partition. Then one has the generating functions
   \begin{equation}
 H[L^S](t) = \prod_{n\in P(S)}  (1-t^n p_n)^{-1}
=\sum_{\lambda\in Par:\lambda_i\in P(S)} t^{|\lambda|} p_\lambda;
\end{equation}
%
\begin{equation} H[\omega(L^S)^{alt}](t)=\prod_{n\in P(S)} (1+t^n p_n)
=\sum_{\lambda\in D\!Par:\lambda_i\in P(S)} t^{|\lambda|} p_\lambda;
\end{equation}
%
\begin{equation}H[L^{\bar{S}}](t)=\prod_{n\in P(\bar{S})}  (1-t^n p_n)^{-1}
=\sum_{\lambda\in Par:\lambda_i\in P(\bar{S})} t^{|\lambda|} p_\lambda;
\end{equation}
%
\begin{equation} H[\omega(L^{\bar{S}})^{alt}](t)=\prod_{n\in P(\bar{S})} (1+t^n p_n)
=\sum_{\lambda\in D\!Par:\lambda_i\in P(\bar{S})} 
t^{|\lambda|} p_\lambda;
\end{equation}
%
\end{thm}

When $S=\emptyset,$  equations (3.3), (3.4) and (3.5) reduce to  known formulas of Thrall \cite{T}, Cadogan \cite{C}, and Solomon \cite{So}, respectively. See also \cite{St4EC2}. %Write $\pi_n=\omega(Lie_n).$
%\begin{thm} 
\begin{equation}  (Thrall)\qquad H[\sum_{n\geq 1} Lie_n](t)=(1-tp_1)^{-1}\iff \sum_{\lambda\vdash n} H_\lambda[Lie]=p_1^n, \quad n\geq 1.
\end{equation}
 \begin{equation} (Cadogan)\qquad  H[\sum_{n\geq 1} (-1)^{n-1} \omega(Lie_n)](t)
=1+tp_1 \iff \sum_{\lambda\vdash n} H_\lambda[\omega(Lie)^{alt}]=0, \quad n\geq 1.\end{equation} 
\begin{equation} (Solomon )\qquad H[\sum_{n\geq 1}  Conj_n](t)=\prod_{n\geq 1} (1-t^np_n)^{-1}
\iff \sum_{\lambda\vdash n} H_\lambda[Conj]=\sum_{\lambda\vdash n} p_\lambda, \quad n\geq 1.
\end{equation}
%\end{thm}


In similar fashion, by computing the values of $L_n^S(-1)$ and $L_n^{\bar
S}(-1)$ and invoking equations (2.5) and (2.7) of Theorem \ref{T1}, we
obtain the exterior power analogue of Theorem \ref{T5}.  

