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\newcommand{\ket}[1]{#1 \big\rangle}
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%\usepackage{lipsum}

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\title{A unipotent realization of the chromatic quasisymmetric function}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Lucas Gagnon\thanks{\href{mailto:lgagnon@yorku.ca}{lgagnon@yorku.ca}.}\addressmark{1}}

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\address{\addressmark{1}Department of Mathematics and Statistics, York University, Toronto ON, Canada}

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\received{\today}

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\abstract{
The chromatic quasisymmetric function is a $t$-analogue of Stanley's chromatic symmetric function, and has recently been at the center of a number of exciting developments in algebraic combinatorics.  
This extended abstract contributes to this trend, describing a novel realization of certain chromatic quasisymmetric functions as characters of the finite general linear group $\mathrm{GL}_{n}(\mathbb{F}_{q})$.  
Additional results tie these characters to other aspects of the chromatic quasisymmetric function: point counting in Hessenberg varieties over $\mathbb{F}_{q}$, realizing the plethystic connection with unicellular LLT polynomials, and re-interpreting positivity conjectures.
}

%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

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\keywords{Chromatic quasisymmetric function; unicellular LLT polynomial; unipotent group; combinatorial Hopf algebra}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
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\usepackage[backend=bibtex]{biblatex}
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\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The chromatic symmetric function sits at a nexus of disparate areas of mathematics.  
At face value, this symmetric function encodes the coloring problem of a graph as an analogue of the chromatic polynomial~\cite{Sta}.  
However, through a well-known equivalence between the ring of symmetric functions and the representation theory of the symmetric groups (see e.g.~\cite{Mac}), some chromatic symmetric functions are also complex characters of the symmetric group~\cite{Gash}.   
Moreover, by way of a $t$-analogue known as the chromatic quasisymmetric function, Brosnan and Chow~\cite{BrosChow} and Guay-Paquet~\cite{GP16} independently proved that the characters corresponding to indifference graphs are afforded by symmetric group modules on the cohomology rings of regular semisimple Hessenberg varieties, as predicted by a conjecture of Shareshian and Wachs~\cite{ShWa}.  
Thus, certain questions about graphs, representation theory, and algebraic geometry coincide in the structure of these symmetric functions, and vice versa.

Based on the paper~\cite{Gagb}, this extended abstract describes yet another connection to the chromatic quasisymmetric function, this time from the general linear group $\GL_{n}(\FF_{q})$ over the finite field with $q$ elements.  
The maximal unipotent subgroup $\UT_{n}(\FF_{q})$ of $\GL_{n}(\FF_{q})$ has a family of characters indexed by indifference graphs, and \cref{thm:CQSbijection} shows that and up to a factor of $(q-1)^{n}$ the induction functor gives a map
\[
\left\{ \begin{array}{c} \text{indifference graph} \\ \text{indexed characters} \end{array}\right\} 
\xrightarrow{\;\; \ind^{\GL_{n}(\FF_{q})}_{\UT_{n}(\FF_{q})} \;\;}  
\left\{ \begin{array}{c}  \text{chromatic quasisymmetric functions for} \\ \text{indifference graphs evaluated at $t = q$} \end{array} \right\}
\]
where unipotently supported class functions of $\GL_{n}(\FF_{q})$ are identified with symmetric functions via the Hall algebra.  
\cref{sec:mainresult1} describes this result and its proof using a Hopf algebraic interpretation of induction established in a second paper of the author~\cite{Gaga}.  

The remaining sections of this abstract explore implications of \cref{thm:CQSbijection}.  % for the theory of chromatic quasisymmetric functions.  %, broadly speaking.  
While the theorem is reminiscent of the Brosnan--Chow--Guay-Paquet theorem, the underlying association between symmetric functions and $\GL_{n}(\FF_{q})$ characters in this paper is quite different and offers a complementary perspective.  
Moreover, the relevant characters are elementary in nature and may be more accessible from a module theoretic standpoint.

\cref{sec:Hessenberg} gives a geometric interpretation of the characters in \cref{thm:CQSbijection}: their values count the points of Hessenberg varieties associated to an ad-nilpotent ideal over $\FF_{q}$.  
Even over $\CC$, Hessenberg varieties associated to ad-nilpotent ideals are markedly different from the ones in the Brosnan--Chow--Guay-Paquet theorem, and there is no known module structure on their cohomology rings.  
However, Precup and Sommers~\cite{PrecupSommers} have given a geometric connection between the Poincar\'{e} polynomials of these Hessenberg varieties and the chromatic quasisymmetric function, and \cref{cor:adnilpotentpoincare} shows how \cref{thm:CQSbijection} can be seen as a representation theoretic manifestation of this phenomenon.

\cref{sec:mainresult2} revisits a well-known relationship between chromatic quasisymmetric functions and the family of unicellular LLT polynomials~\cite{CM} from the perspective of $\GL_{n}(\FF_{q})$.  
The unipotent characters of $\GL_{n}(\FF_{q})$ give another realization of symmetric functions, and any character can be projected onto its unipotent summands to produce a symmetric function.  
For the characters in  \cref{thm:CQSbijection}, this association turns out to be a twisted version of the relationship between chromatic quasisymmetric functions and unicellular LLT polynomials, and \cref{thm:LLTbijection} gives a map
\[
\left\{ \begin{array}{c} \text{indifference graph} \\ \text{indexed characters} \end{array}\right\} \xrightarrow{\;\; \omega \circ \operatorname{projection} \circ \ind^{\GL_{n}(\FF_{q})}_{\UT_{n}(\FF_{q})} \;\;}  \left\{ \begin{array}{c}  \text{unicellular LLT polyno-} \\ \text{mials evaluated at $t = q$} \end{array} \right\}.
\]
No previous connection between LLT polynomials and $\GL_{n}(\FF_{q})$ representation theory was known, though similar associations exist for quantum groups~\cite{LLT}, affine Hecke algebras~\cite{GrojHai}, and the symmetric groups~\cite{GP16, HHL}.

Finally, \cref{sec:positivity} describes the $\GL_{n}(\FF_{q})$ representation theoretic meaning of two ``positivity conjectures'' about the chromatic quasisymmetric function and unicellular LLT polynomial.  
Each conjecture postulates that when the symmetric function in question is expressed in a chosen basis, each coefficient will be nonnegative. 
For the chromatic quasisymmetric functions, the modified Stanley--Stembridge conjecture~\cite[Conjecture 1.3]{ShWa} (see also~\cite{StaStem}) concerns positivity in the elementary symmetric function basis, and is entirely open.  
For the LLT polynomials, positivity in the Schur basis has been established by Grojnowski and Haiman~\cite{GrojHai}, but no positive combinatorial formula for the coefficients is known~\cite{HaglundBook}.  
While no immediate progress is made on either conjecture in this abstract, the $\GL_{n}(\FF_{q})$ interpretations may be a useful starting point for future work.  

