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\title[Unit Interval Orders and Hessenberg Varieties]{Unit Interval Orders and the Dot Action on the Cohomology of Regular Semisimple Hessenberg Varieties}

\author{Patrick Brosnan\thanks{\href{mailto:pbrosnan@umd.edu}{pbrosnan@umd.edu}. Partially supported by NSF grant DMS-1361159.}\addressmark{1} \and Timothy Y. Chow\thanks{\href{mailto:tchow@alum.mit.edu}{tchow@alum.mit.edu}}\addressmark{2}}

\address{\addressmark{1}Department of Mathematics, University of Maryland, College Park, MD 20742, USA \\ \addressmark{2}Center for Communications Research, 805 Bunn Drive, Princeton, NJ 08540, USA}

\received{October 17, 2016}

\abstract{
Motivated by a 1993 conjecture of Stanley and Stembridge,
Shareshian and Wachs conjectured
that the characteristic map takes the dot action of the symmetric group
on the cohomology of a regular semisimple Hessenberg variety
to $\omega X_G(t)$, where $X_G(t)$ is the chromatic quasisymmetric
function of the incomparability graph~$G$ of the
corresponding natural unit interval order,
and $\omega$ is the usual involution on symmetric functions.
We prove the Shareshian--Wachs conjecture.
Our proof uses the local invariant cycle theorem
of Beilinson--Bernstein--Deligne to obtain a surjection from
the cohomology of a regular Hessenberg variety of Jordan type~$\lambda$
to a space of local invariant cycles;
as $\lambda$ ranges over all partitions, these spaces
collectively contain all the information
about the dot action on a regular semisimple Hessenberg variety.
Using a palindromicity argument,
we show that in our case the surjections are actually isomorphisms,
thus reducing the Shareshian--Wachs conjecture to computing the cohomology
of a regular Hessenberg variety.  But this cohomology has already
been described combinatorially by Tymoczko; we give a bijective proof
(using a generalization of a combinatorial reciprocity theorem of Chow)
that Tymoczko's combinatorial description coincides with
the combinatorics of the chromatic quasisymmetric function.
}

\keywords{chromatic quasisymmetric function, indifference graph, local invariant cycles, palindromic}


\begin{document}

\maketitle

\section{Introduction}

Let $G$ be the incomparability graph of a unit interval order
(also known as an \emph{indifference graph}),
i.e., a finite graph whose vertices are closed unit intervals
on the real line, and whose edges join overlapping unit intervals.
It is a longstanding conjecture \cite{stanley-stembridge}
related to various deep conjectures about immanants
that if $G$ is such a graph,
then the so-called \emph{chromatic symmetric function}
$X_G$ studied by Stanley~\cite{stanley-xg} is $e$-positive,
i.e., a nonnegative combination of elementary symmetric functions.
(In fact, Stanley and Stembridge conjectured something seemingly
more general, but Guay-Paquet~\cite{guay-paquet} has reduced
their conjecture to the one stated here.)
Early on, Haiman~\cite{haiman} proved that the expansion
of~$X_G$ in terms of Schur functions
has nonnegative coefficients, and Gasharov~\cite{gasharov}
showed that these coefficients enumerate certain combinatorial
objects known as \emph{$P$-tableaux}.
It is well known that if $\chi$ is a character of
the symmetric group~$S_n$, then the image of~$\chi$
under the so-called characteristic map~$\ch$
\begin{equation}
\label{eq:ch}
\ch \chi := \frac{1}{n!} \sum_{\sigma\in S_n} \chi(\sigma) \,
    p_{\cycletype(\sigma)}
\end{equation}
(where $p$ here denotes the power-sum symmetric function)
is a nonnegative linear combination of Schur functions,
with the coefficients giving the multiplicities of
the corresponding irreducible characters of~$S_n$.
One may therefore suspect that $X_G$ is the image under~$\ch$
of the character of some naturally occurring representation of~$S_n$,
but until recently, there was no candidate, even conjecturally,
for such a representation.


