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\newcommand{\Hess}{\mathrm{Hess}}
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\title{Higher Specht Polynomials and Tableaux Bijections for Hessenberg Varieties}

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\author{Kyle Salois\thanks{\href{mailto:kyle.salois@colostate.edu}{kyle.salois@colostate.edu}. Partially supported by NSF DMS award number 2054391.}\addressmark{1}}

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\address{\addressmark{1}Department of Mathematics, Colorado State University, Fort Collins, CO, USA}

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\received{November 15, 2023}

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\abstract{The cohomology rings of regular semisimple Hessenberg varieties are only completely understood in some cases. One such case is when the Hessenberg function is $h=(h(1),n,\ldots,n)$, and is described by Abe, Horiguchi, and Masuda in 2017. We define an alternative basis for the cohomology ring in this case, which is a higher Specht basis. We give combinatorial bijections between the monomials in this basis and sets of $P$-tableaux, motivated by the work of Gasharov in 2008 and Shareshian and Wachs in 2016. This bijection illustrates the connection between the symmetric group action on these cohomology rings and the Schur expansion of chromatic symmetric functions. We further use the inversion formula for $P$-tableaux to give a new combinatorial proof of the known Poincar\'{e} polynomial for these Hessenberg varieties.}


%% put your keywords here, or comment this out if you don't have them yet
\keywords{Hessenberg varieties, $P$-tableaux, higher Specht bases}

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\begin{document}

\maketitle

%%%%%%%%%%%%%%%%%%%%%% Introduction

\section{Introduction}

In this extended abstract, we exhibit new connections between the combinatorics of Hessenberg varieties, $P$-tableaux, and chromatic symmetric functions, and illustrate their use in proving geometric results using combinatorial tools. In particular, when $S$ is a regular semisimple matrix and $h=(h(1),n,\ldots,n)$, we construct a higher Specht basis for the cohomology ring $H^{*}(\Hess(S,h))$, display combinatorial bijections between these higher Specht basis elements and sets of tableaux, and use a new combinatorial method to find the Poincar{\'e} polynomial for $\Hess(S,h)$. Full proofs of the results in this paper are forthcoming in \cite{KyleThesis}. 

Hessenberg varieties, initially defined and studied in \cite{DePrSh, DeSh88}, are linear subvarieties of the full flag variety $Fl(\C^n)$. They are connected to chromatic symmetric and quasisymmetric functions due to the action of the symmetric group $S_n$ on their cohomology rings defined by Tymozcko in \cite{Ty07}. The geometry of Hessenberg varieties has been -- and continues to be -- extensively studied, including in \cite{AbHaHoMa, AHM, HHMPT, HaPr19, InPr19, Ty05}.   

%%%%%%%%%%%%%%%%%%%%%%%%%% Background Section

\subsection{Background on Hessenberg Varieties and Specht Modules}\label{sec:Background}

Given a proper coloring $\kappa:V\to\mathbb{N}$ of a finite simple graph $G=(V,E)$ on a totally-ordered vertex set $V$, define an \textbf{ascent} to be an edge $\{v,w\}$ such that $v<w$ and $\kappa(v)<\kappa(w)$. Define $\asc(\kappa)$ to be the total number of ascents in $\kappa$. In this paper, we consider graphs with $V=[n]=\{1,2,\ldots,n\}$. In \cite{SW}, Shareshian and Wachs defined the chromatic quasisymmetric function, a graded analogue of Stanley's chromatic symmetric function \cite{StanCSF}, using the ascent statistic:

\begin{definition}
Given a finite simple graph $G=(V,E)$, the \textbf{chromatic quasisymmetric function} for $G$ is \begin{equation*}
    X_{G}(\mathbf{x};q) = \sum_{\kappa:V\to \mathbb{N}} \left( \prod_{i\in V} x_{\kappa(i)} \right) q^{\asc(\kappa)}
\end{equation*} where the sum ranges over all proper colorings $\kappa$ of the vertices of $G$.
\end{definition} 

Recall that the \textbf{(full) flag variety} of $\C^n$ is the variety $Fl(\C^n)$ whose points are flags $F_{\bullet} = F_0\subset F_1\subset \cdots \subset F_n$ such that $\dim(F_i) = i$. We will define Hessenberg varieties to be a subvariety of the flag variety, given a matrix $X:\C^n\to\C^n$ and a function $h$. First, a \textbf{Hessenberg function} is a function $h:[n]\to[n]$ such that for all $i$, we have $i\leq h(i)$, and $h(i)\leq h(i+1)$. We usually denote this as a vector $h=(h(1),h(2),\ldots, h(n))$.