The remainder of this abstract is organized as follows.  \cref{sec:prelims} gives general background material.  \cref{sec:mainresult1} contains the main result, \cref{thm:CQSbijection}, and \cref{sec:Hessenberg} relates this result to Hessenberg varieties.  \cref{sec:mainresult2} concerns the unicellular LLT polynomial.  Finally, \cref{sec:positivity} discusses the aforementioned positivity conjectures.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries}
\label{sec:prelims}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This section contains preliminary material on Hopf algebras and symmetric functions (\cref{sec:QSym}), chromatic quasisymmetric functions (\cref{sec:Catalan}), the character theory of $\UT_{n}(\FF_{q})$ (\cref{sec:sct}), and its relation to $\GL_{n}(\FF_{q})$ characters (\cref{sec:repHopf}).

%%%%%%%%%%%
\subsection{(Quasi-)Symmetric functions and combinatorial Hopf algebras}
\label{sec:QSym}
\label{sec:CHAs}
%%%%%%%%%%%

This section describes the Hopf algebraic techniques required to prove the main result.  Throughout, the term \emph{Hopf algebra} will refer to a graded connected $\CC$-vector space 
\[
H_{\bullet} = \bigoplus_{n \ge 0} H_{n} \qquad\text{such that}\qquad H_{0} \cong \CC,
\]
equipped with a $\CC$-bialgebra structure and a compatible antipode map.  
This extra structure is not used prominently in this work, which instead focuses on Hopf algebra homomorphisms; these can be thought of as a well-behaved subclass of graded linear maps.

Two particularly important examples now follow.  
A \emph{composition} of $n \in \ZZ_{\ge 0}$ is a sequence of positive integers $\alpha = (\alpha_{1}, \ldots, \alpha_{k})$ with $\alpha_{1} + \cdots + \alpha_{k} = n$.  Call each $\alpha_{i}$ a \emph{part} of $\alpha$.  The \emph{monomial quasisymmetric function} associated to the composition $\alpha$ is 
\[
M_{\alpha} = \sum_{i_{1} < \cdots < i_{\ell}} x_{i_{1}}^{\alpha_{1}} x_{i_{2}}^{\alpha_{2}} \cdots x_{i_{\ell}}^{\alpha_{\ell(\alpha)}} \in \CC[[\mathbf{x}]].
\]
where $\mathbf{x} = \{x_{1}, x_{2}, \ldots \}$ is an infinite, totally ordered set of indeterminates.  With grading given by degree, the \emph{Hopf algebra of quasisymmetric functions} is
\[
\QSym= \CC\spanning\{M_{\alpha} \;|\; \text{$\alpha$ is a composition}\}.
\]

A \emph{partition of $n$} is a composition of $n$ with non-increasing parts. Let 
\[
\PPP = \bigsqcup_{n \ge 0} \PPP_{n} \qquad \text{with}\qquad \PPP_{n} = \{\text{partitions of $n$}\}.
\]
The \emph{Hopf algebra of symmetric functions} is the subspace
\[
\Sym = \CC\spanning\{m_{\lambda}\;|\; \lambda \in \PPP\} \subseteq \QSym \qquad \text{with}\qquad m_{\lambda} = \hspace{-0.5em} \sum_{\operatorname{sort}(\alpha) = \lambda} \hspace{-0.5em} M_{\alpha},
\]
where $\operatorname{sort}(\alpha)$ is the partition obtained by listing the parts of $\alpha$ in non-increasing order.  

This abstract will use several standard bases of $\Sym$ found in~\cite{Mac}: the elementary symmetric functions $\{e_{\lambda} \;|\; \lambda \in \PPP\}$~\cite[\nopp I.2]{Mac}, the Schur functions $\{s_{\lambda} \;|\; \lambda \in \PPP\}$~\cite[\nopp I.3]{Mac}, and the degree-shifted Hall-Littlewood symmetric functions $\{\widetilde{P_{\lambda}}(\mathbf{x}; t) \;|\; \lambda \in \PPP\}$~\cite[\nopp IV.4]{Mac}, which depend on an additional parameter $t$.  We will also  use the involution $\omega: \Sym \shortto \Sym$, given by $\omega(s_{\lambda}) = s_{\lambda'}$, where $\lambda'$ denotes the transpose of $\lambda$. 

The paper~\cite{AgBerSot} gives classification of all homomorphisms from an arbitrary Hopf algebra $H$ to $\QSym$ via certain algebra homomorphisms $\zeta: H \shortto \CC$, called \emph{linear characters} herein in order to avoid confusion with the group characters in this work.  The Hopf algebra $\QSym$ has a linear character known as the \emph{first principal specialization},
\[
\begin{array}{rrcl}
\operatorname{ps}_{1}\colon & \QSym & \to & \;\; \CC \\[0.1em]
& M_{\alpha} & \mapsto & \begin{cases} 1 & \text{if $\alpha = ()$ or $(n)$ for $n \ge 0$,} \\ 0 & \text{otherwise.} \end{cases}
\end{array}
\]

\begin{thm}[\cite{AgBerSot} Theorem 4.1]
\label{thm:universalhopfmap}
Let $H$ be a Hopf algebra.  Then the map
\[
\begin{array}{rcl}
\{\text{homomorphisms $H \shortto \QSym$}\} & \to & \{ \text{linear characters of $H$} \} \\
\Psi & \mapsto & \operatorname{ps}_{1} \circ \Psi
\end{array}
\]
is a bijection.  
In particular, for each linear character $\zeta$ of $H$, there is a unique homomorphism $\Psi: H \shortto \QSym$ for which $\zeta = \operatorname{ps}_{1} \circ \Psi$.
\end{thm}

%%%%%%%%%%%
\subsection{Indifference graphs and the chromatic quasisymmetric function}
\label{sec:Catalan}
%%%%%%%%%%%

This section will describe the chromatic quasisymmetric function of a graph, and go on to define a special class of graphs for which this function is particularly well behaved.