Meanwhile, independently and seemingly unrelatedly,
De~Mari, Procesi, and Shayman~\cite{demari-procesi-shayman}
inaugurated the study of \emph{Hessenberg varieties}.
Let $\bm = (m_1, m_2, \ldots, m_{n-1})$ be a weakly
increasing sequence
of positive integers satisfying $i\le m_i\le n$ for all~$i$,
and let $s:\mathbb{C}^n \to \mathbb{C}^n$ be a linear transformation.
The (type~A) Hessenberg variety $\sH(\bm,s)$ is defined by
\begin{equation}
\label{eq:hessenberg}
\sH(\bm,s) := \{
\mbox{complete flags $F_1 \subseteq F_2 \subseteq \cdots \subseteq F_n :
	sF_i \subseteq F_{m_i}$ for all $i$}\}.
\end{equation}
The geometry of a Hessenberg variety depends on the Jordan form of~$s$.
If the Jordan blocks have distinct eigenvalues then we say that
$\sH(\bm, s)$ is \emph{regular},
and if $s$ is diagonalizable then we say that
$\sH(\bm, s)$ is \emph{semisimple}.
Of particular interest to us is that
there is a representation, called the \emph{dot action}, of~$S_n$
on the cohomology of
regular semisimple Hessenberg varieties.  To the best of our
knowledge, this dot action was first defined by Tymoczko, who
asked for a complete description of it~\cite{tymoczko};
e.g., one can ask if there is a combinatorial formula for
the multiplicities of the irreducible representations
and/or for the character values.
(Note that Tymoczko defines the dot action in terms of
something called the \emph{moment graph};
the moment graph, and hence the dot action,
depends only on~$\bm$ and not on the 
choice of regular semisimple $s$.)

A connection between these two apparently unrelated topics has
been conjectured by
Shareshian and Wachs~\cite{shareshian-wachs-0,shareshian-wachs}.
Motivated by the $e$-positivity conjecture,
they have generalized $X_G$ to something they call the
\emph{chromatic quasisymmetric function} $X_G(t)$ of a graph,
which is a polynomial in~$t$ with power series coefficients
that reduces to~$X_G$ when $t=1$.
They also noted that if we are given a sequence~$\bm$ as above,
and we let $G(\bm)$ be the undirected
graph on the vertex set $\{1, 2, \ldots, n\}$
such that $i$ and~$j$ are adjacent if $i<j \le m_i$,
then $G(\bm)$ is an indifference graph,
and moreover that every indifference graph
is isomorphic to some $G(\bm)$.
They then made the following conjecture.
Let $\omega$ denote the usual involution
on symmetric functions~\cite[Section 7.6]{stanley-ec2}.
\begin{conjecture}
\label{conj:sw}
If $\chi_{\bm,d}$ denotes the dot action
on the cohomology group $H^{2d}$ of the regular semisimple
Hessenberg variety $\sH(\bm, s)$,
then $\ch \chi_{\bm,d}$ equals
the coefficient of $t^d$ in
$\omega X_{G(\bm)}(t)$.
\end{conjecture}
This conjecture is intriguing not only because it would
answer Tymoczko's question, but it would open up the possibility
of proving the $e$-positivity conjecture by geometric techniques.

The main result of the present paper is a proof of
\cref{conj:sw}.
The linchpin of our proof is the following result.

\begin{thm}
\label{thm:linchpin}
Let $\lambda$ be a partition of~$n$.
Let $s$ be a regular element with Jordan type~$\lambda$,
and let $S_\lambda := S_{\lambda_1} \times \cdots\times S_{\lambda_\ell}$
be a Young subgroup of the symmetric group~$S_n$.
Consider the restriction of $\chi_{\bm,d}$ to $S_\lambda$.
Then the dimension of the subspace fixed by~$S_\lambda$
equals the Betti number $\beta_{2d}$ of $\sH(\bm,s)$.
\end{thm}

