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\title[ { \texorpdfstring{$(q,t)$}{(q,t)}}-tau functions and path operators]{{ \texorpdfstring{$(q,t)$}{(q,t)}}-tau functions and path operators}

\author[H. Ben
  Dali, V. Bonzom, and M. Do\l\k{e}ga]{\texorpdfstring{Houcine Ben
  Dali\thanks{\href{mailto:bendali@math.harvard.edu}{bendali@math.harvard.edu}}\addressmark{1}
  Valentin Bonzom\thanks{\href{mailto:valentin.bonzom@univ-eiffel.fr}{valentin.bonzom@univ-eiffel.fr}. V.~B.~was
    partially supported by the ANR-23-CE48-0018 CartesEtPlus, and the ANR-20-CE48-0018 3DMaps and the
    ANR-21-CE48-0017 LambdaComb.}\addressmark{2},
    \and Maciej Do\l\k{e}ga\thanks{\href{mailto:mdolega@impan.pl}{mdolega@impan.pl}. This research was funded in whole or in part by {\it Narodowe Centrum Nauki}, grant 2021/42/E/ST1/00162. For the purpose of Open Access, the author has applied a CC-BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.}\addressmark{3}}{Houcine Ben
  Dali
  Valentin Bonzom,
    and Maciej Do\l\k{e}ga}}

\address{\addressmark{1}Department of Mathematics, Harvard University, Cambridge, MA 02138, USA\\ \addressmark{2}LIGM, CNRS UMR 8049, Université Gustave Eiffel, Champs-sur-Marne, France\\ \addressmark{3}Institute of Mathematics, Polish Academy of Sciences}

\received{\today}

\abstract{Motivated by weighted Hurwitz theory and its connection to integrability, we introduce a $(q,t)$-tau function that deforms the classical case of hypergeometric tau functions using Macdonald polynomials, while simultaneously generalizing several important series that have already appeared in enumerative geometry, gauge theory, and integrability. We prove that this function is uniquely characterized by a family of differential equations and demonstrate a positive combinatorial expansion of these PDEs in terms of a new family of operators encoded by alternating paths. As a byproduct of our techniques, we establish a connection between path operators and the Delta conjecture.}

\keywords{Macdonald polynomials, Hurwitz series, vertex operators, delta conjecture.}

\usepackage[backend=bibtex, giveninits]{biblatex}
\addbibresource{biblio2015.bib}

\begin{document}

\maketitle

\section{Motivation and main results}
\subsection{The \texorpdfstring{$(q,t)$}{}-tau functions and the first main result}

Hurwitz theory is a branch of enumerative geometry that focuses on
counting branched coverings of the sphere. It has been found that
various generating functions in Hurwitz theory can be expressed, using
monodromy and representation theory of symmetric groups, as explicit
infinite series involving Schur symmetric functions. This perspective
proved highly effective, revealing many beautiful connections between
the old and somewhat forgotten Hurwitz theory and other modern fields such as
Gromov--Witten theory, integrable probability, integrable systems, and matrix models,
which gave a new life to this old topic~\cite{CavalieriMiles2016}. In particular, these series are often called tau
functions due to the fact that they are tau functions of certain
integrable hierarchies such as the KP-hierarchy, or more generally the
2D Toda hierarchy~\cite{Guay-PaquetHarnad2017}.

Schur symmetric functions have several interesting deformations, such
as Jack symmetric functions (a one-parameter deformation) and more generally, Macdonald
polynomials (a two-parameter deformation). These symmetric functions can furthermore be used to deform the tau functions of Hurwitz theory, by replacing occurences of Schur functions with Jack or Macdonald polynomials. This leads to a natural question: \emph{what are the structural
properties of these ``deformed'' tau functions?} In the case of
Jack symmetric functions, it was recently proven
in~\cite{ChapuyDolega2022} that such a deformation can
still be interpreted as a generating function for geometric objects in
an extended version of Hurwitz theory that includes real
(non-oriented) branched coverings. A primary novelty in these findings
was the construction of an explicit PDE satisfied by the deformed tau
function, which exhibited a rich combinatorial structure. It also allowed to discover further
connections to the Whitakker vector of $\mathcal W$-algebras and topological recursion~\cite{ChidambaramDolegaOsuga2024}. Inspired by those results, we here study a $(q,t)$-deformed tau function, and prove analogous
PDEs in this more general setting. Let us now introduce this $(q,t)$-deformed tau function.\medskip

Consider two infinite sequences of variables $u_0,u_1,\dotsc$ and $v_0,v_1,\dotsc$, and define $\KK$ as the field of formal Laurent series in these variables with coefficients in $\QQ(q,t)$ 
\begin{align}
  \KK:=\QQ(q,t)((u_0,u_1,\dots,v_0,v_1,\dots)).  
\end{align}
We denote $\Lambda$ the space of symmetric functions with coefficients in $\KK$. If $X = x_1 + x_2 + \dotsb, Y = y_1+y_2 + \dotsb$ are two alphabets, then $\Lambda_X$ and
$\Lambda_Y$ will be the spaces of symmetric functions in $X$ and $Y$, respectively.

