%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2021}
\articlenumber{57}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

%% note that the class file already loads {amsmath, amsthm, amssymb}

%%---Packages---%%
%\usepackage{afterpage}
\makeatletter
\setlength{\@fptop}{0pt plus 1fil}
\setlength{\@fpbot}{0pt plus 1fil}
\makeatother
\usepackage{amsfonts}
%\usepackage{amsmath}
\allowdisplaybreaks[4]
%\usepackage{amssymb}
%\usepackage{amsthm}
\usepackage{array}
\usepackage{bbm}
\usepackage[justification=centering]{caption}
\usepackage[capitalize,nameinlink,noabbrev]{cleveref}
%\usepackage{comment}
%\usepackage{enumerate}
\usepackage{enumitem}
\usepackage{float}
\restylefloat{figure}
\usepackage{graphicx}
\usepackage{mathrsfs}
\usepackage[framemethod=tikz,ntheorem,xcolor]{mdframed}
\usepackage{parcolumns}
\usepackage{tabularx}
\usepackage{tikz}\pdfpageattr{/Group <</S /Transparency /I true /CS /DeviceRGB>>}
\usetikzlibrary{arrows,calc,intersections,matrix,positioning,through}
\usepackage{tikz-cd}
\tikzset{commutative diagrams/.cd,every label/.append style = {font = \normalsize}}
\usepackage[all]{xy}
\usepackage[aligntableaux=center]{ytableau}
\ytableausetup{boxsize=16pt}

%%---Theorem Environments---%%
\numberwithin{equation}{section}

\newtheorem{thm}[equation]{Theorem}
\AfterEndEnvironment{thm}{\noindent\ignorespaces}
\newtheorem{conj}[equation]{Conjecture}
\AfterEndEnvironment{conj}{\noindent\ignorespaces}
\newtheorem{cor}[equation]{Corollary}
\AfterEndEnvironment{cor}{\noindent\ignorespaces}
\newtheorem{lem}[equation]{Lemma}
\AfterEndEnvironment{lem}{\noindent\ignorespaces}
\newtheorem{prob}[equation]{Problem}
\AfterEndEnvironment{prob}{\noindent\ignorespaces}
\newtheorem{prop}[equation]{Proposition}
\AfterEndEnvironment{prop}{\noindent\ignorespaces}

\theoremstyle{definition}
\newtheorem{defn}[equation]{Definition}
\AfterEndEnvironment{defn}{\noindent\ignorespaces}
\newtheorem*{pf_no_qed}{Proof}
\newenvironment{pf}[1][]{\begin{pf_no_qed}[#1]\renewcommand{\qedsymbol}{$\blacksquare$}\pushQED{\qed}}{\popQED\end{pf_no_qed}}
\AfterEndEnvironment{pf}{\noindent\ignorespaces}
\newtheorem{eg_no_qed}[equation]{Example}
\newenvironment{eg}[1][]{\begin{eg_no_qed}[#1]\renewcommand{\qedsymbol}{$\Diamond$}\pushQED{\qed}}{\popQED\end{eg_no_qed}}
\AfterEndEnvironment{eg}{\noindent\ignorespaces}
\newtheorem{rmk}[equation]{Remark}
\AfterEndEnvironment{rmk}{\noindent\ignorespaces}

\theoremstyle{remark}
\newtheorem*{claim}{Claim}
\AfterEndEnvironment{claim}{\noindent\ignorespaces}
\newtheorem*{claimpf_no_qed}{Proof of Claim}
\newenvironment{claimpf}[1][]{\begin{claimpf_no_qed}[#1]\renewcommand{\qedsymbol}{$\square$}\pushQED{\qed}}{\popQED\end{claimpf_no_qed}}
\AfterEndEnvironment{claimpf}{\noindent\ignorespaces}

%%---Commands---%%
\DeclareMathOperator{\alt}{alt}
\DeclareMathOperator{\AND}{\; AND \;}
\DeclareMathOperator{\Ann}{Ann}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Buch}{Buch}
\newcommand{\bx}[1]{\, \begin{ytableau}#1\end{ytableau}\, }
\newcommand{\ceil}[1]{\left\lceil#1\right\rceil}
%\newcommand{\cd}{\cdots\!}
\DeclareMathOperator{\ch}{char}
\DeclareMathOperator{\Cl}{Cl}
\DeclareMathOperator{\cl}{cl}
\newcommand{\colex}{\text{colex}}
\DeclareMathOperator{\conv}{conv}
\DeclareMathOperator{\Diag}{Diag}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\FALSE}{FALSE}
\newcommand{\Fl}{\textnormal{Fl}}
\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor}
\DeclareMathOperator{\FP}{FP}
\DeclareMathOperator{\FS}{FS}
\newcommand{\GL}{\textnormal{GL}}
\DeclareMathOperator{\gl}{\mathfrak{gl}}
\newcommand{\Gr}{\textnormal{Gr}}
\DeclareMathOperator{\heig}{ht}
\DeclareMathOperator{\HJ}{HJ}
\DeclareMathOperator{\hj}{hj}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\hull}{hull}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\im}{im}
\renewcommand{\Im}{\operatorname{Im}}
\DeclareMathOperator{\Ind}{Ind}
\DeclareMathOperator{\interior}{int}
\DeclareMathOperator{\Inv}{Inv}
\DeclareMathOperator{\Jac}{Jac}
\DeclareMathOperator*{\lcm}{lcm}
\newcommand{\lex}{\text{lex}}
\DeclareMathOperator{\Newt}{Newt}
\DeclareMathOperator{\Noninv}{Noninv}
\DeclareMathOperator{\NOT}{NOT \;}
\renewcommand{\O}{\operatorname{O}}
\DeclareMathOperator{\OG}{OG}
\DeclareMathOperator{\op}{op}
\DeclareMathOperator{\OR}{\; OR \;}
\DeclareMathOperator{\per}{per}
%\DeclareMathOperator{\pr}{pr}
\DeclareMathOperator{\rank}{rank}
\renewcommand{\Re}{\operatorname{Re}}
\DeclareMathOperator{\relint}{relint}
\DeclareMathOperator{\Res}{Res}
\newcommand{\rf}[1]{\hyperref[#1]{(\ref*{#1})}}
\DeclareMathOperator{\RFl}{RFl}
\DeclareMathOperator{\RG}{RG}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\SL}{SL}
\newcommand{\smallbx}[1]{\ytableausetup{smalltableaux}\, \begin{ytableau}#1\end{ytableau}\, \ytableausetup{nosmalltableaux}}
\DeclareMathOperator{\soc}{soc}
\DeclareMathOperator{\Sp}{Sp}
\DeclareMathOperator{\Spec}{Spec}
%\DeclareMathOperator{\spn}{span}
\DeclareMathOperator{\spr}{spr}
\DeclareMathOperator{\SSYT}{SSYT}
\DeclareMathOperator{\SYT}{SYT}
\DeclareMathOperator{\Stab}{Stab}
\DeclareMathOperator{\supp}{supp}
%\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\symp}{sp}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\TRUE}{TRUE}
\DeclareMathOperator{\type}{type}
\DeclareMathOperator{\var}{var}
\DeclareMathOperator{\wt}{wt_q}
\DeclareMathOperator{\Wr}{Wr}

