text:
\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 ${\mathbf{m}} = (m_1, m_2, \ldots, m_{n-1})$ be a monotonically
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 ${\mathscr{H}}({\mathbf{m}},s)$ is defined by
\begin{equation}
\label{eq:hessenberg}
{\mathscr{H}}({\mathbf{m}},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
${\mathscr{H}}({\mathbf{m}}, s)$ is \emph{regular},
and if $s$ is diagonalizable then we say that
${\mathscr{H}}({\mathbf{m}}, s)$ is \emph{semisimple}.
Hessenberg varieties have many interesting properties,
but 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.

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~${\mathbf{m}}$ as above,
and we let $G({\mathbf{m}})$ 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 $G({\mathbf{m}})$ is an indifference graph,
and moreover that every indifference graph
is isomorphic to some $G({\mathbf{m}})$.
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 $\chi_{{\mathbf{m}},d}$ denotes the dot action
on the cohomology group $H^{2d}$ of the regular semisimple
Hessenberg variety ${\mathscr{H}}({\mathbf{m}}, s)$,
then $\ch \chi_{{\mathbf{m}},d}$ equals
the coefficient of $t^d$ in
$\omega X_{G({\mathbf{m}})}(t)$.
\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
Conjecture~\ref{conj:sw} (Theorem~\ref{thm:sw}).
The linchpin of our proof is the following result
(which is stated more formally later as Theorem~\ref{dsf}).

\begin{theorem}
\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 $\chi_{{\mathbf{m}},d}$ to $S_\lambda$.
Then the dimension of the subspace fixed by~$S_\lambda$
equals the Betti number $\beta_{2d}$ of ${\mathscr{H}}({\mathbf{m}},s)$.
\end{theorem}

What Theorem~\ref{thm:linchpin} does is to reduce the problem of
computing the dot action on a regular semisimple Hessenberg variety
to computing 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 Theorem~\ref{thm:linchpin} in hand,
all that remains to prove Conjecture~\ref{conj:sw} is
to give a bijection between Tymoczko's combinatorial description
and the combinatorics of $\omega X_{G({\mathbf{m}})}(t)$.
More precisely, it is a standard fact (Proposition~\ref{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 $\ch \chi$.
So the first step of our proof is to compute the
coefficients $c_{d,\lambda}({\mathbf{m}})$ of $t^d m_\lambda$
in the monomial symmetric function expansion
of $\omega X_{G({\mathbf{m}})}(t)$.
We do this with a generalization of
a combinatorial reciprocity theorem of Chow (Theorem~\ref{thm:reciprocity}).
This yields a description of $c_{d,\lambda}({\mathbf{m}})$ 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 (Theorem~\ref{thm:betti}).
As a corollary (Corollary~\ref{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
$\omega X_{G({\mathbf{m}})}(t)$ is palindromic.

The idea behind the proof of Theorem~\ref{thm:linchpin} is to show
that Tymoczko's dot action coincides with the monodromy action for the
family ${\mathscr{H}}^{\mathrm{rs}}({\mathbf{m}})\to\mathfrak{g}^{\mathrm{rs}}$ of Hessenberg varieties
over the space of regular semisimple $n\times n$ matrices
(Theorem~\ref{tdot}).  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, which is stated in our context as
Theorem~\ref{thm: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.  In
Theorem~\ref{t1}, 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. Then, in Theorem~\ref{tiy}, 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
ingredients of the proof of Theorem~\ref{tdot} were inspired by
 Ng{\^o}'s paper on the Hitchin fibration~\cite{NgoHitchin}.)

 Here is a brief description of the contents by
 section. Section~\ref{prelim} mainly fixes notation and gives preliminary
 results.  Section~\ref{reciprocity} proves the combinatorial reciprocity
 theorem, Theorem~\ref{thm:reciprocity}, mentioned above.
 Section~\ref{sbetti} proves Theorem~\ref{thm:betti} on the Betti numbers
 of regular Hessenberg varieties, and derives palindromicity as a corollary
 of a theorem of Shareshian and Wachs.
 Section~\ref{lmlfg} reviews the concept of local monodromy and
 the related notion of a good fundamental system of neighborhoods to a
 point in topological space.
 Section~\ref{sbn} proves Theorem~\ref{t1} on the local invariant cycle map.
 Section~\ref{gc} proves Proposition~\ref{pgal}
 on local monodromy of a Galois cover, which is applied later
 (in Lemma~\ref{myoung}) to
 compute the local monodromy near a matrix $s$ of type~$\lambda$.
 Section~\ref{sghs} introduces the family ${\mathscr{H}}({\mathbf{m}})\to\mathfrak{g}$ of
 Hessenberg varieties.  Finally, Section~\ref{mtd} shows that the monodromy
 action coincides with Tymoczko's dot action, and uses this fact to
 prove Theorem~\ref{thm:sw}, which is a restatement of
 Conjecture~\ref{conj:sw}.

\subsection{Previous work and acknowledgments}

Prior to our work, Conjecture~\ref{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 (personal communication)
proved that the multiplicity of the trivial representation is indeed as
predicted by Conjecture~\ref{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 Conjecture~\ref{conj:sw} in that case from the computation.

 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 literature relevant to
 section~\ref{lmlfg}, 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 ${\mathbf{m}} = (m_1, \ldots, m_{n-1})$
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 ${\mathbf{m}}$, let $P({\mathbf{m}})$ denote 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}.
The \emph{incomparability graph} $G({\mathbf{m}})$ 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~$P({\mathbf{m}})$.
In other words, if $i<j$ then $i$ and~$j$ are adjacent in $G({\mathbf{m}})$
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 monotonically decreasing sequence of
positive integers that sum to~$n$.
Each $\lambda_i$ is a \emph{part} of~$\lambda$,
and the number of parts of~$\lambda$ is denoted by $\ell(\lambda)$.
The \emph{Young diagram} of~$\lambda$
comprises $\ell$ rows of boxes, left-justified,
with $\lambda_i$ boxes in the $i$th row from the top.

A \emph{composition}
$\alpha = (\alpha_1, \alpha_2, \ldots, \alpha_\ell)$ of
a positive integer~$n$ is a sequence (not necessarily monotonic)
of positive integers that sum to~$n$.
Each $\alpha_i$ is a \emph{part} of~$\alpha$,
and the number of parts of~$\alpha$ is denoted by $\ell(\alpha)$.
It can be useful to 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 are then the numbers of objects between successive bars.
Motivated by the equivalence between compositions and sets of bars,
we define:
\begin{itemize}
\item $|\alpha|$ for the number of bars of~$\alpha$;
\item $\overline \alpha$ for the composition that has bars in precisely
the positions where $\alpha$ does \emph{not} have bars;
\item $\alpha \cup \beta$ for the composition whose bars comprise
the union of the bars of $\alpha$ and the bars of~$\beta$; and
\item $\alpha \le \beta$ if the bars of~$\alpha$
are a subset of the bars of~$\beta$.
\end{itemize}

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} ${q}$ of a finite set~$S$ of cardinality~$n$
is a bijective map ${q}:[n] \to S$.
It is helpful to think of ${q}$ as the sequence ${q}(1), \ldots, {q}(n)$
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
$x_1^{\lambda_1}x_2^{\lambda_2}\cdots x_\ell^{\lambda_\ell}$.  For example,
\[ m_{2,1,1} = x_1^2 x_2 x_3 + x_2^2 x_1 x_3 + x_3^2 x_1 x_2 +
               x_1^2 x_3 x_4 + x_3^2 x_1 x_4 + x_4^2 x_1 x_3 +  \cdots\]
There is an unfortunate conflict between
our notation for monomial symmetric functions and
our notation~${\mathbf{m}}$ for Hessenberg functions.
It should be clear from context which is meant since
the subscript of a monomial symmetric function is a partition,
whereas the entries of~${\mathbf{m}}$ have integer subscripts.

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
monotonically 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}

\begin{proof}
Let $\langle \cdot,\cdot\rangle$ denote the
standard inner product on symmetric functions
\cite[Section 7.9]{stanley-ec2},
and let $\chi{\mathord{\downarrow}}^{S_n}_{S_\lambda}$ denote the
restriction of~$\chi$ to~$S_\lambda$.
Let $d_\lambda$ be
the dimension of the subspace fixed by~$S_\lambda$.
Then $d_\lambda$ equals
the multiplicity of the trivial representation~$\mathbf{1}$
in $\chi{\mathord{\downarrow}}^{S_n}_{S_\lambda}$, i.e.,
$d_\lambda = \langle \mathbf{1}, \chi{\mathord{\downarrow}}^{S_n}_{S_\lambda}\rangle$.
By Frobenius reciprocity~\cite[Theorem 1.12.6]{sagan},
\begin{equation}
\langle \mathbf{1}, \chi{\mathord{\downarrow}}^{S_n}_{S_\lambda}\rangle
= \langle \mathbf{1}{\mathord{\uparrow}}^{S_n}_{S_\lambda}, \chi\rangle,
\end{equation}
where $\mathbf{1}{\mathord{\uparrow}}^{S_n}_{S_\lambda}$ is the induction
of $\mathbf{1}$ from~$S_\lambda$ up to~$S_n$.
But $\mathbf{1}{\mathord{\uparrow}}^{S_n}_{S_\lambda}$ is just
the  homogeneous symmetric function $h_\lambda$
\cite[Corollary 7.18.3]{stanley-ec2}.
The monomial symmetric functions and the complete
homogeneous symmetric functions are dual bases
\cite[Equation (7.30)]{stanley-ec2},
so $d_\lambda = \langle h_\lambda, \chi\rangle = c_\lambda$.
\end{proof}

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 ${\mathbf{m}}$ 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*}
{\mathscr{H}}({\mathbf{m}},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
${\mathscr{H}}({\mathbf{m}}, s)$ is \emph{regular}, if $s$ is diagonalizable then we say
that ${\mathscr{H}}({\mathbf{m}}, s)$ is \emph{semisimple}, and if $s$ is nilpotent then
we say that ${\mathscr{H}}({\mathbf{m}}, s)$ is \emph{nilpotent}.  Since ${\mathscr{H}}({\mathbf{m}},s)$ can
equal ${\mathscr{H}}({\mathbf{m}},s')$ for $s\neq s'$ (e.g., if $s'-s$ is a constant),
this is a (very minor) abuse of terminology.
We adopt the convention of writing
$y$ for~$s$ in the regular semisimple case.

\begin{remark}
  The Hessenberg varieties are defined on affine open subsets of the
  complete flag variety by fairly obvious equations.  So they are
  closed subschemes of the complete flag variety in a natural way.  In
  general, they are not irreducible.  For example, the regular
  semisimple Hessenberg variety corresponding to the function
  $\Bell=(1,2,\ldots, n-1)$ is a collection of $n!$ distinct
  points.  They are also not always reduced.  For example, when $n=2$
  and ${\mathbf{m}}=\Bell$ as above, the regular nilpotent Hessenberg
  variety is defined by the equation $x^2=0$ in $\mathbb{A}^1$.
  (See~\cite[Theorem 7.6]{dt} for a much more general statement.)
  Hartshorne defines an abstract variety to be an integral separated
  scheme of finite type over an algebraically closed
  field~\cite[p.105]{hartshorne}.  So, perhaps, it is unfortunate that
  Hessenberg varieties are called varieties as they are not, in
  general, integral.  However, it happens that we are only introduced in Betti
  cohomology in this paper.  So the non-reduced structure will not
  play a role.  Moreover, we will reserve the term ``Hessenberg
  \emph{scheme}'' for the families discussed in \S\ref{sghs}.  So we will
  stick with tradition and continue to call the schemes ${\mathscr{H}}({\mathbf{m}},s)$
  Hessenberg \emph{varieties}.
\end{remark}

\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}

\begin{proof}
Given a coloring $\kappa\in C(G)$, let $\sigma_i$ be the set of vertices
that are assigned the $i$th smallest color.
Then it is immediate that
\begin{enumerate}
\item $\sigma=(\sigma_1, \ldots, \sigma_\ell)$ is an ordered partition
of the vertex set of~$G$;
\item $\sigma_i$ is a stable set for all~$i$; and
\item $\asc \kappa = \asc\sigma$.
\end{enumerate}
It is easy to see that if we sum $\mathbf{x}_\kappa$ over all
$\kappa\in C(G)$ that yield the same ordered partition~$\sigma$,
then we obtain the monomial quasisymmetric function $M_\alpha$
where the $i$th part~$\alpha_i$ of the composition~$\alpha$
is the cardinality $|\sigma_i|$ of~$\sigma_i$.
The proposition follows.
\end{proof}

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

It is possible to derive a combinatorial interpretation for
$c_{d,\lambda}({\mathbf{m}})$ by using
the reciprocity theorem of Shareshian and Wachs
\cite[Theorem 3.1]{shareshian-wachs}.
However, as we now explain, we shall take a different route.