Let $G_1(\hbar) = \sum_{i=0}^\infty u_i \hbar^i, G_2(\hbar) =  \sum_{i=0}^\infty v_i \hbar^i\in\QQ(q,t)[\hbar]((u_0,u_1,\dots,v_0,v_1,\dots))$, and define $G(\hbar) = \frac{G_1(\hbar)}{G_2(\hbar)}$. In particular, $G(q^{j} t^{i})\in\KK$ is well-defined for all $i,j\geq 0$. The $G$-weighted $(q,t)$-tau function is the series in $\Lambda_X\otimes\Lambda_Y\llbracket z\rrbracket$ defined by:
\begin{equation}
  \label{eq:GWeightedDoubleTau}
  \tau_G(z,X,Y) :
= \sum_{\lambda\in\mathbb
  Y}z^{|\lambda|}\frac{\tH_\lambda[X]\tH_\lambda[Y]}{\normH{\lambda}}\prod_{(i,j)\in\lambda} G(q^{j-1} t^{i-1}),
\end{equation}
where $\tH_\lambda$ are modified Macdonald polynomials, see \Cref{sub:notation}.

Beyond it being a natural $(q,t)$-deformation of the tau
functions for the $G$-weighted $b$-Hurwitz numbers of \cite{ChapuyDolega2022}, this function has appeared as an important generating series in enumerative geometry and
mathematical physics for specific cases of the weight $G$. When $G(\hbar) = \prod_{i=1}^n(1-\hbar u_i)$, the plethystic
logarithm of $\tau_G(z,X,Y)$ was conjectured by
Hausel--Lettelier--Rodriguez-Villegas~\cite{HauselLetellierRodriguezVillegas2011} to be the generating series of
the mixed Hodge polynomials of character varieties of the Riemann
sphere. When $G(\hbar) = (1-u\hbar)^{-1}$, the specialization $Y =
1$ corresponds to the Whittaker vector for the
deformed Virasoro algebra from the $5D$ $\mathcal{N} = 1$ pure
$\SU(2)$ case of the AGT conjecture~\cite{Yanagida2016}. Finally, the
case $G(\hbar)=\hbar$ has appeared in the recent work of
Bourgine and Garbali~\cite{BourgineGarbali2024} as a tau function of a $(q,t)$-extension of the
2D Toda hierarchy\footnote{In
  \cite{BourgineGarbali2024,Yanagida2016} the authors worked with the
  $P$-version of Macdonald polynomials. We define our $(q,t)$-tau
  function in terms of the modified Macdonald polynomials, as such a function seems
to have much more interesting combinatorial structure that is
partially supported by the conjecture of Hausel--Lettelier--Rodriguez-Villegas.}. 

Our first main theorem gives explicit PDEs 
that allow to obtain $\tau_G(z,X,Y)$ as their unique solution. Our family of differential operators is defined using the classical operator $D_0$,
\begin{align}\label{eq:def_D}
  D_0:=\sum_{k\geq 0}(-1)^{k} e_{k}[X]h_k^\perp[MX],
\end{align}
where $M = (1-q)(1-t)$ and $h_k^\perp$ is the adjoint of $h_k$ for the Hall inner product, see \Cref{sub:notation}.
We then define the following operators, obtained from $D_0$ and\footnote{Here, we think of $e_1$ as the operator acting on $\Lambda$ by multiplying by $e_1$.} $e_1$ by commutation operations. 

Throughout this paper, the series $F$ will be $F=G_1$ or $F=G_2$, with an expansion $F(\hbar)=\sum_{i \geq 0}a_i \hbar^i$ ($a_i=u_i$ in the first case and $a_i=v_i$ in the second one). We then define
\[ \A_F := \sum_{i \geq 0}a_i \ad^i_{-\frac{D_0}{M}}(-e_1[X]),\quad
  \A^{(1)}_F := \ad_{-\frac{D_0}{M}} \left(\A_F\right),\quad \A^{(\ell)}_F =
  \ad^{\ell-1}_{\A_F/M}(\A^{(1)}_F),\,\, \ell \geq 2,\]
where $\ad_{A}(B) := [A,B] = AB-BA$, for any operators $A$ and $B$. The operators $\A_{F}^{(\ell)}$ are differential when we think of symmetric functions as functions in $\left(h_i\right)_{i\geq 1}$ for example. 

\begin{thm}\label{thm:dif_eq}
    For any $\ell\geq 1$ we have
    \begin{equation}\label{eq:diff_eq}
      z^\ell\A^{(\ell)}_{G_1}(X)\cdot\tau_G(z,X,Y)=\left(\A^{(\ell)}_{G_2}(Y)\right)^*\cdot\tau_G(z,X,Y).  
    \end{equation}
        Moreover, $\tau_G(z,X,Y)$ is the unique function $\Phi_G\in\Lambda_X\otimes\Lambda_Y\llbracket z\rrbracket$ which satisfies these equations and such that $[z^0]\Phi_G(z,X,Y)=1$, where $\left(\A_{G_2}^{(\ell)}\right)^*$ is the adjoint of $\A_{G_2}^{(\ell)}$ with respect to the star scalar product (see \Cref{sub:notation}).
\end{thm}
      
\subsection{Path operators and the second main theorem}
      Note that the operators $\A^{(\ell)}_{F}$ appearing in \Cref{thm:dif_eq} are defined through multiple applications of the adjoint
      action, and it is not clear at all whether they possess a
      combinatorial interpretation. A key part
      of~\cite{ChapuyDolega2022} that enabled a
      combinatorial/topological interpretation in the case of the tau
      function involving Jack symmetric functions was the explicit
      expression of analogous operators, defined by the adjoint
      action, as a positive sum of operators defined combinatorially
      using lattice paths. Our second main result demonstrates that a similar statement holds in the Macdonald case. The key idea is the following:  we write the operators $\A_F^{(\ell)}$ by \textit{alternating} multiplicative  and derivative  operators ($(-1)^ke_k$ and $h^\perp_k$).
      We start by defining the underlying combinatorial objects: alternating paths.


