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\title{Wreath Macdonald operators}

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\author[D. Orr, M. Shimozono, and J.J. Wen]{Daniel Orr\thanks{\href{dorr@math.vt.edu}{dorr@math.vt.edu}. D.O. was supported by the Simons Foundation (Collaboration Grant for Mathematicians, 638577) and the Max Planck Institute for Mathematics (MPIM Bonn). }\addressmark{1,2}, 
Mark Shimozono\thanks{\href{mailto:mshimo@math.vt.edu}{mshimo@math.vt.edu}.}\addressmark{2},
\and Joshua Jeishing Wen\thanks{\href{mailto:j.wen@northeastern.edu}{j.wen@northeastern.edu}. J.J.W. was supported by NSF-RTG grant ``Algebraic Geometry and Representation Theory at Northeastern University'' (DMS-1645877).}\addressmark{3} }

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\address{
\addressmark{1}Max Planck Institut f\"{u}r Mathematik, Vivatsgasse 7,
53111 Bonn,
Germany\\
\addressmark{2}Department of Mathematics, Virginia Tech, 460 McBryde Hall, Virginia Tech,
	225 Stanger St.,
	Blacksburg, VA 24061 USA \\ 
\addressmark{3}Department of Mathematics, Northeastern University, 463 Lake Hall, 43 Leon St, Boston, MA 02115 USA}

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\abstract{We construct a novel family of difference-permutation operators and prove that they are diagonalized by the wreath Macdonald $P$-polynomials. 
%The eigenvalues are written in terms of elementary symmetric polynomials. 
Our operators arise from 
%integral formulas for 
the action of the horizontal Heisenberg subalgebra in the vertex representation of the quantum toroidal algebra.}

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\keywords{Macdonald polynomials, 
%difference operators, 
quantum toroidal algebras, shuffle algebras}

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\begin{document}

\maketitle
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\section{Introduction}
Let $X_{N}=\left\{ x_{1},\ldots, x_N \right\}$ be a set of variables and let $\Y_N$ be the set of partitions with at most $N$ parts.
The \textit{Macdonald polynomials} $\left\{ P_\lambda[X_N;q,t]\mid \la\in\Y_N \right\}$ are a basis of the ring of symmetric polynomials $\Q(q,t)[X_N]^{\Sym_N}$ that can be characterized as eigenfunctions of a commuting family of difference operators, the \textit{Macdonald operators}: for $1\le n\le N$,
\begin{align}
\label{OldMacOp}
D_n(X_N ;q,t)&:=t^{\frac{n(n-1)}{2}}\sum_{\substack{J\subset\{1,\ldots, N\}\\|J|=n}}\left(\prod_{\substack{i\in J \\j\not\in J}}\frac{tx_i-x_j}{x_i-x_j}\right)\prod_{i\in J} T_{q,x_i}\\
\label{OldEigen}
D_n(X_N ;q,t) P_\lambda[X_N;q,t] &= e_n(q^{\la_1} t^{N-1},q^{\la_2} t^{N-2},\dotsc,q^{\la_N}) P_\lambda[X_N ;q,t]
\end{align}
Here $T_{q,x_i}$ is the $q$-shift operator
$
T_{q,x_i} x_j=q^{\delta_{i,j}}x_j
$
and $e_n$ is the $n$th elementary symmetric polynomial.
% and $u_\la:\Q(q,t)[X_N]\to \Q(q,t)$ is the $\Q(q,t)$-algebra evaluation homomorphism sending 
%$x_i$ to $q^{\la_i}t^{N-i}$ for $1\le i \le N$.
%The Macdonald operators are themselves distinguished as Hamiltonians of the quantum trigonometric \textit{Ruijsenaars-Schneider} integrable system.

This paper is concerned with \textit{wreath} Macdonald polynomials, a generalization of the Macdonald polynomials defined by Haiman \cite{H2}.
Let $r$ be a positive integer and let $I=\Z/r\Z$. Fix an $r$-core partition $\gamma$ and a vector $\Nd=(N_0,N_1,\dotsc,N_{r-1})\in\Z_{\ge0}^I$ that is compatible with $\gamma$ in the sense given by \eqref{Compatibility} below. Letting $N= |\Nd|:=\sum_{i\in I} N_i$, we partition the variables $x_1,\ldots, x_N$ into $r$ subsets:
\begin{align}\label{E:variables}
X_{N_\bullet}:=\bigsqcup_{i=0}^{r-1}\left\{x^{(i)}_{l}\mid 1 \le l \le N_i \right\}=\left\{ x_1,\ldots,x_N \right\}.
\end{align}
We call the index $i$ the \textit{color} of $x^{(i)}_{l}$. 
The product of symmetric groups $\Sym_{N_\bullet}:=\prod_{i\in I}\Sym_{N_i}$ acts on the 
polynomial ring $\Q(q,t)\left[ X_{N_\bullet} \right]$ whereby $\Sym_{N_i}$ only permutes the variables of color $i$.
Let $\Y_{N,\gamma}$ be the set of partitions with $r$-core $\gamma$ and at most $N$ parts.
The wreath Macdonald polynomials $\{P_\la[X_{\Nd};q,t]\mid \la\in \Y_{N,\gamma}\}$
are a basis of the space $\Q(q,t)\left[ X_{N_\bullet} \right]^{\Sym_{N_\bullet}}$ of color-symmetric polynomials.
The original Macdonald polynomials $P_\la(X_N;q,t)$ \cite[VI.4]{Mac} are the case $r=1$.