\begin{thm}\label{T6}
Let $S$ be a set of  primes, and let $L_n^S, L_n^{\bar{S}}$ be as defined
in Theorem \ref{T5}.  Then 
 \begin{equation} E[L^S(t)]=
\begin{cases} \prod_{ n\in P(S)} (1-t^n p_n)^{-1} \prod_{n\, even, \frac{n}{2}\in P(S)} (1-t^np_n), & 2\notin S\\
\prod_{n\, odd, \, n\in P(S) } (1-t^np_n)^{-1}, & 2\in S.
\end{cases}
\end{equation}
(Note $n$ is necessarily odd in the first product of the case $2\notin S.$)
\begin{equation} \omega(E[L^S(t)])=
 \prod_{ n\in P(S)} (1-t^n p_n)^{-1}\prod_{n\, even, \frac{n}{2}\in P(S)} (1+t^np_n),  \text{ provided } 2\notin S
\end{equation}
\begin{equation} E[\omega(L^S)^{alt}(t)]=
\begin{cases} \prod_{ n\in P(S)} (1+t^n p_n) \prod_{n\, even, \frac{n}{2}\in P(S)} (1+t^np_n)^{-1}, & 2\notin S\\
\prod_{n\, odd, \, n\in P(S) } (1+t^np_n), & 2\in S.
\end{cases}
\end{equation}
\begin{equation} E[L^{\bar{S}}(t)]=
\begin{cases} \prod_{n\ odd,\, n\in P(\bar{S})} (1-t^np_n)^{-1}, & 2\notin S\\
\prod_{ n\in P(\bar{S})} (1-t^np_n)^{-1} \prod_{n\text{ even, } \frac{n}{2}\in P(\bar{S})} (1-t^np_n), & 2\in S .
\end{cases}
\end{equation}
(Note $n$ is necessarily odd in the first product of the case $2\in S,$ as is $\frac{n}{2}$ in the second product.)
\begin{equation} \omega(E[L^{\bar{S}}(t)])=
\prod_{ n\in P(\bar{S})} (1-t^np_n)^{-1} \prod_{n\text{ even, } \frac{n}{2}\in P(\bar{S})} (1+t^np_n), \text{ provided } 2\in S .
\end{equation}
\begin{equation} E[\omega(L^{\bar{S}})^{alt}(t)]=
\begin{cases} \prod_{n\ odd,\, n\in P(\bar{S})} (1+t^np_n), & 2\notin S\\
\prod_{ n\in P(\bar{S})} (1+t^np_n) \prod_{n\text{ even, } \frac{n}{2}\in P(\bar{S})} (1+t^np_n)^{-1}, & 2\in S .
\end{cases}
\end{equation}
\end{thm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For any subset $T_n$ of partitions of $n,$ denote by $P_{T_n}$ the sum of power-sum symmetric functions $\sum_{\lambda\in T_n} p_\lambda.$  Since $L_n^S$ and $L_n^{\bar{S}}$ are Schur-positive, so are their symmetric and exterior powers.  Hence we deduce the following  from the preceding two theorems.
 
\begin{thm}\label{T7}
For a fixed set of primes $S,$ the  sums $P_{T_n}$ are Schur-positive for the following choices of $T_n$:
\begin{enumerate}
\item $T_n=\{\lambda\vdash n: \lambda_i\in P(S)\};$
%\item $T_n=\{\lambda\vdash n: \lambda_i\in P(\bar{S})\};$
\item If $2\in S,$ $T_n=\{\lambda\vdash n: \lambda_i\, odd,\, \lambda_i\in P(S)\};$
\item If $2\notin S,$ $T_n$ consists of all partitions $\lambda$ of $n$ such that the parts are (necessarily odd and) in $P(S),$ or the parts are twice an odd number in $P(S)$, the even parts occurring at most once.
%\item If $2\notin S,$ $T_n=\{\lambda\vdash n: \lambda_i\, odd,\, \lambda_i\in P(\bar{S})\};$ 
%\item If $2\in S,$ $T_n$ consists of all partitions $\lambda$ of $n$ such that the parts are (necessarily odd and) in $P(\bar{S}),$ or the parts are twice an (odd) number in $P(\bar{S})$, the even parts occurring at most once.
\end{enumerate}
\end{thm}

\section{The case $q=2:$ A comparison of 
$Lie_n$ and $Lie_n^{(2)}$} 

In this section we will describe some remarkable properties of the
$S_n$-module $Lie_n^{(2)}$.  Theorem \ref{T4} implies that  $Lie^{(2)}_2=h_2$ and $Lie^{(2)}_n=Lie_n$ if $n$ is odd.
 We use Theorems \ref{T1} and \ref{T2}, and plethystic computations with
 the identities of Theorems \ref{T5} and \ref{T6}, to show that the representation $Lie_n^{(2)}$ has properties that curiously parallel those of $Lie_n.$   We write $Lie$ for the sum of symmetric functions $\sum_{n\geq 1} Lie_n$ and $Lie^{(2)}$ for the sum 
of symmetric functions $\sum_{n\geq 1} Lie_n^{(2)}.$

