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\title[Homology of subword order]{The  homology representation of subword order}

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\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, USA }

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\abstract{We investigate the homology representation of the symmetric group on rank-selected subposets of subword order.  We show that the homology module for  words of bounded length, over an alphabet of size $n,$  decomposes into a sum of tensor powers of the $S_n$-irreducible $S_{(n-1,1)}$ indexed by the partition $(n-1,1),$  recovering, as a special case,  a theorem of Bj\"orner and Stanley  for words of length at most $k.$   For arbitrary ranks we show that the  homology  is an integer combination of  positive tensor powers of the reflection representation $S_{(n-1,1)}$,  and conjecture that this combination is nonnegative.  We uncover a curious duality in homology in the case when one rank is deleted. 

We prove that the action on the rank-selected chains of subword order is a nonnegative integer combination of tensor powers of $S_{(n-1,1)}$, and  show that its Frobenius characteristic is $h$-positive and supported on the set 
$T_{1}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 1\}.$

Our most definitive result describes the Frobenius characteristic of the homology for an arbitrary set of ranks, plus or minus one copy of  $S_{(n-1,1)},$  as an integer combination of the set 
 $T_{2}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 2\}.$  We conjecture that 
 this combination is nonnegative, establishing this fact for particular cases.}

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\keywords{Subword order, reflection representation, $h$-positivity, Whitney homology, Kronecker product, internal product, Stirling numbers.}

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\begin{document}

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\section{Introduction}

Let $A^*$ denote the free monoid of words of finite length in an alphabet $A.$ 
Subword order is defined on $A^*$ by setting $ u\le v$ if $u$ is a subword of $v,$ that is, 
the word $u$ is obtained by deleting letters of the word $v.$  This makes  $(A^*, \leq)$ into a graded poset 
with rank function given by the length  $|w|$ of a word $w,$ the number of letters in $w$.  The topology of this poset was first studied  by Farmer (1979) 
% and Viennot (1983) for combinatorial considerations, 
and then by Bj\"orner, who showed in \cite[Theorem 3]{Bj1} that any interval of this poset admits a dual CL-shelling. The intervals  are thus homotopy Cohen--Macaulay, as well as  all rank-selected subposets 
%(in the sense of Stanley \cite{St1}) 
obtained by considering only words whose rank belongs to
a finite set S \cite{BjShellableTAMS,BjWachs}.   
Suppose now that the alphabet $A$ is finite, of cardinality $n$. The symmetric group
$S_n $ acts  on $A$, and thus on $A^*.$  To avoid trivialities we will assume $n\ge 2.$

In this paper we examine the homology representation of  rank-selected subposets of $A^*$, using the Whitney homology technique and other methods developed in \cite{Su0}.  We refer the reader to \cite{St1} for general facts about rank-selection.  
Theorem~\ref{Bordeaux1991mai} shows that for intervals $[r, k]$ of consecutive ranks in $A^*$, the unique nonvanishing 
homology 
 decomposes as  a direct sum of copies of $r$ consecutive tensor powers of the reflection 
representation of $S_n$, that is, the irreducible representation $S_{(n-1,1)}$ indexed by the partition $(n-1,1).$  This  generalises a result in \cite{Bj1}, conjectured by Bj\"orner and proved by Stanley.  %) on the homology representation of the poset of all words of length at most $k.$ 
  We establish similar results for the Whitney and dual Whitney homology modules.   Both turn out to be  permutation modules in each degree, with pleasing  orbit stabilisers. Theorem~\ref{rank-deletion} establishes the nonnegative decomposition into tensor powers of $S_{(n-1,1)}$   for the case when one rank is deleted from the interval $[1,k],$  and leads to   a curious  homology isomorphism (Proposition~\ref{rank-deletion-duality}), suggesting an equivariant homotopy equivalence between the simplicial complexes associated to the rank sets $[1,k]\backslash{\{r\}}$ and $[1,k]\backslash{\{k-r\}},$ for fixed $r, 1\le r\le k-1.$  

 More generally, we show in Theorem~\ref{RankSelectedChains} that for any nonempty subset $T$ of ranks $[1,k],$ the homology representation of $S_n$ may be written as an integer combination of positive tensor powers of the reflection representation.  Based on our determination of this and other cases of rank-selection, we propose the following conjecture:

\begin{conj}\label{Conj:tensorpowers}  Let $A$ be an alphabet of size $n\ge 2.$ Then the $S_n$-action on the homology of any finite nonempty rank-selected subposet of subword order on $A^*$ is a \textit{nonnegative} integer combination of positive tensor powers of the irreducible indexed by the partition $(n-1,1).$ 
\end{conj}

These considerations lead us to examine the tensor powers of the reflection representation, and the question of how many tensor powers are linearly independent characters.  In answering these questions, we are led to a decomposition (Theorem~\ref{ReflRepMain}) showing that the $k$th tensor power of $S_{(n-1,1)}$ plus or minus one copy of $S_{(n-1,1)},$  has Frobenius characteristic equal to a nonnegative integer combination of the homogeneous  symmetric functions $\{h_{(n-r, 1^r)}: r\ge 2\}.$  It is \lq\lq almost" an $h$-positive  permutation module. Inspired by this phenomenon, we prove, in Theorem~\ref{Homologyhbasis}, that in fact for each rank subset $T$ and associated rank-selected subposet $A_{n,k}(T),$
the homology module $\tilde{H}(A_{n,k}(T))$ 
has the property that $\tilde{H}(A_{n,k}(T)) + (-1)^{|T|} S_{(n-1,1)} $ has Frobenius characteristic equal to an \textit{integer} combination of the homogeneous symmetric functions $\{h_{(n-r, 1^r)}: r\ge 2\}.$  Theorem~\ref{Summary} establishes the truth of the following conjecture for all the rank-selected homology computed in this paper.  
%
\begin{conj}\label{Conj:almost-h-pos} Let $A$ be an alphabet of size $n\ge 2.$ Then the  homology of any finite nonempty rank-selected subposet of subword order on $A^*$, plus or minus one copy of the reflection representation of $S_n,$ is a permutation module. In fact its Frobenius characteristic is $h$-positive and supported on the set  $T_{2}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 2\}.$
\end{conj}
%
%
%Note that the first conjecture  implies the second, since   the irreducible indexed by the partition $(n-1,1)$  restricts to a permutation action on a set of cardinality $(n-1).$
%
We give a simple criterion for when  Conjecture~\ref{Conj:tensorpowers} will imply Conjecture~\ref{Conj:almost-h-pos} in Lemma~\ref{RefRephPos}.

The  main results of this paper are summarised below.  Let $A$ be an alphabet of size $n\ge 2,$ and $T\subseteq [1,k]$ a subset of ranks.  
Let $A_{n,k}(T)$  denote the corresponding rank-selected subposet.
%added 2020 July 14
\begin{thm}\label{SummaryChains}  The action of $S_n$ on the maximal chains of the rank-selected subposet $A_{n,k}(T)$ of $A^*$ of words with lengths in $T,$  is a nonnegative integer combination of tensor powers of the reflection representation $S_{(n-1,1)}.$  If $|T|\ge 1,$  this action  has $h$-positive Frobenius characteristic supported on the set $T_{1}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 1\}.$ 
\end{thm}

\begin{thm}\label{Summary}  The homology module $\tilde{H}(A_{n,k}(T))$ of words with lengths in $T$ is an integer combination of positive tensor powers of the reflection representation $S_{(n-1,1)},$
with the property that $\tilde{H}(A_{n,k}(T)) + (-1)^{|T|} S_{(n-1,1)} $ has Frobenius characteristic equal to an integer combination of the homogeneous symmetric functions $\{h_{(n-r, 1^r)}: r\ge 2\}.$ 

Both integer combinations  are  \textit{nonnegative} 
when $T$ is one of the following  rank sets:   

$ (1)\ [r,k], k\ge r\ge 1; \quad (2)\ [1,k]\backslash\{r\}, k\ge r\ge 1;\quad  
(3)\ \{1\le s_1<s_2\le k\}.$

%\begin{enumerate}
%\item $T=[r,k], k\ge r\ge 1.$ 
%\item $T=[1,k]\backslash\{r\}, k\ge r\ge 1.$ 
%\item $T=\{1\le s_1<s_2\le k\}.$
%\end{enumerate}

\begin{comment}
Then
  \begin{description}
  \item[(A)] The rank-selected homology $\tilde{H}_{k-2}(A^*_{n,k}(T))$ is a nonnegative integer combination of positive tensor powers  $S_{(n-1,1)};$ 
  \item[(B)] The module $V_T=\tilde{H}_{k-2}(A^*_{n,k}(T))+ (-1)^{|T|} S_{(n-1,1)}$ is a nonnegative integer combination of transitive permutation modules with orbit stabilisers 
of the form $S_1^d\times S_{n-d}, d\ge 2.$  Equivalently, the Frobenius characteristic of $V_T$ is supported on the set 
 $T_{2}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 2\}$ with nonnegative integer coefficients. 
 \end{description}
\end{comment}
  \end{thm}
  
\section{Subword order}

The subword order poset $A^*$ has a unique least element at rank 0, namely the empty word $\emptyset$ of length zero.
%
In this section we collect the main facts on subword order from \cite{Bj1} that we will need.  For general facts about posets, M\"obius functions, etc. we refer the reader to \cite{St3EC1}.