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~$c_{d,\lambda}({\mathbf{m}})$.
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 $({q},\beta)$
such that ${q}$ is a sequencing of~$V$,
$\beta = (\beta_1, \ldots, \beta_\ell)$
is a composition of $n := |V|$, and
\[ {q}(\beta_{i-1}+1)\to {q}(\beta_{i-1}+2)\to \cdots\to {q}(\beta_i)\]
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_{({q},\beta)} t^{\asc {q}}\, M_\beta
\end{equation}
where the sum is over all ordered path covers $({q},\beta)$ of~$D$
and $\asc {q}$ 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~${q}$ than $u$ does.
\end{enumerate}
\end{definition}

For brevity, we sometimes write $\Xi_D(t)$ for $\Xi_D(\mathbf{x},t)$.
The chromatic quasisymmetric function and the path quasisymmetric function
coincide for posets.  More precisely, we have the
following proposition.

\begin{proposition}
\label{prop:xequalsxi}
Let $P$ be a
poset whose vertex set~$V$ is a finite subset of~$\mathbb{P}$.
Let $D(P)$ be the digraph on~$V$
that has an edge $u\to v$ if and only if $v\prec u$ in~$P$.
Let $G(P)$ be the incomparability graph of~$P$.
Then $\Xi_{D(P)}(\mathbf{x},t) = X_{G(P)}(\mathbf{x}, t)$.
\end{proposition}

\begin{proof}[Proof (sketch)]
The proof is mostly a routine verification
that the two definitions coincide in this special case.
Only a few points require some attention.
First, if $S$ is a stable subset in~$G(P)$,
then $S$ is a totally ordered subset of~$P$,
and hence there is exactly one directed path in~$D(P)$
through the vertices of~$S$.
Hence ordered partitions~$\sigma$ of~$V$
such that every $\sigma_i$ is a stable set of~$G(P)$
are in bijective correspondence with
ordered path covers $({q},\alpha)$ of~$D(P)$.
Second, because $P$ is a poset, it is not possible
for $u\prec v$ and $v\prec u$ simultaneously,
so the condition that ``either $u\to v$ and $v\to u$
are both edges of~$D(P)$
or neither one is'' is equivalent to adjacency in~$G(P)$.
Third, one might worry that $\asc{q}$ counts some pairs
$\{u,v\}$ where $v$ appears later in the \emph{sequencing}
but in the same \emph{path} while $\asc\sigma$ counts only
pairs from different parts, but in fact this cannot happen
because vertices in the same path are part of the same
totally ordered subset of~$P$ and thus have a directed edge
between them in exactly one direction.
\end{proof}

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}

\begin{proof}
Applying $\omega$ to Equation~(\ref{eq:fundamental})
and invoking Equation~(\ref{eq:omega}) yields
\[
   \omega F_\alpha = \sum_{\beta\ge\alpha} \omega M_\beta
   = \sum_{\beta\ge\alpha} (-1)^{|\beta|} \sum_{\gamma\le\beta} M_\gamma.
\]
So the coefficient of~$M_\gamma$ in $\omega F_\alpha$ is
\[
   \sum_{\beta: \text{($\beta\ge\alpha$ and $\beta\ge\gamma$})}
     (-1)^{|\beta|}
   = \sum_{\beta \ge \alpha \cup \gamma} (-1)^{|\beta|} =
\begin{cases}
1, &\text{if $\overline{\alpha \cup \gamma} = \varnothing$;}\\
0, &\text{otherwise.}
\end{cases}
\]
But $\overline{\alpha\cup \gamma} = \varnothing$ is equivalent to
$\gamma\ge\overline\alpha$, so
$\omega F_\alpha = \sum_{\gamma\ge\overline\alpha} M_\gamma =
F_{\overline\alpha}$.
\end{proof}

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

\begin{theorem}
\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{theorem}

\begin{proof}
Let us apply $\omega$ to both sides of Equation~(\ref{eq:xidt})
and invoke the definition of~$\omega$.
\[
\omega \Xi_D(\mathbf{x},t) = \sum_{({q},\beta)} t^{\asc {q}} \omega M_\beta
   = \sum_{({q},\beta)} t^{\asc {q}} (-1)^{|\beta|}
      \sum_{\alpha\le\beta} M_\alpha.
\]
Now we interchange the order of summation; i.e., we want to
compute the coefficient of $M_\alpha$ in $\omega \Xi_D(t)$.
This involves a sum over all ordered path covers $({q},\beta)$
such that $\beta\ge\alpha$.
The summands involving a fixed sequencing~${q}$ are
\begin{equation}
\label{eq:alternatingsum}
   \sum_{\beta\ge\alpha} t^{\asc {q}} (-1)^{|\beta|}
    = t^{\asc {q}} \sum_{\beta\ge\alpha} (-1)^{|\beta|}.
\end{equation}
Now note that if $({q},\alpha)$ is an ordered path cover
and $\beta$ is any composition such that $\beta\ge\alpha$,
then $({q},\beta)$ is also an ordered path cover,
because deleting an edge from a directed path simply
subdivides it into two smaller directed paths.
Therefore the alternating sum in Equation~(\ref{eq:alternatingsum})
is zero unless the only $\beta\ge\alpha$
for which $({q},\beta)$ is an ordered path cover
is the maximal composition ($\beta_i = 1$ for all~$i$),
in which case the alternating sum equals one.
But this condition is equivalent to the condition
that there is no directed edge in~$D$ between
any consecutive vertices in the sequencing~${q}$
that are in the same segment of~$\alpha$,
i.e., that $({q},\alpha)$ is an ordered path cover of~$\overline D$.
Finally, note that the definition of $\asc {q}$
is invariant under taking complements of the digraph.
The theorem follows.
\end{proof}

Theorem~\ref{thm:reciprocity} gives us
a nice combinatorial interpretation of $c_{d,\lambda}({\mathbf{m}})$.

\begin{corollary}
\label{cor:asc}
Let ${\mathbf{m}}$ be a Hessenberg function, and
let $D({\mathbf{m}})$ 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~$\lambda$, $c_{d,\lambda}({\mathbf{m}})$ equals
the number of ordered path covers $({q},\alpha)$ of $\overline{D({\mathbf{m}})}$
with $\asc {q} = d$.
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:xequalsxi},
we know that $X_{G({\mathbf{m}})}(t) = \Xi_{D({\mathbf{m}})}(t)$.
Shareshian and Wachs prove~\cite[Theorem 4.5]{shareshian-wachs} that
$X_{G({\mathbf{m}})}(t)$ is actually a \emph{symmetric} function (whose coefficients
are polynomials in~$t$).
Therefore $\omega X_{G({\mathbf{m}})}(t) = \omega\Xi_{D({\mathbf{m}})}(t)$
is also a symmetric function, and the coefficient of $t^d m_\lambda$
equals the coefficient of $t^d M_\alpha$ for any composition~$\alpha$
whose parts are a permutation of the parts of~$\lambda$.
The result then follows from Theorem~\ref{thm:reciprocity}.
\end{proof}

\begin{corollary}
\label{cor:des}
For a sequencing~${q}$ of a digraph
whose vertex set is a subset of~$\mathbb{P}$,
let the definition of $\des {q}$ be the same as
the definition of $\asc {q}$ except with
``$u<v$'' replaced by ``$v<u$.''
then Corollary~\ref{cor:asc} holds with $\des {q}$ in place of $\asc {q}$.
\end{corollary}

\begin{proof}
For a proper coloring $\kappa$ of a graph
whose vertex set is a subset of~$\mathbb{P}$,
let the definition of $\des\kappa$ be the same as
the definition of $\asc\kappa$ except with
``$\kappa(u)<\kappa(v)$'' replaced by ``$\kappa(u) > \kappa(v)$.''
Shareshian and Wachs prove~\cite[Corollary 2.7]{shareshian-wachs}
that the value of $X_G(t)$ is unchanged if ``asc'' is replaced by ``des.''
It is readily checked that the proofs of
Proposition~\ref{prop:xequalsxi} and Theorem~\ref{thm:reciprocity}
go through if ``asc'' is replaced by ``des'' everywhere.
\end{proof}

As we remarked before, Corollaries \ref{cor:asc} and~\ref{cor:des}
can be derived from Shareshian--Wachs
\cite[Theorem 3.1]{shareshian-wachs}, but
we have taken our approach because we believe that
Theorem~\ref{thm:reciprocity} is of independent interest.

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

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

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{theorem}[Tymoczko]
\label{thm:tymoczko}
Let ${\mathscr{H}}({\mathbf{m}},s)$ be a regular Hessenberg variety
and let the partition $\lambda$ encode the sizes of the Jordan blocks of~$s$.
Then ${\mathscr{H}}({\mathbf{m}},s)$ 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{theorem}

It remains for us to establish a correspondence between the combinatorics
of Theorem~\ref{thm:tymoczko} and the combinatorics of~$\omega X_{G({\mathbf{m}})}(t)$,
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{theorem}
\label{thm:betti}
Let ${\mathscr{H}}({\mathbf{m}},s)$ be a regular Hessenberg variety and let the Jordan
type of~$s$ be~$\lambda$.  Then the Betti number $\beta_{2d}$ of
${\mathscr{H}}({\mathbf{m}},s)$ equals $c_{d,\lambda}({\mathbf{m}})$, and $\beta_i=0$ for $i$ odd.
\end{theorem}

\begin{proof}
As Tymoczko~\cite[Proposition 2.2]{tymoczko-linear} mentions,
it is well known that if we have a paving by affines,
then $\beta_{2d}$ is just the number of nonempty cells
with dimension~$d$.
Furthermore, $\beta_i=0$ for $i$ odd
\cite[Corollary 6.2]{tymoczko-linear}.
On the other hand, by Corollaries \ref{cor:asc} and~\ref{cor:des},
we know that $c_{d,\lambda}({\mathbf{m}})$ is the number of
ordered path covers $({q},\alpha)$ of $\overline{D({\mathbf{m}})}$
with $\des {q} = d$,
where we may take the parts of the composition~$\alpha$
to be any permutation of the parts of~$\lambda$.
So it suffices to show, firstly,
that there is a bijection between
nonempty cells and ordered path covers $({q},\alpha)$,
and secondly, that under this bijection,
the dimension of the nonempty cell is equal to $\des{q}$.

First we should specify~$\alpha$.
If $\lambda$ has $\ell$ parts $\lambda_1, \ldots, \lambda_\ell$,
we set $\alpha_i := \lambda_{\ell+1-i}$.
That is, the parts of~$\alpha$ are the parts of~$\lambda$
in \emph{reverse} order.