\begin{thm}\label{T8}  %We have the following identities:
%\begin{center}$
%H[Lie]=(1-p_1)^{-1}; \qquad\qquad E [Lie^{(2)}]=(1-p_1)^{-1}$
%\end{center}
%\text{or equivalently,  in the notation of Theorem 1}
\begin{equation}%\text{or equivalently:  }
\sum_{\lambda\vdash n}H_\lambda[Lie]=p_1^n; \qquad\qquad \sum_{\lambda\vdash n}E_\lambda[Lie^{(2)}]=p_1^n;
\end{equation}
\begin{equation}
H[\sum_{n\geq 1} (-1)^{n-1}\omega(Lie_n)]=1+p_1;
\qquad E[\sum_{n\geq 1} (-1)^{n-1}\omega(Lie^{(2)}_n)]=1+p_1
\end{equation}
%\begin{center}$
%(\sum_{r\geq 1}(-1)^{r-1} e_r)[Lie]=p_1; \qquad\qquad (\sum_{r\geq 1}(-1)^{r-1} h_r) [Lie^{(2)}]=p_1$
%\end{center}
%or equivalently,
\begin{equation}%\text{or equivalently:  }
\text{If }n\geq 2:\ \sum_{\lambda\vdash n}(-1)^{n-\ell(\lambda)}E_\lambda[Lie]=0; \qquad \sum_{\lambda\vdash n}(-1)^{n-\ell(\lambda)}H_\lambda[Lie^{(2)}]=0;
\end{equation}
%
%\begin{center}
%$E[Lie]=(1-p_2)(1-p_1)^{-1};\qquad\qquad H[Lie^{(2)}]=\prod_{k\geq 0} (1-p_{2^k})^{-1}$
%\end{center}
%or equivalently,
\begin{equation}%\text{or equivalently:  }
\text{If }n\geq 2:\ \sum_{\lambda\vdash n}E_\lambda[Lie]=2e_2 p_1^{n-2}; \qquad \sum_{\lambda\vdash n}H_\lambda[Lie^{(2)}]=\sum_{\lambda\vdash n, \lambda_i=2^{a_i}} p_\lambda.
\end{equation}

\end{thm}

We now discuss the implications of Theorem \ref{T8}. 

In (4.1) of the preceding theorem, the second equation  gives a decomposition of the regular representation of $S_n$ as a sum of {\it exterior} powers of induced modules, whereas the first equation is precisely  Thrall's theorem \cite{T}, restated in (3.7), that the regular representation of $S_n$ decomposes into a sum of {\it symmetrised} Lie modules.  

In (4.2), the first equation contains the known result of Cadogan \cite{C} (see (3.8)) giving the plethystic inverse of the homogeneous symmetric functions $\sum_{n\geq 1} h_n$, and the second equation is a new result, giving the plethystic inverse of the elementary symmetric functions $\sum_{n\geq 1} e_n.$ 

The equations in  (4.3) and (4.4) are particularly significant.  It is well known that the degree $n$ term in $e_{n-r}[Lie]$ is the Frobenius characteristic of the $r$th Whitney homology $W\!H_r(\Pi_n)$ of the partition lattice $\Pi_n,$ tensored with the sign (see \cite[Remark 1.8.1]{Su0}),  and hence of the sign-twisted $r$th cohomology of the pure configuration space arising as the complement of the braid arrangement. The $r$th Whitney homology also coincides as an $S_n$-module with the $r$th cohomology of the pure braid group, see \cite{HL}.
The first equation in (4.3) therefore restates the acyclicity of Whitney homology for the partition lattice \cite{Su0}. 
Writing $W\!H_{odd}(\Pi_n)$ for $\oplus_{k=0}^{n/2} W\!H_{2k+1}(\Pi_n),$ and $W\!H_{even}(\Pi_n)$ for $\oplus_{k=0}^{n/2} W\!H_{2k}(\Pi_n),$ we have the isomorphism of $S_n$-modules
\begin{equation} W\!H_{odd}(\Pi_n)\simeq W\!H_{even}(\Pi_n).\end{equation}