%\begin{defn}\cite{F} A word $\alpha$ in $A^*$ is  \textit{normal} if no two consecutive letters of $\alpha$ are equal.  
%\end{defn}
%For example, $aabbccaabbcc$ is not normal, while $abcabc$ is normal.  Normal words are also called \textit{Smirnov} words in the recent literature.   Observe that the number of normal words of length $i$ is $n(n-1)^{i-1}.$ 
Farmer computed the M\"obius number of an arbitrary interval $(\hat 0, \alpha)$ and also showed: 
\begin{thm}[{\cite[Theorem 5 and preceding Remark]{F}}] \label{Farmer}  Let $|A|=n$ and let $A^*_{n,k}$ denote the subposet of $A^*$ consisting of the first $k$ nonzero ranks and the empty word, i.e. of words of length at most $k,$ with an artificially appended top element $\hat 1.$ Then $\mu(A^*_{n,k})=\mu(\hat 0, \hat 1) = (-1)^{k-1} (n-1)^k$ and $A^*_{n,k}$ has the homology of a wedge of $(n-1)^k$ spheres of dimension $(k-1).$
%(, see also Viennot \cite{V}) 
%\begin{enumerate}
%\item
%Let $\alpha$ be any word in $A^*$.  Then the M\"obius function of subword order satisfies $\mu(\hat 0, \alpha)
%=\begin{cases} (-1)^{|\alpha|}, & \text{ if }\alpha \text{ is a normal word}\\
%0, & \text{ otherwise}.
%\end{cases}$
%\item 
%\begin{equation}\label{MobiusNumber}\mu(A^*_{n,k})=\mu(\hat 0, \hat 1) = (-1)^{k-1} (n-1)^k.\end{equation}
%\item \cite[Theorem 5 and preceding Remark]{F} $A^*_{n,k}$ has the homology of a wedge of $(n-1)^k$ spheres of dimension $(k-1).$
%\end{enumerate}
%
\end{thm}

%\begin{cor}\label{MobiusNumber} (\cite{F}, \cite{V}) Let $|A|=n$ and let $A^*_{n,k}$ denote the subposet of $A^*$ consisting of the first $k$ nonzero ranks and the empty word, i.e. of words of length at most $k,$ with an artificially appended top element $\hat 1.$ Then 
%$\mu(A^*_{n,k})=\mu(\hat 0, \hat 1) = (-1)^{k-1} (n-1)^k.$
%\end{cor}

Bj\"orner generalised Farmer's computation of the M\"obius function $\mu(\hat 0, \alpha)$  to give a  formula for the M\"obius function of an arbitrary interval $(\beta, \alpha)$, and established the following generating functions.
%
%
\begin{comment}
\begin{defn} \cite{Bj1}
Given a word $\alpha= a_1a_2\ldots a_n$ in $A^*$, its repetition set is $R(\alpha)=\{i: a_{i-1}=a_i\}.$   An embedding of $\beta$ in $\alpha$ is a sequence $1\le i_1<i_2<\ldots<i_k\le n$ such that $\beta =a_{i_1} a_{i_2}\ldots a_{i_k}.$ It is called a \textit{normal embedding} if in addition $R(\alpha)\subseteq \{i_1,i_2,\ldots,i_k\}.$

Denote by ${\alpha \choose \beta}$ the number of embeddings of $\beta$ in $\alpha,$ and by ${\alpha \choose \beta}_n$ the number of normal embeddings of $\beta$ in $\alpha$. 

\end{defn}

\begin{thm} \cite[Theorem 1]{Bj1} For all $\alpha, \beta\in A^*,$ 
\[\mu(\beta,\alpha)=(-1)^{|\alpha|-|\beta|} {\alpha \choose \beta}_n.\]
\end{thm}
Observing that the word $\alpha$ is normal if and only if its repetition set $R(\alpha)$ is empty, one sees that this generalises  Farmer's formula for $\mu(\hat 0, \alpha).$
\end{comment}
%
%
Recall that the zeta function \cite{St3EC1} of a poset is defined by 
$\zeta(\beta, \alpha)=1$ if $\beta\le \alpha,$ and equals zero otherwise. 
\begin{thm}[\cite{Bj1}] \label{Bjgf} Let $A$ be an alphabet of size $n,$ and $\beta$  a word in $A^*$ of length $k.$  The following generating functions hold:
\begin{enumerate}
\item \cite[Theorem~2~(i)]{Bj1} 
 For the  M\"obius function of subword order: 
\[\sum_{\alpha\in A^*} \mu(\beta,\alpha)t^{|\alpha|} 
= \dfrac{t^k(1-t)}{(1+(n-1)t)^{k+1}}.\]
%\item \cite[3.~Remark.]{Bj1}   The number of words of length $p$ in the interval $[\beta, \infty]$ depends only on  
%the length $k$ of $\beta, $ and equals 
%\[\sum_{i=0}^{p-k} {p\choose i} (n-1)^i.\]
%
 %Proof: (Bj\"orner first establishes, by means of the following simple  bijection, that the number of $p$-length words depends only on the length of $\beta.$ 
% Let $\beta=b_1\ldots b_k,$ and $\gamma=c_1\ldots c_k.$  Let $
%\alpha =\alpha_0 b_1\alpha_1 b_2\ldots b_k\alpha_k$ where the 
%$\alpha_i$ are arbitrary words, and $\alpha$ is a word of length 
%$p\ge k.$. Define 
%$\phi\alpha=\hat \alpha_0 c_1\hat\alpha_1 c_2\ldots c_k\hat\alpha_k, $ %where $\hat \alpha_i$ is obtained from $\alpha_i$ by replacing every 
%$c_j$ in $\alpha_i$ by $b_j.$ This is a one-to-one map from every %length $p$ superword $\alpha$ of $\beta$ onto a length $p$ superword %$\phi\alpha$ of $\gamma.$ ) 
%Now  take $\beta=bb\ldots b\, (k \text{times})$. Then 
%subtract the $p$-length words with exactly $j$ $b$'s, $0\le j\le k-1.$  %This gives 
% \begin{multline*} 
 %\sum_{j=0}^p \binom{p}{j} (n-1)^{p-j} - \sum_{j=0}^{k-1} 
 %\binom{p}{j} (n-1)^{p-j}\\ =\sum_{j=k}^p \binom{p}{j}(n-1)^{p-j}
 %=\sum_{i=0}^{p-k} \binom{p}{p-i} (n-1)^{i},\end{multline*}
 %since the first sum of binomials is just $n^p=(1+(n-1))^p.$
\item \cite[3.~Remark (i)]{Bj1} For the zeta function of subword order:
\[\sum_{\alpha\in A^*} \zeta(\beta,\alpha)t^{|\alpha|} 
= \dfrac{t^k}{(1-nt)(1-(n-1)t)^{k}}.\]
\end{enumerate}
\end{thm}