Let $\gamma$ be a simple, undirected graph with vertex set $[n]$ and edge set $E(\gamma)$.  A \emph{coloring} of $\gamma$ is a function $\kappa: [n] \shortto \ZZ_{>0}$.  A coloring $\kappa$ of $\gamma$ is \emph{proper} if $\kappa(i) \neq \kappa(j)$ for all $\{i, j\} \in E(\gamma)$.  The \emph{$\gamma$-ascent number} of a coloring $\kappa$ is
\begin{equation}
\label{eq:ascstat}
\asc_{\gamma}(\kappa) = \big|\big\{\{i, j\} \in E(\gamma) \;|\; \text{$i < j$ and $\kappa(i) < \kappa(j)$}\big\}\big|.
\end{equation}
For example, if $\kappa: [5] \shortto \ZZ_{> 0}$ is given by $\kappa(1) = 2$, $\kappa(2) = 5$, $\kappa(3) =1$, and $\kappa(4) = 5$,
\[
\asc_{\hspace{-0.25em}\begin{tikzpicture}[scale = 0.35, baseline = 0.35*-0.25cm]
\draw[fill] (0, 0) circle (2pt) node[inner sep = 1pt] (1) {};
\draw[fill] (1, 0) circle (2pt) node[inner sep = 1pt] (2) {};
\draw[fill] (2, 0) circle (2pt) node[inner sep = 1pt] (3) {};
\draw[fill] (3, 0) circle (2pt) node[inner sep = 1pt] (4) {};
\node[below] at (1) {$\scriptscriptstyle 1$};
\node[below] at (2) {$\scriptscriptstyle 2$};
\node[below] at (3) {$\scriptscriptstyle 3$};
\node[below] at (4) {$\scriptscriptstyle 4$};
\draw (1) to[] (2);
\draw (2) to[]  (3);
\draw (1) to[out = 45, in = 135]  (3);
\draw (3) to[]  (4);
\end{tikzpicture}\hspace{-0.25em}}(\kappa) = \big|\big\{\{1, 2\}, \{3, 4\}\big\}\big| = 2.
\]
In this example, $\kappa$ is a proper coloring of the given graph.

The \emph{chromatic quasisymmetric function} of $\gamma$ is
\[
X_{\gamma}(\mathbf{x}; t) = \sum_{\substack{\kappa: [n] \shortto \ZZ_{>0} \\ \text{proper}}} t^{\asc_{\gamma}(\kappa)} x_{\kappa(1)} x_{\kappa(2)} \dots x_{\kappa(n)} \in \QSym[t],
\]
so that $X_{\gamma}(\mathbf{x}; t)$ is a polynomial in an indeterminate $t$ whose coefficients---by properties of the ascent statistic---are quasisymmetric functions.  For example,
\begin{equation}
\label{eq:CQSexample}
X_{\hspace{-0.25em}\begin{tikzpicture}[scale = 0.35, baseline = 0.35*-0.35cm]
\draw[fill] (0, 0) circle (2pt) node[inner sep = 1pt] (1) {};
\draw[fill] (1, 0) circle (2pt) node[inner sep = 1pt] (2) {};
\draw[fill] (2, 0) circle (2pt) node[inner sep = 1pt] (3) {};
\node[below] at (1) {$\scriptscriptstyle 1$};
\node[below] at (2) {$\scriptscriptstyle 2$};
\node[below] at (3) {$\scriptscriptstyle 3$};
\draw (1) -- (2);
\draw (2) --  (3);
\end{tikzpicture}\hspace{-0.25em}}(\mathbf{x}; t) = t \; M_{(2, 1)} + t \; M_{(1, 2)} + (t^{2}+4t+1) \; M_{(1^{3})}.
\end{equation}  
Evaluating the indeterminate $t$ in $X_{\gamma}(\mathbf{x}; t)$ at a complex number gives an actual quasisymmetric function.  For example, taking $t= 1$ gives the ordinary chromatic symmetric function of $\gamma$ as defined by Stanley in~\cite{Sta}, and the main result of this paper concerns the evaluation of $X_{\gamma}(\mathbf{x}; t)$ at $t = q$, the order of the finite field $\FF_{q}$.

The coefficients of $X_{\gamma}(\mathbf{x}; t)$ are known to be symmetric under certain conditions.  An \emph{indifference graph} of size $n \ge 0$ is a simple, undirected graph $\gamma$ on the vertex set $[n] = \{1, \ldots, n\}$ with edge set $E(\gamma)$ satisfying
\[
\text{for each $\{i, l\} \in E(\gamma)$:} \qquad \big\{\{j, k\} \;|\; i \le j < k \le l\big\} \subseteq E(\gamma).
\]
The empty graph on $\emptyset$ is the unique indifference graph of size zero.  
Let
\[
\III\GGG = \bigsqcup_{n \ge 0} \III\GGG_{n}\qquad\text{with}\qquad \III\GGG_{n} = \{\text{indifference graphs on $[n]$}\}.
\]
For example,
\[
\gamma = 
\begin{tikzpicture}[scale = 0.45, baseline = 0.45*-0.3cm]
\draw[fill] (0, 0) circle (2pt) node[inner sep = 1pt] (1) {};
\draw[fill] (1, 0) circle (2pt) node[inner sep = 1pt] (2) {};
\draw[fill] (2, 0) circle (2pt) node[inner sep = 1pt] (3) {};
\draw[fill] (3, 0) circle (2pt) node[inner sep = 1pt] (4) {};
\draw[below] (1) node {$\scriptstyle 1$};
\draw[below] (2) node {$\scriptstyle 2$};
\draw[below] (3) node {$\scriptstyle 3$};
\draw[below] (4) node {$\scriptstyle 4$};
\draw (1) -- (2);
\draw (2) -- (3);
\draw (1) to[out = 45, in = 135] (3);
\draw (3) -- (4);
\end{tikzpicture}
\in \III\GGG_{4}
\qquad\text{but}\qquad
\sigma = 
\begin{tikzpicture}[scale = 0.45, baseline = 0.45*-0.3cm]
\draw[fill] (0, 0) circle (2pt) node[inner sep = 1pt] (1) {};
\draw[fill] (1, 0) circle (2pt) node[inner sep = 1pt] (2) {};
\draw[fill] (2, 0) circle (2pt) node[inner sep = 1pt] (3) {};
\draw[fill] (3, 0) circle (2pt) node[inner sep = 1pt] (4) {};
\draw[below] (1) node {$\scriptstyle 1$};
\draw[below] (2) node {$\scriptstyle 2$};
\draw[below] (3) node {$\scriptstyle 3$};
\draw[below] (4) node {$\scriptstyle 4$};
\draw (1) -- (2);
\draw (2) -- (3);
\draw (1) to[out = 45, in = 135] (3);
\draw (1) to[out = 55, in = 125] (4);
\end{tikzpicture}
\notin \III\GGG_{4},
\]
as $\{1, 4\} \in E(\sigma)$ but $\{3, 4\} \notin E(\sigma)$.