Let $N=(x_N,y_N)$ be a point of $(\ZZ_{\geq 0})^2$. We call \emph{a path} $\gamma$ from the origin $(0,0)$ to $N$, a sequence of points $(w_{0}, w_{1},\dotsc,w_{x_N})$ in $(\ZZ_{\geq 0})^2$ such that $w_j=(j,y_j)$, with $w_{0}=(0,0)$ and $w_{x_N}=N$. Such a path is uniquely encoded by its sequence of \emph{steps} $\gamma_j \coloneqq y_j-y_{j-1} \in\ZZ$. The length of $\gamma$ is $x_N$.

We say that a path $\gamma=(\gamma_1,\dotsc, \gamma_n)$ from $(0,0)$ to $N$ is \emph{alternating},
 if $\gamma_{2j-1}\geq 0$ and $\gamma_{2j}\leq 0$ for any $0\leq j\leq n$. 
In other terms, odd steps are up steps and even steps are down steps,
with the convention that a flat step is considered either an up or a down
step according to parity.

We say that a point $V=(x_V,y_V)$ of $\gamma$ is a \emph{valley} of $\gamma$ if $x_V$ is even, it
means that $V$ is preceded by a down step and followed by an up step. Similarly, we say that $P=(x_P,y_P)$ is a \emph{peak} if $x_P$ is odd.

\begin{defi}
	Fix two integers $\ell \geq 1$ and $n\geq 0$ and let $\gamma$
        be an alternating path of length $2n+2$ starting at $(0,0)$ and ending at $(2n+2,\ell)$. Let $\bfu:=(u_1, \dots,u_{\ell})$ be a non-decreasing
        sequence of integers satisfying $1=u_1\leq u_2 \leq\dotsb \leq
        u_{\ell}\leq n+1$.
For $j=1, \dotsc, n$ we define $m_j$ as the largest integer $k$ for which
$u_{k}\leq j$, we also set $m_0:=0$. We define the \emph{height} of a valley $V=(2j, y_{2j})$ by
	\[\height(V):=y_{2j}-m_j.\]
	We say that  $\bfgamma := (\gamma,\bfu)$ is a
        \textbf{decorated alternating path} if all valleys of $\gamma$ have non-negative heights. The integer $\ell$ will be called the \emph{degree} of the path.
      \end{defi}

      Throughout the rest of the paper, we will think of these paths as alternating paths whose peaks are decorated by particles as follows: the integers
      $(u_j)_{1 \leq j \leq \ell}$ indicate the positions of $\ell$ particles labelled $1,2,\dots,
\ell$, such that the $j$-th particle sits on the $u_j$-th peak (the peak of $x$-coordinate $2u_j-1$). The integer $m_j$ is then precisely the number of
particles to the left of the valley at $x$-coordinate $2j$. Note that the number of particles is always equal to the $y$-coordinate of the end point of the path. As a consequence, the first and the last valleys of the path (which correspond to the origin and end points) have always height 0.
\begin{figure}[ht]
	\centering
	\includegraphics[width = 0.5\linewidth]{Path4.pdf}
	\caption{A decorated alternating path of length 8 and degree 4. The five valleys are represented in red.}
	\label{fig:alternating_path}
      \end{figure}

      \begin{example}
         We give in \Cref{fig:alternating_path} an example of a decorated alternating path
        $\bfgamma := (\gamma,\bfu)$ of length $8$ and of degree 4. The 4 particles are
        represented by the turquoise points above the peaks. The particle positions are given by
        $\bfu = (u_1,u_2,u_3,u_4)=(1,2,2,4)$. The heights of the five valleys from left to right are equal to $0,1,0,1,0$.
      \end{example}

      We are almost ready to define the path operators, the main object of our second main theorem. Roughly, these operators are  weighted generating functions of decorated
      alternating paths, where the weight associated with each path is given by the following definition.

\begin{defi}\label{def:decorated_paths}
	
	Let $\bfgamma := (\gamma,\bfu)$ be a decorated alternating
        path. We then associate to $\bfgamma$ the \textit{valley weight}
\begin{equation}\label{eq:valley_weight_PMAP}
\vweight(\bfgamma)=\prod_{V\in\Valley(\bfgamma)}(qt)^{\height(V)},  
	\end{equation}
	the product being taken over the valleys of $\bfgamma$. We associate to $\gamma$ the operator $\mcO(\gamma)$ defined as follows: if $\gamma=(\gamma_1,\dots,\gamma_{2n+2})$ then 
	\begin{equation}\label{eq:decorated_path}
\mcO(\bfgamma):=\vweight(\bfgamma)\mcO(\gamma_1)\dots \mcO(\gamma_{2n+2}), 
	\end{equation}
where for any integer $m$ we define
    \begin{equation}\label{eq:def_step_operator}
		\mcO(m):=\left\{
		\begin{array}{ll}
			(-1)^me_{m}[X]=h_{m}[-X]   & \text{ if } m>0  \\
			h^\perp_{-m}[MX]   & \text{ if } m<0\\
			1 &\text{ if } m=0.
		\end{array}
		\right.  
	\end{equation}
      \end{defi}