Farmer's result on the homology of $A^*_{n,k}$   was strengthened by Bj\"orner, who  showed the following (see \cite{BjShellableTAMS, BjWachs}): %and also \cite{Wa} for a survey of lexicographic shellability):
\begin{thm}[Bj\"orner {\cite[Theorem 3, Corollary 2]{Bj1}}]\label{BjCLshellable} Every interval $(\beta, \alpha)$ in the subword order poset $A^*$ is dual CL-shellable, and hence homotopy Cohen--Macaulay.  In particular, for a finite alphabet $A,$ the poset $A^*_{n,k}$ of nonempty words of length at most $k,$ which may be viewed as the result of rank-selection from an appropriate interval of $A^*$, is also dual CL-shellable and hence also homotopy Cohen--Macaulay. 
\end{thm}

It follows from the general theory of shellability \cite{BjShellableTAMS, BjWachs} 
that all rank-selected subposets of $A^*$ are homotopy Cohen--Macaulay. 
\section{Rank-selection in $A^*$}

In this section we will assume the alphabet $A$ is finite of size $n.$ 

We follow the standard convention as in \cite{St1}, \cite{St3EC1}: By the homology of a poset $P$ with greatest element $\hat 1$ and least element $\hat 0,$ we mean the reduced homology $\tilde{H}(P)$ of the simplicial complex whose faces are the chains of 
$P\backslash\{\hat 0, \hat 1\}.$
%
%Note that by the Bj\"orner-Wachs theory of shellability \cite{BjShellableTAMS,BjWachs}, rank-selection preserves CL-shellability and hence the property of being homotopy Cohen--Macaulay.
In order to determine the homology representation of $S_n$ on rank-selected subposets of $A^*_{n,k},$ we will use the techniques developed in \cite{Su0}.  %For an elementary treatment of these and more general methods, see \cite{SuJer}. 

The Whitney homology of a poset was originally defined by Baclawski \cite{BaWhitneyHom}.   Bj\"orner showed \cite{Bj0WhitneyHom} that  the $i$th Whitney homology of a graded Cohen--Macaulay poset $P$ with  least  element $\hat 0$ is given by the isomorphism
\begin{equation}\label{WHBjdef}W\!H_i(P)\simeq\bigoplus_{x:\mathrm{rank}(x)=i} \tilde{H}_{i-2}(\hat 0, x).\end{equation}
Note that if $P$ has a top element $\hat 1,$ then the top Whitney homology coincides with the top homology of $P.$
%
\begin{comment}
If $G$ is a group acting on the poset $P$, then $W\!H_i(P)$ is also a (possibly virtual) $G$-module.
The present author observed  that the isomorphism~(\ref{WHBjdef})  is in fact group-equivariant, and also established the  equivariant acyclicity of Whitney homology   (see \cite{Su0}).  Thus  
(\ref{WHBjdef}) becomes an effective tool for computing both the $G$-module structure of Whitney homology as well as the homology of the full poset, as an equivariant analogue of the inherently recursive structure in the M\"obius function.  This technique was then  exploited 
in \cite{Su0} and \cite{SuJer} to study group actions on the homology of  posets.  
\end{comment}
The present author observed  that the isomorphism~(\ref{WHBjdef})  is in fact group-equivariant, leading to a powerful technique, the equivariant acyclicity of Whitney homology, to determine group actions on the homology of  posets \cite{Su0}.  

%If $G$ is a group acting on the poset $P$, and $P$ and all its intervals 
%$(\hat 0,x)$ have unique nonvanishing homology (e.g.\ if $P$ is Cohen--Macaulay), then each $W\!H_i(P)$ is also a true $G$-module and the isomorphism ~\eqref{WHBjdef} is $G$-equivariant, by a result of \cite{Su0}.

It is also computationally useful to consider the \textit{dual} Whitney homology of the Cohen--Macaulay poset $P$ when  $P$ has a top element $\hat 1$, that is, the Whitney homology of the dual poset $P^*$, which we denote by $W\!H^*(P).$ Note that we now have an equivariant isomorphism %(\cite{Su0})
\begin{equation}\label{WHdualBjdef}W\!H^*_i(P)\simeq\bigoplus_{x:\mathrm{rank}(x)=r-i} \tilde{H}_{i-2}( x, \hat 1), 0\leq i\leq r.\end{equation}
Here $r$ is the length of the longest chain from $\hat 0$ to $\hat 1.$

%See \cite{Su0} and \cite{SuJer} for a more general version of the following  theorem (for arbitrary posets).
\begin{thm}[{\cite[Lemma~1.1, Theorem 1.2, Proposition~1.9]{Su0}}]\label{WHmainSS} Let $P$ be a graded Cohen--Macaulay poset of rank $r$  carrying an action of a group $G.$ Then the unique nonvanishing top homology of $P$ coincides with the top Whitney homology module $W\!H_r(P),$ and  as a $G$-module, can be computed as an alternating sum of Whitney homology modules:
\begin{equation}\label{Acyclicity}\tilde{H}_{r-2}(P)\simeq \bigoplus_{i=0}^{r-1} (-1)^i W\!H_{r-1-i}(P).\end{equation}
In particular, if $P(\underline{k})$ denotes the subposet consisting of the first $k$ nonzero ranks, with a bottom and top element attached, then one has the $G$-module decomposition 
\begin{equation}\label{ConsecRanks}
\tilde{H}_{k-2}(P(\underline{k-1}))\oplus \tilde{H}_{k-1} (P(\underline{k}))
\simeq W\!H_k(P), r\geq k\geq 1.\end{equation}
Note that $W\!H_0(P)$ is the trivial $G$-module, while $W\!H_r(P)$ gives the reduced top homology of the poset $P$.
\end{thm}

Denote by $S_\lambda$ the irreducible representation of the symmetric group $S_n$ indexed by the partition $\lambda$ of $n$, and write $S_\lambda
^{\otimes i}$ for the $i$th tensor power of the module $S_\lambda$.

%Using the generating function for the M\"obius function of $A^*$ given in Theorem~\ref{Bjgf}, 
The theorem below computes all but the top Whitney homology $S_n$-modules for subword order.  The proof  requires a key formula, which we derive from the generating function for the M\"obius function of $A^*$ given in Theorem~\ref{Bjgf}. 

\begin{thm}\label{WHsubword} Consider the subword order poset $A^*_{n,k},$ with $|A|=n.$ As $S_n$-modules, the Whitney homology $W\!H(A^*_{n,k})$  and the dual Whitney homology $W\!H^*(A^*_{n,k}),$ 
for $1\le i\le k,$ are as follows. Note that $W\!H_0(A^*_{n,k})=S_{(n)}=W\!H_{k+1}^*(A^*_{n,k})$ (the trivial $S_n$-module).
\begin{equation}\label{Whitney}
W\!H_{i}(A^*_{n,k})=S_{(n-1,1)}^{\otimes i} \oplus S_{(n-1,1)}^{\otimes (i-1)};\end{equation}
\begin{equation}\label{dualWhitney} W\!H^*_{k+1-i}(A^*_{n,k})={k\choose i} S_{(n-1,1)}^{\otimes (k-i)} \otimes (S_{(n-1,1)}\oplus S_{(n)})^{\otimes i}\\
=  \bigoplus_{j=0}^i{k\choose i}{i\choose j} S_{(n-1,1)}^{\otimes j+(k-i)}.\end{equation}
%Also $W\!H_0(A^*_{n,k})=S_{(n)}=W\!H_{k+1}(A^*_{n,k})$ (the trivial $S_n$-module).
\end{thm}

Since the sum from ~\eqref{Whitney} telescopes,  acyclicity of Whitney homology, Theorem~\ref{WHmainSS}, immediately allows us to deduce the top homology:

\begin{cor}[{Bj\"orner--Stanley \cite[Theorem 4]{Bj1}}] The top homology of $A^*_{n,k}$ as an $S_n$-module is given by $S_{(n-1,1)}^{\otimes k}.$
%\begin{enumerate}
%\item (Bj\"orner-Stanley \cite[Theorem 4]{Bj1}) 
%\[S_{(n-1,1)}^{\otimes k}.\]
%\item 
%It is also equal to the alternating sums
%\[\sum_{i=0}^{k}(-1)^{k-i}\left(S_{(n-1,1)}\oplus S_{(n)}^{\otimes i}\otimes{k\choose i} S_{(n-1,1)}^{\otimes (k-i)}  \right)\]
%\[=\sum_{i=0}^k (-1)^{k-i} \bigoplus_{j=0}^i{k\choose i}{i\choose j} S_{(n-1,1)}^{\otimes j+(k-i)}.\]
%\end{enumerate}
\end{cor}
%\begin{proof} The first result is an immediate consequence of the fact that the alternating sum of Whitney homology modules in 
%\eqref{Whitney} telescopes to a single term, namely $S_{(n-1,1)}^{\otimes k}.$
%
%The second and third expressions are simply the alternating sums of dual Whitney homology modules in \eqref{dualWhitney}.
%\end{proof}