%%---Flexible macros---%%
\newcommand{\arrowsum}{\mathfrak{m}}
\newcommand{\bigspn}[1]{\left\langle {#1}\right\rangle}
\newcommand{\block}[2]{#1[#2]}
\newcommand{\Bor}{B^+}
\newcommand{\Boxes}{X}
\newcommand{\col}[1]{\binom{n}{2} - \ell({#1})}
\DeclareMathOperator{\colsum}{\mathsf{colsum}}
\newcommand{\con}[1]{#1^\top}
\DeclareMathOperator{\cont}{\mathsf{c}}
\renewcommand{\d}[2]{\mathsf{d}(#1,#2)}
\newcommand{\dimcomp}[2]{\mathsf{d}(#1,#2)}
\newcommand{\dec}[1]{V_{#1}}
\newcommand{\decnil}[2]{V^{\sharp}_{#1,#2}}
\newcommand{\dom}{\textnormal{dom}}
\newcommand{\double}[1]{{#1}^{\sharp}}
\newcommand{\dualset}{\mathcal{H}}
\newcommand{\evac}{\operatorname{\mathsf{evac}}}
\newcommand{\field}{\Bbbk}
\newcommand{\flags}[1]{\mathsf{Flags}(#1)}
\newcommand{\indec}[2]{V_{#1,#2}}
\DeclareMathOperator{\JF}{\mathsf{JF}^{\hspace*{-0.5pt}\raisebox{-1pt}{$\scriptstyle\top$\hspace*{-1.5pt}}}}
\DeclareMathOperator{\JFold}{\mathsf{JF}}
\DeclareMathOperator{\mat}{\mathsf{mat}}
\newcommand{\mc}{\mathsf{m}}
\newcommand{\n}[1]{\mathsf{n}(#1)}
\newcommand{\ncon}[1]{\tilde{\mathsf{n}}(#1)}
\newcommand{\Nil}{\mathfrak{u}^+}
\newcommand{\nil}{N}
\newcommand{\normalizing}[2]{Z_{#1,#2}}
\newcommand{\pa}[1]{\field{#1}}
\newcommand{\Par}{P^+}
\newcommand{\pav}[1]{e_{#1}}
\newcommand{\perm}[1]{\mathsf{w}_{#1}}
\newcommand{\permdot}[1]{\dot{\mathsf{w}}_{#1}}
\newcommand{\Perms}[1]{\mathsf{W}(#1)}
\newcommand{\ppa}[1]{\Pi(#1)}
\newcommand{\ppareps}[2][]{\mathcal{V}_{#2}^{#1}}
\newcommand{\PFl}{\mathcal{P}}
\renewcommand{\Pr}{\mathbb{P}}
\newcommand{\pr}{\mathsf{p}}
\newcommand{\Ptab}[1]{\mathsf{P}(#1)}
\newcommand{\qbinom}[2]{\begin{bmatrix}#1 \\ #2\end{bmatrix}_q}
\newcommand{\qbinomsmall}[2]{\scalebox{0.7}{$\begin{bmatrix}{#1} \\ {#2}\end{bmatrix}$}_q}
\newcommand{\qF}{\mathbb{F}_q}
\newcommand{\qfac}[1]{[#1]_q!}
\newcommand{\qn}[1]{[#1]_q}
\newcommand{\qinvbinom}[2]{\begin{bmatrix}#1 \\ #2\end{bmatrix}_{1/q}}
\newcommand{\qinvbinomsmall}[2]{\scalebox{0.7}{$\begin{bmatrix}{#1} \\ {#2}\end{bmatrix}$}_{1/q}}
\newcommand{\qinvF}{\mathbb{F}_{1/q}}
\newcommand{\qinvfac}[1]{[#1]_{1/q}!}
\newcommand{\qinvn}[1]{[#1]_{1/q}}
\newcommand{\Qtab}[1]{\mathsf{Q}(#1)}
\newcommand{\relpos}[1]{\xrightarrow{\makebox[12pt]{\scriptsize{$#1$}}}}
\newcommand{\rev}[1]{\overline{#1}}
\DeclareMathOperator{\rowsum}{\mathsf{rowsum}}
\newcommand{\RPP}[2]{\mathsf{RPP}(#1,#2)}
\newcommand{\RPPbig}[2]{\mathsf{RPP}\!\left(#1,#2\right)}
\newcommand{\socf}[2]{\soc^{#2}(#1)}
\newcommand{\socRPP}[1]{\mathsf{RPP}(#1)}
\newcommand{\source}[1]{h(#1)}
\newcommand{\spn}[1]{\langle #1\rangle}
\DeclareMathOperator{\Sym}{\mathfrak{S}}
\newcommand{\Tab}[2]{T(#1,#2)}
\newcommand{\target}[1]{t(#1)}
\newcommand{\transpose}[1]{#1^\top}
\newcommand{\triples}[2][]{\mathcal{N}_{#2}^{#1}}
\newcommand{\Uni}{U^+}
\newcommand{\ydiagramsmall}[1]{\ytableausetup{smalltableaux}\, \ydiagram{#1} \, \ytableausetup{nosmalltableaux}}
\newcommand{\ytableausmall}[1]{\ytableausetup{smalltableaux}\, \begin{ytableau}#1\end{ytableau}\, \ytableausetup{nosmalltableaux}}
\newcommand{\hugh}[1]{{\color{blue}#1}}
\newcommand{\steven}[1]{{\color{magenta}#1 -Steven}}

%% define your title in the usual way
\title[$q$-Whittaker functions, finite fields, and Jordan forms]{$q$-Whittaker functions, finite fields, and Jordan forms}

%\author[Optional shorter names]{Longer names me\thanks{\href{mailto:hello@world.c}{hello@world.c}. Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}\addressmark{1}, \and you\addressmark{2}}

\author{Steven N.\ Karp\thanks{S.N.K.\ was supported by an NSERC postdoctoral fellowship.}\addressmark{1} \and Hugh Thomas\thanks{H.T.\ was partially supported by an NSERC Discovery Grant and the Canada Research Chairs program.}\addressmark{1}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Laboratoire de combinatoire et d'informatique math\'{e}matique, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, CP 8888, Succ.\ Centre-ville, Montr\'{e}al, QC H3C 3P8, Canada}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{The $q$-Whittaker function $W_\lambda(\mathbf{x};q)$ associated to a partition $\lambda$ is a $q$-analogue of the Schur function $s_\lambda(\mathbf{x})$, and the $t=0$ specialization of the Macdonald polynomial $P_\lambda(\mathbf{x};q,t)$. We give a new formula for $W_\lambda(\mathbf{x};q)$ in terms of partial flags compatible with a nilpotent endomorphism over the finite field of size $1/q$, analogous to a well-known formula for the Hall--Littlewood functions. We show that considering pairs of partial flags and taking Jordan forms leads to a probabilistic bijection between nonnegative-integer matrices and pairs of semistandard tableaux of the same shape, which we call the {\itshape $q$-Burge correspondence}. In the $q\to 0$ limit, we recover a description of the classical Burge correspondence (also known as column RSK) due independently to Gansner (1981), Spaltenstein (1982), and Steinberg (1988) for permutation matrices, and to Rosso (2012) in general. Finally, we apply the $q$-Burge correspondence to prove enumerative formulas for certain modules over the preprojective algebra of a path quiver.}
%\resume{}