Haiman's definition characterizes $P_\lambda[X_{N_\bullet};q,t]$ using a pair of triangularity conditions.
In contrast with the usual Macdonald theory, we \textit{a priori} do not have an analogous characterization as the joint eigenfunction of an explicit family of difference operators.
The present work remedies this situation.
%: we produce a novel family of difference-permutation operators that are diagonalized by the wreath Macdonald polynomials and whose eigenvalues are written in terms of the elementary symmetric polynomials.
%In addition to the degree $n$, they also carry a color parameter $p\in I$. 
For each $p\in I$ and positive integer $n$, define the operators $D_{p,n}$ on $\Q(q,t)[X_\Nd]$ by
\begin{equation}
\begin{aligned}
&D_{p,n}(X_{N_\bullet};q,t):=
\frac{(-1)^{\frac{n(n-1)}{2}}}{\prod_{k=1}^n(1-q^{k}t^{-k})}\\
&\times\sum_{\mathbf{\Ju}\in Sh_p^{[n]}(X_{N_\bullet})}
\overset{\curvearrowright}{\prod_{a=1}^n}
\left\{(1-qt^{-1})^{|\Ju_a|}\left(\frac{x^{(r-1)}_{\Ju_a^\triangledown}}{x^{(p)}_{\Ju_a}}\right)
\frac{\displaystyle
\prod_{\substack{l=1\\x^{(p)}_{l}\not\in|\mathbf{\Ju}|_{\ge a}}}^{N_p}\left( tx^{(p-1)}_{\Ju_a^\triangledown}-x^{(p)}_{l} \right)}
{\displaystyle
\prod_{\substack{l=1\\x^{(p)}_{l}\not\in|\mathbf{\Ju}|_{\le a}}}^{N_p}\left( x^{(p)}_{\Ju_a}-x^{(p)}_{l} \right)}\right.\\
&\times
\left.
\left( \prod_{\substack{i\in I\backslash\{p\}}}\, 
\prod_{\substack{l=1\\x^{(i)}_{l}\not=x^{(i)}_{\Ju_a^\triangledown}}}^{N_i}\frac{\left(tx^{(i-1)}_{\Ju_a^\triangledown}-x^{(i)}_{l}\right)}
{\left( x^{(i)}_{\Ju_a^\triangledown}-x^{(i)}_{l} \right)}
\right)
\left( \prod_{i\in J_a\backslash \left\{ p \right\}} 
\frac{q^{-1}t T_{\Ju_a}x^{(i)}_{\Ju_a}}
{\left( x^{(i)}_{\Ju_a}-T_{\Ju_a}x^{(i)}_{\Ju_a} \right)}
\right) T_{\Ju_a}\right\}.
\end{aligned}
\label{DpnDef}
\end{equation}
The notation used in this formula is defined in \cref{S:operators}. Our main result is the following,