It is a standard fact (\cref{prop:frobenius}
below) from the representation theory of~$S_n$
that the dimension of the subspace fixed by $S_\lambda$
in a representation~$\chi$ is the coefficient of $m_\lambda$
in the monomial symmetric function expansion of $\ch \chi$,
and therefore these dimensions completely determine~$\chi$.
So \cref{thm:linchpin} reduces the
computation of the dot action on
the cohomology of a regular semisimple Hessenberg variety
to the computation of the cohomology of
regular (but not necessarily semisimple) Hessenberg varieties.
However, this latter task has already been largely carried out
by Tymoczko~\cite{tymoczko-linear},
who has given a combinatorial description of the Betti numbers $\beta_{2d}$
for all Hessenberg varieties in type~A.
So with \cref{thm:linchpin} in hand,
all that remains to prove \cref{conj:sw} is
to give a bijection between Tymoczko's combinatorial description
and the combinatorics of $\omega X_{G(\bm)}(t)$.
To do this, we first compute the
coefficients $c_{d,\lambda}(\bm)$ of $t^d m_\lambda$
in the monomial symmetric function expansion
of $\omega X_{G(\bm)}(t)$.
We do this with a generalization of
a combinatorial reciprocity theorem of Chow (\cref{thm:reciprocity}).
This yields a description of $c_{d,\lambda}(\bm)$ that is almost,
but not quite, identical to Tymoczko's description of $\beta_{2d}$;
we show that the descriptions are equivalent
by describing a explicit bijection
between the two (\cref{thm:betti}).
As a corollary (\cref{pbetti}), we derive the fact that the
Betti numbers of regular Hessenberg varieties
form a palindromic sequence
(even though the varieties are not smooth),
because Shareshian and Wachs have proved that
$\omega X_{G(\bm)}(t)$ is palindromic.

In this extended abstract, we give complete definitions and
theorem statements (without proofs, which of course are in the
full version) of the combinatorial ingredients described above.
There is no space to provide details of the geometric ingredients,
or even a complete definition of the dot action,
so we limit ourselves to the following brief comments
about the proof of \cref{thm:linchpin}.
The idea is to show
that Tymoczko's dot action coincides with the monodromy action for the
family $\sH^{\mathrm{rs}}(\bm)\to\mathfrak{g}^{\mathrm{rs}}$ of
Hessenberg varieties
over the space of regular semisimple $n\times n$ matrices.
This allows us to apply results from the theory
of local systems and perverse sheaves to questions involving the dot
action.  In particular, the local invariant cycle theorem of
Beilinson--Bernstein--Deligne
\cite[Corollaire 6.2.9]{bbd}
implies that there is a surjective map from
the cohomology of a regular Hessenberg variety to the space of local
invariants of the monodromy action near a regular element $s$ in the
space $\mathfrak{g}$ of all $n\times n$-matrices.
We prove a general theorem showing that
the local invariant cycle map is an
isomorphism if and only if the Betti numbers of the special fiber are
palindromic in a suitable sense;
but the Betti numbers \emph{are} palindromic in our case
(by \cref{pbetti}), 
so the local invariant cycle map is indeed an isomorphism in our case.
Then we show
that the local invariant cycles near a regular element $s$ with Jordan
type~$\lambda$ coincide with
$S_{\lambda}$ invariants of the dot action on the regular semisimple
Hessenberg variety.  The latter fact is proved by a monodromy argument
that uses the Kostant section.  (This Kostant section argument and some
other ingredients of the proof were inspired by
Ng{\^o}'s paper on the Hitchin fibration~\cite{NgoHitchin}.)

\subsection{Previous work and acknowledgments}

Prior to our work, \cref{conj:sw} was already known
for some graphs~$G$:
a complete graph (trivial),
a complete graph minus an edge~\cite{teffthesis},
a complete graph minus a path of length three (Tymoczko, unpublished),
and a path (by piecing together known results as
explained in~\cite{shareshian-wachs}).
Teff also showed that it would suffice to prove the conjecture
for all \emph{connected} graphs~$G$.
In a different direction,
Abe, Harada, Horiguchi and Masuda~\cite{ahhm2}
proved that the multiplicity of the trivial representation is indeed as
predicted by \cref{conj:sw}.  (Hearing about this development
and reading the last paragraph of~\cite{ahhm}, which explains how to
compute the multiplicity of the trivial representation in terms of the
regular nilpotent Hessenberg variety, partially inspired our own proof.)
They also computed the ring structure on regular semisimple
Hessenberg varieties of type $(m_1,n,\ldots, n)$, and
deduced \cref{conj:sw} in that case from the computation.

Shortly after this work was completed, Guay--Paquet~\cite{guay-paquet2}
announced an independent proof of \cref{conj:sw} using
completely different methods.