\keywords{$q$-Whittaker function, finite field, Jordan form, partial flag variety, Burge correspondence, RSK correspondence, preprojective algebra, socle filtration.}

%% 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
%% regardless of how you do it, please use numerical citations, ie. [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex]{biblatex}
\addbibresource{57Karp.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\begin{document}
\maketitle

\section{Introduction}\label{sec_introduction}

The {\itshape Cauchy identity} for the Schur functions $s_\lambda(\mathbf{x})$
\begin{align}\label{cauchy_equation_intro}
\prod_{i,j \ge 1}\frac{1}{1-x_iy_j} = \sum_{\lambda}s_\lambda(\mathbf{x})s_\lambda(\mathbf{y})
\end{align}
plays an important role in the theories of symmetric functions and Schur processes. A bijective proof of \eqref{cauchy_equation_intro} is provided by the celebrated {\itshape Robinson--Schensted--Knuth correspondence}, or its column-insertion variant, the {\itshape Burge correspondence} \cite{burge74} (which will be more relevant for us). Namely, we expand the left-hand side of \rf{cauchy_equation_intro} as a weighted sum over nonnegative-integer matrices, and expand the right-hand side as a weighted sum over pairs of semistandard tableaux of the same shape, and these correspondences give a weight-preserving bijection between such matrices and pairs of tableaux. For example, the coefficient of $x_1x_2y_1y_2$ on each side of \rf{cauchy_equation_intro} is $2$, and the Burge correspondence acts by
\begin{align}\label{burge_equation_intro}
\scalebox{0.8}{$\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$} \mapsto \left(\ytableausmall{1 \\ 2},\ytableausmall{1 \\ 2}\right), \qquad \scalebox{0.8}{$\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$} \mapsto \left(\ytableausmall{1 & 2},\ytableausmall{1 & 2}\right).
\end{align}

The goal of this paper is to give a combinatorial and algebraic understanding of the following $q$-analogue of \rf{cauchy_equation_intro} \cite[VI.(4.13)]{macdonald95}:
\begin{align}\label{q-cauchy_equation_intro}
\prod_{\substack{i,j \ge 1,\\ d \ge 0\hphantom{,}}}\frac{1}{1-x_iy_jq^d} = \sum_{\lambda}\frac{(1-q)^{-\lambda_1}}{\prod\limits_{i\ge 1}\qfac{\lambda_i - \lambda_{i+1}}}W_\lambda(\mathbf{x};q)W_\lambda(\mathbf{y};q).
\end{align}
Here the Schur function $s_\lambda(\mathbf{x})$ has been replaced by the {\itshape $q$-Whittaker function} $W_\lambda(\mathbf{x};q)$, which is the $t=0$ specialization of the Macdonald polynomial $P_\lambda(\mathbf{x};q,t)$; see \cref{sec_whittaker_background} for a precise definition and other notation. We can expand the left-hand side of \rf{q-cauchy_equation_intro} as a $q$-weighted sum over nonnegative-integer matrices, and expand the right-hand side as a $q$-weighted sum over pairs of semistandard tableaux of the same shape. For example, extracting the coefficients of $x_1x_2y_1y_2$ in \rf{q-cauchy_equation_intro} and rescaling by $(1-q)^2$, we obtain\setlength{\tabcolsep}{3pt}
$$
\begin{tabular}{ccccccc}
$1$ & $+$ & $1$ & $=$ & $(1-q)$ & $+$ & $(1+q).$ \\[8pt]
\scalebox{0.8}{$\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$} & & \scalebox{0.8}{$\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$} & & $\left(\ytableausmall{1 \\ 2},\ytableausmall{1 \\ 2}\right)$ & & $\left(\ytableausmall{1 & 2},\ytableausmall{1 & 2}\right)$
\end{tabular}
$$
\setlength{\tabcolsep}{6pt}We see that for $q\neq 0$, there is no $q$-weight preserving bijection between the two matrices labeling the terms of the left-hand side and the two pairs of tableaux labeling the terms of the right-hand side. Nevertheless, we can define a {\itshape probabilistic bijection}, for example one which acts by
\begin{align}\label{eg_n=2_equation}
\scalebox{0.8}{$\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$} \mapsto \begin{cases}
\left(\ytableausmall{1 \\ 2},\ytableausmall{1 \\ 2}\right), & \text{with probability $1-q$}; \\[6pt]
\left(\ytableausmall{1 & 2},\ytableausmall{1 & 2}\right), & \text{with probability $q$},
\end{cases} \qquad \scalebox{0.8}{$\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$} \mapsto \left(\ytableausmall{1 & 2},\ytableausmall{1 & 2}\right).
\end{align}

Such a probabilistic bijection proving \rf{q-cauchy_equation_intro} was given by Matveev and Petrov \cite{matveev_petrov17} via a $q$-row insertion algorithm, building on work of Borodin and Petrov \cite{borodin_petrov16}; we refer to \cite[Section 1]{aigner_frieden} for further discussion and references. We prove \rf{q-cauchy_equation_intro} by introducing a new probabilistic bijection which we call the {\itshape $q$-Burge correspondence}, which is defined when $1/q$ is a prime power via the geometry of nilpotent endomorphisms over the finite field $\qinvF$. In the limit $q\to 0$, when we replace $\qinvF$ by an infinite field, we recover a description of the classical Burge correspondence \cite{burge74}, due independently to Gansner \cite{gansner81}, Spaltenstein \cite{spaltenstein82}, and Steinberg \cite{steinberg76,steinberg88} in the case of permutation matrices, and due to Rosso \cite{rosso12} in general.

In order to state our result, we introduce some terminology. If $\nil$ is a nilpotent endomorphism of $\field^n$, we say that a partial flag of subspaces
$$
F: 0 = F_0 \subseteq F_1 \subseteq \cdots \subseteq F_k = \field^n
$$
in $\field^n$ is {\itshape strictly compatible} with $\nil$ if
\begin{align}\label{compatible_intro}
\nil(F_i) \subseteq F_{i-1} \quad \text{ for } 1 \le i \le k.
\end{align}
Restricting $\nil$ to $F_0, \dots, F_k$ gives a sequence of nilpotent endomorphisms whose (conjugated) Jordan form types define a semistandard tableau $\JF(\nil;F)$ of size $n$ on the alphabet $\{1, \dots, k\}$. If $F = (F_0, \dots, F_k)$ and $F' = (F'_0, \dots, F'_l)$ are two partial flags in $\field^n$, we define the {\itshape relative position of $(F,F')$} as the $k\times l$ nonnegative-integer matrix $M$ satisfying
$$
\dim(F_i\cap F'_j) = \sum_{1\le i'\le i,\, 1\le j'\le j}M_{i',j'} \quad \text{ for all } 0 \le i \le k,\, 0 \le j \le l.
$$