Now consider (4.4).  Denote by $W\!H(\Pi_n)$ the sum of all the graded pieces of the Whitney homology of $\Pi_n.$ 
%it is well known that this coincides with the cohomology of the complement of hyperplanes for the root system $A_n$. 
The first equation in (4.4) says (recall that we have tensored with the sign representation) that 
 $W\!H(\Pi_n)=2 h_2 p_1^{n-2},$ 
 a result originally due to Lehrer \cite[Proposition 5.6 (i)]{Le}.  We may rewrite this in our notation as 
\begin{equation} {\rm ch\,}W\!H(\Pi_n)={\rm ch\,}(W\!H_{odd}(\Pi_n)\oplus W\!H_{even}(\Pi_n))=2 h_2 p_1^{n-2}.\end{equation}
(Note that it then follows that ${\rm ch\,}W\!H(\Pi_{n+1})=p_1 \cdot {\rm ch\,}W\!H(\Pi_{n}), n\geq 2.$  )

By combining this with (4.5), we obtain
\begin{equation}{\rm ch\,}(W\!H_{odd}(\Pi_n)) ={\rm
ch\,}(W\!H_{even}(\Pi_n))=h_2 p_1^{n-2},
\end{equation}
yielding the decomposition of the regular representation which appears in the recent paper \cite{HL} of Hyde and Lagarias:
\begin{equation} {\rm ch\ }W\!H_{odd}(\Pi_n) + \omega({\rm ch\ }W\!H_{odd}(\Pi_n))=\sum_{r=0}^{n-1}\omega^r({\rm ch\ }W\!H_r(\Pi_n))=p_1^n.\end{equation}

We will now describe analogous results arising from  the representations $Lie_n^{(2)}.$ 
Define a new module $V\!h_r(n)$ whose Frobenius characteristic is the
degree $n$ term in $h_{n-r}[Lie^{(2)}].$ By Theorem \ref{T4}, this is a
true $S_n$-module. The second equation of (4.3) can now be interpreted as
an acyclicity statement:
$$V\!h_n(n)-V\!h_{n-1}(n) + V\!h_{n-2}(n)-\ldots +(-1)^r V\!h_r(n) +\ldots=0,$$
and hence, in analogy with (4.5), letting $V\!h_{odd}(n)=\oplus_{k=0}^{n/2} V\!h_{2k+1}$ and $V\!h_{even}=\oplus_{k=0}^{n/2} V\!h_{2k}:$
\begin{equation} V\!h_{odd}(n)\simeq V\!h_{even}(n).\end{equation}

The second equation in (4.4) gives, similarly, 
\begin{equation} {\rm ch\,}(V\!h_{odd}(n)\oplus V\!h_{even}(n))=
\sum_{\lambda\vdash n; \lambda_i=2^{a_i}} p_\lambda
\end{equation}
Hence, combining this with (4.9) we have established the following results, showing that the modules $Vh_r(n)$ share the same features as the Whitney homology modules and hence of the cohomology of the configuration space for the braid arrangement:

\begin{thm}\label{T9}
The following symmetric functions are Schur-positive with integer coefficients:
\begin{equation}{\rm ch\,}(V\!h_{odd}(n))={\rm ch\,} (V\!h_{even}(n))=
\frac{1}{2}\sum_{\lambda\vdash n; \lambda_i=2^{a_i}} p_\lambda
\end{equation}
\begin{equation}{\rm ch\,} V\!h(n)={\rm ch\,}(V\!h_{odd}(n))+{\rm ch\,} (V\!h_{even}(n))=
\sum_{\lambda\vdash n; \lambda_i=2^{a_i}} p_\lambda
\end{equation}
\begin{equation}%\frac{1}{2}({\rm ch\,} V\!h(n)+\omega({\rm ch\,} V\!h(n))) =
{\rm ch\,}V\!h_{odd}(n)+ \omega({\rm ch\,}V\!h_{odd}(n))=
\sum_{\lambda\vdash n; n-\ell(\lambda) {\text even };\lambda_i=2^{a_i}} p_\lambda
\end{equation}
Also, ${\rm ch\,} V\!h(2n+1)=p_1\cdot {\rm ch\,} V\!h(2n).$
\end{thm}