Note that there are two ways of describing the positions of the particles in a decorated path: either by the sequence of distances between them, or by the sequence counting the number of particles on every peak. We introduce then the following definitions:
\begin{itemize}
\item
for any
vector $\alpha \in \mathbb{Z}_{\geq 0}^\ell$ with $|\alpha| \coloneqq
\sum_i \alpha_i = n$, we define $\bQ_{\alpha}$
as the set of all decorated paths $\bfgamma=(\gamma,\bfu)$  of length $2n+2$ and degree $\ell$ such that $\bfu=(u_1,\dotsc,u_{\ell})$ satisfies $\alpha_i=u_{i+1}-u_{i}$, for any $1\leq i\leq \ell-1$ and $\alpha_\ell=n+1-u_{\ell}$. In other terms, $2\alpha_i$ is the distance between the $i$-th and the $i+1$-th particles, and $2\alpha_{\ell}$ is the distance between the last particle and the last peak (the distance here is defined as the difference between the $x$-coordinates).
\item
for any
vector $\alpha \in \mathbb{Z}_{> 0}\times\mathbb{Z}_{\geq 0}^{n}$, we
define $\bQ^{\alpha}$
as the set of all decorated paths $\bfgamma$ of length $2n+2$ and degree $\ell:=|\alpha|$, with $\alpha_i$ particles sitting on the $i$-th peak.
\end{itemize}

Finally, we define  
\begin{equation}\label{eq:def_path_operators}
\mcQ_{\alpha}:=\sum_{\bfgamma\in\bQ_{\alpha}}\mcO\left(\bfgamma\right),\qquad
\mcQ^{\alpha}:=\sum_{\bfgamma\in\bQ^{\alpha}}\mcO\left(\bfgamma\right).
\end{equation}
\begin{example}
  The operator associated to the path $\bfgamma$ of \Cref{fig:alternating_path} is given by
        \[\mcO(\bfgamma)=(qt)^{1+0+1} \cdot(-1)^{3+2+1+1}e_{3}[X]h^\perp_{1}[MX]e_{2}[X]h^\perp_{1}[MX]e_{1}[X]e_{1}[X]h^\perp_{1}[MX].\]
        Moreover $\bfgamma \in
\bQ_{(1,0,2,0)}=\bQ^{(1,2,0,1)}$.
      \end{example}

We are now ready to state the seond main result of this paper.
      \begin{thm}\label{thm:A_Pk}
 If $F(\hbar) := \sum_{i \geq 0}a_i \hbar^i\in\{G_1,G_2\}$, then
\begin{equation}\label{eq:A_Pk}
  \A_F^{(\ell)}=\sum_{\alpha \in \mathbb{Z}_{\geq 0}^\ell}a_{\alpha}\mcQ_{\alpha},\end{equation}
where $a_{\alpha} := \prod_{i=1}^\ell a_{\alpha_i}$.
\end{thm}


\subsection{Explicit formulas for the functions \texorpdfstring{$\mcQ^{\bfalpha}\cdot 1$}{}}
As we will see in \Cref{eq:tau_G_formula}, the $(q,t)$-tau function admits a simple expansion on a basis of symmetric functions built upon consecutive actions of the operators $\A_G^{(\ell)}$. From \Cref{thm:A_Pk}, these functions are in turn linear combinations of symmetric functions of the form
\begin{equation}\label{eq:Q^alpha.1}
  \mcQ^{\bfalpha}\cdot1:=\mcQ^{\alpha^{(1)}}\cdot
\mcQ^{\alpha^{(2)}}\dotsm\mcQ^{\alpha^{(r)}}\cdot 1  
\end{equation}
 where
$\bfalpha=\left(\alpha^{(1)},\alpha^{(2)},\dots,\alpha^{(r)}\right)=(\alpha_1,\dots,\alpha_\ell)$
is a list of $r$ vectors of respective lengths $\ell_1,\dotsc,\ell_r$ of total size $n$. 
In our last main theorem, we give explicit formulas for these functions, by extracting the coefficients of some rational function with respect to a second alphabet $Z=z_1 + \dotsb+z_\ell$. We start by introducing some notation.

Let $\mcNI(\bfalpha)$ be the set of \textit{non-consecutive indices}
of $\bfalpha$ defined by: 
\begin{multline}
  \mcNI(\bfalpha):=
  \left\{(i,j)|1\leq i<j\leq \ell, \text{ such that }i<j-1 \right.\\
  \left.\text{ or } (i=j-1 \text{ and }i\leq L_k<j \text{ for some }1\leq k\leq r)\right\},  
\end{multline}
  where we denote the partial sums $L_k=\sum_{1\leq i\leq k}\ell_i$. Note that $\mcNI(\bfalpha)$ depends only on the sequence $(\ell_1,\ell_2,\dots,\ell_r)$ and not $\alpha$ itself.
\begin{example}
    If $\bfalpha=((2,0,0),(3))$, then $\mcNI(\bfalpha)=\left\{\right(1,3),(1,4),(2,4),(3,4)\}.$
\end{example}