\begin{thm}\label{cauchy_intro}
Given a $k\times l$ nonnegative-integer matrix $M$ whose entries sum to $n$, let $(F,F')$ be a pair of partial flags in $\qinvF^n$ with relative position $M$. If $T$ and $T'$ are semistandard tableaux of size $n$ of the same shape, we define $\pr_M(T,T')$ as the probability that a uniformly random nilpotent endomorphism $\nil$ of $\field^n$ which is strictly compatible with both $F$ and $F'$ satisfies
$$
\JF(\nil;F) = T \quad \text{ and } \quad \JF(\nil;F') = T'.
$$
(This definition does not depend on the choice of $(F,F')$.)
\begin{enumerate}[label=(\roman*), leftmargin=*]
\item\label{cauchy_intro_bijection} The map $\pr$ defines a probabilistic bijection proving \rf{q-cauchy_equation_intro} whenever $1/q$ is a prime power.\footnote{This establishes \rf{q-cauchy_equation_intro} for all $q$, since it is sufficient to verify equality at infinitely many values of $q$.}
\item\label{cauchy_intro_symmetry} Transposing the matrix corresponds to swapping the tableaux: $\pr_M(T,T') = \pr_{\transpose{M}}(T',T)$.
\item\label{cauchy_intro_specialization} As $q\to 0$, the probability $\pr_M(T,T')$ converges to $1$ if the Burge correspondence sends $M$ to $(T,T')$, and converges to $0$ otherwise.
\end{enumerate}
\end{thm}

\begin{eg}\label{eg_cauchy_intro}
Let us show that $\pr$ as defined in \cref{cauchy_intro} leads to \rf{eg_n=2_equation}. For $M = \scalebox{0.8}{$\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$}$, we can take $F = F'$ to be the partial flag
$$
0 = F_0 \subseteq F_1 = \langle e_1\rangle \subseteq F_2 = \langle e_1,e_2\rangle.
$$
Then the nilpotent endomorphisms of $\qinvF^2$ which are strictly compatible with $F = F'$ are precisely of the form
$$
\nil = \begin{bmatrix}0 & a \\ 0 & 0\end{bmatrix} \quad (a\in\qinvF).
$$
If $a\neq 0$ (which occurs with probability $1-q$), then we obtain the pair of tableaux $\left(\ytableausmall{1 \\ 2},\ytableausmall{1 \\ 2}\right)$, while if $a=0$ (which occurs with probability $q$), we obtain the pair of tableaux $\left(\ytableausmall{1 & 2},\ytableausmall{1 & 2}\right)$.

On the other hand, for $M = \scalebox{0.8}{$\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$}$, we can take $F$ and $F'$ to be
$$
0 = F_0 \subseteq F_1 = \langle e_1\rangle \subseteq F_2 = \langle e_1,e_2\rangle \quad \text{ and } \quad 0 = F'_0 \subseteq F'_1 = \langle e_2\rangle \subseteq F'_2 = \langle e_1,e_2\rangle.
$$
Then the only nilpotent endomorphism of $\qinvF^2$ which is strictly compatible with both $F$ and $F'$ is the zero endomorphism, and so (with probability $1$) we obtain the pair of tableaux $\left(\ytableausmall{1 & 2},\ytableausmall{1 & 2}\right)$. In summary, we have
\begin{gather*}
\pr_{\scalebox{0.8}{$\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$}}\left(\ytableausmall{1 \\ 2},\ytableausmall{1 \\ 2}\right) = 1-q, \quad \pr_{\scalebox{0.8}{$\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$}}\left(\ytableausmall{1 & 2},\ytableausmall{1 & 2}\right) = q, \quad \pr_{\scalebox{0.8}{$\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}$}}\left(\ytableausmall{1 & 2},\ytableausmall{1 & 2}\right) = 1.\qedhere
\end{gather*}

\end{eg}

In general, as in \cref{eg_cauchy_intro}, given $M$ we can always choose the pair of partial flags $(F,F')$ in \cref{cauchy_intro} so that the nilpotent endomorphisms $\nil$ compatible with both $F$ and $F'$ have the following form: each entry is either zero or an arbitrary element of $\qinvF$.

An important open problem is to determine whether $\pr_M(T,T')$ defines a polynomial in $q$, and if so, to find a combinatorial interpretation of it. We remark that the combinatorics of $\pr$ is different from the $q$-column insertion of O'Connell and Pei \cite{o'connell_pei13} and the $q$-row insertion of Borodin and Petrov \cite{borodin_petrov16} and Matveev and Petrov \cite{matveev_petrov17}.

Along the way to proving \cref{cauchy_intro}, we find a new formula for $W_\lambda(\mathbf{x};q)$:
\begin{thm}\label{whittaker_expansion_intro}
Let $\lambda$ be a partition of $n$, and fix a nilpotent endomorphism $\nil$ of $\qinvF^n$ whose (conjugated) Jordan form type is $\lambda$. Then for any weak composition $(\alpha_1, \dots, \alpha_k)$ of $n$, the coefficient of $x_1^{\alpha_1}\cdots x_k^{\alpha_k}$ in $W_\lambda(\mathbf{x};q)$ equals $q^{\sum_{i\ge 1}\binom{\lambda_i}{2} - \sum_{i=1}^k\binom{\alpha_i}{2}}$ times the number of partial flags
$$
0 = F_0 \subseteq F_1 \subseteq \cdots \subseteq F_k = \qinvF^n,\quad \dim(F_i) = \alpha_1 + \cdots + \alpha_i,
$$
which are strictly compatible with $\nil$.
\end{thm}

\begin{eg}\label{eg_whittaker_expansion_intro}
Let us use \cref{whittaker_expansion_intro} to find the coefficient of $x_1^2x_2^2$ in $W_{(3,1)}(\mathbf{x};q)$, when $\lambda := (3,1)$ and $\alpha := (2,2)$. We may take $\nil$ to be the nilpotent endomorphism of $\qinvF^4$ given by
$$
\nil := \begin{bmatrix}
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}.
$$
We must enumerate partial flags
$$
0 = F_0 \subseteq F_1 \subseteq F_2 = \qinvF^4, \quad \dim(F_1) = 2,
$$
which are strictly compatible with $\nil$. The condition $\nil(F_2)\subseteq F_1$ means that $F_1$ contains $e_1$, and the condition $\nil(F_1)\subseteq F_0$ means that $F_1 \subseteq \langle e_1, e_2, e_3\rangle$. Therefore $F_1$ is the direct sum of $\langle e_1\rangle$ and an arbitrary $1$-dimensional subspace of $\langle e_2, e_3\rangle$, of which there are $\frac{1}{q} + 1$ in total. Therefore the coefficient of $x_1^2x_2^2$ in $W_{(3,1)}(\mathbf{x};q)$ equals
\begin{gather*}
q^{\binom{3}{2} + \binom{1}{2} - \binom{2}{2} - \binom{2}{2}}(\textstyle\frac{1}{q} + 1) = 1+q.\qedhere
\end{gather*}

\end{eg}

We note that since $W_\lambda(\mathbf{x};q)$ is a symmetric function, the enumeration in \cref{whittaker_expansion_intro} does not depend on the order of the parts of the composition $\alpha$, but this is far from obvious. We also remark that there is a well-known formula for the {\itshape Hall--Littlewood functions} which is similar to \cref{whittaker_expansion_intro} (see, e.g., \cite[Chapter II]{macdonald95} or \cite[Corollary 2.13]{mellit20}), where we instead enumerate partial flags which are {\itshape weakly} compatible with $\nil$ (i.e.\ we only require $\nil(F_i) \subseteq F_i$ in \rf{compatible_intro}). However, we do not know how to deduce one formula from the other.