Instead of nonempty cells, we use the tableaux~$T$
of Theorem~\ref{thm:tymoczko} to describe our bijection.
Given an ordered path cover $({q},\alpha)$ of $\overline{D({\mathbf{m}})}$,
take the elements of the $i$th path
\[ {q}(\alpha_{i-1}+1)\to {q}(\alpha_{i-1}+2)\to \cdots\to {q}(\alpha_i)\]
and place them from left to right in the $i$th row (from the bottom) of~$T$.
We need to verify that Tymoczko's condition $k\le m_j$
is equivalent to the condition that $k\to j$ is
a directed edge in $\overline{D({\mathbf{m}})}$.
By definition, there is a directed edge $k\to j$ in $\overline{D({\mathbf{m}})}$
if and only if there is \emph{not} a directed edge $k\to j$ in~$D({\mathbf{m}})$,
i.e., if and only if either $k$ and~$j$ are incomparable in~$P({\mathbf{m}})$,
or $k\prec j$ in~$P({\mathbf{m}})$.
The only way this property can fail is if $j\prec k$ in~$P({\mathbf{m}})$,
i.e., if $k > m_j$.  So indeed the conditions are equivalent.

Let us call a pair $i,k$ satisfying the conditions
in Theorem~\ref{thm:tymoczko} a ``T-inversion.''
Using the above bijection, we can think of T-inversions
as certain pairs $i, k$ in an ordered path cover $({q},\lambda)$.
The statistic $\des$ can also be thought of as counting certain pairs
$i, k$ of $({q},\lambda)$, namely those satisfying
\begin{enumerate}
\item either $i\to k$ and $k\to i$ are both edges of~$\overline{D({\mathbf{m}})}$
or neither is,
\item $i>k$, and
\item $k$ appears later in the sequencing~${q}$ than $i$ does.
\end{enumerate}
Call such a pair an ``SW-inversion.''
We claim that for any ordered path cover,
the number of T-inversions equals the number of SW-inversions.
This will prove the theorem.

First let us note that the condition $k<i$ implies
that $k\le m_i$ (since ${\mathbf{m}}$ is a Hessenberg function)
and therefore, by the argument we gave above,
$k\to i$ is an edge of $\overline{D({\mathbf{m}})}$.
That is, if $k<i$ then it is not possible
for neither $i\to k$ nor $k\to i$ to be an edge of $\overline{D({\mathbf{m}})}$,
so in fact both must be, and in particular we must have $i\to k$,
or in other words $i\le m_k$.
Therefore an SW-inversion can be redefined as a pair $i,k$ such that
\begin{enumerate}
\item $i$ appears earlier in the sequencing~${q}$ than $k$ does, and
\item $k < i \le m_k$.
\end{enumerate}
It is now immediate that if
$i$ and~$k$ are in different paths then
$i,k$ is a T-inversion if and only if $i,k$ is an SW-inversion,
because by construction, $i$ appearing in an earlier path than~$k$
is equivalent to appearing in a lower row than~$k$ in the tableau.

If $i$ and $k$ are in the same path then the situation
is more complicated because T-inversions and SW-inversions
do not necessarily coincide.
However, we now give a bijection from the set of SW-inversions
to the set of T-inversions, thereby showing that they are equinumerous.

Given an SW-inversion $i,k$, let
$k_1, k_2, \ldots, k_r$ denote the remaining elements, in order,
that succeed $k$ in the path.
For convenience, set $k_0 := k$ and $k_{r+1} := \infty$.
Now let $j$ be the smallest number such that $i\le m_{k_{j+1}}$.
Then we claim that $i, k_j$ is a T-inversion, and that this is a bijection.

First let us verify that $i, k_j$ is a T-inversion.
Condition 1(d) is satisfied almost by definition
because what the construction is doing is scanning to the right
\emph{until} condition 1(d) is satisfied,
and it will always succeed, since we just take $j=r$ in the worst case.
So we just need to verify that $i>k_j$.  If $j=0$ then we are done,
because $(i,k_0) = (i,k)$ is an SW-inversion by assumption,
and in particular $i>k$.  Otherwise, by minimality of~$j$,
we know that $i>m_{k_j} \ge k_j$.

Thus the construction scans rightwards from~$k$
\emph{until the first T-inversion $i, k_j$ is reached.}

To see that this map is injective, observe that by minimality of~$j$,
we have $i>m_{k_{j'}}$ for every $0\le j' \le j$, so
$(i,k_{j'})$ is \emph{not} an SW-inversion.
Thus, as we scan rightwards from~$k$ in search of the first T-inversion
$i, k_j$, we do not encounter any other SW-inversions en route.
If more than one SW-inversion were mapped to the same T-inversion,
then the leftmost one would have to cross over the other ones en route.

To see that the map is surjective, we can define an inverse map,
that scans \emph{leftwards} from a T-inversion until it finds a pair
that satisfies $i\le m_k$.  Such a scan always succeeds because
in the worst case it ends up at the successor $i'$ of~$i$, and
$i\le m_{i'}$ because they are consecutive elements of a path.
Then by minimality, if we arrive at a pair $i,k$ with $k'$
being the successor of~$k$, we must have $k>m_{k'}\ge k$,
so what we have arrived at is indeed an SW-inversion.
\end{proof}

Let us remark that our proof shows that at least in the
case of regular Hessenberg varieties,
the two cases of Theorem~\ref{thm:tymoczko} can be unified,
namely that the dimension is just the number of pairs $i,k$
such $i$ appears to the left of~$k$ or in a lower row than~$k$,
and $k<i\le m_k$.

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

\begin{proof}
First note that $|{\mathbf{m}}|$ (as defined in Equation~(\ref{eq:mdim}))
is number $|E|$ of edges in the incomparability
graph $G=G({\mathbf{m}})$ of $P({\mathbf{m}})$.  This follows directly from the
description of $G({\mathbf{m}})$ given in \S\ref{gn}.
By \cite[Corollary 4.6]{shareshian-wachs},
$X_G(\mathbf{x},t)$ is palindromic.  More precisely, we have
$X_G(\mathbf{x},t)=t^{|{\mathbf{m}}|} X_G(x,t^{-1})$.   Therefore, for each partition
$\lambda$, we have
$$
\sum_d c_{d,\lambda}({\mathbf{m}}) t^{d} = t^{|{\mathbf{m}}|} \sum_{d} c_{d,\lambda} t^{-d}.
$$
So
\begin{align*}
  q(t)&=\sum_i \beta_i t^{i-|{\mathbf{m}}|}= \sum_{d} c_{d,\lambda}({\mathbf{m}}) t^{2d-|{\mathbf{m}}|}\\
&=t^{-|{\mathbf{m}}|}\sum_{d} c_{d,\lambda}({\mathbf{m}})  t^{2d}=
t^{-|{\mathbf{m}}|}t^{2|{\mathbf{m}}|}\sum_{d} c_{d,\lambda}({\mathbf{m}}) t^{-2d}\\
&=t^{|{\mathbf{m}}|}\sum_{d} c_{d,\lambda}({\mathbf{m}}) t^{-2d}
=\sum_d c_{d,\lambda}({\mathbf{m}}) t^{|{\mathbf{m}}|-2d}\\
&=q(t^{-1}).
\end{align*}
\end{proof}

\section{Local monodromy and local fundamental groups}\label{lmlfg}

\subsection{Local homotopy type}
We begin by defining local homotopy type.  This material is probably
well-known, but we feel that it will be convenient to review it.  Our
treatment follows the book by Looijenga~\cite{Looijenga}, a paper by
Kumar~\cite{Kumar93} and another paper by Prill~\cite{Prill}.

Suppose $X$ is a topological space and $x\in X$.  A \emph{fundamental
  system of neighborhoods} $\mathscr{U}$ of $x$ is a system of open
neighborhoods such that any open neighborhood $V$ of $x$ contains a
$U\in\mathscr{U}$.

The following Lemma is~\cite[Lemma 1.1.]{Kumar93}.

\begin{lemma}\label{lm1}
  Suppose $X$ is a CW complex, $x\in X$ and $Y$ is a closed subcomplex
  of $X$ containing $x$.  Then there exists a fundamental system
  $\{U\}_{U\in\mathscr{U}}$ of open neighborhoods of $x$ in $X$ such
  that the following condition is satisfied:
\begin{equation}
\parbox{25em}{For any $U,V\in\mathscr{U}$ with $V\subset U$, the inclusion $V\setminus Y\hookrightarrow U\setminus Y$ is a homotopy  equivalence.}
\end{equation}
\end{lemma}

A system of neighborhoods $\mathscr{U}$ as in Lemma~\ref{lm1} is
called a \emph{good fundamental system of neighborhoods relative to
  $Y$}.

\begin{lemma}\label{lm1.5}
  Suppose $\mathcal{C}$ is a category and
$$
A\stackrel{f}{\to} B\stackrel{g}{\to} C\stackrel{h}{\to} D
$$
is a sequence of morphisms.  Assume that $g\circ f$ and $h\circ g$
are isomorphisms.  Then $f, g$ and $h$ are all isomorphisms.
\end{lemma}
\begin{proof}
Easy exercise.
\end{proof}

We have adapted the proof of the following Proposition from
Looijenga's~\cite[p. 114]{Looijenga}, and Prill's Proposition
2~\cite{Prill}.

\begin{proposition}\label{pgood}
  Suppose $\{\mathscr{U}_{\alpha}\}_{\alpha\in I}$ is a non-empty collection
of good fundamental systems of neighborhoods as in Lemma~\ref{lm1}.
Then so is $\mathscr{U}:=\cup_{\alpha\in I} \mathscr{U}_{\alpha}$.
Consequently, the union of all good fundamental systems of neighborhoods
is itself a good fundamental system of neighborhoods.
\end{proposition}

\begin{proof}
Take $U\in\mathscr{U}_{\alpha}$ and $V\in\mathscr{U}_{\beta}$ with $V\subset U$.
We can find $U'\in\mathscr{U}_{\alpha}$ such that $U'\subset V$,
and $V'\in
    \mathscr{U}_{\beta}$ such that $V'\subset U'$.  Then apply Lemma~\ref{lm1.5}
to the sequence of inclusions
$$
V'\setminus Y\to
U'\setminus Y
\to V\setminus Y
\to U\setminus Y.
$$
This shows that the inclusion $V\setminus Y\to U\setminus Y$ is a homotopy
equivalence.
\end{proof}

\begin{definition}\label{lhtdef}
  Suppose $X$ is a CW complex and $Y$ is a closed subcomplex
  containing a point $x$.  We say an open neighborhood $U$ of $x$ is
  \emph{good relative to $Y$} if $U$ is an element of a good
  fundamental system of neighborhoods.   The \emph{local homotopy type
    of $X\setminus Y$ at $x$} is the homotopy type of $U\setminus Y$
  where $U$ is any good neighborhood.
\end{definition}

Suppose $U$ is a good neighborhood of $x$ and $W$ is an arbitrary (not
necessarily good) open neighborhood of $x$ contained in $U$.  Then we
can find a good neighborhood $V$ such that $V\subset W$.
Since $V$ is good, Proposition~\ref{pgood} shows that the composition
$$
V\setminus Y\to W\setminus Y\to U\setminus Y
$$
is a homotopy equivalence.  In other words, the local homotopy type of
$X\setminus Y$ at $x$ is a retract of the homotopy type of $W\setminus
Y$.

Now suppose $X$ is an analytic space, $Y$ is a Zariski closed subspace
and $x\in Y$.  We can find an open neighborhood $W$ of $x$ in $X$ such
that $W$ has the topological structure of a CW complex with $W\cap Y$
a subcomplex.  (See, for example, ~\cite{Lojasiewicz64}).
Consequently, there exist good neighborhoods of $x$ in $X$ relative to
$Y$.