\begin{thm}\label{thm:explicit_formula}
    Fix $\bfalpha=(\alpha^{(1)},\alpha^{(2)},\dots,\alpha^{(r)})=(\alpha_1,\dots,\alpha_\ell)$ of total size $n$. 
    For any partition $\lambda$ of size $n$ and for the alphabet $Z =
    z_1+\dotsm+z_\ell$ we have
    \begin{align*}
      (-1)^n[m_\lambda[X]] &\mcQ^{\bfalpha}\cdot 1
        =\left[z^{\bfalpha}\right]
        e_{\lambda}[Z]\prod_{1\leq i<j\leq \ell}
        \frac{1-z_j/z_i}{(1-qz_j/z_i)(1-tz_j/z_i)}\prod_{(i,j)\in\mcNI(\bfalpha)}
           (1-qtz_j/z_i),  \\
(-1)^n [s_\lambda[X]] &\mcQ^{\bfalpha}\cdot 1=\left[z^{\bfalpha}\right]
        s_{\lambda'}[Z]\prod_{1\leq i<j\leq \ell}
        \frac{1-z_j/z_i}{(1-qz_j/z_i)(1-tz_j/z_i)}\prod_{(i,j)\in\mcNI(\bfalpha)}
                                                       (1-qtz_j/z_i),
           \\
            (-1)^n[e_\lambda[X]] &\mcQ^{\bfalpha}\cdot 1
                                   \\
      =\frac{1}{|\Aut(\lambda)|}&\sum_{\sigma\in\mathfrak
        S_{\ell}}\left[z^{\bfalpha-\sigma(\lambda)}\right]\prod_{1\leq i<j\leq \ell}
        \frac{1-z_j/z_i}{(1-qz_j/z_i)(1-tz_j/z_i)}\prod_{(i,j)\in\mcNI(\bfalpha)}
        (1-qtz_j/z_i),  
    \end{align*}
    where $\lambda'$ is the transpose of $\lambda$, $\sigma(\lambda_1,\dots,\lambda_\ell) = (\lambda_{\sigma(1)},\dots, \lambda_{\sigma(\ell)})$, and 
    the rational functions in the RHS should be expanded as formal series in $z_j/z_i$ for $j>i$.
  \end{thm}

  
Interestingly, the functions of \Cref{eq:Q^alpha.1} turned out to be related to some other important problems in the field, namely the Shuffle and the Delta conjectures.

\subsection{Connection to the Delta conjecture}
For $f\in\Lambda$, the operators $\Delta'_{f}$ are defined by the equations
\begin{equation}\label{eq:Delta_operator}
	\Delta'_f \tH_\lambda[X] \coloneqq f\Bigg[ \sum_{(i,j)\in\lambda} q^{j-1} t^{i-1}-1\Bigg] \tH_\lambda[X].
\end{equation}
The Delta conjecture, formulated in \cite{HaglundRemmelWilson2018} and proved by D'Adderio and Mellit in \cite{DAdderioMellit2022} states that $\Delta'_{e_k}\cdot e_n$ is positive in the monomial basis, and the coefficients count Dyck paths with weights given by the \textit{area} and the \textit{dinv} statistics. The Shuffle conjecture corresponds to the case $k=n-1$. It turns out that these problems are closely related to our path operators. Indeed, we prove the following
\begin{equation}\label{eq:Delta'}
    \Delta'_{e_k}\cdot
    e_n=(-1)^{n}\sum_{\substack{\ell(\alpha)=k+1\\{|\alpha|=n}}}\mcQ^{\alpha}\cdot 1.
\end{equation}
Actually, we obtain a more general formula for $\Delta_{h_l} \Delta'_{e_k}\cdot e_n$  also in terms of path operators. This last quantity is the subject of the extended delta conjecture. 

As a byproduct of \Cref{thm:explicit_formula}, we obtain explicit formulas for $\Delta_{h_l} \Delta'_{e_k}\cdot e_n$, from which we obtain \cite[Theorem 4.4.1.]{BlasiakHaimanMorsePun2023} of Blasiak 
{\it et al.}  used to prove the extended delta conjecture. While the proof of this result is based in \cite{BlasiakHaimanMorsePun2023} on a connection with the Schiffmann algebra, we are able to derive it here completely combinatorially using path operators. 
We then hope that the formulas of \Cref{thm:explicit_formula} might give a new approach to understand the combinatorics of the delta conjecture and other problems related to it.

\section{Elements of proofs}
\subsection{Notation and preliminaries}\label{sub:notation}

We denote $\YY$ the set of integer partitions. For $i\geq 1$ and $\lambda\in\YY$, let $m_i(\lambda)$ be the number of parts of size $i$ in $\lambda$. We then set 
\[z_\lambda:=\prod_{i\geq1}m_i(\lambda)!i^{m_i(\lambda)}.\]
The \emph{Hall scalar product}  $\langle.,.\rangle$ on $\Lambda$  is defined by $\langle p_\mu,p_\lambda\rangle=z_\lambda\delta_{\mu,\lambda}$, for any $\lambda,\mu\in\YY$,
where $\delta_{\mu,\lambda}$ is the Kronecker delta, and $p_\mu$ are power-sum symmetric functions.
The \emph{star scalar product} $\langle.,.\rangle_*$ is the deformation of the Hall scalar product given by 
$$\langle p_\mu,p_\lambda\rangle_*=z_\mu\delta_{\mu,\lambda}p_\mu[-M]=(-1)^{\ell(\mu)}z_\mu\delta_{\mu,\lambda}\prod_{1\leq i\leq \ell(\mu)}\left(1-q^{\mu_i}\right)\left(1-t^{\mu_i}\right),~ \text{for any $\mu,\lambda\in\YY$.} $$
The modified  Macdonald polynomials are then orthogonal with respect to this product, $\langle \tH_\mu, \tH_\lambda\rangle_* = \delta_{\mu, \lambda} \normH{\lambda}$.