\begin{prop}[{\cite[Theorem 4.5]{ShWa}}]
For an indifference graph $\gamma \in \III\GGG_{n}$, the coefficients of each power of $t$ in $X_{\gamma}(\mathbf{x}; t)$ is a symmetric function.
\end{prop}

The indifference graphs of size $n \ge 0$ are enumerated by the $n$th Catalan number, as seen in the following bijection, found in~\cite[Sol.~187]{StaCat}.  Associate to each $\gamma \in \III\GGG_{n}$ the southeast lattice path from $(0, 0)$ to $(n, -n)$ which lies directly above the diagonal $x = -y$ and the unit squares centered at $(j - \frac{1}{2}, \frac{1}{2} - i)$ for each $\{i, j\} \in E(\gamma)$.  For example,
\begin{equation}
\label{eq:DyckGraphExample}
\begin{tikzpicture}[scale = 0.5, baseline = 0.5*-0.3cm]
\draw[fill] (0, 0) circle (2pt) node[inner sep = 1pt] (1) {};
\draw[fill] (1, 0) circle (2pt) node[inner sep = 1pt] (2) {};
\draw[fill] (2, 0) circle (2pt) node[inner sep = 1pt] (3) {};
\draw[below] (1) node {$\scriptstyle 1$};
\draw[below] (2) node {$\scriptstyle 2$};
\draw[below] (3) node {$\scriptstyle 3$};
\draw (1) -- (2);
\draw (2) -- (3);
\end{tikzpicture}
\qquad
\longleftrightarrow
\qquad
\begin{tikzpicture}[scale = 0.55, baseline = 0.55*-1.7cm]
\draw[gray] (-0.2, 0.2) grid (3.2, -3.2);
\draw[dashed, gray] (-0.2, 0.2) -- (3.2, -3.2);
\draw[very thick] (0, 0) -- (2, 0) -- (2, -1) -- (3, -1) -- (3, -3);
\draw[fill, red] (1.5, -0.5) node{$\scriptscriptstyle \{1, 2\}$};
\draw[fill, red] (2.5, -1.5) node {$\scriptscriptstyle \{2, 3\}$};
\end{tikzpicture} = (EESESS).
\end{equation}

%%%%%%%%%%%
\subsection{Linear algebraic groups and class functions}
\label{sec:sct}
%%%%%%%%%%%

For a finite group $G$, the space of complex valued class functions on $G$ is
\[
\cf(G) = \{\psi : G \to \CC \;|\; \text{$\psi(g) = \psi(hgh^{-1})$ for all $g, h \in G$} \}.
\]
Under the usual inner product $\langle \cdot, \cdot \rangle: \cf(G) \otimes \cf(G) \shortto \CC$, there are two orthogonal bases of $\cf(G)$: the irreducible characters of $G$ and the conjugacy class identifier functions, 
\begin{equation}
\label{eq:classindicator}
\{\delta_{K} \;|\; \text{$K \in \mathtt{Cl}$}\}
\qquad\text{with}\qquad
\delta_{K}(g) = \begin{cases} 1 & \text{if $g \in K$.} \\ 0 & \text{otherwise.} \end{cases}
\end{equation}

The remainder of this section will construct a subspace of class functions of a particular family of groups.  
Fix a prime power $q$, let $\FF_{q}$ denote the field with $q$ elements, and let $\GL_{n} = \GL_{n}(\FF_{q})$.  The \emph{unipotent upper triangular group} is the subgroup
\[
\UT_{n} = \{ g \in \GL_{n} \;|\; \text{ $(g - 1_{n})_{i, j} \neq 0$ only if $i < j$ } \}
\]
where $1_{n} \in \GL_{n}$ is the identity matrix.  The set $\III\GGG_{n}$ indexes a family of normal subgroups in $\UT_{n}$ known as \emph{normal pattern subgroups}~\cite[Lemma~4.1]{Mar}: for $\gamma \in \III\GGG_{n}$, let
\[
\UT_{\gamma} = \{ g \in \UT_{n} \;|\; \text{ $g_{i, j} = 0$ if $\{i, j\} \in E(\gamma)$ } \},
\]
where $E(\gamma)$ denotes the edge set of $\gamma$.  
If $\pi$ is the Dyck path corresponding to $\gamma$, then $\UT_{\gamma}$ is the subset of elements of $\UT_{n}$ with nonzero entries occurring only on the diagonal or above the path $\pi$.  
For example using the graph and Dyck path from Equation~\eqref{eq:DyckGraphExample},
\[
\UT_{\hspace{-0.25em} \begin{tikzpicture}[scale = 0.35, baseline = 0.35*-0.3cm]
\draw[fill] (0, 0) circle (2pt) node[inner sep = 1pt] (1) {};
\draw[fill] (1, 0) circle (2pt) node[inner sep = 1pt] (2) {};
\draw[fill] (2, 0) circle (2pt) node[inner sep = 1pt] (3) {};
\draw[below] (1) node {$\scriptscriptstyle 1$};
\draw[below] (2) node {$\scriptscriptstyle 2$};
\draw[below] (3) node {$\scriptscriptstyle 3$};
\draw (1) -- (2);
\draw (2) -- (3);
\end{tikzpicture}\hspace{-0.25em} }
= 
\begin{tikzpicture}[scale = 0.45, baseline = 0.45*-1.7cm]
\draw[gray] (0, 0) grid (3, -3);
\draw[gray, thick] (0, 0) -- (3, 0) -- (3, -3) -- (0, -3) -- cycle;
\draw[very thick] (0, 0) -- (2, 0) -- (2, -1) -- (3, -1) -- (3, -3);
%%
\foreach \x in {1, 2, 3}{\draw (\x - 0.5, -\x + 0.5) node {$1$};}
%
\foreach \x in {1}{\foreach \y in {2, 3}{\draw (\x - 0.5, -\y + 0.5) node {$0$};}}
\foreach \x in {2}{\foreach \y in {3}{\draw (\x - 0.5, -\y + 0.5) node {$0$};}}
%
\foreach \x in {2}{\foreach \y in {1}{\draw (\x - 0.5, -\y + 0.5) node {$0$};}}
\foreach \x in {3}{\foreach \y in {1}{\draw (\x - 0.5, -\y + 0.5) node {$\ast$};}}
\foreach \x in {3}{\foreach \y in {2}{\draw (\x - 0.5, -\y + 0.5) node {$0$};}}
%
\begin{scope}[xshift = -0.05cm]
\draw (0.2, 0.085) -- (-0.15, 0.085);
\draw[very thick] (-0.1, 0.1) -- (-0.1, -3.1);
\draw (-0.15, -3.085) -- (0.2, -3.085);
\end{scope}
%
\begin{scope}[xshift = 0.05cm]
\draw (2.8, 0.085) -- (3.15, 0.085);
\draw[very thick] (3.1, 0.1) -- (3.1, -3.1);
\draw (3.15, -3.085) -- (2.8, -3.085);
\end{scope}
%
\end{tikzpicture}.
\]