Brosnan would like to thank Jim Carrell for discussions about
Hessenberg varieties and equivariant cohomology and Mark Goresky for
mentioning Springer's Bourbaki article~\cite{springer}, which helped
convince us to look for a description of Tymoczko's dot action in
terms of monodromy.  He would also like to thank Najmuddin Fakhruddin
and Nero Budur for suggestions about relevant literature,
and Tom Haines for discussions about
Grothendieck's simultaneous resolution.  Finally, he thanks the
Institute for Advanced Study for hospitality in Princeton, NJ during
the Fall of 2014 when this work started.

\section{Preliminaries}\label{prelim}
We fix some notation that will be used throughout the paper.

\subsection{General notation}\label{gn}

We let $\mathbb{P}$ denote the positive integers.
If $n\in \mathbb{P}$, we let $[n]$ denote the set $\{1, 2,\ldots, n\}$.

The vector $\bm = (m_1, \ldots, m_{n-1})$
will always denote a \emph{Hessenberg function},
by which we mean a sequence of positive integers satisfying
\begin{enumerate}
\item $m_1\le m_2\le \cdots \le m_{n-1} \le n$, and
\item $m_i \ge i$ for all $i$.
\end{enumerate}
We also define
\begin{equation}
\label{eq:mdim}
\left|\mathbf{m}\right| := \sum_{i=1}^{n-1} (m_i-i).
\end{equation}
Given $\bm$, let $P(\bm)$ be the poset
on the vertex set $[n]$ whose order relation $\prec$ is given by
\[ i\prec j \Longleftrightarrow j \in \{m_i+1, m_i+2, \ldots, n\}.\]
Such a poset is called a \emph{natural unit interval order}.
(It is a theorem that
every unit interval order in the sense defined in the introduction
is isomorphic to some natural unit interval order.)
The \emph{incomparability graph} $G(\bm)$ is the undirected
graph on the vertex set $[n]$ in which $i$ and~$j$ are adjacent
if and only if $i$ and~$j$ are incomparable in~$P(\bm)$.
In other words, if $i<j$ then $i$ and~$j$ are adjacent in $G(\bm)$
if and only if $j\le m_i$.

An \emph{integer partition}
$\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_\ell)$ of
a positive integer~$n$ is
a weakly decreasing sequence of
positive integers that sum to~$n$,
while a \emph{composition}
$\alpha = (\alpha_1, \alpha_2, \ldots, \alpha_\ell)$ of
a positive integer~$n$ is any sequence
of positive integers that sum to~$n$.
We visualize a composition of~$n$
by drawing vertical bars in some subset of the $n-1$ spaces between
consecutive objects in a horizontal line of $n$ objects;
the parts $\alpha_i$ are then the numbers of objects between successive bars.
Motivated by the equivalence between compositions and sets of bars,
we define the following (possibly not quite standard) notation:
$|\alpha|$ for the number of bars of~$\alpha$;
$\overline \alpha$ for the composition that has bars in precisely
the positions where $\alpha$ does \emph{not} have bars; and
$\alpha \le \beta$ if the bars of~$\alpha$
are a subset of the bars of~$\beta$.

We write $S_n$ for the symmetric group.
If $S_n$ acts in the usual way on a set of size~$n$,
and $\alpha$ is a composition of~$n$,
then the \emph{Young subgroup}~$S_\alpha$
is the subgroup
\begin{equation}
\label{eq:young}
S_{\alpha_1} \times S_{\alpha_2} \times \cdots \times S_{\alpha_\ell}
  \subseteq S_n
\end{equation}
comprising all the permutations that permute the first $\alpha_1$
elements among themselves, the next $\alpha_2$ elements among themselves,
and so on.

An \emph{ordered (set) partition}
$\sigma = (\sigma_1, \sigma_2, \ldots, \sigma_\ell)$ of
a finite set~$S$ is a sequence of pairwise disjoint subsets of~$S$
whose union is~$S$.

A \emph{sequencing} $\sq$ of a finite set~$S$ of cardinality~$n$
is a bijective map $\sq:[n] \to S$.
It is helpful to think of $\sq$ as the sequence $\sq(1), \ldots, \sq(n)$
of elements of~$S$.

By a \emph{digraph} we mean a finite directed graph with no loops or
multiple edges but that may have bidirected edges, i.e., it
may contain both $u\to v$ and $v\to u$ simultaneously.
If $D$ is a digraph, we write $\overline D$ for the
\emph{complement} of~$D$, i.e., the digraph with the same vertex set
as~$D$ but with a directed edge $u\to v$ if and only if
there does \emph{not} exist a directed edge $u\to v$ in~$D$.