We outline the proofs of our results, which will appear in the full version of this paper. We approach \cref{whittaker_expansion_intro} by proving a refined version, showing that the contribution of a semistandard tableau $T$ to $W_\lambda(\mathbf{x};q)$ is given by enumerating partial flags $F$ satisfying $\JF(\nil;F) = T$. We proceed by induction on $k$, which reduces the proof to enumerating subspaces $F_{k-1}$ of $\qinvF$ which contain $\nil(\qinvF)$ and satisfy certain rank conditions; we are able to explicitly carry out this calculation. For part \ref{cauchy_intro_bijection} of \cref{cauchy_intro}, we must show that in \rf{q-cauchy_equation_intro}, the contribution to the right-hand side from a fixed pair of tableaux $(T,T')$ of the same shape $\lambda$ equals the contribution to the left-hand side from all nonnegative-integer matrices $M$, each weighted by $\pr_M(T,T')$. To do so, we show that each of these contributions equals, up to an explicit scalar factor, the number of triples $(F,F',\nil)$, such that $\nil$ is a nilpotent endomorphism of $\qinvF^n$ which is strictly compatible with both $F$ and $F'$ and satisfies $\JF(\nil;F) = T$ and $\JF(\nil;F') = T'$. We enumerate such triples in two ways. For the left-hand side, we first enumerate all $F$, then all $F'$ such that $(F,F')$ has given relative position $M$, and finally all $\nil$ satisfying the conditions above (which involves $\pr_M(T,T')$). For the right-hand side, we first enumerate all $\nil$ with $\JF(\nil) = \lambda$ (which is well-known), and then, using the refined version of \cref{whittaker_expansion_intro}, separately enumerate all $F$ and all $F'$ satisfying the conditions above. Part \ref{cauchy_intro_symmetry} of \cref{cauchy_intro} follows from the definitions, and we deduce part \ref{cauchy_intro_specialization} using the work of Rosso \cite{rosso12}.

The remainder of this abstract is organized as follows. In \cref{sec_whittaker_background}, we define the $q$-Whittaker functions $W_\lambda(\mathbf{x};q)$ and introduce other notation. In \cref{sec_burge}, we review the classical Burge correspondence, which is the $q=0$ specialization of our $q$-Burge correspondence. Finally, in \cref{sec_socle}, we apply the $q$-Burge correspondence to prove enumerative results about modules over the preprojective algebra of a path quiver. The idea is that each triple $(F,F',\nil)$ defines such a module, where $M$ records its isomorphism type over the path algebra and the pair $(T,T')$ records its socle filtration.

\section{\texorpdfstring{$q$}{q}-Whittaker functions}\label{sec_whittaker_background}

In this section we define some notation used in \cref{sec_introduction}. We refer to \cite{bergeron} for background on $q$-Whittaker functions, and to \cite{macdonald95,stanley99} for background on symmetric functions.
\begin{defn}\label{defn_q}
Given nonnegative integers $n\ge k$, we define
\begin{gather*}
\qn{n} := 1 + q + \cdots + q^{n-1}, \quad \qfac{n} := \qn{n}\qn{n-1}\cdots\qn{1}, \quad \text{ and } \quad \qbinom{n}{k} := \frac{\qfac{n}}{\qfac{k}\qfac{n-k}}.
\end{gather*}

\end{defn}

\begin{defn}[{cf.\ \cite[VI.(7.13')]{macdonald95}}]\label{defn_whittaker}
Let $\lambda$ be a partition of $n$. A {\itshape semistandard tableau} $T$ is a filling of the boxes of $\lambda$ with positive integers which is weakly increasing along rows and strictly increasing along columns. For $i\ge 1$ and $j \ge 0$, we let $T^{(j)}$ denote the partition formed by the entries $1, \dots, j$ of $T$, and let $T^{(j)}_i$ denote the $i$th part of $T^{(j)}$. We define the {\itshape q-weight of $T$} as
$$
\wt(T) := \prod_{i,j\ge 1}\qbinom{T^{(j)}_i - T^{(j)}_{i+1}}{T^{(j)}_i - T^{(j-1)}_i} \in \mathbb{N}[q].
$$
Given indeterminates $\mathbf{x} = (x_1, x_2, \dots)$, we define the {\itshape $q$-Whittaker function}
$$
W_\lambda(\mathbf{x};q) := \sum_T\wt(T)\mathbf{x}^T,
$$
where the sum is over all semistandard tableaux $T$ of shape $\lambda$, and $\mathbf{x}^T := \prod\limits_{i\ge 1}x_i^{\#\text{$i$'s in $T$}}$.
\end{defn}

\begin{eg}
\setlength{\tabcolsep}{3pt}Let $\lambda := (2,2)$. Then $W_\lambda(\mathbf{x};q)$ equals
\begin{gather*}
\begin{tabular}{cccccccccc}
$\displaystyle\sum_{1 \le i_1 < i_2}\hspace*{-4pt}$ & $x_{i_1}^2x_{i_2}^2$ & $+$ & $\displaystyle\sum_{1 \le i_1 < i_2 < i_3}\hspace*{-2pt}\Big(\hspace*{-6pt}$ & $(1+q)x_{i_1}^2x_{i_2}x_{i_3}$ & $+$ & $(1+q)x_{i_1}x_{i_2}^2x_{i_3}$ & $+$ & $(1+q)x_{i_1}x_{i_2}x_{i_3}^2$ & $\hspace*{-4pt}\Big)$ \\[14pt]
& $\ytableausmall{i_1 & i_1 \\ i_2 & i_2}$ & & & $\ytableausmall{i_1 & i_1 \\ i_2 & i_3}$ & & $\ytableausmall{i_1 & i_2 \\ i_2 & i_3}$ & & $\ytableausmall{i_1 & i_2 \\ i_3 & i_3}$
\end{tabular}\hspace*{36pt} \\[12pt]
\hspace*{120pt}\begin{tabular}{cccccc}
$+$ & $\displaystyle\sum_{1 \le i_1 < i_2 < i_3 < i_4}\hspace*{-2pt}\Big(\hspace*{-6pt}$ & $(1+q)^2x_{i_1}x_{i_2}x_{i_3}x_{i_4}$ & $+$ & $(1+q)x_{i_1}x_{i_2}x_{i_3}x_{i_4}$ & $\hspace*{-4pt}\Big).$ \\[14pt]
& & $\ytableausmall{i_1 & i_2 \\ i_3 & i_4}$ & & $\ytableausmall{i_1 & i_3 \\ i_2 & i_4}$ & 
\end{tabular}\qedhere
\end{gather*}
\setlength{\tabcolsep}{6pt}
\end{eg}

\begin{rmk}
Given a partition $\lambda$, let $P_\lambda(\mathbf{x};q,t)$ and $\widetilde{H}_\lambda(\mathbf{x};q,t)$ denote the {\itshape Macdonald polynomial} and {\itshape modified Macdonald polynomial}, respectively, and let $\omega$ denote the standard involution on symmetric functions. Then we have \cite[Section 3]{bergeron}
$$
W_\lambda(\mathbf{x};q) = P_\lambda(\mathbf{x};q,0) = q^{\deg(\widetilde{H}_\lambda)}\omega(\widetilde{H}_\lambda(\mathbf{x};1/q,0)).
$$
In particular, we have the following specializations of $W_\lambda(\mathbf{x};q)$ \cite[Section VI.4]{macdonald95}:
\begin{gather*}
W_\lambda(\mathbf{x};0) = s_\lambda(\mathbf{x}), \quad W_\lambda(\mathbf{x};1) = e_{\con{\lambda}}(\mathbf{x}).
\end{gather*}