We now have at least four decompositions of the regular representation, namely the two in (4.1) and two from (4.8) (tensoring the latter with the sign representation gives two),  into sums of modules indexed by the conjugacy classes,  each module  obtained by  inducing a linear character from a centraliser of $S_n.$  We write these out  for $S_4.$ %to show that they are indeed all distinct.

\begin{ex}\label{Ex10}
The first two decompositions are from (4.1)  of Theorem \ref{T8}; the third is from (4.8).
In all cases, of course,  the four pieces each have the same dimension, equal to the size of the corresponding conjugacy class, namely,  $1,6,11,6.$  That these four decompositions are all distinct is clear, since each has a distinguishing feature.  Both copies of the irreducible for the partition $(2^2)$ appear only in one piece for {\bf [PBW]}, while the natural representation is a submodule of one piece only in the third.  
\begin{description}
\item[{\small PBW}:] $h_4[Lie]=(4);\ h_3[Lie]=(3,1)+(2,1^2);$

   $h_2[ Lie]=(3,1)+2 (2^2)+(2,1^2)+(1^4);$
 $\ h_1[Lie]=Lie_4=(3,1)+(2,1^2).$
\item[{\small Ext.} $Lie^{(2)}$:] $e_4[Lie^{(2)}]=(1^4);\ e_3[Lie^{(2)}]=(3,1)+(2,1^2);\ $

$\qquad e_2[Lie^{(2)}]=2(3,1)+(2^2)+ (2,1^2);\ 
 e_1[Lie^{(2)}]=Lie_4^{(2)}=(4)+(2^2)+(2,1^2).$
\item[{\small Whitney}:] $\omega(WH_0)=(1^4);\  WH_1=(4)+(3,1)+(2^2);$ 

$\qquad\omega(WH_2)=(3,1)+ 2(2,1^2)+ (2^2);\ WH_3=Lie_4=(3,1)+(2,1^2).$
\end{description}
\end{ex}

There is yet another analogy between $WH_k(\Pi_n)$ and the modules
$V_k(n)$  arising from the identities of Theorem \ref{T8}.  In \cite{Su0}, it
was shown that the Whitney homology of the partition lattice (and more
generally of any Cohen--Macaulay poset) has the following important property:

\begin{thm}[{\cite[Proposition 1.9]{Su0}}]\label{T11}
For $0\leq k\leq n-1,$ the truncated alternating sum 
$$W\!H_k(\Pi_n)-W\!H_{k-1}(\Pi_n)+\ldots+(-1)^k W\!H_0(\Pi_n)$$ is a true $S_n$-module, and is isomorphic as an $S_n$-module to the unique nonvanishing homology of the rank-selected subposet of $\Pi_n$ obtained by selecting the first $k$ ranks. Equivalently, 
the degree $n$ term in the plethysm 
$$(e_{n-k}-e_{n-k+1}+\ldots +(-1)^{k} e_n)[Lie]$$ is Schur-positive. In particular, 
the $k$th Whitney homology decomposes into a sum of two $S_n$-modules as follows:
$${\rm ch\,} W\!H_k(\Pi_n)=\omega\left(e_{n-k}[Lie]|_{\text{deg }n}\right)=\beta_n([1,k])+\beta_n([1,k-1]),$$ 
where  $\beta_n([1,k])$ denotes  the Frobenius characteristic of this rank-selected homology (of the first $k$ ranks of $\Pi_n$) as in \cite[Proposition 1.9]{Su0}.
\end{thm}

We conjecture that a similar decomposition exists for the $S_n$-modules $V\!h_k(n).$ More precisely, we have 