For any operator $\mcO$ on $\Lambda$, we denote the $\mcO^\perp$ (resp. $\mcO^*$) the adjoint of $\mcO$ with respect to the Hall scalar product (resp. the star scalar product). In particular, if $f\in \Lambda$ then $f^\perp$ is the adjoint of the multiplication by $f$.

We define the linear operator $\Pi_G$ on $\Lambda_X$ by its action on modified Macdonald polynomials $\tH_\lambda$ 
\begin{equation}\label{eq:Pieri_Operator}
	\Pi_G \cdot\tH_\lambda[X] \coloneqq \tH_\lambda[X] \prod_{(i,j)\in\lambda} G(q^{j-1} t^{i-1}).
      \end{equation}
One may notice that the $G$-weighted tau function is obtained by applying this operator on the Macdonald Cauchy kernel:
\begin{equation}\label{eq:Cauchy_kernel}
  \tau_G(z,X,Y)=\Pi_G\cdot\sum_{\lambda\in\mathbb
  Y}z^{|\lambda|}\frac{\tH_\lambda[X]\tH_\lambda[Y]}{\normH{\lambda}}.  
\end{equation}


We think of the adjoint $\ad_A(\cdot)=[A,\cdot]$ as a linear operator on the space of operators on $\Lambda$. In particular, $\ad^i_A(B)$ corresponds to iterating the commutation by $A$ $i$ times, and if $F(\hbar)=\sum_{i\geq 0} a_i\hbar^i$, then 
$F(\ad_A):=a_0+a_1 \ad_A+a_2 \ad^2_A+\dotsb$.
  

\subsection{Proof of the equations of \texorpdfstring{\ref{thm:dif_eq}}{}: Pieri rule and the operator~\texorpdfstring{$D_0$}{}}
We will derive the equations and prove uniqueness separately. In order to prove the equations, we need to find how the operators $\A^{(\ell)}_F$ act on modified Macdonald polynomials, starting with $\ad^i_{D_0/M}(-e_1[X])$. The action of $e_1[X]$ (by multiplication) is given by the following Pieri rule, \cite[Chapter VI, Equation 6.7]{Macdonald1995} (see also {\cite[I.11, I.12]{GarsiaTesler1996}}).
\begin{thm}[Pieri Rule, \cite{Macdonald1995}]
	\label{theo:Pieri}
	$$-e_1[X]\tH_\mu=\sum_{\mu\nearrow\lambda}d^{\mu,\lambda}\tH_\lambda,$$
	where the sum is taken over partitions $\lambda$ obtained from $\mu$  by adding one cell, and $d^{\mu,\lambda}\in \QQ(q,t)$.
\end{thm}
Remarkably, no explicit expression of $d^{\mu,\lambda}$ is required in our work, only the fact that they vanish if $\lambda$ is not obtained from $\mu$ by the addition of a box. Moreover, $D_0$ acts diagonally on Macdonald polynomials.
\begin{thm}[{\cite[Theorem 1.2] {GarsiaHaimanTesler1999}}]\label{thm:D_0}
We have
	\[D_0\cdot \tH_{\lambda}=\biggl(1-M\sum_{(i,j)\in\lambda} q^{j-1} t^{i-1}\biggr)\tH_{\lambda},\quad \text{for any $\lambda\in \YY$.}\]
\end{thm}

\begin{lem}\label{lem:DeltaG_A+}
	For any $\ell\geq 1$, we have	\[\Pi_G\cdot \A^{(\ell)}_{G_2}\cdot \Pi_G^{-1}=\A^{(\ell)}_{G_1}.\] 
\end{lem}
\begin{proof}
	From \Cref{theo:Pieri}, \Cref{thm:D_0} and the definition $\A_F = F(\ad_{-D_0/M})\cdot (-e_1)$ we find
	\begin{equation} \label{eq:A_PonMacdonald}
		\A_F \tH_\mu = \sum_{\mu\nearrow\lambda} d^{\mu,\lambda} F(c_{q,t}(\lambda/\mu)) \tH_\lambda,
              \end{equation}
              where $c_{q,t}(\lambda/\mu) := q^{j-1}t^{i-1}$, with $(i,j)$ being the only cell of $\lambda/\mu$.
	With the definition of $\Pi_G$, this gives directly the lemma for $\ell=1$. Then for $\ell\geq 2$, one has $\A^{(\ell)}_F = \frac{1}{M}[\A_F, \A_F^{(\ell-1)}]$, so the lemma is obtained by induction.
\end{proof}