We can now state the main result of this section, which generalises the preceding corollary
%and \cite[Theorem 4]{Bj1} 
 to  the rank-set $[r,k]$ consisting of the interval of consecutive ranks $r, r+1, \ldots, k.$ To do this, we must rewrite the partial alternating sums of terms appearing in the dual Whitney homology \eqref{dualWhitney}  as a nonnegative linear combination rather than a signed sum. 
 The poset of words in an alphabet of size $n,$ with lengths bounded above by $k$ and below by $r,$ has homology as follows.

\begin{thm}\label{Bordeaux1991mai} 
%Fix $k\geq 1$ and let $S$ be the interval of consecutive ranks $[k-r,k]$ for $0\le r\le k-1.$ Then the rank-selected subposet $A^*_{n,k}(S)$ has unique nonvanishing homology in degree $r,$ and the $S_n$-homology representation on $\tilde{H}_r(A^*_{n,k}(S))$ 
%is given by the decomposition
%$\oplus_{i=1+r}^k  b_i(S_{(n-1,1)})^{\otimes i},$  where \[b_i={k\choose i}{i-1\choose r}, i=1+r, \ldots, k.\]
%%%%%%%%%%
Fix $k\geq 1$ and let $T$ be the interval of consecutive ranks $[r,k]$ for $1\le r\le k.$ Then the rank-selected subposet $A^*_{n,k}(T)$  has unique nonvanishing homology in degree $k-r,$ and the $S_n$-homology representation on $\tilde{H}_{k-r}(A^*_{n,k}(T))$ 
is given by the decomposition
\begin{equation}\label{epicstruggle}\bigoplus_{i=1+k-r}^k  b_i\,S_{(n-1,1)}^{\otimes i},  \text{ where }b_i={k\choose i}{i-1\choose k-r}, i=1+k-r, \ldots, k.\end{equation}
\end{thm}

The technically difficult proof of the preceding result, which we have omitted, also establishes the following combinatorial identity, which is instrumental in the proof of Theorem~\ref{Summary}. 
\begin{cor}\label{NewBinomialIdentity?}The alternating sum $\sum_{i=0}^{k+1-r} (-1)^i \mathrm{dim}\, W\!H^*_{k+1-(r+i)}(A^*_{n,k})$ equals  %\begin{multline*} \sum_{i=0}^{k+1-r} (-1)^i \mathrm{dim}\, W\!H^*_{k+1-(r+i)}(A^*_{n,k})\\
\[\sum_{i=0}^{k-r} (-1)^i {k\choose r+i}n^{r+i}(n-1)^{k-(r+i)}+(-1)^{k+1-r}
=\sum_{i=1+k-r}^k {k\choose i}{i-1\choose k-r} (n-1)^i.\]
%
%\end{multline*}
\end{cor}

\section{A curious isomorphism of homology}

In this section we will determine the homology representation of the rank-selected subposet $A^*_{n,k}(S)$ of $A^*_{n,k}$ when $S$ is obtained by deleting one rank from the interval $[1,k].$   In this special case   the computation will reveal a curious duality  in homology. 

We use a method developed in \cite[Theorem 1.10]{Su0}
% \cite{SuJer})
 which is particularly useful for  computation of the Lefschetz homology, i.e. the alternating sum of homology modules,  when the deleted set is an antichain.  %The  version  below is the special case when homology is concentrated in a single degree.
%
%\begin{thm}\label{antichain} (\cite[Theorem 1.10]{Su0}, \cite{SuJer}) Let $P$ be a Cohen--Macaulay poset of rank $r$, $G$ a group of automorphisms of $P$ and let $Q$ be a subposet obtained by deleting a $G$-invariant antichain $T$ in $P$.  Then $Q$ is also $G$-invariant.  Assume $Q$ is graded and has homology concentrated in the highest degree $rank(Q)-2$. Then one has the $G$-equivariant decomposition
%\begin{multline}(-1)^{r-rank(Q)}\tilde{H}(Q)-\tilde{H}_{r-2}(P)
%=\bigoplus_{x\in T/G}(-1)\cdot(\tilde{H}(\hat 0, x)_P\otimes \tilde{H}(x,\hat 1)_P)\uparrow_{stab(x)}^G,\\
% \text{ where the sum runs over one element } x\in T \text{ in each orbit of }G.
%\end{multline}
%Here $stab(x)$ denotes the stabiliser subgroup of $G$ which fixes the element $x.$
%\end{thm}
%
Applying this technique to the poset $A^*_{n,k}$ and the rank-set $T=[1,k]\backslash\{r\},$ removing all words of length $r,$ for a fixed $r$ in $[1,k],$ 
we obtain:

\begin{thm}\label{rank-deletion} As an $S_n$-module, we have 
\[\tilde{H}_{k-2}(A^*_{n,k}(T))\simeq \left[{k\choose r}-1\right]S_{(n-1,1)}^{\otimes k}
\oplus {k\choose r} S_{(n-1,1)}^{\otimes k-1}.\]
\end{thm}
In particular, Conjecture~\ref{Conj:tensorpowers} is true for this case of rank-selection.

%\textcolor{red}{IS THIS A PERM MODULE??} No. See results of Sec 6.

An immediate and intriguing corollary is the following.
\begin{prop}\label{rank-deletion-duality}  Let $|A|=n.$ Fix a rank $r\in [1,k-1].$ Then the homology modules of the subposets $A^*_{n,k}({[1,k]\backslash\{r\}})$ and 
$A^*_{n,k}({[1,k]\backslash\{k-r\}})$ are $S_n$-isomorphic.
%
\end{prop}

It would be interesting to explain this isomorphism topologically. More precisely:
\begin{qn} Is there an $S_n$-homotopy equivalence between the simplicial complexes associated to the subposets $A^*_{n,k}({[1,k]\backslash\{r\}})$ and 
$A^*_{n,k}({[1,k]\backslash\{k-r\}})$?
\end{qn}

\section{Chains and arbitrary rank-selected homology} 


 Assume $|A|=n.$ 
  For a subset $S\subseteq [1,k],$ denote by $\alpha_n(S)$ the permutation module of $S_n$ afforded by the maximal chains of the rank-selected subposet $A^*_{n,k}(S).$   In this section we derive a recurrence for the action, and hence an explicit formula.  The proof uses Part (2) of 
  Theorem~\ref{Bjgf}.
  
  \begin{thm}\label{RankSelectedChains} For any subset $T\subseteq [1,k],$ the action of $S_n$ on the maximal chains of the rank-selected subposet $A^*_{n,k}(T)$ is a nonnegative integer combination of tensor powers of the irreducible indexed by $(n-1,1).$ Hence the $S_n$-action on the homology of the rank-selected subposet $A^*_{n,k}(T), T\ne \emptyset,$ is an integer combination of positive tensor powers of the irreducible indexed by $(n-1,1).$ The highest tensor power that can occur is the $m$th, where $m=\max(T).$ 
  \end{thm}
  
  Thus Theorem~\ref{RankSelectedChains}   supports Conjecture~\ref{Conj:tensorpowers}.
%  Note that it is easy to concoct signed integer combinations of tensor powers that may or may not be true $S_n$-modules. For instance,  the integer combination $S_{(n-1,1)}^{\otimes 2}-2\, S_{(n-1,1)}$ decomposes into $S_{(n)}+S_{(n-2,2)}S_{(n-2,1,1)}-S_{(n-1,1)},$ while $S_{(n-1,1)}^{\otimes 2}-\, S_{(n-1,1)}=S_{(n)}+S_{(n-2,2)}+S_{(n-2,1,1)}$ is a true $S_n$-module. % See Theorem~\ref{myBrauer} below.
  