\begin{thm}
Let $\Nd$ be compatible with $\gamma$ and $\la\in \Y_{N,\gamma}$. Then the number of $1\le i\le N$ with 
$i-\la_i-1\equiv p \,\text{mod}\,r$ is equal to $N_p$ and $P_\lambda[X_{N_\bullet};q,t]$ satisfies the eigenfunction equation
\begin{equation}
D_{p,n}(X_{N_\bullet};q,t)P_\lambda[X_{N_\bullet};q,t]
=e_n\left[\sum_{\substack{i=1 \\ i-\la_i-1\equiv p\,\text{mod}\, r}}^N q^{\la_i}t^{-(N-i)}\right] P_\lambda[X_{N_\bullet};q,t].
\label{WreathEigen}
\end{equation}
\end{thm}
The eigenvalue uses plethystic notation: the $N_p$ variables of the elementary symmetric polynomial are replaced by the summands of the form $q^at^b$.
%In earlier work \cite{OS}, the first two authors constructed  first order operators $D_{p,1}^*$ which are the dual version of the operators $D_{p,1}$ and obtained their eigenfunction equation.
%We write down the degree one operators explicitly:
%\begin{align*}
%	%D_{p,1}^*(X_{N_\bullet};q,t^{-1})
%	%&:=
%	%\frac{1}{1-q^{-1}t^{-1}}
%	%\sum_{\Ju\in Sh_p(X_{N_\bullet})}
%	%\left( 1-q^{-1}t^{-1} \right)^{|J|}
%	%\frac{x^{(p+1)}_{\Ju^\triangle}}{x_{\Ju^\triangle}^{(0)}}\\
%	%&\times
%	%\left(\prod_{i\in I}
%	%\prod_{\substack{l=1\\x_{l}^{(i)}\not= x^{(i)}_{\Ju^\triangle}}}^{N_i}
%	%\frac{\left(t x^{(i+1)}_{\Ju^\triangle}-x^{(i)}_{l} \right)}
%	%{\left( x^{(i)}_{\Ju^\triangle}-x^{(i)}_{l} \right)}\right)
%	%\left(\prod_{i\in J\backslash\left\{ p \right\}}
%	%\frac{qtT_\Ju^{-1}(x^{(i)}_{\Ju})}{\left( x^{(i)}_{\Ju}-T_\Ju^{-1}(x^{(i)}_{\Ju}) \right)}\right)
%	%T_\Ju^{-1}\\
%	D_{p,1}(X_{N_\bullet}; q, t^{-1})
%	&:=
%	\frac{1}{1-qt}
%	\sum_{\Ju\in Sh_{p}(X_{N_\bullet})}\left( 1-qt \right)^{|J|}
%	\frac{x^{(r-1)}_{\Ju^\triangledown}}{x^{(p)}_{\Ju}}\\
%	&\times
%	\left(\prod_{i\in I}
%	\prod_{\substack{l=1\\x^{(i)}_{l}\not= x^{(i)}_{\Ju^\triangledown}}}^{N_i}
%	\frac{\left( t^{-1}x^{(i-1)}_{\Ju^\triangledown}-x^{(i)}_{l} \right)}
%	{\left( x^{(i)}_{\Ju^\triangledown}-x^{(i)}_{l} \right)}\right)
%	\left(\prod_{i\in J\backslash\left\{ p \right\}}
%	\frac{q^{-1}t^{-1}T_\Ju(x^{(i)}_{\Ju})}{\left( x^{(i)}_{\Ju}-T_\Ju (x^{(i)}_{\Ju}) \right)}\right)
%	T_\Ju.
%\end{align*}
%Observe that when $r=1$, $D_{0,1}(x_{0,\bullet};q,t)$ is the first Macdonald operator.
%
%The eigenvalues of $\left\{ D_{p,n}(X_{N_\bullet};q,t) \right\}_{p\in I}$ on $\left\{ P_\lambda[X_{N_\bullet};q,t] \right\}$ are nondegenerate.
%Therefore, we have
%\begin{cor}\label{CharCor}
%	Let $\gamma$ be an $r$-core, $\Nd$ compatible with $\gamma$, and $\la\in\Y_{N,\gamma}$,
%	the one-dimensional subspace $\Q(q,t) P_\lambda[X_{N_\bullet};q,t]$ of $\La^I$ is characterized by the eigenfunction equations \eqref{WreathEigen} for $n=1$ ranging over all $p\in I$.
%\end{cor}
Our operators \eqref{DpnDef} are much more complicated than the original Macdonald operators \eqref{OldMacOp}.
In the case $r=1$, we do indeed obtain \eqref{OldMacOp} after some simplification.	
%In contrast with the $n=1$ case, it is less obvious that the operators \eqref{DpnDef}  yield the higher order Macdonald operators with $t$ inverted when $r=1$.
%	When $r=1$, note that our sum is over \textit{ordered} $n$-tuples of distinct shift operators, whereas the usual formula for the $n$th Macdonald operator is over \textit{unordered} $n$-tuples.
%	Summing over the orderings for a given $n$-tuple, the numerator will contain a factor that is antisymmetric, while the denominator will contain a Vandermonde determinant.
%	The quotient of these two will yield $(-t)^{\pm \frac{n(n-1)}{2}}$ times the $(qt)^{\mp 1}$-generating function of lengths of elements in $\Sym_n$.
%	After consolidating all constants, one is indeed left with the $n$th Macdonald operator.
%
When $r>1$, the $q$-shift operator $T_{q,x_i}$ is replaced with what we call a \textit{cyclic-shift operator} $T_{\Ju_a}$, which cyclically permutes variables of different colors in addition to multiplying by a power of $q$.
%Because of this extra permutation, the cyclic-shift operators do not generally commute.
%Note the composition of operators in \eqref{DpnDef}. We expect the formula to simplify meaningfully after taking to account the noncommutativity of the constituent cyclic-shift operators, commuting them to the right past the multiplication operators.