The characters appearing in the main results below span a subspace of $\cf(\UT_{n})$ which was initially constructed in~\cite{AlTh20}.  For $\gamma \in \III\GGG_{n}$, let
\[
\permchar^{\gamma} = \ind^{\UT_{n}}_{\UT_{\gamma}}(\mathbbm{1}),
\]
the character of the $\UT_{n}$-module $\CC[\UT_{n}/\UT_{\gamma}]$, so that %as functions%for $g \in \UT_{n}$, 
\begin{equation}
\label{eq:permtoind}
\permchar^{\gamma} = q^{|E(\gamma)|} \delta_{\UT_{\gamma}}. 
\end{equation}
The characters $\permchar^{\gamma}$ are linearly independent, and span a self-dual subspace
\[
\scf(\UT_{n}) = \CC\spanning\{ \permchar^{\gamma} \;|\; \gamma \in \III\GGG_{n} \},
\]
known as the subspace of \emph{superclass functions}.  

\begin{rem}
\label{rem:supercharacters}
The space $\scf(\UT_{n})$ comes from a \emph{supercharacter theory} of $\UT_{n}$~\cite{AlTh20}, and has a number of interesting bases indexed by $\III\GGG_{n}$.  
One of these---the ``supercharacter basis''---is typically denoted by $\chi^{\gamma}$ and has the property that $\permchar^{\gamma} = \sum_{\sigma \subseteq \gamma} \chi^{\sigma}$, from which the notation $\permchar^{\gamma}$ is derived.   
More details can be found in the extended version of this abstract~\cite[Section~2.3]{Gagb}.  
\end{rem}


%%%%%%%%%%%
\subsection{Homomorphisms between Hopf algebras of class functions}
\label{sec:repHopf}
%%%%%%%%%%%

In~\cite[\nopp III]{Zel}, Zelevinsky defines a graded connected Hopf algebra on the space
\[
\cf(\GL_{\bullet}) = \bigoplus_{n \ge 0} \cf(\GL_{n}),
\]
with structure maps coming from the parabolic induction and restriction functors.  
The paper~\cite{Gaga} defines a similar Hopf structure on the spaces
\[
\scf(\UT_{\bullet}) = \bigoplus_{n \ge 0} \scf(\UT_{n}),
\qquad\text{and}\qquad
\cf(\UT_{\bullet}) = \bigoplus_{n \ge 0} \cf(\UT_{n}),
\]
in which the former is a sub-Hopf algebra of the latter.  This section will describe several homomorphisms involving these Hopf algebras.

In~\cite[Section 6]{GP16}, Guay-Paquet defines a $\CC[t]$-Hopf algebra on the free $\CC[t]$-module $\CC[t][\III\GGG]$, and specializing $t \shortmapsto q^{-1}$ gives a Hopf algebra over $\CC$; see~\cite[Section~7]{Gaga}.

\begin{thm}[{\cite[Corollary~7.6]{Gaga}}]
\label{thm:scfNG}
The map $\gamma \shortmapsto q^{-|E(\gamma)|} \permchar^{\gamma}$ is an isomorphism from Guay-Paquet's specialized Hopf algebra to $\scf(\UT_{\bullet})$.
\end{thm}

A second map comes from the induction functors $\ind_{\UT_{n}}^{\GL_{n}} : \cf(\UT_{n}) \shortto \cf(\GL_{n})$;  let
\[
\ind_{\UT}^{\GL} = \bigoplus_{n \ge 0} \ind_{\UT_{n}}^{\GL_{n}}: \cf(\UT_{\bullet}) \to \cf(\GL_{\bullet}).
\]

\begin{thm}[{\cite[Theorem~6.1]{Gaga}}]
The map $\ind^{\GL}_{\UT}$ is a Hopf algebra homomorphism.
\end{thm}

The image of $\ind_{\UT}^{\GL}$ is a sub-Hopf algebra, the \emph{class functions with unipotent support}:
\[
\cfunsupp(\GL_{\bullet}) = \ind^{\GL}_{\UT}(\cf(\UT_{\bullet})) \subseteq \cf(\GL_{\bullet}).
\]
The space $\cfunsupp(\GL_{\bullet})$ can also be constructed more explicitly, as follows.  An element $g \in \GL_{n}$ is \emph{unipotent} if $g$ is conjugate to an element of $\UT_{n}$.  The conjugacy classes of unipotent elements in $\GL_{n}$ are indexed by $\PPP_{n}$: the partition $\lambda = (\lambda_{1}, \lambda_{2}, \ldots, \lambda_{\ell})$ corresponds to the conjugacy class $O_{\lambda}$ of the Jordan matrix
\[
J_{\lambda} = J_{\lambda_{1}} \oplus J_{\lambda_{2}} \oplus \cdots \oplus J_{\lambda_{\ell}} 
\qquad \text{with} \qquad 
(J_{k})_{i, j} = \begin{cases} 1 & \text{if $j \in \{i, i+1\}$} \\ 0 & \text{otherwise} \end{cases} 
\]
Thus, writing $\delta_{\lambda} = \delta_{O_{\lambda}}$, standard properties of induction imply that
\[
\cfunsupp(\GL_{\bullet}) = \CC\spanning\{ \delta_{\lambda} \;|\; \lambda \in \PPP\}.
\]
Zelevinsky~\cite[\nopp 10.13]{Zel} (see also \cite[\nopp IV.4.1]{Mac}) constructs a Hopf algebra isomorphism
\begin{equation}
\label{eq:canoUCmap}
\begin{array}{rcl}
\canoUS\colon \cfunsupp(\GL_{\bullet}) & \to & \Sym \\[0.5em]
\delta_{\lambda} & \mapsto & \widetilde{P_{\lambda}}(\mathbf{x}; q) 
\end{array}
\end{equation}
where $\widetilde{P_{\lambda}}$ is the degree shifted Hall--Littlewood polynomial mentioned in \cref{sec:QSym}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Realizing $X_{\gamma}(\mathbf{x}; t)$ as a $\mathrm{GL}_{n}$ character}
\label{sec:mainresult1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This section gives the main result.  Recall the definitions of $\III\GGG_{n}$ and $X_{\gamma}(\mathbf{x}; t)$ from \cref{sec:Catalan}, the groups $\UT_{\gamma}$ %characters $\permchar^{\gamma}$ 
from \cref{sec:sct}, and the maps $\ind^{\GL}_{\UT}$ and $\canoUS$ from \cref{sec:repHopf}.