The following fact is certainly well known (see, e.g.,~\cite[Corollary
1]{Prill}), but we give a proof because it is short.

\begin{fact}\label{lmfact}
  Suppose $X$ is complex manifold and $Y$ is a closed, nowhere dense,
  analytic subspace of $X$ containing a point $x$.  Then $U\setminus
  Y$ is a non-empty and connected for any good neighborhood of $x$.
\end{fact}

\begin{proof}
  Let $U$ be a good neighborhood of $x$ and let $V$ be any connected
  neighborhood of $x$ contained in $U$.  Then $V\setminus Y$ is
  connected (for example, by the Criterion for Connectedness on page
  133 of~\cite{GR}).  It is also non-empty.  But the homotopy type of
  $U\setminus Y$ is a retract of the homotopy type of $V\setminus Y$.  So $U\setminus Y$ is connected
  and non-empty as well.
\end{proof}

In fact, if $X$ is smooth at $x$,  we can find a contractible good
neighborhood $U$ of $x$.  (See~\cite{Prill}.)  In fact, we can
take a sufficiently small ball as in the following theorem, which
follows from Theorem 5.1 of Dimca's~\cite{dimca}.

\begin{theorem}
  Suppose $X$ is smooth of dimension $n$  at $x$ and $Y$ is a closed analytic
subspace of $X$ containing $x$.  For each positive real
number $r$, write $B_r$ for the ball of radius $r$ centered at $0$ in $\mathbb{C}^n$. Then there exists a good neighborhood $U$
of $x$ relative to $Y$ and biholomorphism $\varphi:U\to B_1$ such that the following holds:
For each $r\in (0,1)$, $\varphi^{-1}B_{r}$ is a good neighborhood of $x$.
\end{theorem}

\subsection{Local fundamental group}

Fact~\ref{lmfact} leads to the following definition.

\begin{definition}\label{dlfg}
  Suppose $X$ is a complex manifold $x$, and suppose $Y$ is a closed,
  nowhere dense, analytic subspace of $X$ containing a point $x$.
  Then the local fundamental group of $X\setminus Y$ at $x$ is the
  isomorphism class of the group $\pi_1(U,p)$ where $U$ is any good
  neighborhood of $x$ (with respect to $Y$) and $p\in U$.
\end{definition}

Since the smoothness of $X$ in the Definition~\ref{dlfg} implies that
$U$ is connected, the isomorphism class of $\pi_1(U,p)$ is indeed
well-defined.  But, since we have not given a way to fix a base point,
it is only defined up to a non-canonical isomorphism.

On the other hand, suppose $X$ as in Definition~\ref{dlfg} is
connected.  Pick any point $q\in X\setminus Y$. Given a good
neighborhood $U$ of $x$ relative to $Y$, we can find a point $p\in
U\setminus Y$ and a path $\gamma$ from $p$ to $q$.  From this, we get
a group homomorphism
$$
\varphi_{\gamma}:\pi_1(U\setminus Y,p)\to \pi_1(X\setminus Y,q).
$$
Changing $\gamma$ has the effect of conjugating $\varphi_{\gamma}$ by
an element of $\pi_1(X\setminus Y,q)$.   So the conjugacy class of the
subgroup $\varphi_{\gamma}(\pi_1(U\setminus Y,p))$
is independent of $\gamma$.

\begin{proposition}\label{psec}
  Suppose $f:X'\to X$ is a morphism of complex analytic spaces
  admitting a section $\epsilon:X\to X'$.  Let $Y$ be a closed,
  nowhere dense, analytic subspace of $X$ containing $x\in X$, and set
  $Y':=f^{-1}Y$.  Then the local homotopy type of $X\setminus Y$ at
  $x$ is a retract of the local homotopy type of $X'\setminus Y'$ at
  $\epsilon(x)$.  In particular, if $X$ and $X'$ are complex
  manifolds, then the local fundamental group of $X\setminus Y$ at $x$
  is a retract of the local fundamental group of $X'\setminus Y'$ at
  $\epsilon(x)$.
\end{proposition}
\begin{proof}
  Pick a good neighborhood $U$ of $x$.  Then find a good neighborhood
  $V$ of $\epsilon(x)$ contained in $f^{-1}U$.  Finally, find a good
  neighborhood $U'$ of $x$ contained in $\epsilon^{-1}V$.  We then
  have a composition
$$
U'\setminus Y\stackrel{\epsilon}{\to} V\setminus Y'\stackrel{f}{\to} U\setminus Y
$$
which is a homotopy equivalence.  The result follows.
\end{proof}

\subsection{Local systems and local invariant cycles}
Suppose $X$ is a CW complex containing a closed subcomplex $Y$ which
contains a point $x$, and $\mathcal{L}$ is a local system of complex
vector spaces on $X\setminus Y$.  For any two good neighborhoods $U_1$
and $U_2$ of $x$ and any integer $k$, the sheaf cohomology groups
$H^k(U_i\setminus Y,\mathcal{L}), i=1,2$ are canonically isomorphic.
To see this, take a good neighborhood $V\subset U_1\cap U_2$, and note
that the restriction maps $H^k(U_i\setminus Y,\mathcal{L})\to
H^k(V\setminus Y,\mathcal{L})$ are isomorphisms.  So we write
$H^k(x,\mathcal{L})$ for the group $H^k(U\setminus Y,\mathcal{L})$
where $U$ is any good neighborhood of $x$.   It is isomorphic to the
group $\colim H^k(U,\mathcal{L})$ where the colimit is taken
over all open neighborhoods of $x$.
The group $H^0(x,\mathcal{L})$ is
called the space of \emph{local invariants}.

If $X$ is a complex manifold and $Y$ is a nowhere dense analytic subspace, then
$U\setminus Y$ is connected for any good neighborhood of $x$ relative
to $Y$.  Pick a basepoint $p\in U\setminus Y$.  Then the data of the
local system $\mathcal{L}$ defines an action of $\pi_1(U\setminus
Y,p)$ on the fiber $\mathcal{L}_p$ at $p$.  Moreover, the space of
local invariants is given by the invariants of the action:
\begin{equation}
  \label{fgi}
  H^0(x,\mathcal{L})=\mathcal{L}_p^{\pi_1(U\setminus Y,p)}.
\end{equation}

\begin{corollary}
  Suppose $X$ is smooth and $B$ is any connected neighborhood of $x$
contained in a good neighborhood $U$.  Then $H^0(x,\mathcal{L})=
H^0(B\setminus Y,\mathcal{L})$.
\end{corollary}

\begin{proof}
Pick a point $b\in B\setminus Y$.
We have $H^0(B\setminus Y,\mathcal{L})=\mathcal{L}_b^{\pi_1(B\setminus Y,b)}$.
But $\pi_1(U\setminus Y,b)$ is a retract of $\pi_1(B\setminus Y, b)$.
So $\mathcal{L}_b^{\pi_1(B\setminus Y,b)}=\mathcal{L}^{\pi_1(U\setminus Y,b)}=
H^0(x,\mathcal{L})$.
\end{proof}

We say a connected neighborhood of a point $x$ in a complex manifold
$X$ is \emph{sufficiently small} if it is contained a good
neighborhood.

We can also describe the space $H^k(x,\mathcal{L})$ sheaf
theoretically.  Write $j:X\setminus Y\to X$ for the inclusion.  Then
the group $H^k(X,\mathcal{L})$ is naturally isomorphic to
$(R^kj_*\mathcal{L})_x$; i.e., to the stalk at $x$ of the $k$-th
higher direct image $R^kj_*\mathcal{L}$.

The following is certainly well-known, but we sketch a short proof.

\begin{lemma}\label{lsurj}
  Suppose $X$ is a connected complex manifold and $Y$ is
  a nowhere dense closed analytic subspace.  Then, for $p\in X\setminus Y$, the
  homomorphism $\pi_1(X\setminus Y,p)\to \pi_1(X,p)$ is surjective.
\end{lemma}

\begin{proof}[Sketch]
Let $\pi:\tilde{X}\to X$ denote the universal cover of $X$.  Then
$\pi^{-1}(X-Y)=\tilde{X}\setminus \pi^{-1}(Y)$ is connected because $\tilde{X}$
is a complex manifold and $\pi^{-1}(Y)$ is a closed, nowhere dense,
complex analytic subspace~\cite[p.~133]{GR}.  Therefore the
map of fundamental groups is surjective.
\end{proof}

Now, suppose $X$ is a connected, complex manifold, $Y$ is a closed,
nowhere dense, analytic subspace, $\mathcal{L}$ is a local system on
$X\setminus Y$, $x\in Y$ and $q\in X\setminus Y$.  The \emph{monodromy
  group of $\mathcal{L}$} is the image $M$ of the group homomorphism
$\pi_1(X\setminus Y,q)\to \mathbf{GL}(\mathcal{L}_q)$.  By
Lemma~\ref{lsurj}, $M$ is independent of $Y$.  That is, if $Y'$
contains $Y$ (but not $q$), then the image of the homomorphism
$\pi_1(X\setminus Y',q)$ is also $M$.

If $p$ is a point in $Y$, then the \emph{local monodromy group of
  $\mathcal{L}$ at $p$} is the image $H=H(p)$ of the composition
$$
\pi_1(U\setminus Y,p)\stackrel{\varphi_{\gamma}}{\to} \pi_1(X\setminus Y,q)\to M
$$
where $\gamma$ is a path from $p$ to $q$.  It depends on the choice of $\gamma$
and $p$, but only up to conjugacy by an element of $M$.
Like $M$ itself, $H$ is independent of $Y$ in the sense that enlarging $Y$
does not change $H$.

\section{Symmetric Betti numbers and the local invariant cycle
theorem}\label{sbn}

A crucial tool in our argument is the
local invariant cycle theorem of
Beilinson, Bernstein, and Deligne, which we state here in the
generality relevant to this paper.

\begin{theorem}[{\cite[Corollaire 6.2.9]{bbd}}]
\label{thm:bbd}
Suppose $f:X\to Y$ is a projective morphism of smooth complex
varieties.   For $y\in Y$, set $X_y:=f^{-1}(y)$ and
suppose that there is a Zariski dense open subset $U$ in $Y$ such
that, for all $y\in U$, $X_y$ is smooth of dimension $d$.
Then for every sufficiently small
connected open neighborhood $B(y)$ of a closed point $y\in Y$,
there is a surjective map
\begin{equation}\label{e.lic}
  H^i(X_y,\mathbb{C})\twoheadrightarrow
H^0(B(y)\cap U, R^if_*\mathbb{C}).
\end{equation}
\end{theorem}

The vector space $H^0(B(y)\cap U, R^if_*\mathbb{C})$ is called the
space of \emph{local invariant cycles}.  As Beilinson, Bernstein and
Deligne point out, it is independent of $U$.  Moreover, as we have
seen above, for a fixed $U$, the spaces $H^0(B(y)\cap U,
R^if_*\mathbb{C})$ are isomorphic for $B(y)$ sufficiently small.