%  \textcolor{red}{WAIT!!! IS EVERY IRREDUCIBLE IN THE INTEGER SPAN OF THE TENSOR POWERS???  NOT TRUE FOR n=3 BUT SEEMS TRUE OTHERWISE.  TRUE FOR n=3 IF WE INCLUDE 0th TENSOR POWER AS THE TRIVIAL REP.  HUH??? This makes no sense.  Need at least p(n) powers to get the p(n) irreps.  But YES -- a result of Burnside says that a sufficiently large power of any faithful representation must contain all irreducibles...see Bessenrodt's review of Goupil-Chauve}
  
  It is worth pointing out the special case  for the full poset $A^*_{n,k}.$
  
   \begin{thm}\label{Chains} The action of $S_n$ on the maximal chains of ${A^*_{n,k}}$ decomposes into the sum 
  \[ S_{(n)}\oplus \bigoplus_{j=1}^k c(k+1,j) S_{(n-1,1)}^{\otimes k+1-j},\]
  where $c(k+1,j)$ is the number of permutations in $S_{k+1}$ with exactly $j$ cycles in its disjoint cycle decomposition.
  \end{thm}

\begin{comment}  
  By expanding the expression for $\alpha_n(S)$ in ~\eqref{ChainRec}, we have the following  observation. Although $S_{(n-1,1)}$ is a quotient of two permutation modules, it is not clear how to deduce this corollary directly.  Later in the paper we will examine these tensor powers more carefully; see Lemma~\ref{RefRephPos}.
  
  \begin{cor} The $S_n$-module $S_{(n)}\oplus \bigoplus_{j=1}^k c(k+1,j) S_{(n-1,1)}^{k+1-j}$ is in fact a permutation module.  
  More generally, for any subset $S=\{1\le s_1<\ldots<s_p\le k\}$ of $[1,k],$ the $S_n$-module 
  \[\bigotimes_{r=1}^p\left(\bigoplus_{i=0}^{s_r-s_{r-1}}{s_r\choose i} S_{(n-1,1)}^{\otimes i}\right), s_0=1,\]
  is a permutation module.
  \end{cor}
  
  \begin{proof} The expression gives the $S_n$-action on the chains of the rank-selected subposet 
  $A^*_{n,k}(S)$ and is therefore a permutation module. \end{proof}
  
\end{comment}
  
\begin{comment}
 We have the following two descriptions of the action on chains between two ranks. 
  \begin{prop}\label{2chains} For $S=\{1\le s_1<s_2\le k\},$ 
  $\alpha_n(\{s_1<s_2\})$ is given by 
  \begin{enumerate}
  
  \item \[\bigoplus_{j=0}^{s_1} \binom{s_1}{j} 
  S_{(n-1,1)}^{\otimes j}
  \otimes \left( \bigoplus_{i=0}^{s_2-s_1} \binom{s_2}{i} S_{(n-1,1)}^{\otimes i}\right);\]
%
  \item \[ (S_{(n-1,1)}\oplus S_{(n)})^{\otimes s_1}
  \otimes \bigoplus_{i,j\ge0, i+j=s_2-s_1}  \binom{s_1+j-1}{j}S_{(n-1,1)}^{\otimes j}\otimes
  (S_{(n-1,1)}\oplus S_{(n)})^{\otimes i}\]
   %
  \end{enumerate}
  \end{prop}
\end{comment}
 
%  We can now show that Conjecture~\ref{Conj:tensorpowers} is true for rank sets of size 2.
 
\begin{comment}
  \begin{thm}\label{TwoRanks}  Let $S=\{1\le s_1<s_2\le k\}$ be a rank-set of size 2 in $A^*_{n,k}.$ There are nonnegative integers $d_v$ such that the  homology representation of $A^*_{n,k}(S)$ is given by 
 \[ \oplus_{v=1}^{s_2}d_v\, S_{(n-1,1)}^{\otimes v}.\]
%  \[\sum_{v=1}^{s_1} S_{(n-1,1)}^{\otimes v} (c_v -\binom{s_1}{v}) \oplus \sum_{v=1+s_1}^{s_2} S_{(n-1,1)}^{\otimes v} c_v,\]
%  where $c_v$ is the following positive integer:
%  \[c_v
 % = \sum_{j=1}^{\min(v, s_2-s_1)}\binom{s_2-j}{v-j}\binom{s_1+j-1}{j} .\]
%  Moreover $c_v\ge \binom{s_1}{v}$ when $v\le s_1,$ and hence 
 % the homology is a nonnegative integer combination of tensor powers of $S_{(n-1,1)}.$
  \end{thm}
\end{comment}
  \section{The Whitney homology and the tensor powers $S_{(n-1,1)}^{\otimes k}$}\label{PowersOfRefl}

In this section we explore the tensor powers $S_{(n-1,1)}^{\otimes k}.$  
%The paper \cite{CG} gives a combinatorial model for determining the multiplicity of an irreducible in the $k$th tensor power, and an explicit formula in the case when $n$ is sufficiently larger than $k.$  We give general formulas that apply to the case of arbitrary tensor powers.
%
\begin{comment}
Burnside proved that given a faithful representation $V$ of a finite group $G$, every $G$-irreducible occurs in some tensor power of $V.$  A simple and beautiful proof of a generalisation of this was given by Brauer in \cite{BrauerR}. In the present context, it  states that since $S_{(n-1,1)}$ is a faithful representation whose character takes on $n$ distinct values (viz. $-1,0,1,\ldots,n-1,$ but not $n-2$), every irreducible $S_\lambda$ occurs in at least one of the $n$  tensor powers $S_{(n-1,1)}^{\otimes j}, 0\le j \le n-1.$ 

% A representation $\rho:G\rightarrow GL(V)$ is faithful if $\ker(\rho)=\{1\}.$ The fact that $\ker(\rho)=\{g\in G: \chi_\rho(g)=\chi_\rho(1)=\dim (V)\}$ (see Isaacs, Lemma 2.19) implies   that $\rho$ is faithful if and only if 
 %$\chi_\rho(g)=\chi_\rho(1)=\dim (V)$ implies $g=1.$ That is, $\rho$ is faithful if and only if $\dim(V)$ appears exactly once among the character values $\chi_\rho(g)$ of $\rho.$

In view of these results, the next fact, which we obtain by mimicking Brauer's elegant argument,
 is interesting.  We were unable to find it in the literature. 

\begin{thm}\label{myBrauer}  Let $G$ be any finite group and $X$ any character of $G$. Suppose $X$ takes on $k$ distinct nonzero values $b_i.$
% such that $b_i^m\ne b_j^m$ if  $i\ne j$, for $1\le m\le k.$ (Fr example this holds if all the $b_i$ are real).  NO, I don't need this!
Then the first $k$ tensor powers of $X$ are linearly independent functions on $G$, and form a basis for the subspace of class functions spanned by all the positive tensor powers.   
If $X^{k+1}=\sum_{i=1}^k c_i X^i,$ then the polynomial 
$P(t)=t^{k+1}-\sum_{i=1}^k c_i t^i$ has the factorisation 
$t\prod_{i=1}^k (t-b_i).$ 

Let $X^0=1_G$ denote the trivial character of $G.$  Then
$X^{k}=\sum_{i=1}^{k} c_i X^{i-1}$ if and only if the character $X$ never takes the value zero.
\end{thm}
\end{comment}
%
We use symmetric functions to describe some of the results that follow. See \cite{M} and \cite[Chapter 7]{St4EC2}. The homogeneous symmetric function $h_n$ is the Frobenius characteristic, denoted $\mathrm{ch},$ of the trivial representation of $S_n.$ Also let $*$ denote the internal product on the ring of symmetric functions, so that the Frobenius characteristic of the inner tensor product, or Kronecker product,  of two $S_n$-modules is the internal product of the two characteristics.