\begin{thm}
\label{thm:CQSbijection}
For $n \ge 0$ and $\gamma \in \III\GGG_{n}$,
\[
\ind^{\GL_{n}}_{\UT_{\gamma}}(\mathbbm{1})
= (q-1)^{n} \, \canoUS^{-1}\big(X_{\gamma}(\mathbf{x}; q)\big).
\]
\end{thm}

A full proof of \cref{thm:CQSbijection} is given in~\cite[Section 3]{Gagb}, but we will give a short sketch below, after a few preliminaries.  
\cref{thm:universalhopfmap} states that the homomorphism
\[
\canoCQS: \scf(\UT_{\bullet}) \xrightarrow{\;\; \ind^{\GL}_{\UT} \;\;} \cfunsupp(\GL_{\bullet}) \xhookrightarrow{\canoUS} \Sym \xrightarrow{\;\; \operatorname{inclusion}  \;\;} \QSym
\]
is uniquely determined by the linear character $\operatorname{ps}_{1} \circ \canoUS \circ \ind^{\GL}_{\UT}$ of $\scf(\UT_{\bullet})$.
The proof of \cref{thm:CQSbijection} amounts to showing that this linear character also corresponds to the  homomorphism
\begin{equation}
\label{eq:CQSmap}
\begin{array}{rcl}
\scf(\UT_{\bullet}) & \to & \QSym \\
\permchar^{\gamma} &\mapsto& (q-1)^{n} X_{\gamma}(\mathbf{x}; q),
\end{array}
\end{equation}
where for each $\gamma \in \III\GGG_{n}$, $\permchar^{\gamma} = \ind^{\UT_{n}}_{\UT_{\gamma}}(\mathbbm{1})$ as in \cref{sec:sct}.

A relative of the map in Equation~\eqref{eq:CQSmap} has been studied by Guay-Paquet in~\cite{GP16}.  Translating through the isomorphism of \cref{thm:scfNG} and a $q \leftrightarrow q^{-1}$ interchange property established in~\cite[Proposition 2.6]{ShWa}, we have the following.

\begin{thm}[{\cite[Theorem~57]{GP16}}]
\label{thm:GPhomresults}
The map $\permchar^{\gamma} \shortmapsto X_{\gamma}(\mathbf{x}; q)$ is a Hopf algebra homomorphism from $\scf(\UT_{\bullet})$ to $\QSym$.  Writing $\zeta_{0}: \scf(\UT_{\bullet}) \shortto \CC$ for the corresponding linear character,
\[
\zeta_{0}(\permchar^{\gamma}) = \begin{cases} q^{ |E(\gamma)| } & \text{if $\gamma = \UTnInd$,} \\ 0 & \text{otherwise.}  \end{cases}
\]
\end{thm}

We now relate $\zeta_{0}$ to the linear character of $\canoCQS$.  Zelevinsky~\cite[\nopp 10.8]{Zel} has shown that for a unipotently supported class function $\psi \in \cfunsupp(\GL_{\bullet})$,
\[
\operatorname{ps}_{1} \circ \canoUS(\psi) = \psi(J_{(n)}),
\]
where $J_{(n)}$ is the unipotent Jordan matrix corresponding to the partition $(n)$, as in \cref{sec:repHopf}.  Taking $\psi = \ind^{\GL}_{\UT}(\permchar^{\gamma})$, standard facts about induction and the conjugacy class of $J_{(n)}$ give the following result, proved in~\cite[Proposition~3.12]{Gagb}.

\begin{prop}
\label{prop:CQSchar}
Let $\gamma$ be an indifference graph of size $n \ge 0$.  Then
\[
\operatorname{ps}_{1} \circ \canoUS \circ \ind^{\GL}_{\UT} (\permchar^{\gamma}) 
= \begin{cases} (q-1)^{n} q^{|E(\gamma)|} & \text{if $\gamma = \UTnInd$,} \\ 0 & \text{otherwise.} \end{cases}
\]
\end{prop}

\begin{proof}[Proof sketch of \cref{thm:CQSbijection}]
Starting from \cref{thm:GPhomresults}, it can be verified directly that the map in Equation~\eqref{eq:CQSmap} is a Hopf algebra homomorphism.   By \cref{prop:CQSchar}, this map has the same linear character as $\canoCQS$, so by \cref{thm:universalhopfmap}, they are the same.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Connections to Hessenberg varieties}
\label{sec:Hessenberg}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This section will describe a relationship between the characters in \cref{thm:CQSbijection} and Hessenberg varieties.  Take $n \ge 0$ and let $\KK$ be a field.  For each element $A \in \Mat_{n}(\KK)$ and subspace $M \subseteq \Mat_{n}(\KK)$ which is stable under conjugation by the subgroup of upper triangular matrices $B_{n}(\KK) \subseteq \GL_{n}(\KK)$, the \emph{Hessenberg variety} associated to $A$ and $M$ is
\[
\BBB^{M}_{A} = \{g B_{n}(\KK) \in \GL_{n}(\KK)/B_{n}(\KK) \;|\; g^{-1} A g \in M\}.
\]

The following results exclusively concern Hessenberg varieties associated to strictly upper triangular subspaces known as \emph{ad-nilpotent ideals}.  For $\gamma \in \III\GGG_{n}$, let
\begin{align*}
\mathfrak{ut}_{\gamma}(\KK) &= \{A \in \Mat_{n}(\KK) \;|\; \text{$A_{i, j} \neq 0$ only if $i < j$ and $\{i, j\} \notin E(\gamma)$}\} %\\
%&
= \UT_{\gamma}(\KK) - 1_{n}.%,
\end{align*}
Some key examples of Hessenberg varieties of the form $\BBB^{\mathfrak{ut}_{\gamma}(\KK)}_{A}$ are well-known, but a specific study of these varieties is quite recent; see~\cite{PrecupSommers} and the references therein.