\begin{definition}
    Given a matrix $X:\C^n\to \C^n$ and a Hessenberg function $h:[n]\to[n]$, define the \textbf{Hessenberg variety} to be: \begin{equation}
        \Hess(X,h) = \left\{ F_{\bullet}\in Fl(\C^n) \,|\, X(F_i)\subseteq F_{h(i)} \, \textrm{ for all }\, 1\leq i\leq n \right\}
    \end{equation}
\end{definition}

In \cite{Ty07}, Tymoczko defined an action of the symmetric group on the cohomology ring $H^{*}(\Hess(S,h))$ when $S$ is a regular semisimple matrix, allowing us to study the structure of this ring as an $S_n$-module.  For each Hessenberg function, we can also construct a poset $P_h$ on $[n]$, using $h$ to determine which elements are comparable: We say that $i<_{P_h} j$ if and only if $h(i)<j$. Let $G_h$ be the incomparability graph of $P_h$, which has an edge $\{i,j\}$ whenever $i$ and $j$ are incomparable in $P_h$. Notably, posets formed by this construction are $(3+1)$- and $(2+2)$-avoiding, so their incomparability graphs are relevant to the following conjecture of Stanley and Stembridge on chromatic symmetric functions.

\begin{conjecture}[\cite{StSt} Conjecture 5.1]\label{conj:StanStem}
    If $P$ is a $(3+1)$-free poset, then $X_{\mathrm{inc}(P)}(\mathbf{x})$ is $e$-positive, that is, it can be written with positive coefficients when expanded in the elementary basis of symmetric functions.
\end{conjecture}

In \cite{GP13}, Guay-Paquet proved that it suffices to prove the conjecture for posets which are $(3+1)$- and $(2+2)$-avoiding. Shareshian and Wachs conjectured that there is a connection between chromatic quasisymmetric functions and the graded cohomology ring of regular semisimple Hessenberg varieties, using the ascent formula for the incomparability graph $G_h$. This connection, stated below, was proven by Brosnan and Chow in \cite{BC18} and separately by Guay-Paquet in \cite{GP16}.

\begin{proposition}[\cite{BC18,GP16}]\label{prop:BCGP}
    Let $S$ be a regular semisimple $n\times n$ matrix and $h:[n]\to[n]$ be a Hessenberg function. Let $G_h$ be the incomparability graph for $P_h$. Then
    \begin{equation*}
        \omega X_{G_h}(\mathbf{x};q) = \sum_{k=0}^{|E|} \mathrm{Frob}(H^{2k}(\Hess(S,h)))q^k
    \end{equation*}
\end{proposition}

Above, $\omega$ is the standard involution on symmetric functions which sends the Schur function $s_{\lambda}$ to  $s_{\lambda'}$, where $\lambda'$ is the transpose of $\lambda$, and $\mathrm{Frob}$ is the Frobenius characteristic map which sends the irreducible $S_n$-module $V_{\lambda}$ to the Schur function $s_{\lambda}$.

\begin{definition}[\cite{Gasharov}]
    Let $P$ be a poset and $\lambda$ be a partition of $n$. A \textbf{$P$-tableau of shape $\lambda$} is a filling of the Young diagram of $\lambda$ with entries from $P$ such that:
    \begin{itemize}
        \item Each entry in $P$ is used at most once.
        \item Adjacent entries in rows are $P$-increasing from left to right.
        \item Adjacent entries in columns are $P$-nondecreasing from bottom to top. 
    \end{itemize}
\end{definition}

\begin{figure}\label{fig:PTabEx}
    \centering
    \begin{multicols}{2}
    \includegraphics[width=0.3\textwidth]{Poset h34455.png}

    $\begin{ytableau} 2 \\ 4 \\ 5 \\ 1 \\ 3 \end{ytableau} \hspace{1cm} \begin{ytableau} 4 \\ 3 \\ 2 \\ 1 & 5 \end{ytableau} \hspace{1cm} \begin{ytableau} 3 \\ 1 & 4 \\ 2 & 5 \end{ytableau}$
    \end{multicols}
    \caption{On the left, the poset $P_h$ for $h=(3,4,4,5,5)$. On the right, three $P_h$-tableaux (in French notation). The leftmost tableau has $4$ inversions given by the pairs $(1,3),(2,3),(2,4)$, and $(4,5)$, but does not have the inversion $(2,5)$ since $2<_{P_h}5$.}
\end{figure}