\end{rmk}

\begin{defn}\label{defn_JF}
Let $\nil$ be an endomorphism of the $n$-dimensional vector space $V$ over $\field$. We say that $\nil$ is {\itshape nilpotent} if some power of $\nil$ is zero. If so, we can choose a basis of $V$ so that $\nil$ is represented by the $n\times n$ block-diagonal matrix
$$
\begin{bmatrix}
J_{\mu_1} & 0 & \cdots & 0 \\
0 & J_{\mu_2} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & J_{\mu_k}
\end{bmatrix}, \text{ where $\mu_1 \ge \cdots \ge \mu_k$ and $J_l$ is the $l\times l$ matrix} \begin{bmatrix}
0 & 1 & 0 & \cdots \\
0 & 0 & 1 & \cdots \\
0 & 0 & 0 & \cdots \\
\vdots & \vdots & \vdots & \ddots
\end{bmatrix}.
$$
Let $\lambda := \con{\mu}$ denote the conjugate partition of $\mu$; we call $\lambda$ the {\itshape (conjugated) Jordan form partition} of $\nil$, denoted $\JF(\nil)$. Equivalently, we have
$$
\dim(\ker(\nil^i)) = \lambda_1 + \cdots + \lambda_i \quad \text{ for all } i \ge 0.
$$
\end{defn}
For example, if $\nil$ is the zero endomorphism of $\field^n$, then $\JF(\nil) = (n)$.
\begin{lem}
Let $\nil$ be a nilpotent endomorphism of $\field^n$ which is strictly compatible with the partial flag $F = (F_0, \dots, F_k)$ (see \rf{compatible_intro}). Let $T$ be the tableau of shape $\JF(\nil)$ such that $T^{(j)} = \JF(\nil|_{F_j})$ for $0 \le j \le k$, where $\nil|_{F_j}$ denotes the endomorphism $\nil$ restricted to $F_j$. Then $T$ is a semistandard tableau, which we denote by $\JF(\nil;F)$.
\end{lem}

\section{Burge correspondence}\label{sec_burge}

In this section we review the Burge correspondence \cite{burge74}, following \cite[Appendix A]{fulton97}.
\begin{defn}[{\cite[Appendix A]{fulton97}}]\label{defn_column_insertion}
Let $T$ be a semistandard tableau, and let $i\in\mathbb{Z}_{>0}$. Then the {\itshape column insertion of $i$ into $T$} is the semistandard tableau obtained from $T$ by performing the following steps. If $i$ is greater than all the entries of $T$ in column $1$, we place $i$ at the end of column $1$. Otherwise, we replace the smallest entry $j\ge i$ in column $1$ with $i$, and insert $j$ into column $2$ by the same procedure, and so on.

Given a $k\times l$ matrix $M$ of nonnegative integers, we define semistandard tableaux $\Ptab{M}$ and $\Qtab{M}$ of the same shape by the following procedure, starting with $\Ptab{M}$ and $\Qtab{M}$ as the empty tableau. For $j = 1, \dots, l$ and $i = k, \dots, 1$ (i.e.\ we read $M$ column by column, left to right, and within each column from bottom to top), we column-insert $i$ into $\Ptab{M}$ and record $j$ in the new box of $\Qtab{M}$ a total of $M_{i,j}$ times, so that at each step $\Ptab{M}$ and $\Qtab{M}$ have the same shape. The map $M\mapsto (\Ptab{M},\Qtab{M})$ is called the {\itshape Burge correspondence}.
\end{defn}

\begin{eg}\label{eg_burge}
Let $M := \scalebox{0.8}{$\begin{bmatrix}1 & 0 & 1 \\ 2 & 1 & 1 \\ 0 & 1 & 1\end{bmatrix}$}$. Then $\Ptab{M}$ is obtained by column-inserting $2$, $2$, $1$, $3$, $2$, $3$, $2$, and $1$ into the empty tableau, and $\Qtab{M}$ is obtained by recording $1$, $1$, $1$, $2$, $2$, $3$, $3$, and $3$. We get
\begin{gather*}
\Ptab{M} = \ytableausmall{1 & 1 & 2 & 2 & 2 \\ 2 & 3 \\ 3} \quad \text{ and } \quad \Qtab{M} = \ytableausmall{1 & 1 & 1 & 3 & 3 \\ 2 & 2 \\ 3}.\qedhere
\end{gather*}

\end{eg}

\begin{eg}\label{eg_burge_from_cauchy}
In the case of $2\times 2$ permutation matrices, the Burge correspondence acts by \rf{burge_equation_intro}. This is consistent with \cref{cauchy_intro}\ref{cauchy_intro_specialization} and the calculation in \cref{eg_cauchy_intro}.
\end{eg}

\begin{rmk}\label{RSK_to_burge}
The {\itshape Robinson--Schensted--Knuth (RSK) correspondence} is a map defined similarly to the Burge correspondence, where we use {\itshape row insertion} instead of column insertion, and read the columns of the given matrix $M$ from top to bottom rather than bottom to top. The relationship between the RSK and Burge correspondences is explained in \cite[Section A.4.2]{fulton97}. Namely, let $(T,T')$ be the pair of tableaux corresponding to $M$ under the RSK correspondence. If $M$ is a permutation matrix, then
$$
T = \transpose{\Ptab{M}} \quad \text{ and } \quad T' = \transpose{\Qtab{M}}.
$$
In general, let $M_{\text{rev}}$ (respectively, $M^{\text{rev}}$) denote the matrix obtained from $M$ by reversing the order of its rows (respectively, columns). Then
$$
T = \Ptab{M^\text{rev}} = \evac(\Ptab{M_\text{rev}}) \quad \text{ and } \quad T' = \Qtab{M_{\text{rev}}} = \evac(\Qtab{M^{\text{rev}}}),
$$
where $\evac$ is the {\itshape evacuation} map (also known as the {\itshape Sch\"{u}tzenberger involution}).
\end{rmk}

\section{The preprojective algebra and socle filtrations}\label{sec_socle}

In this section, we apply the $q$-Burge correspondence to the enumeration of quiver representations. In order to keep the exposition brief, we will not define all the necessary terms. We refer to \cite{derksen_weyman17,kirillov16} for background.