\noindent 
We obtain \Cref{thm:dif_eq} by combining \Cref{lem:DeltaG_A+} and the following lemma, which is a consequence of the orthogonality of Macdonald polynomials. 
\begin{lem}\label{lem:diff_eq}
	Let $\mcO_1$ and $\mcO_2$ be two operators on $\Lambda$ which have the same homogeneous degree $\ell\geq0$. 
Then
	\begin{equation}\label{eq:diff_DeltaG}
		z^\ell\mcO_2(X)\cdot \tau_G(z,X,Y)=\mcO_1^{*}(Y)\cdot \tau_G (z,X,Y)\ \Longleftrightarrow\ \Pi_G\cdot \mcO_1\cdot \Pi_G^{-1}=\mcO_2.  
	\end{equation}
\end{lem}

\subsection{Proof of uniqueness in \texorpdfstring{\Cref{thm:dif_eq}}{}: the basis \texorpdfstring{$\aaa_{F,\lambda}$}{}}

For $F \in \{G_1,G_2\}$, and any partition $\lambda$, we define the  symmetric function of degree $\lambda$ 
$$\aaa_{F,\lambda} \coloneqq \A_F^{(\lambda_1)}\cdot \A_F^{(\lambda_2)}\dotsm \A_F^{(\lambda_{\ell(\lambda)})}\cdot 1.$$

\begin{prop}\label{prop:basis_a_lambda}
	The family  $(\aaa_{F,\lambda})_{\lambda\in \YY}$ is a basis of $\Lambda$.
\end{prop}
\textit{Idea of the proof.}
It is enough to prove that $\aaa_{F,\lambda}$ is a basis when we take the specialization $q=1$. In this case, the derivative operators ($\mcO(m)$ in \Cref{eq:def_step_operator}	for $m$ negative) become trivial, and all the path operators are multiplicative. The proposition follows then by a triangularity argument.\medskip

Let us now explain how this proposition implies uniqueness in \Cref{thm:dif_eq}.  For any partition $\lambda$, the differential equations \Cref{eq:diff_eq} allow us to write
\begin{multline*}
	z^{|\lambda|}\A_{G_1}^{(\lambda_1)}(X)\cdot \A_{G_1}^{(\lambda_2)}(X)\dotsm \A_{G_1}^{(\lambda_{\ell(\lambda)})}(X)\cdot \tau_G(z,X,Y)\\
	=\Bigl(\A_{G_2}^{(\lambda_{\ell(\lambda)})}(Y)\Bigr)^* \dotsm  \Bigl(\A_{G_2}^{(\lambda_2)}(Y)\Bigr)^* \cdot\Bigl(\A_{G_2}^{(\lambda_1)}(Y)\Bigr)^*\cdot\tau_G(z,X,Y).  
\end{multline*}
We now extract the coefficient of $z^{|\lambda|}p_\emptyset[Y] =
z^{|\lambda|}$ (the constant term in $Y$). On the left-hand side we simply get $\aaa_{G_1,\lambda}(X)$. On the right-hand side, we obtain 
\begin{align*}
	\left[z^{|\lambda|} \right]&\Bigl(\A_{G_2}^{(\lambda_{\ell(\lambda)})}(Y)\Bigr)^* \dotsm  \Bigl(\A_{G_2}^{(\lambda_2)}(Y)\Bigr)^* \cdot\Bigl(\A_{G_2}^{(\lambda_1)}(Y)\Bigr)^*\cdot\tau_G(z,X,Y)\\
	&=\left\langle p_\emptyset[Y], \left[z^{|\lambda|}\right]\Bigl(\A_{G_2}^{(\lambda_{\ell(\lambda)})}(Y)\Bigr)^* \dotsm  \Bigl(\A_{G_2}^{(\lambda_2)}(Y)\Bigr)^* \cdot\Bigl(\A_{G_2}^{(\lambda_1)}(Y)\Bigr)^*\cdot\tau_G(z,X,Y)\right\rangle_*\\
	&=\left\langle \A_{G_2}^{(\lambda_1)}(Y)\cdot \A_{G_2}^{(\lambda_2)}(Y)\cdots \A_{G_2}^{(\lambda_{\ell(\lambda)})}(Y)\cdot 1, \left[z^{|\lambda|}\right]\tau_G(z,X,Y)\right\rangle_*\\
	&=\left\langle \aaa_{G_2,\lambda}(Y), \left[z^{|\lambda|}\right]\tau_G(z,X,Y)\right\rangle_*,
\end{align*}
where the scalar product is taken with respect to the alphabet $Y$. Let $\bbb_{G_2,\lambda}$ denote the dual basis of $\aaa_{G_2,\lambda}$ with respect to the star scalar product:
$\left\langle\aaa_{G_2,\lambda}, \bbb_{G_2,\mu}\right\rangle_*=\delta_{\lambda,\mu},$ for any $\lambda$ and $\mu$.
Then the differential equations of \Cref{thm:dif_eq} imply that for any $\lambda$
	\[\aaa_{G_1,\lambda}(X)=\left[z^{|\lambda|}\bbb_{G_2,\lambda}(Y)\right]\tau_G(z,X,Y),\]
or equivalently 
\begin{equation}\label{eq:tau_G_formula}
\tau_G(z,X,Y)=\sum_{\lambda\in\YY}z^{|\lambda|}\aaa_{G_1,\lambda}(X)\bbb_{G_2,\lambda}(Y).
\end{equation}
In particular, the differential equations of \Cref{thm:dif_eq} characterize the function $\tau_G$. Note that when $G_1=G_2$, 
the last equation corresponds to the Macdonald--Cauchy identity. 