We say a \textbf{$P$-inversion} in a $P$-tableau is a pair of entries $(i,j)$ such that $i<j$ as integers, $i$ is in a higher row than $j$, and $i$ and $j$ are incomparable in $P$. Define $\inv(T)$ to be the number of $P$-inversions in $T$. In \cite{Gasharov}, Gasharov used $P$-tableaux to show that the chromatic symmetric functions of incomparability graphs of $(3+1)$-free posets are Schur-positive. Using this inversion statistic on $P$-tableaux, Shareshian and Wachs extended this result to the chromatic quasisymmetric case, as stated below.

\begin{proposition}[\cite{SW} Theorem 6.3]\label{prop:SW}
    Let $G$ be the incomparability graph of a $(3+1)$-free poset $P$, and let $PT(\lambda)$ be the set of $P$-tableaux of shape $\lambda$. Then we have:
    \begin{equation*}
        X_G(\mathbf{x};q) = \sum_{\lambda\vdash n} \left( \sum_{T\in PT(\lambda)} q^{\inv(T)}\right) s_{\lambda}
    \end{equation*}
\end{proposition}

Combining the results of Propositions~\ref{prop:BCGP} and \ref{prop:SW}, we can connect the graded cohomology of $\Hess(S,h)$ with $P$-tableaux in the following way. If $S$ is a regular semisimple matrix, and $h$ is a Hessenberg function with poset $P_h$ and incomparability graph $G_h$, then
\begin{equation}\label{eqn:main}
    \sum_{k=0}^{|E|} \mathrm{Frob}(H^{2k}(\Hess(S,h)))q^k = \omega X_{G_h}(\mathbf{x};q) = \sum_{\lambda\vdash n} \left( \sum_{T\in PT(\lambda)} q^{\inv_h(T)}\right) s_{\lambda'}
\end{equation} where $\lambda'$ is the transpose partition of $\lambda$, and $\inv_h$ is the inversion statistic for $P_h$. 

The formula above gives us a nice way of understanding the decomposition of the $S_n$-module $H^{*}(\Hess(S,h))$ into irreducible modules. Irreducible $S_n$-modules are isomorphic to the Specht modules, which have a basis indexed by standard tableaux:

\begin{definition}
    Given a standard tableau $T$ of shape $\lambda$, define the \textbf{Specht polynomial} to be \begin{equation*}
        F_T = \prod_{C\in \lambda} \left( \prod_{i<j\in C} (x_j-x_i)\right) 
    \end{equation*}
    where the first product is over all columns in the Young diagram, and the second product is over all pairs of entries $i<j$ in the column $C$. If $SYT(\lambda)$ is the set of all standard tableaux of shape $\lambda$, then the \textbf{Specht module} $V_{\lambda}$ is the subspace of $\mathbb{Q}[x_1,\ldots, x_n]$ generated by $\{F_T\}_{T\in SYT(\lambda)}$. 
\end{definition}

An immediate consequence of this definition is that the dimension of the Specht module $V_{\lambda}$ is the number of standard tableaux of shape $\lambda$, which we denote $\#SYT(\lambda)$. We define a higher Specht basis for an $S_n$-module as follows.

\begin{definition}[\cite{GR21}, Definition 1.5]
    If $R$ is an $S_n$-module which decomposes into irreducible $S_n$-modules as $$R = \bigoplus_{\lambda} c_{\lambda}V_{\lambda}\,,$$ then a \textbf{higher Specht basis of $R$} is a set of elements $\mathcal{B}$ with a decomposition $\mathcal{B} = \bigcup_{\lambda} \bigcup_{i=1}^{c_\lambda} \mathcal{B}_{i,\lambda}$ such that the elements of $\mathcal{B}_{i,\lambda}$ are a basis of the $i$-th copy of $V_{\lambda}$ in the decomposition of $R$.  
\end{definition}

Hence, higher Specht bases of $S_n$-modules are a natural way to understand the action of $S_n$, and allow us to more easily identify the decomposition into irreducible modules. 