We will devote the remainder of this section to proving the following
result.

\begin{theorem}\label{t1}
  Suppose that $f:X\to Y$ is a  projective morphism
 between smooth, quasi-projective, complex
  schemes; and let $y$ be a closed point of $Y$.
Assume that $X$ and $Y$ are equidimensional with dimensions $d_X$ and
$d_Y$ respecitively, and set $d=d_X-d_Y$.
Then the local
  invariant cycle map~\eqref{e.lic} is an isomorphism for all $i\in\mathbb{Z}$ if and only if
$$
\dim H^i(X_y,\mathbb{C})=\dim H^{2d-i}(X_y,\mathbb{C})
$$
for all $i$.
\end{theorem}

It will be convenient to introduce some notation concerning
palindromic polynomials.

\begin{definition}
  Suppose $p\in \mathbb{R}[t,t^{-1}]$ is a Laurent polynomial.  We say that
$p$ is \emph{palindromic} if $p(t)=p(t^{-1})$.
\end{definition}

\begin{lemma}\label{l1}
  Suppose $q=\sum_{\ell\geq 0} p_{\ell}t^{\ell}$ where $q$ and the
  $p_{\ell}$ are palindromic Laurent polynomials.  Assume that, for
  $\ell>0$, the coefficients of the $p_{\ell}$ are non-negative.  Then
  $p_{\ell}=0$ for $\ell>0$.  In particular, $q=p_0$.
\end{lemma}

\begin{proof}
  Note that a Laurent polynomial $f$ is palindromic only if $f'(1)=0$.
Now,
$$
\frac{d}{dt}p_{\ell}t^{\ell}\bigg|_{t=1}=
\ell p_{\ell}(1).
$$
It follows that $0=q'(1)=\sum_{\ell>0} \ell p_{\ell}(1)$.  Since the coefficients
of the polynomials $p_{\ell}$ are non-negative for $\ell>0$, it follows that
$p_{\ell}=0$ for $\ell>0$.
\end{proof}

In the proof of Theorem~\ref{t1}, we
are going to use the language of Beilinson, Bernstein and
Deligne~\cite{bbd} regarding perverse sheaves and the derived category
$D^b_c(X,\mathbb{C})$ of sheaves with constructible cohomology (with
complex coefficients) on a complex scheme $X$.  In particular, if $K$
is a complex in $D^b_c(X,\mathbb{C})$ and $i\in\mathbb{Z}$, then
$K[i]$ denotes the shift of $K$ by $i$ units to the left.

\begin{proof}[Proof of Theorem~\ref{t1}]
By the decomposition theorem of~\cite{bbd}, we have
$Rf_*\mathbb{Q}[d_X]= \oplus E_{i,j}[-i]$ where the $E_{i,j}$ are
perverse sheaves on $Y$ supported strictly in codimension $j$.
Moreover, by the Hard Lefschetz theorem of~\cite[Theor\'eme 6.2.10]{bbd}, there are
isomorphisms $E_{i,j}\cong E_{-i,j}$ for all $i$.

We can write
$$
E_{ij}=\oplus g(ijh)_{!*}\mathcal{F}_{ijh}[d_Y-j]
$$
where, for each $h$, $\mathcal{F}_{ijh}$ is a local system on a
locally closed subscheme $Z_{ijh}$ of codimension $j$ in $X$ and $g(ijh):Z_{ijh}\to X$ is the
inclusion.

For any complex $K$ of sheaves on $Y$, write
$H^*(y,K)$ for the cohomology of the stalks of $K$ at $y$.  Write
$\IH^*(y,\mathcal{F}_{ijh})$ for the local intersection cohomology of
$\mathcal{F}_{ijh}$ at $y$.   So $\IH^k(y,\mathcal{F}_{ijh})=0$ for $k<0$ and
\begin{equation}
  \label{EF}
  H^k(y,E_{ij})=\oplus\IH^{k+d_Y-j}(y,\mathcal{F}_{ijh}).
\end{equation}

Now, write
\begin{equation}
  \label{ppoly}
  q:=\sum t^k\dim H^k(X_y,\mathbb{Q}[d]).
\end{equation}
So, we have $\dim H^i(X_y,\mathbb{C})=\dim H^{2d-i}(X_y,\mathbb{C})$
for all $i$, if and only if $q$ is palindromic.

Now, by proper base change, we have
\begin{align*}
  H^k(X_y,\mathbb{Q}[d])&=H^k(y,Rf_*\mathbb{Q}[d])\\
&=H^k(y,Rf_*\mathbb{Q}[d_X-d_Y])\\
&=H^k(y,\oplus_{ij} E_{ij}[-i-d_Y])\\
&=\oplus_{ij}H^{k-i-d_Y}(y,E_{ij})\\
&=\oplus_{ijh}\IH^{k-i-j}(y,\mathcal{F}_{ijh}).
\end{align*}

Set
\begin{equation}
  b_{ijk}=\dim \IH^k(y,\oplus_h \mathcal{F}_{ijh}).
\end{equation}
We get then that
\begin{equation}
  \dim H^k(X_y,\mathbb{Q}[d])=\sum b_{i,j,k-i-j}.
\end{equation}
So
\begin{align*}
  q&=\sum b_{i,j,k-i-j}t^k\\
   &=\sum b_{i,j,k} t^{k+i+j}.
\end{align*}

Now set
$$
p_{\ell}=\sum_{k+j=\ell} b_{i,j,k} t^i.
$$
Then $q=\sum_{\ell\geq 0} p_{\ell} t^{\ell}$. Moreover,
$b_{i,j,k}=b_{-i,j,k}$, so each $p_{\ell}$ is palindromic. Since each
$b_{i,j,k}$ is a non-negative integer, Lemma~\ref{l1} shows that $q$
is palindromic if and only if $p_{\ell}=0$ for $\ell>0$.
The local invariant cycles are given by $\oplus_{i} \IH^0(E_{i0}[-d_Y])$.
So  $p_{\ell}=0$ for all $\ell>0$ if and only if the local invariant
cycle map is an isomorphism.  This completes the proof of the Theorem~\ref{t1}.
\end{proof}

\section{Galois covers}\label{gc}
The purpose of this section is to prove a lemma about the local
monodromy groups of Galois covers.  We use the concepts of
\S\ref{lmlfg}, but we have changed some of the notation (partially to avoid running out of capital letters
towards the end of the alphabet).

\subsection{Covers and monodromy}
Suppose $U$ is a smooth, connected, complex, quasi-projective variety
and $G$ is a finite group acting freely on $U$.  Let $V=U/G$, and write
$\pi:U\to V$ for the quotient morphism.  For each (closed) point $u\in U$, we
get a surjective group homomorphism
\begin{equation}
  \label{e.GaloisHomomorhpism}
  \psi_u: \pi_1(V,\pi(u))\twoheadrightarrow G.
\end{equation}
If $\gamma:[0,1]\to V$ represents an element of $\pi_1(V,\pi(u))$ and
$\tilde{\gamma}$ is a lift of $\gamma$ to $U$ with
$\tilde{\gamma}(0)=u$, then $\psi_u(\gamma)u=\tilde{\gamma}(1)$.  From
this description, we see that, if $\pi(u')=\pi(u)$, then $\psi_u$ and
$\psi_{u'}$ differ by conjugation by an element of $\pi_1(V,\pi(u))$.
(See~\cite{Hatcher} for a complete discussion of these matters.)

Now, suppose that $V$ is contained as a Zariski open subset of a smooth,
quasi-projective variety $Y$.  Set $Z=Y\setminus V$, and suppose $z$ is a
closed point of $Z$.  Let  $W$ be a good neighborhood of $z$ in $Y$ relative
to $Z$, and let $w$ be a point in $W\setminus Z$.  The choice of a path from
$w$ to $\pi(u)$ gives us a
sequence of group homomorphisms
\begin{equation}\label{lmg}
\pi_1(W\setminus Z,w)\to \pi_1(V,\pi(u))\twoheadrightarrow G.
\end{equation}
Moreover, up to conjugation by an element of $G$, this map is
independent of $u$, the path from $w$ to $\pi(u)$, $W$ and $w$.  We
call the image $H(z)$ of the composition in~\eqref{lmg} the \emph{local
  monodromy subgroup at $z$}.  (The conjugacy class of $H(z)$ is independent
of any choices.)   Note that, if we replace $V$ with a any
non-empty Zariski open subset $V'$ of $V$ containing $\pi(u)$ and we
replace $U$ with $\pi^{-1}(V')$, then $H(z)$ does not change.  This follows from
Lemma~\ref{lsurj}.

\begin{proposition}\label{pgal}
  Let $G$ be a finite group acting on a smooth, quasi-projective
  variety $X$, and suppose $G$ acts freely on a Zariski dense open
  subset $U$ of $X$.  Suppose $\pi:X\to Y$ is the quotient of $X$ by
  $G$ and let $V=\pi(U)$.  Pick a closed point $x\in X\setminus U$, and suppose that $Y$
  is smooth.  Then $H(\pi(x))$ is the stabilizer $G_x$ of the point
  $x$.
\end{proposition}

\begin{proof}
  Take a good neighborhood $B$ of $y:=\pi(x)$ with respect to $Z:=Y\setminus V$.
Pick $b\in B\cap V$ and set $A=\pi^{-1}B$.  Let $A_x$ denote the component of $A$ containing $x$.   There exists $a\in A_x$ such that $\pi(a)=b$.  Let $H$ denote
the image of the composition $\pi_1(B\cap V,b)\to \pi_1(V,b)\twoheadrightarrow G$, where the last homomorphism is $\psi_a$.
Then $H=H(a)$.   The group $G$ acts transitively
on the connected components of $A\cap U$, and the stabilizer of the
component of $A\cap U$ containing $a$ is $H$.  Since $A_x\cap U$ is connected,
$A_x\cap U$ is this component.   So the stabilizer in $G$ of this component
is the same as the stabilizer of the $A_x$.  But, by possibly shrinking $B$, we can arrange that this is just $G_x$.
\end{proof}

\begin{remark}
We use the assumption that $Y$ is smooth because we have only
defined the local fundamental group in that case.
However, since $Y$ is a quotient of a smooth variety, it is automatically
normal.  And good neighborhoods of normal quasi-projective varieties are
connected.  (See Chapter 3 of Mumford's~\cite{mumford-cpv}). It follows that the assumption that $Y$ is smooth
can be dropped.
\end{remark}

\section{Geometry of Hessenberg Schemes}\label{sghs}

In this section, we study the geometry of the family of Hessenberg
varieties over the space of regular matrices.  We also study the
family of maximal tori defined by centralizers of regular, semisimple
matrices.  Ng{\^o}'s paper on the Hitchin fibration~\cite{NgoHitchin}
significantly influenced our thinking about these matters, and we have consequently
borrowed Ng{\^o}'s notation.

\subsection{Regular matrices} Fix a positive integer $n$ and write
$\mathfrak{g}$ for the Lie algebra $\mathfrak{gl}_n$.  Recall that a
matrix $s\in\mathfrak{g}$ is regular if the Jordan blocks of $s$ have
distinct eigenvalues.  A matrix $s$ is regular if and only if its centralizer
has dimension $n$.  As above, we say that $s$ is regular of type
$\lambda$ for a partition $\lambda$ of $n$ if the Jordan blocks of $s$
are of sizes $\lambda_1,\ldots \lambda_r$.  We write
$\mathfrak{g}^{\mathrm{r}}$ for the subset of regular matrices and
$\mathfrak{g}^{\mathrm{r}}_{\lambda}$ for the subset of regular
matrices of type $\lambda$.  We write $\mathfrak{g}^{\mathrm{rs}}$ for the
subset of regular semisimple matrices.   This is a dense open
subset of $\mathfrak{g}$.

\subsection{Hessenberg schemes}\label{hs} Fix a Hessenberg function
$\mathbf{m}=(m_1,\ldots, m_{n-1})$ with $\mathbf{m}(i):=m_i$, and
set $m_n=n$.
 Write
$\mathscr{X}$ for the variety of complete flags in $\mathbb{C}^n$, and set
\begin{equation}
{\mathscr{H}}(\mathbf{m}):=\{(F,s)\in \mathscr{X}\times \mathfrak{g}: sF_i\subset
F_{m_i}\text{ for all } i\}.
\end{equation}
Note that the projection $\pr_1$ on the first factor makes ${\mathscr{H}}(\mathbf{m})$
into a vector bundle of rank $\sum_{i=1}^n m_i$ over ${\mathscr{X}}$.  So ${\mathscr{H}}(\mathbf{m})$ is a smooth,
connected scheme.  Let $\pi:{\mathscr{H}}(\mathbf{m})\to \mathfrak{g}$ denote
the projection on the second factor.  Then the fiber of $\pi$ over a
matrix $s\in\mathfrak{g}$ is simply the Hessenberg variety
${\mathscr{H}}(\mathbf{m},s)$.  Note that $\pi$ is smooth over
$\mathfrak{g}^{\mathrm{rs}}$.