\subsection{Symmetric and quasisymmetric functions}

We mostly follow the notation of Stanley~\cite{stanley-ec2}
for symmetric functions.
For convenience, we recall some of that notation here.
Let $\mathbf{x} = \{x_1, x_2, x_3, \ldots\}$ be a countable set
of independent indeterminates.
If $\kappa:[n] \to \mathbb{P}$ is a map
then we write $\mathbf{x}_\kappa$ for the monomial
$x_{\kappa(1)} x_{\kappa(2)} \cdots x_{\kappa(n)}$.
A formal power series in $\mathbf{x}$ is a \emph{symmetric function}
if it is invariant under any permutation of the variables~$\mathbf{x}$.
If $\lambda=(\lambda_1, \lambda_2, \ldots, \lambda_\ell)$ is
an integer partition,
then the \emph{monomial symmetric function} $m_\lambda$
is the symmetric function of minimal support that contains the monomial
$x_1^{\lambda_1}x_2^{\lambda_2}\cdots x_\ell^{\lambda_\ell}$.

The \emph{characteristic map}~$\ch$
is a function that sends characters~$\chi$
of the symmetric group to symmetric functions via the formula
\begin{equation}
	\ch \chi := \frac{1}{n!} \sum_{\sigma\in S_n} \chi(\sigma) \,
	    p_{\cycletype(\sigma)}
\end{equation}
where $\cycletype(\sigma)$ is the integer partition consisting
of the cycle sizes of~$\sigma$, listed with multiplicity in
weakly decreasing order,
and $p$ denotes the power-sum symmetric function.
As we explained in the introduction,
the following standard fact is an important ingredient in our proof.

\begin{proposition}
\label{prop:frobenius}
Let $\rho$ be a finite-dimensional representation of~$S_n$,
and let~$\chi$ be its character.  Let
$\ch\chi = \sum_\lambda c_\lambda m_\lambda$
be the monomial symmetric function expansion of $\ch\chi$.
Then $c_\lambda$ equals the dimension of the subspace
fixed by any Young subgroup $S_\lambda\subseteq S_n$.
In particular, knowing $c_\lambda$ for all $\lambda$
uniquely determines~$\chi$.
\end{proposition}

Let $\alpha = (\alpha_1, \alpha_2, \ldots, \alpha_\ell)$
be a composition of~$n$.  The \emph{monomial quasisymmetric function}
$M_\alpha$ is the formal power series defined by
\begin{equation}
\label{eqn:Malpha}
M_\alpha := \sum_{i_1<\cdots<i_\ell} x_{i_1}^{\alpha_1} \cdots
   x_{i_\ell}^{\alpha_\ell},
\end{equation}
where the sum is over all strictly increasing sequences
$(i_1, \ldots, i_\ell)$ of positive integers.
A formal power series is a \emph{quasisymmetric function}
if it is a scalar linear combination of monomial quasisymmetric functions.
Note that symmetric functions are always quasisymmetric, but not vice versa.

The \emph{fundamental quasisymmetric function} $F_\alpha$
of Gessel~\cite{gessel} is defined by
\begin{equation}
   \label{eq:fundamental}
   F_\alpha := \sum_{\beta\ge\alpha} M_\beta.
\end{equation}
By inclusion-exclusion,
\begin{equation}
   \label{eq:incexc}
   M_\alpha = \sum_{\beta\ge\alpha} (-1)^{|\beta|-|\alpha|} F_\beta.
\end{equation}

\subsection{Hessenberg varieties}

As mentioned in the introduction,
if $\bm$ is a Hessenberg function and
$s:\mathbb{C}^n \to \mathbb{C}^n$ is a linear transformation,
then we define the \emph{Hessenberg variety} (of type~A,
which is the only type that we consider in this paper) by
\begin{equation*}
\sH(\bm,s) := \{
\mbox{complete flags $F_1 \subseteq F_2 \subseteq \cdots \subseteq F_n :
sF_i \subseteq F_{m_i}$ for all $i$}\}.
\end{equation*}
If the Jordan blocks of~$s$ have distinct eigenvalues then we say that
$\sH(\bm, s)$ is \emph{regular}, if $s$ is diagonalizable then we say
that $\sH(\bm, s)$ is \emph{semisimple}, and if $s$ is nilpotent then
we say that $\sH(\bm, s)$ is \emph{nilpotent}.  Since $\sH(\bm,s)$ can
equal $\sH(\bm,s')$ for $s\neq s'$ (e.g., if $s'-s$ is a constant),
this is a very minor abuse of terminology.