%%%%%%%%%%%%%%%%%%%%%%%%%% Higher Specht Basis Section

\section{Higher Specht basis for the cohomology ring}\label{sec:HigherBasis}

In this section, let $S$ be a regular semisimple $n\times n$ matrix, and $h=(h(1),n,\ldots, n)$ be a Hessenberg function. In \cite{AHM} (Theorem 4.3), Abe, Horiguchi, and Masuda give a presentation of the cohomology ring $H^{*}(\Hess(S,h))$ as a quotient of a polynomial ring in $2n$ variables. Further, in Remark 4.5, they describe a set of basis elements for this ring. We name the two sets of different types of elements below:
\begin{equation}\label{eq:Basis1}
    B_1 = \left\{ x_1^{i_1}x_2^{i_2}\cdots x_n^{i_n} \,\,\,\,  \mbox{ not containing the factor } \prod_{\ell=1}^{h(1)} x_{\ell}\right\}
\end{equation} \vspace{-0.1cm}
\begin{equation}\label{eq:Basis2}
    B_2 = \left\{ x_n^{\ell_1}x_{n-1}^{\ell_2}\cdots x_2^{\ell_{n-1}} y_k \,\,\,\, \mbox{ not containing the factor } \prod_{\ell=h(1)+1}^{n} x_{\ell}\right\}  
\end{equation}
over all $0\leq i_j \leq n-j$ in $B_1$, and over all $0\leq \ell_j\leq n-1-j$, and $1\leq k\leq n-1$ in $B_2$. 

The symmetric group $S_n$ acts on the above monomials by fixing the set of $x_i$ and permuting the set of $y_i$ in the natural way. This group action gives a representation of $S_n$, which decomposes into the direct sum of trivial representations (corresponding to the Specht module $V_{(n)}$) and standard representations (corresponding to the Specht module $V_{(n-1,1)}$). 

We define $B_3$ to be the following set of monomials: 
\begin{equation}\label{eq:Basis3}
    B_3 = \left\{ x_n^{\ell_1}x_{n-1}^{\ell_2}\cdots x_2^{\ell_{n-1}}(y_{k+1}-y_1)\,\,\,\, \mbox{not containing the factor } \prod_{\ell=h(1)+1}^{n} x_{\ell}\right\}
\end{equation}
over all $0\leq \ell_j \leq n-1-j$ and $1\leq k\leq n-1$. 

Notice that there are natural projections from $B_1$ to the Specht module $V_{(n)}$ and from $B_3$ to the Specht module $V_{(n-1,1)}$ given by forgetting the $x_i$ variables.

\begin{theorem}[\cite{KyleThesis}]
    The set $B_1\cup B_3$ forms a higher Specht basis of $H^{*}(\Hess(S,h))$.
\end{theorem}

The proof (see \cite{KyleThesis} for full details) uses the fact that $B_1\cup B_2$ forms a $\mathbb{Z}-$basis of $H^{*}(\Hess(S,h))$, and constructs the transition matrix from $B_1\cup B_2$ to $B_1\cup B_3$ using the relations given in \cite{AHM} to express the new elements in terms of the old basis. Then, we prove that this transition matrix is invertible, which requires the following lemma.

\begin{lemma}[\cite{KyleThesis}]
    If $f(x_1,\ldots, x_n)$ is a homogeneous polynomial in the ring $H^{*}(\Hess(S,h))$, then $f$ can be expressed solely in terms of basis elements from $B_1$. 
\end{lemma}

Knowing that $B_1\cup B_3$ forms a higher Specht basis of $H^{*}(\Hess(S,h))$ allows us to obtain a more direct proof of the following fact, by counting the number of monomials of each type in $B_1$ and $B_3$.

\begin{corollary}
    The dot action of $S_n$ on $H^{*}(\Hess(S,h))$ decomposes into $h(1)(n-1)!$ copies of the trivial representation, and $(n-h(1))(n-2)!$ copies of the standard representation. 
\end{corollary}

%%%%%%%%%%%%%%%%%%%%%%% Bijections Section

\section{Bijections between basis elements and P-tableaux}\label{sec:BasisBijection}

As seen in Section \ref{sec:Background}, there is a bijection between the set of standard tableaux of shape $\lambda$ with basis elements of the Specht module $V_{\lambda}$, given by the construction of the basis elements. Further, from Equation~\ref{eqn:main}, we have an explicit connection between the number of basis elements of $H^{*}(\Hess(S,h))$ of each degree and the set of $P_h$-tableaux with each number of inversions. In particular, there should be bijections between the higher Specht basis elements and the sets of $P_h$-tableaux with shape corresponding to the Specht polynomials in the basis. 