\subsection{Diagonal matrices and characteristics}  Write
$G:=\mathbf{GL}_n$ and write $\mathbf{D}$ for the diagonal subgroup of
$G$.  Write $\mathbf{d}$ for the Lie algebra of $\mathbf{D}$, and
$\mathbf{d}_r$ for the regular elements of $\mathbf{d}$. The symmetric
group $S_n$ acts on $\mathbf{d}=\mathbb{A}^n$ in the obvious way:
$\sigma(x_1,\ldots, x_n)=(x_{\sigma^{-1}(1)},\ldots,
x_{\sigma^{-1}(n)})$.  The quotient is the characteristic variety
$\mathbf{car}=\mathbf{car}_G=\mathbf{d}/S_n$.  We can view $\mathbf{car}$ as the
variety of monic polynomials $t^n+a_{n-1}t^{n-1}+\cdots a_0$ of
degree $n$.  The \emph{Chevalley morphism}
$\chi:\mathfrak{g}\to \mathbf{car}$ is the morphism sending a matrix
$s\in\mathfrak{g}$ to its characteristic polynomial $\chi(s)$.

\subsection{The smallest Hessenberg scheme}\label{smallest}
Let $\Bell$ denote the Hessenberg function $\Bell(i)=i$.
Then the restriction ${\mathscr{H}}^{\mathrm{rs}}(\Bell)\to \mathfrak{g}^{\mathrm{rs}}$ of
$\pi:{\mathscr{H}}(\Bell)\to \mathfrak{g}$ to the inverse image of $\mathfrak{g}^{\mathrm{rs}}$
is an \'etale cover of degree $n!$.  Since ${\mathscr{H}}(\Bell)$ is
connected and smooth, so is ${\mathscr{H}}^{\mathrm{rs}}(\Bell)$.  So,
since $\mathfrak{g}^{\mathrm{rs}}$ is also connected, ${\mathscr{H}}^{\mathrm{rs}}(\Bell)\to
\mathfrak{g}^{\mathrm{rs}}$ is an \'etale cover corresponding to an index $n!$ subgroup of
the fundamental group of $\mathfrak{g}^{\mathrm{rs}}$.

We call ${\mathscr{H}}(\Bell)$
the smallest Hessenberg scheme because, for any Hessenberg function
$\mathbf{m}$, there is a canonical inclusion ${\mathscr{H}}(\Bell)\to
{\mathscr{H}}(\mathbf{m})$ (which is a closed immersion).

We have a morphism $\tilde\chi:{\mathscr{H}}(\Bell)\to \mathbf{d}$
sending a pair $(s,F)$ to the diagonal matrix with $\Gr^F_i s$ in the
$(i,i)$-entry.  This gives rise to a commutative diagram
\begin{equation}\label{gsr}
\xymatrix{
{\mathscr{H}}(\Bell)\ar[r]^{\tilde\chi}\ar[d]^{\pi}   & \mathbf{d}\ar[d]^{\chi_{|\mathbf{d}}}\\
\mathfrak{g}\ar[r]^{\chi}             & \mathbf{car},
}
\end{equation}
which coincides with Grothendieck's simultaneous resolution
of $\chi$. (For a discussion of Grothendieck's resolution, see Springer's~\cite[\S 4.1]{springer} or Slodowy's~\cite[\S 4.7]{slodowy}.)
The restriction of ~\eqref{gsr} to the inverse image of
$\mathbf{car}^{\mathrm{rs}}:=\mathbf{d}_{\mathrm{r}}/S_n$ is a pullback
diagram.  In other words,
${\mathscr{H}}^{\mathrm{rs}}(\Bell)=\mathfrak{g}^{\mathrm{rs}}\times_{\mathbf{car}^{\mathrm{rs}}}
\mathbf{d}_{\mathrm{r}}$.  This shows that
${\mathscr{H}}^{\mathrm{rs}}(\Bell)$ is a (connected) Galois cover of
$\mathfrak{g}^{\mathrm{rs}}$ with Galois group $S_n$.

We can describe this Galois covering a little bit more explicitly if
we introduce the closed subscheme $Z$ of
$(\mathbb{P}^{n-1})^n\times\mathfrak{g}^{\mathrm{rs}}$ consisting of ordered tuples
$([v_1],\ldots, [v_n]; y)$ where the $v_i$ form a basis of
eigenvectors of $y$.  Given a point $z=([v_1],\ldots, [v_n]; y)$ in
$Z$, we can define a complete flag $F(z)$ by setting $F_i=\langle
v_1,\ldots, v_i\rangle$.  This defines a morphisms $Z\to
{\mathscr{H}}^{\mathrm{rs}}(\Bell)$ given by $z\mapsto (F(z),y)$.  Using the fact
that $Z$ and ${\mathscr{H}}^{\mathrm{rs}}(\Bell)$ are both \'etale covers of
$\mathfrak{g}^{\mathrm{rs}}$ of the same degree, it is easy to see that $Z\to
{\mathscr{H}}^{\mathrm{rs}}(\Bell)$ is an isomorphism.  Then $S_n$ acts on $Z$ by
permuting the $v_i$: $\sigma([v_1],\ldots, [v_n];
y)=([v_{\sigma^{-1}(1)}],\ldots , [v_{\sigma^{-1}(n)}], y)$.  It is
easy to see that
the map $\tilde{\chi}:{\mathscr{H}}^{\mathrm{rs}}(\Bell)\to \mathbf{d}_{\mathrm{r}}$
is $S_n$-equivariant.

\subsection{The fundamental groups}
Suppose $z=(F,y)\in{\mathscr{H}}^{\mathrm{rs}}(\Bell)$.
We get a surjection
$\psi_z:\pi_1(\mathfrak{g}^{\mathrm{rs}},y)\twoheadrightarrow S_n$ corresponding
to the Galois covering ${\mathscr{H}}^{\mathrm{rs}}(\Bell)\to \mathfrak{g}^{\mathrm{rs}}$ with Galois group $S_n$.   Similarly, for a regular diagonal matrix $u\in\mathbf{d}_r$, we have a surjection $\psi_u:\pi_1(\mathbf{car}^{\mathrm{rs}},\chi(u))\to S_n$.
Since
$$
\xymatrix{
{\mathscr{H}}^{\mathrm{rs}}(\Bell)\ar[r]^{\tilde\chi}\ar[d] & \mathbf{d}_{\mathrm{r}}\ar[d]^{\chi_{|\mathbf{d}_{\mathrm{r}}}}\\
\mathfrak{g}^{\mathrm{rs}}\ar[r]^{\chi}             & \mathbf{car}^{\mathrm{rs}}.
}
$$
is a pullback diagram of Galois \'etale covers with Galois group $S_n$,
we have $\psi_z=\psi_{\tilde{\chi}(z)}\circ\chi_*$.

\begin{definition}
  We say a polynomial $p\in\mathbf{car}$ is of type $\lambda$ if $p=\prod_{i=1}^{\ell}
(x-x_i)^{\lambda_i}$ where $x_1,\ldots , x_{\ell}$ are distinct.
\end{definition}

\begin{lemma}\label{myoung}
Suppose $p=\prod_{i=1}^r (x-x_i)^{\lambda_i}$ is a polynomial of
type $\lambda$.
Then the local monodromy subgroup $H(p)$ of $S_n$ at $p$ for the $S_n$-cover
$\mathbf{d}^{r}\to\car^{\mathrm{rs}}$ is $S_{\lambda}$.
\end{lemma}

\begin{proof}
Let $\tau$ denote the diagonal matrix
\[\diag(x_1,\ldots, x_1,x_2\ldots, x_2,\ldots, x_r,\ldots, x_r).\]
Then the stabilizer in $S_n$ of $\tau$ is precisely
$S_{\lambda}$.   The results then follows from Proposition~\ref{pgal}.
\end{proof}

\subsection{The Kostant section} The Kostant section is a morphism
$\epsilon:\mathbf{car}\to \mathfrak{g}^{\mathrm{r}}$ which is a section of $\chi$;
i.e., $\chi\circ\epsilon=\id$.  We give the definition of $\epsilon$
following Ng\^o's paper~\cite[Theorem 2.1]{NgoHitchin}.
We remark, however, that, while the general definition makes sense for
any reductive Lie algebra, we only discuss it for $\mathfrak{gl}_n$.

Let $\mathbf{x}_{-}$ (resp. $\mathbf{x}_{+}$) denote the $n\times n$
matrix with $1$'s just below (resp. just above) the diagonal and $0$'s
everywhere else.  Then let $\mathfrak{g}^{\mathbf{x}_+}$ denote the
centralizer of $\mathbf{x}_+$ in $\mathfrak{g}$.  In~\cite{Kostant63},
Kostant showed that the subspace
$\mathbf{x}_{-}+\mathfrak{g}^{\mathbf{x}_{+}}$ is contained in
$\mathfrak{g}^{\mathrm{r}}$.  Moreover, he showed that the restriction of $\chi$
to $\mathbf{x}_{-}+\mathfrak{g}^{\mathbf{x}_+}$ induces an isomorphism
onto $\mathbf{car}$.  The Kostant section is the inverse morphism $\epsilon:
\mathbf{car}\to\mathbf{x}_{-}+\mathfrak{g}^{\mathbf{x}_+}$.  In the case of
$\mathfrak{gl}_n$,
\[
\mathbf{x}_{-}+\mathfrak{g}^{\mathbf{x}_+}=
\left\{
\begin{pmatrix}
x_0 & x_1 & x_2 & \ldots & x_{n-2} & x_{n-1} \\
1 & x_0   & x_1 & \ldots &  x_{n-3}& x_{n-2} \\
0 & 1   & x_0   & \ldots & x_{n-4} & x_{n-3} \\
\vdots & \vdots & \vdots &\ddots &\vdots & \vdots \\
0      & \ldots &           & \ldots & 1 & x_0\\
\end{pmatrix}\right\}.
\]
From this, it is elementary to compute the Kostant section. For example,
for $n=2$, it sends the characteristic polynomial $p=x^2+a_1x+a_0$ to
the matrix of the form above with $x_0=-a_1/2$, $x_1=a_1^2/4-a_0$.

\begin{proposition}\label{ryoung}
  Suppose $s\in\mathfrak{g}^r$ is a regular matrix of type $\lambda$.
Then the local monodromy $H(s)$ at $s$ for the $S_n$-cover
$\sH^{\mathrm{rs}}(\mathrm{l})\to\mathfrak{g}^{\mathrm{rs}}$
is conjugate to the Young subgroup $S_{\lambda}$.
\end{proposition}

\begin{proof}
  We can assume that $s=\epsilon(p)$ for some $p\in\mathbf{car}$.  Then, by
  Proposition~\ref{psec}, the local fundamental group at $p$ is a
  retract of the local fundamental group at $s$.  Since the
  $S_n$-cover $\sH^{\mathrm{rs}}(\Bell)\to\mathfrak{g}^{\mathrm{rs}}$ is a pullback of the $S_n$-cover
  $\mathbf{d}^{r}\to\car^{\mathrm{rs}}$, it follows that the local monodromy
  subgroup at $s$ is equal to the local monodromy subgroup at $p$.  By
  Lemma~\ref{myoung}, this subgroup is $S_{\lambda}$.
\end{proof}

\subsection{The commuting group scheme}\label{cgs}
Write $I:=\{(g,x)\in G\times\mathfrak{g}: \Ad g (x)=x\}$.  The projection
$p:I\to\mathfrak{g}$ is a group scheme in a more or less obvious way.  Write $p^{\mathrm{rs}}:\mathscr{T}\to
\mathfrak{g}^{\mathrm{rs}}$ for the restriction of $p$ to the inverse image of
$\mathfrak{g}^{\mathrm{rs}}$.  Then $\mathscr{T}$ is a torus bundle: the fiber over a point
$y\in\mathfrak{g}^{\mathrm{rs}}$ is the maximal torus in $G$ centralizing $y$.