Hessenberg varieties are projective as they are closed subschemes
of the projective variety of complete flags.
If $s$ is regular semisimple then they are smooth,
but in general they may be singular, and sometimes not reduced.
This explains why we cannot simply cite a duality theorem
to prove palindromicity.

\section{The chromatic quasisymmetric function}\label{reciprocity}

Given a graph $G$ whose vertex set is a subset of~$\mathbb{P}$,
Shareshian and Wachs~\cite{shareshian-wachs} define the
\emph{chromatic quasisymmetric function} $X_G(\mathbf{x},t)$ of~$G$.

\begin{definition}
Let $G$ be a graph whose vertex set~$V$ is a finite subset of~$\mathbb{P}$.
Let $C(G)$ denote the set of all proper colorings of~$G$,
i.e., the set of all maps $\kappa:V\to\mathbb{P}$ such
that adjacent vertices are always mapped to distinct positive integers.
Then
\begin{equation}
\label{eqn:XGt}
X_G(\mathbf{x}, t) := \sum_{\kappa\in C(G)}
t^{\asc\kappa}\,\mathbf{x}_\kappa,
\end{equation}
where
\[ \asc{\kappa} := \left|\left\lbrace \mbox{$\{u,v\}$}
 : \mbox{$\{u,v\}$ is an edge of $G$ and $u<v$ and
    $\kappa(u) < \kappa(v)$}\right\rbrace \right|.
\]
\end{definition}

For brevity, we sometimes write $X_G(t)$ for $X_G(\mathbf{x}, t)$.
It will be convenient for us to restate the definition of~$X_G(t)$
in terms of monomial quasisymmetric functions.

\begin{proposition}
Let $G$ be a graph whose vertex set~$V$ is a finite subset of~$\mathbb{P}$.
Then
\begin{equation}
X_G(\mathbf{x},t) = \sum_{\sigma=(\sigma_1, \ldots, \sigma_\ell)}
t^{\asc\sigma}\, M_{|\sigma_1|, \ldots, |\sigma_\ell|},
\end{equation}
where the sum is over all ordered partitions~$\sigma$ of~$V$
such that every~$\sigma_i$ is a stable set of~$G$
(i.e., there is no edge between any two vertices of~$\sigma_i$),
and $\asc{\sigma}$ is the number of edges $\{u,v\}$ of~$G$ such
that $u<v$ and $v$ appears in
a later part of~$\sigma$ than $u$ does.
\end{proposition}

We remark that if we set $t=1$ then the chromatic quasisymmetric
function specializes to the \emph{chromatic symmetric function}~$X_G$
of Stanley~\cite{stanley-xg}.

\subsection{Reciprocity}

If $f$ is a symmetric function, then a ``reciprocity theorem,''
loosely speaking, is a result that gives a combinatorial interpretation
of $\omega f$, where $\omega$ is a well-known involution
on symmetric functions~\cite[Section 7.6]{stanley-ec2}.
Since \cref{conj:sw} concerns $\omega X_G(t)$
rather than $X_G(t)$ itself, one might expect a reciprocity theorem
to be relevant.  This is indeed the case.
Specifically, the coefficients of the monomial symmetric function
expansion of $\omega X_G(t)$ play an important role in our arguments,
so we now introduce some notation for them.

\begin{definition}
Given a Hessenberg function~$\bm$,
we let $c_{d,\lambda}(\bm)$ be the coefficients defined by
the following expansion of $\omega X_{G(\bm)}(\mathbf{x}, t)$
in terms of monomial symmetric functions:
\begin{equation}
\label{eq:monom}
   \omega X_{G(\bm)}(\mathbf{x}, t) =
         \sum_d t^d \sum_\lambda c_{d,\lambda}(\bm) m_\lambda.
\end{equation}
\end{definition}