\subsection{Regular Nilpotent Hessenberg Varieties}

In the case of regular nilpotent Hessenberg varieties, a polynomial presentation of the cohomology ring is known for any Hessenberg function $h$. 
\begin{proposition}[\cite{HHMPT}, Corollary 7.3]
    Let $N$ be a regular nilpotent matrix, and let $h:[n]\to[n]$ be a Hessenberg function. Then the following set of monomials form an additive basis for $H^*(\textrm{Hess}(N,h))$: \begin{equation*} \mathcal{N}_h := \left\{ x_1^{i_1}\cdots x_{n}^{i_n}\,|\,0\leq i_k \leq h(k)-k\textrm{ for } 1\leq k <n\right\}\end{equation*}
\end{proposition}

In \cite{AbHaHoMa}, Abe et al. show that the cohomology rings of regular nilpotent Hessenberg varieties are isomorphic to the fixed points of the cohomology rings for regular semisimple Hessenberg varieties. In particular, these are the pieces of $H^{*}(\Hess(S,h))$ which decompose into trivial $S_n$-modules, corresponding to the Specht module $V_{(n)}$. 

Define $PT(h,\lambda)$ to be the set of $P_h$-tableaux of shape $\lambda$. Taking the transpose partition for $\lambda=(n)$ (because of Equation~\ref{eqn:main}), we form a map $\varphi$ between $\mathcal{N}_h$ and $PT(h,(1^n))$ for any Hessenberg function $h$. 

\begin{definition}
    Let $x_1^{i_1}\cdots x_n^{i_n}\in \mathcal{N}_h$. 
    \begin{itemize}
        \item Begin with a $P_h$-tableau $T$ of a single box whose entry is $n$.
        \item For each $k=n-1,\ldots 1$:
        \begin{itemize}
            \item If $i_k = 0$, insert $k$ into a new box at the bottom of $T$, so that $k$ occurs directly below some $\ell>k$.
            \item If $i_k>0$, then since $i_k\leq h(k)-k$, we have $k<h(k)$. List the entries $k+1,\ldots ,h(k)$, which already exist in $T$, in order from the lowest to highest row position in $T$. Insert $k$ in a new box directly above the $i_k$-th lowest entry of this list.
        \end{itemize}
    \end{itemize}
    Define $\varphi(x_1^{i_1}\cdots x_n^{i_n})\in PT(h,(1^n))$ to be the resulting tableau from this process.
\end{definition}

\begin{example}
    Let $h=(2,3,5,5,5)$, and consider the monomial $x_1x_3x_4\in \mathcal{N}_h$. We construct $\varphi(x_1x_3x_4)$ as follows.
\begin{equation*}
    \begin{ytableau} \none \\ \none \\ \none \\ \none \\ 5 \end{ytableau} \hspace{1cm}
    \begin{ytableau} \none \\ \none \\ \none \\ 4 \\ 5 \end{ytableau} \hspace{1cm}
    \begin{ytableau} \none \\ \none \\ 4 \\ 3 \\ 5 \end{ytableau} \hspace{1cm}
    \begin{ytableau} \none \\ 4 \\ 3 \\ 5 \\ 2 \end{ytableau} \hspace{1cm}
    \begin{ytableau} 4 \\ 3 \\ 5 \\ 1 \\ 2 \end{ytableau} \hspace{1cm}
\end{equation*}
We start with a single box containing a $5$. Then, to insert $4$ with $i_4=1$ $P_h$-inversion, we insert the $4$ above the $5$. To insert $3$ with $i_3=1$ $P_h$-inversion, we insert the $3$ above the $5$ but below the $4$. Notice that at each step, the number of elements in $P_h$ greater than $k$ that are incomparable to $k$ is $h(k)-k$, which is also the largest possible power $i_k$. 
\end{example}

In \cite{KyleThesis}, we prove that $\varphi$ is a bijection, which is weight preserving in the following way: If $m$ is a monomial of degree $d$, then $\varphi(m)$ has $d$ $P_h$-inversions. 

\begin{theorem}[\cite{KyleThesis}]
    The map $\varphi$ is a well-defined, weight-preserving bijection. 
\end{theorem}

\subsection{Regular Semisimple Hessenberg Varieties}

Now we turn our attention to regular semisimple Hessenberg varieties. In this section, $S$ is a regular semisimple matrix and $h=(h(1),n,\ldots, n)$. Recall the partial set of basis elements $B_1$ defined in Equation~\ref{eq:Basis1}, and recall that $S_n$ fixes these monomials, since they contain no $y_i$ variable. These correspond to basis elements of the trivial Specht module $V_{(n)}$. Again, we take the transpose partition, and define a map $\psi$ between $B_1$ and $PT(h,(1^n))$ as follows. 