Identify the scheme $Z$ from \S\ref{smallest} with $\sH^{\mathrm{rs}}(\Bell)$ and
form a pullback diagram
$$
\xymatrix{
  \mathscr{T}_Z\ar[r]\ar[d] & \mathscr{T}\ar[d]^{p^{\mathrm{rs}}}\\
       Z\ar[r]^{\pi}   & \mathfrak{g}^{\mathrm{rs}}.\\
}
$$
Then $\mathscr{T}_Z$ is equipped with an isomorphism
$\mathscr{T}_Z\to \mathbb{G}_{mZ}^{n}$ to the split torus over
$Z$.  To see this, suppose $z=([v_1],\ldots, [v_n];y)$ and $g\in G$ is
an element commuting with $y$.  Then $g$ preserves the eigenspaces of
$y$.  So for each $i=1,\ldots n$, there is a unique character
$t_i\in X^*(\mathscr{T}_Z)$ such that $gv_i=t_i(g)v_i$.  The
$n$-tuple of characters $\vec{t}:=(t_1,\ldots, t_n):\mathscr{T}_Z\to \mathbb{G}_{m,Z}^{n}$ is easily seen to give an isomorphism.

  Over $\mathfrak{g}^{rs}$, the torus $\mathscr{T}$ is determined
  up to isomorphism by its group of characters $X^*(\mathscr{T})$
  viewed as a local system over $\mathfrak{g}^{\mathrm{rs}}$.
  Moreover, this local system is canonically isomorphic to
  $R^1p^{\mathrm{rs}}_*\mathbb{Z}$.  For any point
  $y\in\mathfrak{g}^{\mathrm{rs}}$, the fundamental group
  $\pi_1(\mathfrak{g}^{\mathrm{rs}},y)$ acts on the fiber of
  $X^*(\mathscr{T})$ lying over $y$ by permuting the characters
  $t_1,\ldots, t_n$.

\section{Monodromy and Tymoczko's dot action}\label{mtd}

\subsection{Fiberwise cohomology of $B\mathscr{T}$}
\label{fcbt}

For each $y\in\mathfrak{g}^{\mathrm{rs}}(\mathbb{C})$, we have a torus $\mathscr{T}_y$ and
its associated classifying space $B\mathscr{T}_y$.  The cohomology of
$B\mathscr{T}_y$ is naturally a polynomial ring
$\mathbb{C}[t_1,\ldots, t_n]=
\mathbb{C}[X^*(\mathscr{T}_y)]$ generated in degree $2$ by the
characters of $\mathscr{T}_y$.  As we vary $y$, these glue together to
form a local system $\mathscr{A}$ of polynomial algebras
over $\mathfrak{g}^{\mathrm{rs}}$.  In fact, since $\mathscr{T}$ is \'etale locally trivial,
we can
construct a fiber bundle $a:B\mathscr{T}\to \mathfrak{g}^{\mathrm{rs}}$ over $\mathfrak{g}^{\mathrm{rs}}$ such that
the fiber over each $y\in\mathfrak{g}^{\mathrm{rs}}$ is $B\mathscr{T}$.
Then we have $\mathscr{A}=\oplus_{k\geq 0}\mathscr{A}^k$
where $\mathscr{A}^k=R^{2k}a_*\mathbb{C}$.

Let $y_0$ denote the regular
semisimple matrices with diagonal entries $1$, $2$, $3,\ldots, n$
(written in order).  Let $F^0$ denote the standard flag
$$\langle e_1\rangle \subset \langle e_1,e_2\rangle \subset \cdots$$
where $e_i$ is the standard basis of $\mathbb{C}^n$.  Set $z_0=(F^0,y_0)\in
{\mathscr{H}}(\Bell,y_0)$.  This gives rise to a surjection
\begin{equation}\label{psifixed}
\psi:\pi_1(\mathfrak{g}^{\mathrm{rs}},y_0)\to S_n,
\end{equation}
where, for simplicity, we write $\psi:=\psi_{z_0}$.

Let $T$ denote the fiber of $\mathscr{T}$ over $y_0$.  So $T$ is simply
the diagonal subgroup of $G$.  Then, by the discussion in \S\ref{cgs}, $\pi_1(\mathfrak{g}^{\mathrm{rs}},y_0)$ acts on $X^*(T)$
by permuting the characters.   Explicitly, if we let $\sigma(t_i)=
t_{\sigma^{-1}(i)}$ for $\sigma\in S_n$, then $\gamma(t_i)=
(\psi(\gamma))(t_i)$.
Consequently, if we let $A=\mathscr{A}_{y_0}=\mathbb{C}[t_1,\ldots, t_n]$, then $\pi_1(\mathfrak{g}^{\mathrm{rs}},y_0)$ acts on the polynomials in $A$ by
$\gamma (p)=(\psi(\gamma))(p)$, where $S_n$ acts on $A$ in the standard way:
\begin{equation}\label{fcbtf}
(\sigma p)(t_1,\ldots, t_n)=p(t_{\sigma(1)},\ldots, t_{\sigma(n)}).
\end{equation}

\subsection{Fiberwise equivariant cohomology of Hessenberg varieties} Now, for each Hessenberg
function $\mathbf{m}$, the torus $\mathscr{T}$ acts on the Hessenberg
scheme $\sH^{\mathrm{rs}}(\mathbf{m})\to \mathfrak{g}^{\mathrm{rs}}$.  So for each $y\in\mathfrak{g}^{\mathrm{rs}}$, we can
take the equivariant cohomology groups
$H^*_{\mathscr{T}_y}({\mathscr{H}}(\mathbf{m}),y)$.  By localization, we know
that $H^*_{\mathscr{T}_y}({\mathscr{H}}(\mathbf{m}),y))$ is a free module of
rank $n!$ over $\mathscr{A}_y= H^*(B\mathscr{T}_y)$.  Moreover, the
canonical inclusion ${\mathscr{H}}(\Bell)\to{\mathscr{H}}(\mathbf{m})$ induces an
inclusion $H^*_{\mathscr{T}_y}({\mathscr{H}}(\mathbf{m},y))\to
H^*_{\mathscr{T}_y}({\mathscr{H}}(\Bell,y))$.  (See Tymoczko's
paper~\cite{tymoczko} for results on localization applied to Hessenberg
varieties.)  The modules $H^*_{\mathscr{T}_y}({\mathscr{H}}(\mathbf{m},y))$ glue
together to form a local system $\mathscr{L}(\mathbf{m})$ over $\mathfrak{g}^{\mathrm{rs}}$ of
$\mathscr{A}$-modules.  This can be seen explicitly using
Tymoczko's description of the equivariant cohomology of Hessenberg varieties
in terms of moment graphs.

\begin{proposition}\label{pgkm}  Write $\pi_{\mathbf{m}}:\sH^{\mathrm{rs}}(\mathbf{m})\to\mathfrak{g}^{\mathrm{rs}}$
for the projection morphism, and let
$\mathscr{A}_+$ denote the sheaf of ideals in $\mathscr{A}$ generated
by the positive degree elements.   Then we have an isomorphism of sheaves
$$
\mathscr{L}(\mathbf{m})/\mathscr{L}(\mathbf{m})\mathscr{A}_+\to
R^*\pi_{\mathbf{m}*}\mathbb{C}.
$$
\end{proposition}
\begin{proof}
This follows from the fact that Hessenberg varieties are GKM spaces.
(See~\cite[\S2 and Proposition 5.4]{tymoczko}).
\end{proof}

For each $\mathbf{m}$, localization induces an inclusion
$\mathscr{L}(\mathbf{m})\to \mathscr{L}(\Bell)$ of
$\mathscr{A}$-modules.  Write $L(\mathbf{m})$ for the fiber, $H^*_T({\mathscr{H}}(\mathbf{m},y_0))$ of
$\mathscr{L}(\mathbf{m})$ over $y_0$.  Then $L(\mathbf{m})$ is free as an
$A$-module, and both $A$ and $L(\mathbf{m})$ are equipped with
compatible actions of $\pi_1(\mathfrak{g}^{\mathrm{rs}},y_0)$.  If we write $A_+$ for the
ideal of positive degree polynomials, then we have
\begin{equation}
  H^*({\mathscr{H}}(\mathbf{m},y_0))=A/A_+L(\mathbf{m}),
\end{equation}
and the monodromy action of $\pi_1(\mathfrak{g}^{\mathrm{rs}},y_0)$ on both sides is compatible.

\begin{proposition}\label{snm}
  The action of $\pi_1(\mathfrak{g}^{\mathrm{rs}},y_0)$ on $L(\mathbf{m})$ factors through
the homomorphism $\psi:\pi_1(\mathfrak{g}^{\mathrm{rs}},y_0)\twoheadrightarrow S_n$.
\end{proposition}
\begin{proof}
  The pullback $\mathscr{T}_Z$ of $\mathscr{T}$ to the $S_n$-cover
  $Z\to\mathfrak{g}^{\mathrm{rs}}$ is a constant group scheme, and the pullback of
  $\sH^{\mathrm{rs}}(\Bell)=Z$ to $Z$ is disjoint union of copies of $Z$
  indexed by elements of $S_n$.  It follows that the action of
  $\pi_1(\mathfrak{g}^{\mathrm{rs}},y_0)$ on $H^*_T({\mathscr{H}}(\Bell,y_0))$ is trivial on the
  image of the map $\pi_1(Z,z_0)\to \pi_1(\mathfrak{g}^{\mathrm{rs}},y_0)$.  In other words,
  the action of $\pi_1(\mathfrak{g}^{\mathrm{rs}},y_0)$ on $H^*_T({\mathscr{H}}(\Bell,y_0))$
  factors through $S_n$.

Since $\mathscr{L}(\mathbf{m})\to\mathscr{L}(\Bell)$ is an inclusion of local systems,
we have a $\pi_1(\mathfrak{g}^{\mathrm{rs}},y_0)$-equivariant inclusion $L(\mathbf{m})\to L(\Bell)$.  The result follows.
\end{proof}

\begin{corollary}\label{cmac}
  The action of $\pi_1(\mathfrak{g}^{\mathrm{rs}},y_0)$ on $H^*(\sH^{\mathrm{rs}}(\mathbf{m},y_0))$
induced from the local system $R^*\pi_{\mathbf{m}*}\mathbb{C}$
factors through $\psi:\pi_1(\mathfrak{g}^{\mathrm{rs}},y_0)\twoheadrightarrow S_n$.
\end{corollary}

\begin{proof}
  This follows directly from Propositions~\ref{snm} and~\ref{pgkm}.
\end{proof}

\begin{definition}
  The action of $S_n$ on $L(\mathbf{m})$ (resp.  $H^*(\sH^{\mathrm{rs}}(\mathbf{m},y_0))$) coming from
  Proposition~\ref{snm} (resp. Corollary~\ref{cmac}) is called the \emph{monodromy action of $S_n$}.
\end{definition}

\subsection{Monodromy action for $\sH^{\mathrm{rs}}(\Bell)$}

To make the monodromy action of $S_n$ on $L(\Bell)$ explicit, recall from
\S\ref{fcbt} that $z_0$ denotes the element of $Z_0:={\mathscr{H}}(\Bell,y_0)$
corresponding to $y_0$ with the standard ordering of its eigenspaces.
So $z_0=([e_1],\ldots, [e_n],y_0)$.  Given $\sigma\in S_n$, we have
$\sigma z_0=([e_{\sigma^{-1}1}],\ldots, [e_{\sigma^{-1}(n)}]; y_0)$.  The
cohomology group $H^*Z_{0}=H^0Z_{0}$ is simply the group of functions
$f:Z_{0}\to\mathbb{C}$.  If, for $w\in S_n$, we write $\delta_w$ for the function taking
$wz_0$ to $1$ and all other elements of $Z_0$ to $0$, then we have
$(\sigma \delta_w)(z)= \delta_w(\sigma^{-1}z)$.  From this it easily
follows that $\sigma\delta_w= \delta_{\sigma w}$.