Our starting point is the observation that
Chow~\cite[Theorem 1]{chow} has proved a reciprocity theorem for
a symmetric function invariant of a digraph called the
\emph{path-cycle symmetric function}~$\Xi_D$.
There is a certain precise sense in which $\Xi_D$
is equivalent to Stanley's $X_G$ in the case of posets,
but the nice thing about reciprocity for $\Xi_D$
is that it naturally yields a combinatorial interpretation for the
coefficients of the monomial symmetric function expansion
of~$\omega \Xi_D$, which is not immediately evident from
Stanley's reciprocity theorem~\cite[Theorem 4.2]{stanley-xg} for~$X_G$.
This fact suggests the following plan: Generalize $\Xi_D$ to $\Xi_D(t)$
(just as Shareshian and Wachs have generalized $X_G$ to $X_G(t)$),
prove reciprocity for $\Xi_D(t)$,
and read off the desired combinatorial interpretation
of~$c_{d,\lambda}(\bm)$.
This plan works, and we now show how to carry it out.

We define the \emph{path quasisymmetric function} $\Xi_D(\mathbf{x},t)$
of a digraph~$D$; as its name suggests,
it enumerates paths only and not cycles
(since for our present purposes we do not care about enumerating cycles),
and it has a definition analogous to that of the chromatic quasisymmetric
function.

\begin{definition}
Let $D$ be a digraph whose vertex set~$V$ is a subset of~$\mathbb{P}$.
An \emph{ordered path cover} of~$D$ is an ordered pair $(\sq,\beta)$
such that $\sq$ is a sequencing of~$V$,
$\beta = (\beta_1, \ldots, \beta_\ell)$
is a composition of $n := |V|$, and
\[ \sq(\beta_{i-1}+1)\to \sq(\beta_{i-1}+2)\to \cdots\to \sq(\beta_i)\]
is a directed path in~$D$ for all $i\in [\ell]$
(adopting the convention that $\beta_0=0$).
Define
\begin{equation}
\label{eq:xidt}
\Xi_D(\mathbf{x},t) := \sum_{(\sq,\beta)} t^{\asc \sq}\, M_\beta
\end{equation}
where the sum is over all ordered path covers $(\sq,\beta)$ of~$D$
and $\asc \sq$ is the number of pairs $\{u,v\}$ of vertices of~$D$
such that
\begin{enumerate}
\item either $u\to v$ and $v\to u$ are both edges of~$D$ or neither one is,
\item $u<v$, and
\item $v$ appears later in the sequencing~$\sq$ than $u$ does.
	\end{enumerate}
\end{definition}

For brevity, we sometimes write $\Xi_D(t)$ for $\Xi_D(\mathbf{x},t)$.

Although we are ultimately interested in expansions in terms of
monomial \emph{symmetric} functions,
it turns out that the proofs are more naturally stated in terms of
monomial \emph{quasisymmetric} functions.
So we need to describe the action of~$\omega$
on monomial quasisymmetric functions.

\begin{definition}
The linear map $\omega$ on quasisymmetric functions is defined by the
following action on monomial quasisymmetric functions.
\begin{equation}
\label{eq:omega}
\omega M_\beta := (-1)^{|\beta|} \sum_{\alpha\le \beta} M_\alpha.
\end{equation}
\end{definition}

It is known (e.g., see the proof of~\cite[Theorem 4.2]{stanley-xg})
that the usual map~$\omega$ is characterized by the
equation $\omega F_\alpha = F_{\overline\alpha}$,
so the following proposition confirms that
our definition of $\omega$ coincides with the standard one.

\begin{proposition}
$\omega F_\alpha  = F_{\overline\alpha}$.
\end{proposition}

We are ready for the reciprocity theorem for~$\Xi_D(t)$.

\begin{thm}
\label{thm:reciprocity}
Let $D$ be a digraph whose vertex set~$V$ is a subset of~$\mathbb{P}$.
Then $\omega\Xi_D(\mathbf{x},t) = \Xi_{\overline D}(\mathbf{x}, t)$.
\end{thm}

\cref{thm:reciprocity} gives us
a nice combinatorial interpretation of $c_{d,\lambda}(\bm)$.