\begin{definition} Let $x_1^{i_1}\cdots x_n^{i_n}\in B_1$.
    \begin{itemize}
        \item Begin with a $P_h$-tableau $T$ of a single box whose entry is $n$.
        \item For each $k=n-1,\ldots, 1$, insert $k$ into $T$ above exactly $i_k$ of the existing entries.
        \item  Let $k'$ be the smallest index in $1,\ldots, h(1)$ such that $i_{k'}=0$, which exists by the definition of $B_1$. By this construction, after inserting $n$ through $1$, $k'$ will be on the bottom of $T$.
        \begin{itemize}
            \item If $k'=1$, then define $\psi(x_1^{i_1}\cdots x_n^{i_n})$ to be $T$.
            \item If $1<k'\leq h(1)$, then slide the entry $k'$ up until it is directly below the $1$, and define $\psi(x_1^{i_1}\cdots x_n^{i_n})$ to be $T$ after this slide.
        \end{itemize}
    \end{itemize}
          
\end{definition}

\begin{example}
Let $h=(3,5,5,5,5)$, and consider the monomial $x_1^2 x_3x_4 \in B_1$. We construct $\varphi(x_1^2 x_3x_4)$ as follows. 
    \begin{equation*}
        \begin{ytableau} \none \\ \none \\ \none \\ \none \\ 5 \end{ytableau} \hspace{1cm}
        \begin{ytableau} \none \\ \none \\ \none \\ 4 \\ 5 \end{ytableau} \hspace{1cm}
        \begin{ytableau} \none \\ \none \\ 4 \\ 3 \\ 5 \end{ytableau} \hspace{1cm}
        \begin{ytableau} \none \\ 4 \\ 3 \\ 5 \\ 2 \end{ytableau} \hspace{1cm}
        \begin{ytableau} 4 \\ 3 \\ 1 \\ 5 \\ 2 \end{ytableau} \hspace{1cm}
        \begin{ytableau} 4 \\ 3 \\ 1 \\ 2 \\ 5 \end{ytableau} 
    \end{equation*}
We start with a single box containing a $5$. Then we insert the $4$ above one existing entry, the $3$ above one existing entry, the $2$ above no existing entries, and the $1$ above two existing entries. Since $1<_{P_h}5$ are comparable, the resulting tableau is not a $P_h$-tableau, so we shift the $2$ (which is incomparable to the $1$) to be directly below the $1$. In the second-to-last tableau, the number of $P_{N}$-inversions with each $k$ as the smaller entry is exactly $i_k$, so reading these inversions returns the monomial $x_1^2x_3x_4$.
\end{example}

In \cite{KyleThesis}, we find the inverse map of $\psi$ to prove the following theorem.

\begin{theorem}[\cite{KyleThesis}]
The map $\psi$ is a well-defined bijection. 
\end{theorem}

We now construct a map for the set of basis elements $B_3$ defined in Equation~\ref{eq:Basis3}. Define $PSPT(h,\lambda)$ to be the set of pairs $(S,T)$ where $S$ is a standard tableau and $T$ is a $P_h$-tableau, both of shape $\lambda$. Since the monomials in $B_3$ correspond to the Specht polynomials in the Specht module $V_{(n-1,1)}$, we construct the map $\tau$ to the set $PSPT(h,(2,1^{n-2})$, with the $x_i$ variables corresponding to the $P_h$-tableau and the $y_i$ variables corresponding to the standard tableau.  

\begin{definition} Let $x_n^{\ell_1}\cdots x_{2}^{\ell_{n-1}}(y_{k}-y_1)\in B_3$.
    \begin{itemize}
        \item Define $S$ to be the unique standard tableau of shape $(2,1^{n-2})$ with entries $1$ and $k$ in the bottom row.
        \item Let $j$ be the largest entry among $\{h(1)+1, \ldots, n\}$ such that the exponent $\ell_{n-j+1}$ on $x_j$ is zero, which exists by the definition of $B_3$.
        \item Initialize a tableau $T$ with a single row of two boxes, containing a $1$ and $j$.
        \item For each $i=2,\ldots, n$, other than $j$, insert $i$ into the left column so that it is under exactly $(i-2)-\ell_{n-i+1}$ of the current entries in the left column.
    \end{itemize}    
      Define $\tau(x_n^{\ell_1}\cdots x_{2}^{\ell_{n-1}}(y_{k}-y_1))\in PSPT(h,(2,1^{n-2})$ to be the pair $(S,T)$ resulting from this construction.  
\end{definition}