The following lemma is an easy exercise in permutation actions. %We sketch a proof for completeness.
%
\begin{lem}\label{Stirling} Let $V_{j,n}$ denote the permutation module obtained from the $S_n$-action on the cosets of the Young subgroup $S_1^j\times S_{n-j}.$ Then the $k$th tensor power of the natural representation $V_{1,n}$ of $S_n$ decomposes into a sum of $S(k,j)$ copies of $V_{j,n},$ where $S(k,j)$ is the Stirling number of the second kind:
\begin{equation}\label{StirlingNat} V_{1,n}^{\otimes k}
=\sum_{j=1}^{\min(n,k)}S(k,j)\, V_{j,n}, \quad \text{ and thus} \quad
(h_1h_{n-1})^{* k}=\sum_{j=1}^{\min(n,k)} S(k,j) \, h_1^j h_{n-j}.
\end{equation}
\end{lem}

Using the observation that the  Frobenius characteristic of 
$(S_{(n-1,1)})^{\otimes k}$  is the $k$-fold internal product of $(h_1 h_{n-1}-h_n) ,$ and standard manipulations in the ring of symmetric functions, we obtain the following description of the top homology.

\begin{thm}\label{StirlingHom} The top homology of $A^*_{n,k}$ has Frobenius characteristic 
\[\sum_{i=0}^{\min(n,k)} h_1^i h_{n-i} \left( \sum_{r=0}^{k-i} 
(-1)^{r}\binom{k}{r}S(k-r,i)\right).\]
%\[ (-1)^k h_n +\sum_{j=1}^k (-1)^{k-j} \binom{k}{j} \sum_{i=1}^{\min(j,n)} S(j,i)\,  p_1^i h_{n-i}.\]
%This may be rewritten
%\[\sum_{j=0}^k (-1)^{k-j} \binom{k}{j} \sum_{i=0}^{\min(j,n)} S(j,i)\,  p_1^i h_{n-i}.\]
%since the first term of the inner sum is zero  for $j\ge 1$ ($S(j,0)=0$ if $j>0.$)
\end{thm}
%
%% if you use biblatex then this generates the bibliography
%% if you use some other method then remove this and do it your own way
%\printbibliography

Theorem~\ref{ReflRepMain}  gives a different description of this module, from which it will be evident that the coefficients of $h_1^ih_{n-i}$ are positive for $i\ge 2.$

This gives the multiplicity of the trivial representation in 
the top homology of $A^*_{n,k}$:
\begin{cor}\label{TrivRep}  Let $n\ge 2.$ The multiplicity of the trivial representation in $S_{(n-1,1)}^{\otimes k}$ equals
\[\sum_{r=0}^k (-1)^r \binom{k}{r} \sum_{i=0}^{\min(n,k)} S(k-r,i).\]
%
%\begin{enumerate}
%\item the multiplicity of the trivial representation in $S_{(n-1,1)}^{\otimes k};$
%the top homology of $A^*_{n,k}$
%\item the multiplicity of the irreducible $S_{(n-1,1)}$  in 
%$S_{(n-1,1)}^{\otimes k-1};$
%the top homology of $A_{k-1}^*$
%\item the number 
%\[\sum_{r=0}^k (-1)^r \binom{k}{r} \sum_{i=0}^{\min(n,k)} S(k-r,i).\]
%\end{enumerate}
When $n\ge k,$ this is  %, which is also the stable multiplicity of the trivial representation in the $k$th tensor power of the irreducible indexed by $(n-1,1),$  
 the number of set partitions of $\{1,\ldots,k\}$ with no singleton blocks.
\end{cor}

\section{Proofs of main results}

In this section we indicate the main ideas leading to the proofs of Theorems~\ref{SummaryChains} and \ref{Summary} announced in the Introduction. 

Theorem~\ref{SummaryChains} is proved by an inductive argument using Theorem~\ref{RankSelectedChains} and Lemma~\ref{Stirling}.

The proof of Theorem~\ref{Summary} relies on the precise homology computations of the preceding sections, as well as Theorem~\ref{ReflRepMain} and Lemma~\ref{RefRephPos} below, which  are crucial.  
We first derive a different expression for the Whitney homology modules of $A^*_{n,k}$, thereby obtaining a new expression for the top homology module as well.
 
% Recall that the ordinary Stirling numbers of the second kind satisfy the recurrence $S(n+1,d)=S(n,d-1)+d S(n,d)$ with initial conditions $S(0,0)=1$ and $S(n,0)=0=S(0,d)$ if $n,d>0.$ %It is easy to verify similarly that the reduced Stirling numbers $S^*(j,d)$ satisfy the recurrence 
 % $ S^*(n+1,d)=S^*(n,d-1)+(d-1)S^*(n,d),$ by examining the possibilities for inserting $(n+1)$ into a partition of $[n]$ into $d$ blocks. A comparison of the recurrences immediately shows that in fact 
 % \[S^*(n+1,d)=S(n,d-1)\quad \text{for all } n\ge 0,d\ge 1.\] 
%  See  \cite{Munagi} for generalisations of these numbers. %(a  reduced Stirling number).      
  %
  Recall that in Theorem~\ref{WHsubword}, the $j$th Whitney homology of $A^*_{n,k}$ $j\ge 2,$ was determined as a sum of two consecutive tensor powers of $S_{(n-1,1)}.$  % From the preceding Lemma and Proposition~\ref{LowerIntTopo} 
  We now have the following surprising result.
  
  \begin{prop}\label{WhPermModule} Each Whitney homology module  of subword order, and hence the sum of two consecutive tensor powers of the reflection representation, has $h$-positive Frobenius characteristic, and in particular it is a permutation module.   We have  $\mathrm{ch\ }W\!H_0=h_n, \mathrm{ ch\ }W\!H_1=h_1 h_{n-1},$ and  for $k\ge j\ge 2,$ the $j$th Whitney homology of $A^*_{n,k}$ has Frobenius characteristic
  \begin{equation}\label{StirlingWh} (h_1h_{n-1})*s_{(n-1,1)}^{*(j-1)}=\sum_{d= 2}^j S(j-1,d-1)\, h_1^d h_{n-d}=\sum_{d=2}^j S^*_{j,d}\,h_1^d h_{n-d}= h_1(h_1h_{n-2})^{*j-1},
  \end{equation}
%  In particular, since $Wh_0=S_{(n)},$ the Whitney homology modules of $A^*_{n,k}$ are all  
a permutation module with orbits whose stabilisers are Young subgroups indexed by partitions of the form $(n-d,1^d), d\ge 0.$  

  \end{prop}
  
 \begin{thm}\label{ReflRepMain} Fix $k\ge 1.$ 
  The $k$th tensor power of the reflection representation  
  $S_{(n-1,1)}^{\otimes k},$ i.e. the homology module $\tilde{H}_{k-1}(A^*_{n,k}),$ has the following property:
  $S_{(n-1,1)}^{\otimes k}\oplus (-1)^{k} S_{(n-1,1)}$ 
  is a permutation module $U_{n,k}$ whose Frobenius characteristic is $h$-positive, and  is supported on the set $\{h_\lambda: \lambda=(n-r, 1^r), r\ge 2\}.$  
  If $k=1,$ then $U_{n,1}=0.$
  %
More precisely, the $k$-fold internal product $s_{(n-1,1)}^{ * k}$ has the following expansion in the basis of  symmetric functions $h_\lambda:$ 
  \begin{equation} \label{RefReptoh}
  \sum_{d=0}^n g_n(k,d)  h_1^d h_{n-d}, \, where
  \end{equation}
  \[g_n(k,0)=(-1)^k=- g_n(k,1), \text{ and }g_n(k,d)=\sum_{i=d}^k (-1)^{k-i} S(i-1, d-1) \text{ for }2\le d\le n.\] 
  
  Hence  $s_{(n-1,1)}^{ * k}= (-1)^{k-1}s_{(n-1,1)}+\mathrm{ch}\,(U_{n,k}),$ where $\mathrm{ch}\,(U_{n,k})=\sum_{d=2}^n g_n(k,d)  h_1^d h_{n-d}.$
  
  The integers $g_n(k,d)$ are independent of $n$ for $k\le n,$  nonnegative for $2\le d\le k,$ and  $g_n(k,d)=0 $ if $d>k.$  
%%  Also:
%%  \begin{enumerate} 
%%  \item $g_n(k,2)=\frac{1+(-1)^k}{2}.$ % 0 if  k\ge 1 odd, 1 if k\ge 2 even.
%%  \item $g_n(k,k-1)=\binom{k-1}{2}-1, k\le n.$
  %OLD=\frac{k(k-3)}{2}
%%  \item $g_n(k,k)=1, k< n.$
 %OLD \item $g_n(n, n-1)=\binom{n-1}{2}-1.$
  %OLD=\sum_{i=n-1}^n (-1)^{n-i}S(i-1,n-2)=
 %OLD \begin{cases} 1, & k=n-1,\\%-1+S(n-1,n-2)
  %OLD                     \binom{n-1}{2}-1, & k=n. 
  % OLD                    \end{cases}$        
%  \end{enumerate}
%In particular the coefficient of $h_1^n$ in the expansion ~\eqref{RefReptoh} of $\mathrm{ch}\ S_{(n-1,1)}^{\otimes k}$ is 