\begin{lemma}\label{ldot0}
As an $A$-module, $L(\Bell)$ is isomorphic to the module $A^{|S_n|}$
of functions from the set $S_n$ to $A$.  The monodromy action of $S_n$
on $L(\Bell)$ is given by
$$
((w p)(v))(t_1,\ldots, t_n)=(p(w^{-1}v))(t_{w(1)},\ldots, t_{w(n)})
$$
where $v,w\in S_n$, $p\in A^{|S_n|}$ and $t_1,\ldots ,t_n$ are variables.
\end{lemma}
\begin{proof}
Under the identification $S_n\to Z_{0}$ given by $w\mapsto wz_0$,
the $\delta_w$ form a $\mathbb{C}$-basis of $H^0(Z_{0})$.  Moreover,
the map $H^0_T(Z_{0})\to H^0(Z_{0})$ is an $S_n$-equivariant isomorphism, and,
under this identification, the $\delta_w$ freely generate $H^*_T(Z_{0})$
as an $A$-module.  The result now follows by direct verification using
the fact that $S_n$ acts on $A$ as in \eqref{fcbtf}.
\end{proof}

\begin{corollary}\label{dotl}
  The monodromy action of $S_n$ agrees with Tymoczko's dot action
of $S_n$ on $H^*_T({\mathscr{H}}(\Bell,y_0))$.
\end{corollary}
\begin{proof}
  This follows immediately by comparing the description of the monodromy action
in Lemma~\ref{ldot0} with Tymoczko's description of the dot action~\cite[\S 3.1]{tymoczko}.
\end{proof}

\begin{theorem}\label{dotm}
  Let $\mathbf{m}$ be a Hessenberg function.
The monodromy action of $S_n$ on $H^*_T({\mathscr{H}}(\mathbf{m},y_0))$ is the same as Tymoczko's dot action.
\end{theorem}

\begin{proof}
 Under Tymoczko's dot action, $H^*_T({\mathscr{H}}(\mathbf{m},y_0))$ is an $S_n$-equivariant $A$-submodule of $H^*_T({\mathscr{H}}(\Bell,y_0))$.  The same is true of the
monodromy action of $S_n$.  Therefore, by Corollary~\ref{dotl}, the two actions must coincide.
\end{proof}

\begin{corollary}\label{tdot}
  Tymoczko's dot action of $S_n$ on the non-equivariant cohomology group
$H^*({\mathscr{H}}(\mathbf{m},y_0))$ coincides with the monodromy action.
\end{corollary}
\begin{proof}
  Tymoczko defines the dot action on $H^*({\mathscr{H}}(\mathbf{m},y_0))$ as the
  dot action on the quotient $L(\mathbf{m})/A_+L(\mathbf{m})$.  The
  monodromy action is also given by this quotient.
\end{proof}

Since Tymoczko's dot action and the monodromy action of $S_n$ coincide, we
will not distinguish between them from now on: it will be the only action
of $S_n$ appearing in the remainder of the paper.

\begin{theorem}\label{tiy}
Let $s\in\mathfrak{g}^{\mathrm{r}}$ be a regular element of type $\lambda$ and let $\pi=\pi_{\mathbf{m}}:{\mathscr{H}}(\mathbf{m})\to\mathfrak{g}$.   Let $B(s)$ be a sufficiently small ball in $\mathfrak{g}$ centered at $s$.    Then, for each $k\in\mathbb{Z}$, there is a $\mathbb{C}$-vector space isomorphism
$$
H^0(B(s)\cap \mathfrak{g}^{\mathrm{rs}},R^k\pi_*\mathbb{C})\cong
H^k({\mathscr{H}}(\mathbf{m},y_0))^{S_{\lambda}}.$$
\end{theorem}

\begin{proof}
By~\eqref{fgi} applied with $\mathcal{L}=R^k\pi_*\mathbb{C}$, we have $$H^0(B(s)\cap \mathfrak{g}^{\mathrm{rs}},R^k\pi_*\mathbb{C})=
H^k({\mathscr{H}}(\mathbf{m},b))^{\pi_1(B(s)\cap\mathfrak{g}^{\mathrm{rs}},b)}$$
where $b$ is any point in $B(s)\cap\mathfrak{g}^{\mathrm{rs}}$.
The last vector space is isomorphic to the invariants of $H^k({\mathscr{H}}(\mathbf{m},y))$
under the local monodromy at $s$.
The result then follows from Proposition~\ref{ryoung}.
\end{proof}

\begin{theorem}\label{dsf}
Suppose $s\in\mathfrak{g}^r$ is a regular element of type $\lambda$.
Then, for each $k\in\mathbb{Z}$,
\begin{equation}
\label{eq:dsf}
\dim H^k({\mathscr{H}}(\mathbf{m},s))=\dim H^k({\mathscr{H}}(\mathbf{m},y_0))^{S_{\lambda}}.
\end{equation}
\end{theorem}
\begin{proof}
We are going to apply Theorem~\ref{t1} to the morphism $\pi:
{\mathscr{H}}({\mathbf{m}})\to\mathfrak{g}$.
 Both the source and the target of $\pi$ are
  smooth, quasi-projective varieties.  Moreover, $\pi$ has relative
  dimension $|{\mathbf{m}}|$.  (One way to check this is to use the fact that the
  projection $\pr_1:{\mathscr{H}}({\mathbf{m}})\to \mathscr{X}$ has relative dimension
  $\sum_{i=1}^n m_i$, while $\dim\mathscr{X}=\sum_{i=1}^{n-1} i$.
  Another way to see it, is to use the fact that the regular
  semisimple Hessenberg varieties have dimension $|{\mathbf{m}}|$.)

By Corollary~\ref{pbetti}, we have
$$\dim H^i({\mathscr{H}}({\mathbf{m}},s),\mathbb{C})=
\dim H^{2|{\mathbf{m}}|-i}({\mathscr{H}}({\mathbf{m}},s),\mathbb{C})$$ for all $i$.

It follows then from Theorem~\ref{t1} that the local invariant cycle
map $$H^i({\mathscr{H}}({\mathbf{m}},s))\to H^0(B(s)\cap\mathfrak{g}^{\mathrm{rs}},R^i\pi_*\mathbb{C})$$ is an
isomorphism, where $B(s)$ is any sufficiently small ball centered at
$s$ in $\mathfrak{g}$.   The result now follows from Theorem~\ref{tiy}.
\end{proof}

Finally we can put all the pieces together to prove
Conjecture~\ref{conj:sw}.

\begin{theorem}
\label{thm:sw}
If $\chi_{{\mathbf{m}},d}$ denotes the dot action
on the cohomology group $H^{2d}$ of the regular semisimple
Hessenberg variety ${\mathscr{H}}({\mathbf{m}}, s)$,
then $\ch \chi_{{\mathbf{m}},d}$ equals
the coefficient of $t^d$ in
$\omega X_{G({\mathbf{m}})}(t)$.
\end{theorem}

\begin{proof}
By Theorem~\ref{thm:betti},
the left-hand side of Equation~(\ref{eq:dsf})
(in Theorem~\ref{dsf}) equals $c_{d,\lambda}({\mathbf{m}})$
when $k=2d$.
On the other hand,
by Proposition~\ref{prop:frobenius},
the right-hand side of Equation~(\ref{eq:dsf})
equals the coefficient of $m_\lambda$ in $\ch \chi_{{\mathbf{m}},d}$.
\end{proof}

\section{Future work}

In this paper, we have considered only Hessenberg varieties
of type~A, but
De~Mari, Procesi, and Shayman defined Hessenberg varieties
in the context of an arbitrary reductive linear algebraic group
\cite{demari-procesi-shayman}.
Precup~\cite{precup} has computed the cohomology of regular
Hessenberg varieties in all types.  We hope that eventually it will
be possible to combine our methods with Precup's result to compute
Tymoczko's dot action on the cohomology of Hessenberg varieties in
all (or, at least, all classical) types.

The work of Haiman~\cite{haiman} has established a connection
between $X_G$ and representation theory,
but in seemingly a very different way.
It would be interesting to find a direct connection
between Haiman's work and ours.

\bibliographystyle{hplain}
\def$'${$'$}

redpajama_set_name: RedPajamaArXiv


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4. What Is The Remedy If I File A Successful Ohio Lemon Law Claim?
You are entitled to a new, replacement vehicle, or the manufacturer will take the car back and refund your full purchase price and all incidental damages. If you have a car loan, the manufacturer will pay off the loan and refund your car payments.
5. How Much Does Attorney Thom Cafferty Charge?
Your award may not be reduced by my attorney feedepending on the vehicle manufacturer and whether your matter is settled or resolved at arbitration. The lemon law provides that the court may require that the manufacturer pay reasonable attorney fees and costs with a successful claim in court. Therefore, it may not cost you anything to have an experienced attorney file a claim on your behalf. I've filed for clients with various manufacturers and successfully negotiated remedies for numerous clients. I understand what a good settlement entails and how to avoid manufacturer red tape.
6. Who Makes The Decision On Whether Or Not To Settle?
No all claims got to court. Many are settled before court and even before the claims are heard by an arbitrator. The decision to settle rests with the you, the consumer. Lemon law claims may be settled if you are satisfied with the manufacturer's offer. Note that any settlement may involve a compromise of the claim, including a portion of the manufacturer's settlement proposal to be applied to fees and costs. I will discuss my fees and offer my honest assessment and opinion of the merits of your lemon law case before we enter into a formal attorney-client relationship.
7. How Much Time Does Filing Take?
This varies from one case to another. It can take as little as 30 days to resolve a matter informally. Other cases may require negotiation and/or arbitration which can take several months.
8. What Does The Lemon Law Filing Process Involve?
2. We will then discuss your goals. I will provide you with an analysis of your potential claim. If it appears that you have a valid lemon law claim, I will file it promptly with the manufacturer and we will usually receive a response within 30 days.
3. If we're unable to negotiate a successful remedy with the manufacturer, your claim may be heard by an arbitrator within a couple of months. This method is described in the statutes as informal dispute resolution.
4. In some circumstances, and typically after participating in informal dispute resolution, you may bring a civil complaint. Because this process takes much more time, most claims are resolved through informal negotiation or arbitration.
9. Are There Other Laws That Protect Me?
Even though your warranty matter may not qualify as a lemon law claim, you may have a claim under the Magnuson-Moss Warranty Act, a federal law that protects consumers when a manufacturer is either unable or unwilling to make required repairs, or the Ohio Uniform Commercial Code. As an experienced consumer protection attorney, I will advise you of other remedies that may be available to you if it appears that your claim may not be covered by the Ohio Lemon Law.
For more information or to schedule a free consultation with me at Thom Cafferty Law Office in Toledo: 419-244-0169. Or email me with your question and I will respond promptly, usually within 24 hours.


DHAKA: Biman Bangladesh Airlines will transport 63 thousand 500 hajj pilgrims in this year's hajj season. This year, 1,27,000 hajj pilgrims are expected to go to Saudi Arabia to perform haj.
According to schedule of hajj operation, recently announced by Biman, the national flag carrier will operate in all 515 flights, including 422 dedicated flights. To bring back hajj pilgrims, Biman will operate 338 flights including 278 dedicated ones.
The hajj flight will start from Monday, 24 July and conclude on Saturday, 26 August. The return flights on the other hand, will start from September 06 and conclude on October 05.
For this year's hajj operation, the airline has decided to employ six wide body B777 aircraft. It addition its own four, Biman is going to take lease of two more similar aircraft. The process of taking lease is in final stage.
It may be recalled, of the total 10,1,827 hajj pilgrims last year, Biman transported 49,545 of them.