\subsection{Proof of \texorpdfstring{\Cref{thm:A_Pk}}{} and \texorpdfstring{\Cref{thm:explicit_formula}}{}}

The proof of \Cref{thm:A_Pk} is based on the two families of commutation relations satisfied by path operators.

\begin{prop}[The first commutation relation]\label{prop:comm1}
    For any $n\geq 0$, 
    $$\mcQ_{n}=\ad_{-D_0/M}(\mcQ_{n-1}),$$
    where $\mcQ_n$ is the operator defined in \Cref{eq:def_path_operators} if $n\geq 0$, and $\mcQ_{-1}:=-e_1[X]$.  
\end{prop}
Note that these commutation relations concern one-particle paths. They allow one to obtain the  length $n$ operators from operators associated to paths of lengths $n-1$ by applying the adjoint of $D_0$. 
The proof is based on the following identities, which are a consequence of commutation relations of vertex operators (see \textit{e.g} \cite[Theorem 1.1 and Proposition 1.2]{GarsiaHaimanTesler1999})
\[\begin{aligned}
	[D_0, \mcO(r)] &= -M \sum_{k\geq 0}\text{ }\sum_{1\leq i\leq \min(k,r)}(1+qt+\dots (qt)^{i-1}) \mcO(k) \mcO(k-i)\mcO(r-i),\\
	[D_0, \mcO(-r)] &= M \sum_{k\geq 0}\text{ }\sum_{1\leq i\leq \min(k,r)}(1+qt+\dots (qt)^{i-1})\mcO(-(r-i)) \mcO(k-i) \mcO(-k).
\end{aligned}\]

The second commutation relation concerns paths decorated by an arbitrary number of particles. We give here a symmetrized version, which is enough to prove \Cref{thm:A_Pk}. 

\begin{prop}[The second commutation relation]\label{prop:comm2}
    Let $\alpha \in \mathbb{Z}_{\geq 0}^\ell$ be a sequence of non-negative integers of length $\ell\geq2$.  
    We then have
	\[
          \sum_{\sigma\in\mathfrak{S}_{\ell}}\mcQ_{\sigma(\alpha)}=\frac{1}{M}\sum_{\sigma\in\mathfrak{S}_{\ell}}\left[\mcQ_{\alpha_{\sigma(\ell)}-1},\mcQ_{\sigma(\alpha_{1},\dots,\alpha_{\ell-1})}\right].\]
\end{prop}
\noindent The proof of this proposition uses the first commutation relation and manipulations on commutators. Let us now use these propositions to prove \Cref{thm:A_Pk} by induction on~$\ell$.


\medskip

From \Cref{prop:comm1}, we get that $\ad_{-D_0/M}^{n+1} \left(-e_1\right) = \mcQ_{n}.$ This implies that for $F(\hbar) = \sum_{n\geq 0} a_n \hbar^n$, we have
$\A_F = \sum_{n\geq 0} a_n \mcQ_{n-1}$.
We deduce that 
$$\A_F^{(1)}= \sum_{n\geq 0} a_n \mcQ_{n},$$ 
giving \Cref{thm:A_Pk} for $\ell=1$.
 
We now assume that \Cref{eq:A_Pk} holds for some $\ell\geq 1$. We then have
\[\A_F^{(\ell+1)} = \frac{1}{M}\left[\A_{F},\A_F^{(\ell)}\right] = \frac{1}{M}\sum_{j\geq 0}\sum_{\alpha \in \ZZ_{\geq 0}^\ell} u_{j} u_\alpha\left[\mcQ_{j-1},\mcQ_{\alpha}\right].\]
By applying \Cref{prop:comm2} it comes
\begin{align*}
  \A_F^{(\ell+1)} 
  =& \frac{1}{M}\sum_{0\leq \nu_1\leq\dots\leq \nu_{\ell+1}}
\frac{1}{|\Aut(\nu)|} u_\nu \sum_{\sigma\in\mathfrak	S_{\ell+1}}\left[\mcQ_{\nu_{\sigma(\ell+1)-1}},\mcQ_{\sigma (\nu_1,\dots,\nu_{\ell})}\right]\\
=&\sum_{0\leq \nu_1\leq\dots\leq \nu_{\ell+1}}\frac{1}{|\Aut(\nu)|}u_\nu\sum_{\sigma\in\mathfrak
	S_{\ell+1}}\mcQ_{\sigma (\nu)}=\sum_{\alpha \in
	\mathbb{Z}_{\geq 0}^{\ell+1}}u_{\alpha}\mcQ_{\alpha},
\end{align*}
where $\Aut(\nu)<\mathfrak{S}_{\ell+1}$ is the stabilizer of the partition $\nu$.
This finishes the proof of \Cref{thm:A_Pk}.

\begin{proof}[Idea of the proof of \Cref{thm:explicit_formula}] We start by taking the normal order form in \Cref{thm:A_Pk}, \textit{i.e} we move the derivative part to the right and the multiplicative part to the left. By inspecting the combinatorics of this formula, we prove that it can be written as a sum over alternating paths for which all valleys have minimal height. This gives the formulas of \Cref{thm:explicit_formula}.
\end{proof}
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\end{document}