 % $\begin{cases} g_n(k,n)+g_n(k,n-1) &\mathrm{ if }\ k> n,\\
 %                       \binom{n-1}{2} &\mathrm{ if }\ k= n,\\
 %                        1 &\mathrm{ if }\ k=n-1,\\
 %                         0 &\mathrm{otherwise}. \end{cases}$
 %OLD \begin{equation}\label{WhPerm}
 %OLD (-1)^{k-1} (-h_n + h_1 h_{n-1})+\sum_{d=2}^{n} h_1^d h_{n-d} \sum_{j=d}^k S(j-1,d-1) (-1)^{k-j}.\end{equation}
%OLD  \[=(-1)^{k-1} S_{(n-1,1)} +\sum_{d=2}^{\min(n,k)} h_1^d h_{n-d} \sum_{j=d}^k S(j-1,d-1) (-1)^{k-j}\]
  \end{thm}
  
   There follows a key lemma connecting Conjectures~1 and 2, namely:

% We need one final observation in order to prove Theorem~\ref{SummaryChains} and Theorem~\ref{Summary}.
  \begin{lem}\label{RefRephPos} Suppose $V$ is an $S_n$-module which can be written as an integer combination $V=\oplus_{k=1}^m c_k S_{(n-1,1)}^{\otimes k}$ of positive tensor powers of $S_{(n-1,1)}.$ 
  %$c_k\ge 0.$
   If 
  $\sum_{k=1}^m  (-1)^{k-1}c_k=0,$ then the Frobenius characteristic of $V$ is %$h$-positive and 
  supported on the set $\{h_\lambda: \lambda=(n-r, 1^r), r\ge 2\}.$  If in addition $c_k\ge 0$ for all $k\ge 2,$ the Frobenius characteristic is $h$-positive and hence $V$ is a permutation module.  
  \end{lem}
As an example of its application, we can deduce from this lemma and Theorem~\ref{WHsubword} that
  
   \begin{cor} The dual Whitney homology modules $W\!H^*_{k+1-i}(A^*_{n,k}), 1<i\le k,$ are permutation modules whose  Frobenius characteristic is a nonnegative integer combination of the set $T_2=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 2\}.$
  \end{cor}



 Applying Stanley's theory of rank-selection \cite{St1} in conjunction with Theorem~\ref{SummaryChains} now establishes the result below.

\begin{thm}\label{Homologyhbasis}  Let $T\subseteq [1,k]$  be any nonempty subset of ranks in $A^*_{n,k}.$ The following statements hold for the Frobenius characteristic $F_{n}(T)$ of the homology representation $\tilde{H}(A^*_{n,k}(T)):$ 
 \begin{enumerate}  
 \item its expansion in the basis of homogeneous symmetric functions is an integer combination supported on the set $T_{1}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 1\}.$
 \item $F_{n}(T) +(-1)^{|T|} s_{(n-1,1)}$ is supported on the set 
 $T_{2}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 2\}.$
 \end{enumerate}
 %
 \end{thm}
%
Theorem~\ref{Summary} describes the cases for which we are able to establish that  the symmetric function $F_n(T)+(-1)^{|T|} s_{(n-1,1)}$ is in fact a \textit{nonnegative} integer combination of $T_{2}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 2\}.$ %we may view $F_n(T)$ as being almost a permutation module, as in the cases of Theorem~\ref{Summary}. %In order to prove the above result,  we need the stronger result of Theorem~\ref{SummaryChains} for the action on the chains. 
%
\begin{comment}
\begin{thm}\label{Chainhbasis}  Let $S\subseteq[1,k].$ If $|S|\ge 1,$  the action of $S_n$ on the space of chains $\alpha_n(S)$ has $h$-positive Frobenius characteristic supported on the set $T_{1}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 1\}.$ % If $|S|=1,$ it is supported on the set $T_{ 1}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 1\}.$
 Furthermore, $h_1 h_{n-1}$ always appears with coefficient 1 in the $h$-expansion of $\alpha_n(S)$.
 \end{thm}
 \end{comment}
 
 The preceding theorem motivates Conjecture~\ref{Conj:almost-h-pos} in the Introduction.  
\begin{comment}
 We will show that Conjecture~\ref{Conj:almost-h-pos} is true in the following cases of rank-selection:

 \begin{thm}\label{AlmosthpositiveInstances} For any nonempty rank set $T\subseteq[1,k],$ consider the  module 
  $V_T=\tilde{H}_{k-2}(A^*_{n,k}(T))+ (-1)^{|T|} S_{(n-1,1)}.$  In each of the following cases, $V_T$  is a nonnegative combination of transitive permutation modules with orbit stabilisers 
of the form $S_1^d\times S_{n-d}, d\ge 2.$  Equivalently, the symmetric function $F_{n,k}(T) +(-1)^{|T|} s_{(n-1,1)}$ is supported on the set 
 $T_{2}(n)=\{h_\lambda: \lambda=(n-r, 1^r), r\ge 2\}$ with nonnegative integer coefficients in each of the following cases:
\begin{enumerate}
\item $T=[r,k], k\ge r\ge 1.$ 
%\item $T=\{r\}, r\ge 1.$
\item $T=[1,k]\backslash\{r\}, k\ge r\ge 1.$ 
%\item $T=\{1<a\}.$
\item $T=\{1\le s_1<s_2\le k\}.$
\end{enumerate}
  \end{thm}
\end{comment}  
  %

\section{Enumerative Consequences}
  We conclude  by pointing out a representation-theoretic consequence, and some enumerative implications, of the expansion ~\eqref{RefReptoh}.
%\begin{comment}
  Fix $n\ge 3$ and consider the $n$ by $n-1$ matrix $D_{n}$ whose $k$th column consists of the coefficients $g_n(n-k,n-d), d=1, \ldots, n-1.$  Thus the $k$th column contains the coefficients in the expansion of $S_{(n-1,1)}^{\otimes n-k}$ in the $h$-basis:  we have 
  $\mathrm{ch}\, S_{(n-1,1)}^{\otimes k}=\sum_{d=1}^n g_n(k, n-d)h_1^{n-d} h_d  , 1\le k\le n-1.$ From Theorem~\ref{ReflRepMain} it is easy to see that the matrix $D_{n}$ has rank $(n-1);$   the last two rows, consisting of alternating $\pm 1$s, differ by a factor of $(-1)$, and  the matrix is lower triangular with 1's on the diagonal, hence it has rank $(n-1).$  
  Similarly the $(n+1)$ by $(n-1)$ matrix obtained by appending to $D_{n} $ a first column consisting of the $h$-expansion of the $n$th tensor power of 
  $S_{(n-1,1)}^{\otimes n}$ also has rank $(n-1)$. %We therefore have a second proof of Theorem~\ref{myBrauer}, for the  modules $S_{(n-1,1)}$. In this special case we can now be more precise about the linear combination of tensor powers:
%\end{comment}
  \begin{thm} In the representation ring of the symmetric group $S_n,$ the first $(n-1)$ tensor powers of $S_{(n-1,1)}$ are an integral basis for the vector space spanned by the positive tensor powers.  The $n$th tensor power of $S_{(n-1,1)}$ is an integer linear combination of the first $(n-1)$ tensor powers:
  \[S_{(n-1,1)}^{\otimes n}=\bigoplus_{k=1}^{n-1} a_k(n) S_{(n-1,1)}^{\otimes k}, \text{  with }a_{n-1}(n)= \tbinom{n-1}{2}.\]
  
  Let $c(n,j)$ be the number of permutations in $S_{n}$ with exactly $j$ disjoint cycles. The coefficients $a_k(n)$ are determined by the polynomial 
   $P(t)=t^n -\sum_{k=1}^{n-1} a_k(n) t^{k},$  
  defined by
 \begin{equation}
  P(t)=\tfrac{t+1}{t-(n-2)}\sum_{j=1}^{n} c(n,j)t^{j} (-1)^{n-j}.
 \end{equation}
% where $c(n,j)$ is the number of permutations in $S_{n}$ with exactly $j$ disjoint cycles.
  
\begin{comment}
  These coefficients satisfy the following equations:
   \begin{align}
  %1&=\sum_{i=1}^{n-1}(-1)^i a_i(n);\\ (redundant)
  \text{If $n$ is odd, then }0&=\sum_{1\le i \le n-1, i \text{ odd}} a_i(n) (-1)^i; \\
  \text{If $n$ is even, then }1&=\sum_{1\le i \le n-1, i \text{ even}} a_i(n) (-1)^i. \end{align}
\end{comment}
   \end{thm}
   
%  The paper \cite{MG} has results of a similar flavour in greater generality, involving the Stirling numbers and a $k$-fold transitive action of a group $G$ on a finite set $X.$ In particular, that author gives \cite[Eqn.~(7')]{MG}, for the multiplicity of the trivial representation in the $k$th tensor power of the permutation module $1\uparrow_{S_1\times S_{n-1}}^{S_n},$ the sum $\sum_{i=1}^{\min(n,k)} S(k,i),$ which can be recovered from our computations above.