\begin{conj}[Verified up to $n=7$]\label{Conj1}
Let $V\!h_k(n)$ be the $S_n$-module whose Frobenius characteristic is the degree $n$ term in the plethysm $h_{n-k}[Lie^{(2)}],$ for $k=0,1,\ldots, n-1.$  Then for $0\leq k\leq n-1,$ the truncated alternating sum 
$V\!h_k(n)-V\!h_{k-1}(n)+\ldots+(-1)^k V\!h_0(n)=U_k(n)$ is a true $S_n$-module, and hence one has the $S_n$-module decomposition
$${\rm ch\,} V\!h_k(n)=h_{n-k}[Lie^{(2)}]|_{\text{deg }n}={\rm ch\,} U_k(n) + {\rm ch\,}U_{k-1}(n).$$
(Here we define $U_{-1}(n)$ to be the zero module and $U_0(n)$ to be the
trivial $S_n$-module.)
Equivalently, the degree $n$ term in the  plethysm
$$(h_{n-k}-h_{n-k+1}+\ldots +(-1)^{k} h_n)[Lie^{(2)}]$$ is Schur-positive
for $0\leq k \leq n-1.$
\end{conj}
 
This conjecture is easily verified for $0\leq k\leq 2;$ in that case there are simple formulas for ${\rm ch\,} V\!h_k(n).$

  Recent work of Hyde and Lagarias \cite{HL} rediscovers the representations $\beta_n([1,k])$ in a cohomological setting.
%, namely as a natural subalgebra of the  cohomology algebra of the pure braid group.  
Our results  suggest the existence of a similar topological context in which the modules $V\!h_k(n)$ and $U_k(n)$ appear. 
If $X$ is any topological space, then the ordered configuration space $PConf_n(X)$ of $n$ distinct points in $X$ is defined to be the set 
$\{(x_1,\ldots, x_n): i\neq j \Longrightarrow x_i\neq x_j\}.$ The
symmetric group $S_n$ acts on $PConf_n(X)$ by permuting coordinates, and
hence induces an action on the cohomology $H^k(PConf_n(X), Q), k\geq 0.$
The following theorem of \cite{SW} recasts the ``Lie'' identities of
Theorem \ref{T8} in the context of $X={\bf R}^d$:

\begin{thm}[{\cite[Theorem 4.4]{SW}}]\label{T12}
The Frobenius characteristic of 
$H^k(PConf_n({\bf R}^2), Q)$ is given by  
$\omega\left(e_{n-k}[Lie]|_{\text{deg }n} \right).$

The Frobenius characteristic of 
$H^{2k}(PConf_n({\bf R}^3), Q)$   is given by 
$
h_{n-k}[Lie]|_{\text{deg }n} .$
\end{thm}

\begin{qn}\label{Q13}
Is there a similar cohomological context for the
``$Lie^{(2)}$'' identities of Theorem \ref{T8}?
\end{qn}

We turn next to the action on fixed-point-free permutations.
If $F=\sum_{n\geq 1} f_n$ is any series of symmetric functions where $f_n$ is of homogeneous degree $n,$ let $F_{\geq 2}$ denote the series $F-f_1=\sum_{n\geq 2} f_n.$ 
  Reiner and Webb study the Cohen--Macaulay complex of injective words,
  and compute the $S_n$-action on its top homology \cite{RW}. Using
  Theorem \ref{T2}, we show that the representations $Lie_n^{(2)}$ make an appearance here as well:

%\begin{thm}
%Let $D_n^k$ denote the degree $n$ term in $h_k[Lie_{\geq 2}],$ and let $D_n=\sum_{k\geq 1} D_n^k.$  Similarly let 
%$\Delta_n^k$ denote the degree $n$ term in $e_k[Lie_{\geq 2}^{(2)}],$ and let $\Delta_n=\sum_{k\geq 1} \Delta_n^k.$
%Then \begin{align}
%&D_n=\sum_{k=0}^n (-1)^k p_1^{n-k} e_k\\
%&\Delta_n=\sum_{k=0}^n (-1)^k p_1^{n-k} h_k
%\end{align}

%\end{thm}

\begin{thm}\label{T14}
Let 
$\Delta_n^k$ denote the degree $n$ term in $e_k[Lie_{\geq 2}^{(2)}],$ and let $\Delta_n=\sum_{k\geq 1} \Delta_n^k.$
Then $\Delta_n$ coincides with the homology representation on the complex of injective words.
\end{thm}