\begin{example}
    Let $h=(3,5,5,5,5)$, and consider the monomial $x_5^2x_3(y_3-y_1) \in B_3$. Note that $j=4$ is the largest index where $x_j$ has an exponent of zero. We construct $\tau(x_5^2x_3(y_2-y_1))$ as follows.
    \begin{equation*}
        S \,=\, \begin{ytableau} 5 \\ 4 \\ 2 \\ 1 & 3 \end{ytableau} \hspace{1cm} \hspace{1cm}
        \begin{ytableau}\none \\ \none \\ \none \\ 1 & 4 \end{ytableau} \hspace{1cm}
        \begin{ytableau}\none \\ \none \\ 2 \\ 1 & 4 \end{ytableau} \hspace{1cm}
        \begin{ytableau}\none \\ 3 \\ 2 \\ 1 & 4 \end{ytableau} \hspace{1cm}
        \begin{ytableau} 3 \\ 5 \\ 2 \\ 1 & 4 \end{ytableau} \,=\,T
\end{equation*}
$S$ is defined to be the unique standard Young tableaux of shape $(2,1^{n-2})$ with a $1$ and $3$ in the bottom row. Then, to form $T$, we start with a single row containing a $1$ and a $4$. We then insert a $2$ in the left column underneath $(2-2)-0 = 0$ entries, a $3$ in the left column underneath $(3-2)-1 = 0$ entries, and a $5$ in the left column underneath $(5-2)-2=1$ entry. 
\end{example}

In \cite{KyleThesis}, we again find the inverse map to prove the following theorem.

\begin{theorem}[\cite{KyleThesis}]
The map $\tau$ is a well-defined bijection.
\end{theorem}

This map is almost weight-reversing, since the exponents on the $x_i$ terms correspond to inversions that are missing in the tableaux, since we insert $i$ so that it forms $(i-2)-\ell_{n-i+1})$ inversions. In future work, we hope to use bijections like these to extrapolate potential bases for $H^{*}(\Hess(S,h))$ in other cases.



%%%%%%%%%%%%%%%%%%%%%%%% Poincare Polynomial Section

\section{Poincar{\'e} polynomials of Hessenberg varieties}\label{sec:PoincarePoly}

Given a graded vector space $V$ over a field $k$, if $V = \bigoplus_{i\in \N} V_i$ with each subspace $V_i$ consisting of vectors of degree $i$ being finite dimensional, then the \textbf{Poincar\'{e} polynomial} of $V$ is 
\begin{equation*}
\Poin(V,q) = \sum_{i\in \N} \dim_k(V_i) q^{i}    
\end{equation*}

Then, for an algebraic variety $X$ with graded cohomology ring $H^{*}(X)$, we define the Poincar{\'e} polynomial of $X$ to be $\Poin(X,q) := \Poin(H^{*}(X),q)$. From Equation~\ref{eqn:main}, we can write the Poincar{\'e} polynomial of a regular semisimple Hessenberg variety in the following way:
\begin{equation}
    \mathrm{Poin}(\Hess(S,h),q) = \sum_{\lambda\vdash n} \left( \sum_{T\in \mathrm{PT}(h,\lambda)} q^{\inv_h(T)}\right) \#\mathrm{SYT}(\lambda)
\end{equation}
since the dimension of the irreducible Specht module $V_{\lambda}$ is the number of standard tableaux of shape $\lambda$. We use this formula to provide an alternate proof of the formula of the Poincar{\'e} polynomial for $\Hess(S,h)$ when $h=(h(1),n,\ldots, n)$, originally calculated by Abe, Horiguchi, and Masuda in \cite{AHM}. Recall that the $q$-analogue of $n$ is $(n)_q = (1+q+\cdots +q^{n-1})$, and the $q$-analogue of $n!$ is $(n)_q! = (n)_q(n-1)_q\cdots(1)_q$. We present the full proof here, as it illustrates the new combinatorial method using $P_h$-tableaux. 