Let $Q(k,l)$ denote the path quiver on $k+l-1$ vertices with a unique sink, with $k-1$ vertices to the right and $l-1$ vertices to the left:
$$
\;\begin{tikzpicture}[baseline=(current bounding box.center)]
\tikzstyle{out1}=[inner sep=0,minimum size=1.2mm,circle,draw=black,fill=black,semithick]
\pgfmathsetmacro{\s}{1.80};
\node[out1](v1)at(0*\s,0)[label={[below=3pt]\vphantom{$k-3$}$-l+1$}]{};
\node[out1](v2)at(1*\s,0)[label={[below=3pt]\vphantom{$k-3$}$-l+2$}]{};
\node[out1](v3)at(2*\s,0)[label={[below=3pt]\vphantom{$k-3$}$-l+3$}]{};
\node[out1](v4)at(3*\s,0)[label={[below=3pt]\vphantom{$k-3$}$-1$}]{};
\node[out1](v5)at(4*\s,0)[label={[below=3pt]\vphantom{$k-3$}$0$}]{};
\node[out1](v6)at(5*\s,0)[label={[below=3pt]\vphantom{$k-3$}$1$}]{};
\node[out1](v7)at(6*\s,0)[label={[below=3pt]\vphantom{$k-3$}$k-3$}]{};
\node[out1](v8)at(7*\s,0)[label={[below=3pt]\vphantom{$k-3$}$k-2$}]{};
\node[out1](v9)at(8*\s,0)[label={[below=3pt]\vphantom{$k-3$}$k-1$}]{};
\path[-latex',thick](v1)edge(v2) (v2)edge(v3) (v4)edge(v5) (v9)edge(v8) (v8)edge(v7) (v6)edge(v5);
\node[inner sep=0]at(2.5*\s,0){$\cdots$};
\node[inner sep=0]at(5.5*\s,0){$\cdots$};
\end{tikzpicture}\;.
$$
Every finite-dimensional representation $V$ of $Q(k,l)$ can be expressed (up to isomorphism) as a direct sum of indecomposable representations, which are indexed by the $\binom{k+l}{2}$ subpaths of $Q(k,l)$. We will restrict our attention to representations $V$ in which the subpath of every indecomposable summand is supported at the sink, or equivalently, representations in which all maps involved are injective. Isomorphism classes of such $V$ are labeled by $k\times l$ matrices $M$ of nonnegative integers, where the index $(i,j)$ corresponds to the indecomposable representation of the path between vertices $k-i$ and $j-l$. Alternatively, we may assume that the vector space over vertex $0$ is $\field^n$, and that all maps involved are identity maps; then the vector spaces at each vertex form two partial flags $F$ and $F'$ inside $\field^n$, and $M$ records the relative position of $(F,F')$. We note that the dimension vector of $V$ can be read off from the row and column sums of $M$.

We now consider ways of extending $V$ to a module $\double{V}$ over the {\itshape preprojective algebra} $\ppa{Q(k,l)}$ of $Q(k,l)$. This amounts to additionally associating a map to the reverse of every arrow of $V$ so that at every vertex $i$ of $Q(k,l)$, the compositions of the maps associated to the two paths of length two from $i$ to itself are equal up to a fixed sign. It turns out that $\double{V}$ is given up to isomorphism by a triple $(F,F',\nil)$ (in general not uniquely), where $(F,F')$ is a pair of partial flags with relative position $M$, and $\nil$ is a nilpotent endomorphism of $\field^n$ which is strictly compatible with both $F$ and $F'$:
\begin{gather*}
\;\begin{tikzpicture}[baseline=(current bounding box.center)]
\tikzstyle{out1}=[inner sep=0,minimum size=1.2mm,circle,draw=black,fill=black,semithick]
\pgfmathsetmacro{\s}{1.80};
\node[out1](v-4)at(0*\s,0)[label={[above=-1pt]\vphantom{$F'_1$}$F'_1$}]{};
\node[out1](v-3)at(1*\s,0)[label={[above=-1pt]\vphantom{$F'_1$}$F'_2$}]{};
\node[out1](v-2)at(2*\s,0)[label={[above=-1pt]\vphantom{$F'_1$}$F'_3$}]{};
\node[out1](v-1)at(3*\s,0)[label={[above=-1pt]\vphantom{$F'_1$}$F'_{l-1}$}]{};
\node[out1](v0)at(4*\s,0)[label={[above=-1pt]\vphantom{$F'_1$}$\field^n$}]{};
\node[out1](v1)at(5*\s,0)[label={[above=-1pt]\vphantom{$F'_1$}$F_{k-1}$}]{};
\node[out1](v2)at(6*\s,0)[label={[above=-1pt]\vphantom{$F'_1$}$F_3$}]{};
\node[out1](v3)at(7*\s,0)[label={[above=-1pt]\vphantom{$F'_1$}$F_2$}]{};
\node[out1](v4)at(8*\s,0)[label={[above=-1pt]\vphantom{$F'_1$}$F_1$}]{};
\path[-latex',thick](v-4)edge[bend left=15] node[above=-1pt]{$\id$}(v-3) (v-3)edge[bend left=15] node[below=-1pt]{$-\nil$}(v-4)
(v-3)edge[bend left=15] node[above=-1pt]{$\id$}(v-2) (v-2)edge[bend left=15] node[below=-1pt]{$-\nil$}(v-3)
(v-1)edge[bend left=15] node[above=-1pt]{$\id$}(v0) (v0)edge[bend left=15] node[below=-1pt]{$-\nil$}(v-1)
(v1)edge[bend right=15] node[above=-1pt]{$\id$}(v0) (v0)edge[bend right=15] node[below=-1pt]{\vphantom{$-\nil$}$\nil$}(v1)
(v3)edge[bend right=15] node[above=-1pt]{$\id$}(v2) (v2)edge[bend right=15] node[below=-1pt]{\vphantom{$-\nil$}$\nil$}(v3)
(v4)edge[bend right=15] node[above=-1pt]{$\id$}(v3) (v3)edge[bend right=15] node[below=-1pt]{\vphantom{$-\nil$}$\nil$}(v4);
\node[inner sep=0]at(2.5*\s,0){$\cdots$};
\node[inner sep=0]at(5.5*\s,0){$\cdots$};
\end{tikzpicture}\;.
\end{gather*}

Provided $k+l\geq 7$, it is an intractable (technically, \emph{wild}) problem  to classify such $\double{V}$ up to isomorphism, but we can associate an interesting combinatorial invariant by taking the {\itshape socle filtration}. This records, for every vertex and height $j \ge 1$, the dimension of the subspace at the vertex which is annihilated after applying any $j$ arrows of $\double{V}$, modulo the subspace annihilated after applying any $j-1$ arrows. After deleting trivial zero entries, we obtain a {\itshape reverse plane partition} $R$ on a $k\times l$ rectangle. By splitting apart the rectangle along its diagonal we obtain two {\itshape Gelfand--Tsetlin patterns} (as in \cite[Section 1]{hopkins14}), which we can translate into a pair $(T,T')$ of semistandard tableaux of the same shape. It turns out that
$$
\JF(\nil;F) = T \quad \text{ and } \quad \JF(\nil;F') = T'.
$$
Then we have the following analogue of \cref{whittaker_expansion_intro} in this setting (where, as in the preceding discussion, we assume all maps of $\double{V}$ directed towards the sink are injective):
\begin{thm}\label{preprojective_enumeration}
Set $\field := \qinvF$. Let $\alpha = (\alpha_1, \dots, \alpha_k)$ and $\beta = (\beta_1, \dots, \beta_l)$ be weak compositions of $n$, and set $c := n+\sum_{i=1}^k\binom{\alpha_i}{2} + \sum_{j=1}^l\binom{\beta_j}{2}$. Let $M$ be a $k\times l$ matrix of nonnegative integers with row sums $\alpha$ and column sums $\beta$, and let $R$ be a reverse plane partition corresponding to the pair of semistandard tableaux $(T,T')$.
\begin{enumerate}[label=(\roman*), leftmargin=*]
\item\label{preprojective_enumeration_term} We have
$$
\sum_{\double{V}}\frac{1}{|\!\Aut(\double{V})|} = \frac{q^c(1-q)^{-n}}{\prod\limits_{1 \le i \le k,\, 1 \le j \le l}\qfac{M_{i,j}}}\pr_M(T,T'),
$$
where the sum is over all $\ppa{Q(k,l)}$-modules $\double{V}$ up to isomorphism which are indexed by $M$ as a representation of $Q(k,l)$ and have socle filtration $R$.
\item\label{preprojective_enumeration_RPP} We have
$$
\sum_{\double{V}}\frac{1}{|\!\Aut(\double{V})|} = \frac{q^c(1-q)^{-\lambda_1}}{\prod\limits_{i\ge 1}\qfac{\lambda_i - \lambda_{i+1}}}\wt(T)\wt(T'),
$$
where the sum is over all $\ppa{Q(k,l)}$-modules $\double{V}$ up to isomorphism which have fixed dimension vector corresponding to $(\alpha,\beta)$ and have socle filtration $R$.
\item\label{preprojective_enumeration_sum} We have
$$
\sum_{\double{V}}\frac{1}{|\!\Aut(\double{V})|} = q^c\cdot\text{(coefficient of $x_1^{\alpha_1}\cdots x_k^{\alpha_k}y_1^{\beta_1}\cdots y_l^{\beta_l}$ in \rf{q-cauchy_equation_intro})},
$$
where the sum is over all $\ppa{Q(k,l)}$-modules $\double{V}$ up to isomorphism which have fixed dimension vector corresponding to $(\alpha,\beta)$.
\end{enumerate}