 %Similarly, our results above can be generalised to the  case of $G$ acting on $X$, by considering tensor powers of the complement of  the fixed space $\mathbb{C}<\sum_{x\in X} x>$  in the linearised permutation module $X.$
  % He defines: The stabilizing character for a group $G$ acting on a set $X$ is defined to be the character whose value on $\sigma$ is the number of elements in $X$ fixed by $\sigma.$   Assuming $G$ is $k$-fold transitive, (i.e. for $k$-subsets A={a_1,..,a_k}, B={b_1,..,b_k} of X, there is a g in G such g(a_i)=b_i for all a_i in A) He looks at the $k$th tensor power of this action and expresses it in terms of Stirling numbers of both kinds.
% So for $G=S_n$ acting on [n], this would be the natural action, hence $h_1 h_{n-1}.$
  
%  different expr for mult triv rep Checked  comb identity in Maple
%  Rank sel in normal word poset? Seems hard; don't know dual Whitney hom
%  \textit{ Also write down mult of triv rep in chains....
%   nonstable case?
 % Dual Whitney in coloured injective poset? (Athanasiadis?)}
 
 %%%%%%%%%%%%%%%%%commenting out the following, since we know the polynomial

The preceding result gives a recurrence for the coefficients $a_k(n)$;
we have
\begin{align*} a_{n-1}(n)&=\tbinom{n-1}{2};\\
(n-2)a_j(n)-a_{j-1}(n)&=(-1)^{n-j}[c(n,j)-c(n,j-1)], 2\le j\le n-1;\\
(n-2) a_1(n)&= c(n,1) (-1)^{n-1}\\
\Longrightarrow a_1(n)&= \tfrac{(n-1)!}{n-2} (-1)^{n-1} =(-1)^{n-1}[(n-2)!+(n-3)!]
\end{align*}

Theorem~\ref{ReflRepMain} has the following interesting corollary.
\begin{cor}\label{TrivRepStableCase} Let $k\ge 2.$  
  \begin{enumerate}
  \item For $\min(n,k)\ge d\ge 2,$ the coefficient of $h_1^d h_{n-d}$ in $s_{(n-1,1)}^{* k}=\mathrm{ch}\, S_{(n-1,1)}^{\otimes k}$   is the nonnegative integer $g_n(k,d)$ given by the  two equal expressions:
  \begin{equation}\label{NewStirlingIdentity?}
  \sum_{j=d}^k (-1)^{k-j} S(j-1,d-1)=\sum_{r=0}^{k-d} (-1)^r \binom{k}{k-r} S(k-r,d).
  \end{equation}
  In particular, when $n\ge k,$ this multiplicity is independent of $n.$
  \item The positive integer $\beta_n(k)=\sum_{d=2}^{\min(n,k)} g_n(k,d)$ is the multiplicity of the trivial representation in $S_{(n-1,1)}^{\otimes k}.$  When $n\ge k,$ it equals the number of set partitions $B_k^{\ge 2}$ of the set $\{1,\ldots,k\}$ with no singleton blocks.  We have $\beta_n(n+1)=B_{n+1}^{\ge 2}-1$ and $\beta_n(n+2)= B_{n+2}^{\ge 2}- \binom{n+1}{2}.$
  \end{enumerate}
  \end{cor}
 
 \begin{qn} The identity ~\eqref{NewStirlingIdentity?} holds for all $d=2,\ldots,k.$ 
 Is there a combinatorial explanation?
 \end{qn}
 
  \begin{qn} For fixed $k$ and $n,$ what do the positive integers $g_n(k,d)$ count?   Is there a combinatorial interpretation for $\beta_n(k)=\sum_{j=d}^{\min(n,k)}g_n(k,d),$  the multiplicity of the trivial representation in the top homology of $A^*_{n,k},$  in the nonstable case $k>n?$  Recall that for $k\le n$ this is the number $B_k^{\ge 2}$ of set partitions of $[k]$ with no singleton blocks, and is sequence OEIS A000296.  
 \end{qn}
 
 \begin{qn} Recall that $a_{n-1}(n)=\binom{n-1}{2}.$ Is there a combinatorial interpretation for the signed integers $a_i(n)$?  There are many interpretations for $(-1)^{n-1} a_1(n)= (n-2)!+(n-3)!,$ see OEIS A001048.  For $n\ge 4$ it is the size of the largest conjugacy class in $S_{n-1}.$  We were unable to find the other sequences $\{a_{i}(n)\}_{n\ge 3} $ in OEIS.
 %Number of {12, 12*, 1*2, 21, 21*}-avoiding signed permutations in the hyperoctahedral group.
%a(n) is the hook product of the shape (n, 1). - Emeric Deutsch, May 13 2004
%From Jaume Oliver Lafont, Dec 01 2009: (Start)
%(1+(x-1)*exp(x))/x = Sum_{k >= 1} x^k/a(k).
%Setting x = 1 yields Sum_{k >= 1} 1/a(k) = 1. [Jolley eq 302] (End)
% For n >= 2, a(n)=n!+(n-1)! is the size of the largest conjugacy class of the symmetric group on n + 1 letters. Equivalently, the maximum entry in each row of A036039. - Geoffrey Critzer, May 19 2013
 \end{qn}
 
\begin{comment}
 \begin{ex} Write $X_n^k$ for  $S_{(n-1,1)}^{\otimes k}.$ Maple computations  with Stembridge's SF package show that 
\begin{enumerate}
\item $X_3^3= X_3^2+2X_3.$
\item $X_4^4= 3X_4^3+X_4^2-3X_4.$
% $X_4^5= 10X_4^3-9X_4.$
\item $X_5^5= 6X_5^4-7X_5^3-6X_5^2 +8 X_5.$
\item $X_6^6=10X_6^5 -30X_6^4+20X_6^3+31X_6^2-30X_6$
\item $X_7^7=15X_7^6 -79X_7^5 +165X_7^4 -64 X_7^3 -180 X_7^2+144 X_7$
\item $X_8^8=21 X_8^7 -168 X_8^6 +630 X_8^5 -1029 X_8^4 +189 X_8^3 +1198 X_8^2 -840 X_8.$
\end{enumerate}
\end{ex}
\end{comment}

\printbibliography

\end{document}
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\end{thebibliography}

\end{document}

\section{A section}

\lipsum[3]

\begin{equation}\label{eqn:eq1}
\int_0^\infty \exp(-x)dx = 1.
\end{equation}

\begin{lem}\label{lem:lem1}
\lipsum[11]
\end{lem}

\lipsum[10]

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{plot}
\caption{A plot of a function.}
\label{fig:plot}
\end{figure}

\subsection{A subsection}

\lipsum[5]

\begin{align*}
1 + 1 + 1 &= 2 + 1 \\ &= 3.
\end{align*}

\subsubsection{A subsubsection}

\lipsum[17]

Here is a citation~\cite{greenwade93} with URL. Here is another
citation~\cite{MR4}, which is the earliest-numbered Math Review with a
functioning DOI: MR0000004. Finally, a reference to
equation~\eqref{eqn:eq1}. And a reference to Figure~\ref{fig:plot}.

\acknowledgements{\lipsum[20]}

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