\begin{prop}
\label{prop:HessenbergPoints}
Let $n \ge 0$ and $\gamma \in \III\GGG_{n}$.  For $A \in \Mat_{n}(\FF_{q})$ with $1_{n} + A \in \GL_{n}(\FF_{q})$,
\[
\ind^{\GL_{n}(\FF_{q})}_{\UT_{\gamma}(\FF_{q})}(\mathbbm{1})(1_{n}+A) = (q-1)^{n} q^{|E(\gamma)|} |\BBB^{\mathfrak{ut}_{\gamma}(\FF_{q})}_{A}|.
\]
\end{prop}
\begin{proof}[Proof sketch]
The left side is equal to $q^{|E(\gamma)|}$ times the number of cosets $h \UT_{n}(\FF_{q}) \in \GL_{n}(\FF_{q})/\UT_{n}(\FF_{q})$ with $h^{-1}(1_{n}+A)h \in \UT_{\gamma}(\FF_{q})$.  Each such coset satisfies $h B_{n}(\FF_{q}) \in \BBB^{\mathfrak{ut}_{\gamma}(\FF_{q})}_{A}$, with $(q-1)^{n}$ cosets $h \UT_{n}(\FF_{q})$ for each each element of $\BBB^{\mathfrak{ut}_{\gamma}(\FF_{q})}_{A}$.
\end{proof}

The paper~\cite{PrecupSommers} gives a similar result for the analogous Hessenberg varieties over $\CC$, which involves the modified Poincar\'{e} polynomial
\[
\mathsf{Poin}_{A}^{\mathfrak{ut}_{\gamma}(\CC)}(t) = \sum_{k \ge 0} \beta_{2k} t^{k},
\]
where $\beta_{i}$ is the $i$th Betti number of $\BBB^{\mathfrak{ut}_{\gamma}(\CC)}_{A}$.  
Recall the Hall--Littlewood symmetric function $\widetilde{P_{\lambda}}(\mathbf{x}; t)$ from \cref{sec:QSym}, and define expressions $d_{\lambda}^{\gamma}(t)$ for each partition $\lambda$ of $n$ by
\begin{equation}
\label{eq:dlambdadef}
X_{\gamma}(\mathbf{x}; t) = \sum_{\lambda \in \PPP_{n}} d_{\lambda}^{\gamma}(t) \widetilde{P_{\lambda}}(\mathbf{x}; t).
\end{equation}
Also note that partitions of $n$ index the similarity classes of nilpotent matrices over any field: $\lambda \in \PPP_{n}$ corresponds to the class of $J_{\lambda} - 1_{n}$, where $J_{\lambda}$ is as defined in \cref{sec:repHopf}.

\begin{thm}[{\cite[Equation (4.7)]{PrecupSommers}}]
For $n \ge 0$, take $\gamma \in \III\GGG_{n}$ and $\lambda \in \PPP_{n}$.  For a nilpotent matrix $A \in \Mat_{n}(\CC)$ in the similarity class indexed by $\lambda$, $\mathsf{Poin}^{\mathfrak{ut}_{\gamma}(\CC)}_{A}(t) = t^{-|E(\gamma)|} d_{\lambda}^{\gamma}(t)$.
\end{thm}

Applying the inverse of the map $\canoUS$ defined in Equation~\eqref{eq:canoUCmap} to Equation~\eqref{eq:dlambdadef}, \cref{thm:CQSbijection} and \cref{prop:HessenbergPoints} give the following.

\begin{cor}
\label{cor:adnilpotentpoincare}
For $n \ge 0$, take $\gamma \in \III\GGG_{n}$ and $\lambda \in \PPP_{n}$.  Let $A \in \Mat_{n}(\FF_{q})$ and $A' \in \Mat_{n}(\CC)$ be nilpotent elements belonging to similarity classes indexed by $\lambda$.  Then $\mathsf{Poin}^{\mathfrak{ut}_{\gamma}(\CC)}_{A'}(q) = |\BBB^{\mathfrak{ut}_{\gamma}(\FF_{q})}_{A}|$.
\end{cor}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Realizing $G_{\gamma}(\mathbf{x}; t)$ as a $\mathrm{GL}_{n}$ character}
\label{sec:mainresult2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For an indifference graph $\gamma \in \III\GGG$, the \emph{unicellular LLT polynomial} associated to $\gamma$ is 
\[
G_{\gamma}(\mathbf{x}; t)  = \sum_{\kappa : [n] \shortto \ZZ_{>0}}  t^{\asc_{\gamma}(\kappa)} x_{\kappa(1)} x_{\kappa(2)} \dots x_{\kappa(n)} \in \CC[[\mathbf{x}]][t],
\]
where the sum is over \emph{all} colorings.  
This is a variation of the  polynomials defined by Lascoux, Leclerc, and Thibon in~\cite{LLT}, and is due to Carlsson and Mellit~\cite{CM} (see also~\cite{HHL}).  
Like $X_{\gamma}(\mathbf{x}; t)$, each $G_{\gamma}(\mathbf{x}; t)$ is a polynomial in $t$ with symmetric function coefficients.

\begin{thm}[{\cite[Proposition 3.5]{CM}}]
\label{thm:carlssonmellit}
For $n \ge 0$, let $\gamma \in \III\GGG_{n}$.  Using plethystic notation,
\[
\textstyle
(t-1)^{n} X_{\gamma}(\mathbf{x}; t)\left[\frac{\mathbf{x}}{t-1}\right] = G_{\gamma}(\mathbf{x}; t).
\]
\end{thm}