\end{thm}

\begin{eg}\label{eg_preprojective_enumeration}
We illustrate \cref{preprojective_enumeration}\ref{preprojective_enumeration_term} for $M := \scalebox{0.8}{$\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$}$, which has row sums $\alpha := (1,1)$ and column sums $\beta := (1,1)$. We have $c = 2$. Every $\ppa{Q(k,l)}$-module $\double{V}$ which is indexed by $M$ as a representation of $Q(k,l)$ is, up to isomorphism, of the form
$$
\double{V} = \;\begin{tikzpicture}[baseline=(current bounding box.center)]
\tikzstyle{out1}=[inner sep=0,minimum size=1.2mm,circle,draw=black,fill=black,semithick]
\pgfmathsetmacro{\s}{3.00};
\node[out1](v-1)at(1*\s,0)[label={[above=-1pt]\vphantom{$F'_1$}$\langle e_1\rangle$}]{};
\node[out1](v0)at(2*\s,0)[label={[above=-1pt]\vphantom{$F'_1$}$\langle e_1,e_2\rangle$}]{};
\node[out1](v1)at(3*\s,0)[label={[above=-1pt]\vphantom{$F'_1$}$\langle e_1\rangle$}]{};
\path[-latex',thick](v-1)edge[bend left=15] node[above=-1pt]{$\id$}(v0) (v0)edge[bend left=15] node[below=-1pt]{$-\nil$}(v-1)
(v1)edge[bend right=15] node[above=-1pt]{$\id$}(v0) (v0)edge[bend right=15] node[below=-1pt]{\vphantom{$-\nil$}$\nil$}(v1);
\end{tikzpicture}\;, \quad \text{ where } \quad \nil = \begin{bmatrix}0 & a \\ 0 & 0\end{bmatrix} \quad (a\in\qinvF).
$$
It turns out that there are two isomorphism classes, corresponding to $a\neq 0$ and $a=0$.

First we consider the case $a\neq 0$; we may assume that $a=1$. The automorphism group of $\double{V}$ (which we may regard as the set of $g\in\GL_2$ which fix the vector spaces at each vertex and satisfy $g\nil g^{-1} = \nil$) is
$$
\left\{\begin{bmatrix}g_{1,1} & g_{1,2} \\ 0 & g_{2,2}\end{bmatrix} : g_{1,1} = g_{2,2} \neq 0\right\}, \quad \text{ with size } \frac{1-q}{q^2}.
$$
\setlength{\tabcolsep}{3pt}\renewcommand{\arraystretch}{0.8}The socle filtration of $\double{V}$ is $\scalebox{0.8}{\begin{tabular}{ccc}
& $1$ & \\
$1$ & & $1$ \\
& $1$ &
\end{tabular}}$; splitting it apart gives the pair of Gelfand--Tsetlin patterns
$$
\left(\scalebox{0.8}{\begin{tabular}{cc}
$1$ & \\
& $1$ \\
$1$ &
\end{tabular}},\scalebox{0.8}{\begin{tabular}{cc}
& $1$ \\
$1$ & \\
& $1$
\end{tabular}}\right), \text{ corresponding to the pair of tableaux } \left(\ytableausmall{1 \\ 2},\ytableausmall{1 \\ 2}\right).
$$
\setlength{\tabcolsep}{6pt}\renewcommand{\arraystretch}{1}Therefore \cref{preprojective_enumeration}\ref{preprojective_enumeration_term} states that
$$
\frac{q^2}{1-q} = q^2(1-q)^{-2}\pr_M\left(\ytableausmall{1 \\ 2},\ytableausmall{1 \\ 2}\right),
$$
which is consistent with the calculation $\pr_M\left(\ytableausmall{1 \\ 2},\ytableausmall{1 \\ 2}\right) = 1-q$ in \cref{eg_cauchy_intro}.

Now we consider the case $a=0$. The automorphism group of $\double{V}$ is
$$
\left\{\begin{bmatrix}g_{1,1} & g_{1,2} \\ 0 & g_{2,2}\end{bmatrix} : g_{1,1}, g_{2,2} \neq 0\right\}, \quad \text{ with size } \frac{(1-q)^2}{q^3}.
$$
\setlength{\tabcolsep}{3pt}\renewcommand{\arraystretch}{0.8}The socle filtration of $\double{V}$ is $\scalebox{0.8}{\begin{tabular}{ccc}
& $0$ & \\
$1$ & & $1$ \\
& $2$ &
\end{tabular}};$ splitting it apart gives the pair of Gelfand--Tsetlin patterns
$$
\left(\scalebox{0.8}{\begin{tabular}{cc}
$0$ & \\
& $1$ \\
$2$ &
\end{tabular}},\scalebox{0.8}{\begin{tabular}{cc}
& $0$ \\
$1$ & \\
& $2$
\end{tabular}}\right), \text{ corresponding to the pair of tableaux } \left(\ytableausmall{1 & 2},\ytableausmall{1 & 2}\right).
$$
\setlength{\tabcolsep}{6pt}\renewcommand{\arraystretch}{1}Therefore \cref{preprojective_enumeration}\ref{preprojective_enumeration_term} states that
$$
\frac{q^3}{(1-q)^2} = q^2(1-q)^{-2}\pr_M\left(\ytableausmall{1 & 2},\ytableausmall{1 & 2}\right),
$$
which is consistent with the calculation $\pr_M\left(\ytableausmall{1 & 2},\ytableausmall{1 & 2}\right) = q$ in \cref{eg_cauchy_intro}.
\end{eg}

\acknowledgments{This work arose as part of a working group at LaCIM during the academic year 2019--2020. We thank the other participants of the working group, namely Florian Aigner, Fran\c{c}ois Bergeron, Benjamin Dequ\^{e}ne, Gabriel Frieden, Christophe Reutenauer, and Franco Saliola, for many enlightening discussions.}

\printbibliography
%\bibliographystyle{plain}
%\bibliography{ref}

\end{document}