Hanlon and Hersh have shown that this representation has a Hodge decomposition \cite{HH}. Writing $D_n^k$ for the degree $n$ term in $h_k[Lie_{\geq 2}],$ their result may be stated as follows:
$$\Delta_n=\sum_{k\geq 1} \omega(D_n^k).$$
Surprisingly, the decomposition of $\Delta_n$ given by Theorem \ref{T14} is different from the Hodge decomposition.  The first nontrivial example appears below. 
%for $n=4,$ and then for $n\geq 6.$
%
\begin{ex}\label{Ex15}
For $n=4,$ we have $\Delta_4=p_1^2h_2-p_1h_3+h_4
=(4)+(3,1)+(2^2)+(2,1^2).$ Also $\Delta_4^2=e_2[h_2]=(3,1),$ 
$\Delta_4^1=Lie_4^{(2)}=(4)+(2^2)+(2,1^2).$ 
The two Hodge pieces, however, each consist of two irreducibles:
$\omega(h_2[Lie_2])=(2^2)+(4)$ and $\omega(h_1[Lie_4])=(3,1)+(2,1^2).$
\end{ex}

 A classical result asserts that the free Lie algebra has the following filtration arising from its derived series \cite[Section 8.6.12]{R}.  
Let $\kappa=\sum_{n\geq 2} s_{(n-1,1)}.$  Then     
%
$Lie_{\geq 2}=\kappa+\kappa[\kappa]+ \kappa[\kappa[\kappa]]+\ldots.$ 

Theorem \ref{T8}  allows us to  deduce a similar   decomposition for $Lie_n^{(2)}:$

\begin{thm}\label{T16}
$Lie^{(2)}_{\geq 2}=\omega(\kappa)+\omega(\kappa)[\omega(\kappa)]+ \omega(\kappa)[\omega(\kappa)[\omega(\kappa)]]+\ldots$
%\begin{align}
%& \quad  Lie_{\geq 2}=Lie[\sum_{n\geq 2} s_{(n-1,1)}]  \\
%&Lie^{(2)}_{\geq 2}=Lie^{(2)}[\sum_{n\geq 2} \omega(s_{(n-1,1)}) ]
%\end{align}
%The decomposition (4.15) is  equivalent to the decomposition (4.17) below of the free Lie algebra obtained from its derived series.  Let $\kappa=\sum_{n\geq 2} s_{(n-1,1)}.$ 
%We also obtain a similar decomposition for the modules $Lie^{(2)}:$
\end{thm}

We conclude with yet another feature of the Lie representation that reappears in $Lie_n^{(2)}$.
Recall that $Lie_{n-1}$ admits a lifting $W_{n}$ which is a true $S_n$-module, the Whitehouse module, appearing in many different contexts \cite{RWh}, \cite[Exercise 7.88 (d)]{St4EC2},  whose Frobenius characteristic  is given by 
${\rm ch\,} W_{n}=p_1 Lie_{n-1} -Lie_{n}.$

One can ask if the same construction for $Lie_n^{(2)}$ yields a true 
$S_{n}$-module.  Clearly one obtains a possibly virtual module which restricts to $Lie_{n-1}^{(2)}$ as an $S_{n-1}$-module. We have the following conjecture, verified in Maple (with Stembridge's  SF package) up to $n=32:$

\begin{conj}\label{Conj2}
The symmetric function $p_1 Lie_{n-1}^{(2)}- Lie_n^{(2)}$ 
is Schur-positive if and only if $n$ is \emph{not} a power of 2.
\end{conj}

One direction of this conjecture is easy to verify.  
Let $n=2^k\geq 2.$ Then 
$n-1$ is odd, so $Lie_{n-1}^{(2)}=Lie_n.$ Also (by Theorem \ref{T4}) $Lie_n^{(2)}=Conj_n$  is just the permutation module afforded by the conjugacy action on the class of $n$-cycles of $S_n.$  Consequently it contains the trivial representation (exactly once). But  $Lie_n$ never contains the trivial representation ($n\geq 2$).  Hence, when $n$ is a power of 2,  the trivial module appears with negative multiplicity $(-1)$ in $p_1 Lie_{n-1}^{(2)}- Lie_n^{(2)}.$ 

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