\begin{corollary}
\label{cor:asc}
Let $\bm$ be a Hessenberg function, and
let $D(\bm)$ denote the digraph on~$[n]$
that has an edge $u\to v$ if and only if $v\prec u$ in~$P$.
Then for any composition $\alpha$ whose parts are a permutation of
the parts of~$\lambda$, $c_{d,\lambda}(\bm)$ equals
the number of ordered path covers $(\sq,\alpha)$ of $\overline{D(\bm)}$
with $\asc \sq = d$.
\end{corollary}

\section{Betti numbers of regular Hessenberg varieties}\label{sbetti}

The main result of this section is that if
$\sH(\bm,s)$ is a regular Hessenberg variety
and $s$ has Jordan type~$\lambda$,
then its Betti number $\beta_{2d}$
equals $c_{d,\lambda}(\bm)$.

Tymoczko~\cite[Theorem 7.1]{tymoczko-linear}
has already done a lot of the work needed to prove
this result, by showing that Hessenberg varieties admit a paving
(or cellular decomposition) by affine spaces,
and obtaining a combinatorial interpretation
of the dimensions of the cells.
For regular Hessenberg varieties,
Tymoczko's theorem simplifies as follows.
If $\lambda$ is an integer partition of~$n$ then
by a \emph{tableau of shape~$\lambda$}
we mean any filling of the boxes of the Young diagram of~$\lambda$
with one copy each of the numbers $1,2,\ldots, n$.

\begin{thm}[Tymoczko]
\label{thm:tymoczko}
Let $\sH(\bm,s)$ be a regular Hessenberg variety
and let the partition $\lambda$ encode the sizes of the Jordan blocks of~$s$.
Then $\sH(\bm,s)$ is paved by affines.
The nonempty cells are in bijection with tableaux~$T$ of shape~$\lambda$
with the property that $k$ appears in the box immediately to the left of~$j$
only if $k\le m_j$.  The dimension of a nonempty cell is the sum of:
\begin{enumerate}
\item the number of pairs $i,k$ in~$T$ such that
\begin{enumerate}
\item $i$ and $k$ are in the same row,
\item $i$ appears somewhere to the left of~$k$,
\item $k<i$, and
\item if $j$ is in the box immediately to the right of~$k$ then $i\le m_j$;
\end{enumerate}
\item the number of pairs $i,k$ in~$T$ such that
\begin{enumerate}
\item $i$ appears in a lower row than $k$, and
\item $k < i \le m_k$.
\end{enumerate}
\end{enumerate}
\end{thm}

It remains for us to establish a correspondence between the combinatorics
of \cref{thm:tymoczko} and the combinatorics of~$\omega X_{G(\bm)}(t)$,
or equivalently (by the results of the previous section) the
combinatorics of ordered path covers.

\begin{definition}
If $X$ is a topological space and $i$ is an integer, we write
$\beta_i$ or $\beta_i(X)$ for the $i$-th Betti number $\dim H^i(X,\mathbb{C})$
of $X$.
\end{definition}

\begin{thm}
\label{thm:betti}
Let $\sH(\bm,s)$ be a regular Hessenberg variety and let the Jordan
type of~$s$ be~$\lambda$.  Then the Betti number $\beta_{2d}$ of
$\sH(\bm,s)$ equals $c_{d,\lambda}(\bm)$, and $\beta_i=0$ for $i$ odd.
\end{thm}

\cref{thm:betti} is proved bijectively, by matching up the
combinatorics of ordered path covers with the above combinatorial
description by Tymoczko.

Shareshian and Wachs proved palindromicity for the chromatic
quasisymmetric function, so we can now deduce the following
palindromicity result for Hessenberg varieties.

\begin{corollary}\label{pbetti}
Let $\sH(\mathbf{m},s)$ be a regular Hessenberg variety with $s$ of
type $\lambda$ as in \cref{thm:betti}.   Set
\[
 q=q_{\sH(\mathbf{m},s)}:=\sum_{i\in\mathbb{Z}} \beta_{i} t^{i-|\mathbf{m}|}.
\]
Then $q(t)=q(t^{-1})$.
\end{corollary}

This concludes the combinatorial portion of our proof.
At this point in the full version of the paper, we continue with the
geometric portion of the proof, but in this extended abstract,
we do not have the space to say anything about the
geometric portion beyond what we already sketched
in the introduction.

\printbibliography



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