Somewhat surprisingly, the plethystic substitution above has a natural meaning for $\GL_{n}(\FF_{q})$.  
An irreducible character of $\GL_{n}$ is \emph{unipotent} if it is a summand of $\ind^{\GL_{n}}_{B_{n}}(\mathbbm{1})$, where $B_{n} = B_{n}(\FF_{q})$ is as defined in \cref{sec:Hessenberg}.   These characters are indexed by the partitions of $n$, and $\chi^{\lambda}$ will denote the irreducible unipotent character corresponding to $\lambda \in \PPP(n)$, with $\chi^{(1^{n})} = \mathbbm{1}$ as in~\cite{Mac}.  
Zelevinksy~\cite[\nopp 9.4]{Zel} defines a homomorphism 
\begin{equation}
\label{eq:canoUCdef}
\begin{array}{rccc}
\canoUC\colon & \cf(\GL_{\bullet}) & \to & \Sym \\[0.25em]
& \psi & \mapsto & \sum_{\lambda} \langle \psi,  \chi^{\lambda} \rangle s_{\lambda},
\end{array}
\end{equation}
so that for any character $\psi$ of $\GL_{n}$, the coefficient of $s_{\lambda}$ in $\canoUC(\psi)$ is the multiplicity of $\chi^{\lambda}$ in $\psi$.   Recalling the map $\canoUS$ from \cref{sec:repHopf}, it is known~\cite[\nopp IV.4]{Mac} that the composite
\[
\Sym \xrightarrow{\;\; \canoUS^{-1} \;\;} \cfunsupp(\GL_{\bullet}) \xhookrightarrow{\qquad} \cf(\GL_{\bullet}) \xrightarrow{\;\; \canoUC \;\;} \Sym
\]
can be expressed in plethystic notation as $f \shortmapsto \omega f[\frac{\mathbf{x}}{t-1}]|_{t = q}$.  With \cref{thm:carlssonmellit}, this implies the following result; see~\cite[Theorem 5.1]{Gagb} for a complete proof.

\begin{thm}
\label{thm:LLTbijection}
Let $\gamma$ be an indifference graph.  Then $\canoUC \circ \ind^{\GL_{n}}_{\UT_{\gamma}}(\mathbbm{1}) = \omega G_{\gamma}(\mathbf{x}; q)$.
\end{thm}
%\begin{proof}
%By \cref{thm:CQSbijection}, $\ind^{\GL_{n}}_{\UT_{\gamma}}(\mathbbm{1}) = (q-1)^{n} \canoUC^{-1}(X_{\gamma}(\mathbf{x}; q))$, so that $\canoUC \circ\ind^{\GL_{n}}_{\UT_{\gamma}}(\mathbbm{1}) = \omega (q-1)^{n} X_{\gamma}(\mathbf{x}; q)[\frac{\mathbf{x}}{t-1}]|_{t = q}$.  Now \cref{thm:carlssonmellit} gives the desired equality. 
%\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Positivity conjectures}
\label{sec:positivity}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Recall the bases of $\Sym$ given in \cref{sec:QSym}.  An element $f \in \Sym[t]$ is respectively \emph{$e$-positive} or \emph{$s$-positive} if there are polynomials $a_{\lambda}(t) \in \ZZ_{\ge 0}[t]$ or $b_{\lambda}(t) \in\ZZ_{\ge 0}[t]$ for which
\[
f = \sum_{\lambda \in \PPP} a_{\lambda}(t) e_{\lambda}
\qquad
\text{or}
\qquad
f = \sum_{\lambda \in \PPP} b_{\lambda}(t) s_{\lambda}.
\]

For the chromatic quasisymmetric functions in \cref{sec:Catalan}, $e$-positivity generalizes the Stanley--Stembridge conjecture~\cite[Conjecture 5.5]{StaStem}, which by~\cite{GP13} is the $t = 1$ case of the statement below.

\begin{conj}[{\cite[Conjecture 1.3]{ShWa}}]
\label{conj:epos}
For each $\gamma \in \III\GGG$, $X_{\gamma}(\mathbf{x}; t)$ is $e$-positive.
\end{conj}

In light of \cref{thm:CQSbijection}, there should be a restatement of \cref{conj:epos} in terms of $\GL_{n}$.  
However, the map $\canoUS$ defined in \cref{sec:repHopf} does not associate $e_{\lambda}$ to a character of $\GL_{n}$, so some interpretation is required.  One possible restatement uses the Steinberg character $\chi^{(n)}$ of $\GL_{n}$ defined in \cref{sec:mainresult2}, which agrees with $\canoUS^{-1}(e_{n})$ on all unipotent elements.  For $\lambda = (\lambda_{1}, \ldots, \lambda_{\ell}) \in \PPP_{n}$, let $\operatorname{St}_{\lambda}$ denote the product $\chi^{\lambda_{1}} \cdots \chi^{\lambda_{\ell}}$ in $\cf(\GL_{\bullet})$.

\begin{conj}
\label{conj:GLepos}
Let $n \ge 0$ and $\gamma \in \III\GGG_{n}$.  There are polynomials $a_{\lambda}^{\gamma}(t) \in \ZZ_{\ge 0}[t]$ such that for each prime power $q$ the character
%\[
$\eta_{\gamma} = \sum_{\lambda \in \PPP_{n}} a_{\lambda}^{\gamma}(q) \operatorname{St}_{\lambda}$ 
%\]
satisfies $(q-1)^{n} \eta_{\gamma}(u) = \ind^{\GL_{n}}_{\UT_{\gamma}}(\mathbbm{1})(u)$ for every unipotent element $u \in \GL_{n}(\FF_{q})$.
\end{conj}

By~\cite[Proposition~6.6]{Gagb}, \cref{conj:GLepos,conj:epos} are equivalent.  Ideally, a proof of \cref{conj:GLepos} would construct a module affording the character $\eta_{\gamma}$. 

For the unicellular LLT polynomials from \cref{sec:mainresult2}, Schur positivity has implications in the study of Macdonald polynomials~\cite{HHL} and is known to hold by~\cite[Corollary 6.9]{GrojHai}.  
Unfortunately, the proof in~\cite{GrojHai} does not explicitly construct the Schur coefficients.

\begin{question}[{\cite[Open Problem 6.6]{HaglundBook}}]
\label{question:LLT}
Find a (manifestly positive) combinatorial formula for the Schur coefficients $b_{\lambda}^{\gamma}(t)$  of $G_{\gamma}(\mathbf{x}; t)$.
\end{question}

\cref{thm:LLTbijection} implies that $b_{\lambda}^{\gamma}(q) = \langle \chi^{\lambda'}, \ind^{\GL_{n}}_{\UT_{\gamma}}(\mathbbm{1}) \rangle$ for each prime power $q$, which is a positive integer, but this not yet imply Schur positivity as defined above.

\begin{question}
Find a combinatorial formula for $ \langle \chi^{\lambda'}, \ind^{\GL_{n}}_{\UT_{\gamma}}(\mathbbm{1}) \rangle$ as a function of $q$.
\end{question}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgments{This abstract is based on the paper~\cite{Gagb} and my Ph.D.~thesis, completed at the University of Colorado Boulder. I am thankful for the support of my advisor N.~Thiem during this time.  
I am also grateful to M.~Precup for a helpful conversation about \cref{sec:Hessenberg}, and to two anonymous referees for suggesting many small improvements to this abstract.}

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