\begin{theorem}[\cite{AHM}, Lemma 3.2]
If $h=(h(1),n,\ldots, n)$, then the Poincar\'{e} polynomial of $\Hess(S,h)$ is given by \begin{align*}
    \mathrm{Poin}(\Hess(S,h),q) &= \frac{1-q^{h(1)}}{1-q}\,\prod_{j=1}^{n-1}\frac{1-q^j}{1-q} + (n-1)q^{h(1)-1}\frac{1-q^{n-h(1)}}{1-q}\,\prod_{j=1}^{n-2}\frac{1-q^j}{1-q} \\
    &= h(1)_q(n-1)_q! + (n-1)q^{h(1)-1}(n-h(1))_q(n-2)_q!
\end{align*}
\end{theorem}

\begin{proof}
From above, we know that $$ \mathrm{Poin}(\Hess(S,h),q) = \sum_{\lambda\vdash n} \left( \sum_{T\in \mathrm{PT}(h,\lambda)} q^{\inv_h(T)}\right) \#\mathrm{SYT}(\lambda). $$
  
Let $h=(h(1),n,\ldots, n)$. All chains in $P_h$ have length two and include the element $1$. Since distinct rows in a $P_h$ tableaux need to contain entries from distinct chains in $P_h$, the only shapes $\lambda$ with a nonzero number of $P_h$-tableaux are $\lambda = (1^n)$ and $\mu = (2,1^{n-2})$. Further, we have that $\#\mathrm{SYT}(\lambda) = 1$ and $\#\mathrm{SYT}(\mu)=n-1$. 

For $\lambda=(1^n)$, we need to count the $P_h$-inversions in the $P_h$ tableaux of this shape. Since the element $1$ is incomparable to $2$ through $h(1)$, it can form between $0$ and $h(1)-1$ inversions as the smaller entry. For each $i=2,\ldots, n$, the entry $i$ can form up to $n-i$ inversions as the smaller entry. Hence, we get that $$ \sum_{T\in \mathrm{PT}(h,\lambda)}q^{\inv_h(T)} = (1+q+\cdots+q^{h(1)-1})(1+q+\cdots+q^{n-2})! = h(1)_q(n-1)_q!.$$

For $\mu=(2,1^{n-2})$, the bottom row of any $P_h$-tableaux of shape $\mu$ must be filled with entries from a chain in $P_h$, so it contains a $1$ and an $i$ for some $i=h(1)+1,\ldots, n$. Then, since $i>1$, it is incomparable with all other $j\neq 1$, so the entry $i$ in the bottom row forms inversions as the larger entry with the entries $2,\ldots, i-1$, of which there are $i-2$. So this entry contributes between $h(1)-1$ and $n-2$ inversions to the $P_h$-tableaux as the larger entry. Then, for the column entries of $j=2,\ldots, n$ and $j\neq i$, if $j<i$, then $j$ forms an inversion with $i$ where $j$ is the smaller entry (which was already counted), and can form an inversion as the smaller entry with the other $n-j-1$ entries larger than $j$. If $j>i$, then $j$ does not form an inversion with $i$, and can form an inversion as the smaller entry with any of the $n-j$ entries larger than $j$. In each case, there is a unique placement for $j$ giving each set of inversions. Hence we have 
\begin{equation*}
\sum_{T\in \mathrm{PT}(h,\mu)} q^{\inv_h(T)} = (q^{h(1)-1}+\cdots+q^{n-1}) (1+\cdots+q^{n-3})! = q^{h(1)-1}(n-h(1))_q (n-2)_q! 
\end{equation*}

Therefore, for $\lambda=(1^n)$ and $\mu=(2,1^{n-2})$, we have the Poincar\'{e} polynomial of $\Hess(S,h)$ as follows:
\begin{align*} \mathrm{Poin}(\Hess(S,h),q) &= \sum_{T\in \mathrm{PT}(h,\lambda)} q^{\inv_h(T)} + (n-1)\sum_{T\in \mathrm{PT}(h,\mu)} q^{\inv_h(T)} \\ &= h(1)_q (n-1)_q! + (n-1)q^{h(1)-1}(n-h(1))_q(n-2)_q! \end{align*}
This completes the proof. \end{proof}

These methods provide a new, combinatorial means of finding Poincar{\'e} polynomials of regular semisimple Hessenberg varieties, which may be useful in further understanding the basis decomposition of their cohomology rings.

%%%%%%%%%%%%%%%%%%%%%%% Acknowledgements and Bibliography

\acknowledgements{I thank my advisor, Maria Gillespie, for pointing me in the direction of this project and providing support during the research and writing process. I also thank Martha Precup and Kyle Celano for insightful conversations.}

\printbibliography

\end{document}