\subsection{Integral formulas}
Our strategy for deriving the eigenoperators \eqref{DpnDef} and establishing the eigenfunction equation \eqref{WreathEigen} uses work of the third author \cite{Wen}. We study the wreath Macdonald polynomials using the \textit{quantum toroidal algebra} $\ddot{U}=U_{\qqq,\ddd}(\ddot{\mathfrak{sl}}_r)$ and its \textit{vertex representation} $W$.
The aforementioned work proves that infinite-variable wreath Macdonald polynomials can be naturally embedded inside $W$ such that they diagonalize a large commutative subalgebra of $\ddot{U}$ known as the \textit{horizontal Heisenberg subalgebra}.
This alone is insufficient for obtaining explicit formulas. We also need work of Negu\cb{t} \cite{Neg} realizing $\ddot{U}$ as a \textit{shuffle algebra}.
The shuffle algebra is a space of rational functions endowed with an exotic product structure, and it is isomorphic to a part of $\ddot{U}$ via a map that is morally (but not precisely) an integration map.
Writing its action on $W$ and then specializing from infinite to finite variables, we obtain integral formulas.
Finally, to pin down the eigenvalues, we use the twisted isomorphism established by Tsymbaliuk \cite{T} between the vertex representation and the \textit{Fock representation}.
We apply this process to the shuffle realizations of well-chosen elements of the horizontal Heisenberg subalgebra which were found in \cite{Wen}.
%Their action can be expressed as an integral whose explicit evaluation leads to \eqref{WreathEigen}.

%The various operators $D_{p,n}$ commute because they give the action of elements of a commutative subalgebra in the vertex representation.

%We also carry this out for its dual counterpart in \cref{ThmEigenHigher}.

%Using other shuffle elements from \cite{Wen}, we obtain similar integral formulas for wreath analogues of the \textit{Noumi-Sano operators} %\cite{NSa}, although we are only able to evaluate the integral and obtain formulas for the operators in degree $n=1$.
%We note that our approach is similar to \cite{FHHSY} in the $r=1$ case, although our \textit{a priori} knowledge %and endgoals are different.
%In \cite{FHHSY}, the authors use the well-known Macdonald operators to study the action of certain shuffle %elements, whereas we use $r>1$ analogues of their shuffle elements to discover new operators.
%In \cite{T2}, Tsymbaliuk has also produced difference operators out of %$U_{\qqq,\ddd}(\ddot{\mathfrak{sl}}_r)$ through very different means.
%The relation between Tsymbaliuk's operators to wreath Macdonald theory does not seem straightforward but %could be interesting.

\subsection{Towards duality}

In the case $r=1$, the eigenfunction equation (\ref{OldEigen}) is particularly interesting when juxtaposed with the \textit{Pieri rules} \cite{Mac}.
To make this apparent, introduce a continuous extension of the discrete parameters $\lambda=(\lambda_1,\ldots,\lambda_{N})$:
$
s_i:=q^{\lambda_i}t^{N-i},\, S_N:=\{s_1,\ldots, s_N\}.
$
We call the variables $X_N$ the \textit{position variables} and $S_N$ the \textit{spectral variables}.
It is natural to interpret the $q$-shift $T_{q,s_i}P_\lambda[X_N;q,t]$ by adding a box to row $i$ of the partition $\lambda$.
For a certain renormalization $\tilde{P}_\lambda[X_N;q,t]$ of $P_{\lambda}[X_{N};,q,t]$, we can write the Pieri rules as
\begin{equation}
e_n(x_1,\ldots, x_N)\tilde{P}_\lambda[X_N ;q,t]
=t^{\frac{n(n-1)}{2}}\sum_{\substack{I\subset\{1,\ldots, N\}\\|I|=n}}\left(\prod_{\substack{i\in I\\j\not\in I}}\frac{ts_i-s_j}{s_i-s_j}\right)\prod_{i\in I} T_{q,s_i}\tilde{P}_\lambda[X_{N} ;q,t].
\label{Pieri}
\end{equation}
The fact that no shift operator $T_{q,s_i}$ appears more than once enforces the well known support condition of the Pieri rules: the $\tilde{P}_\mu[X_N;q,t]$ that appear on the right hand side of (\ref{Pieri}) are such that $\mu\backslash\lambda$ contains no horizontally adjacent boxes.
On the other hand, we can view the eigenfunction equation (\ref{OldEigen}) as describing multiplication by $e_n(s_1,\ldots, s_N)$.
The similarity between (\ref{OldEigen}) and (\ref{Pieri}) is reflective of a symmetry $X_N\leftrightarrow S_N$.
One manifestation of this symmetry is the well-known \textit{evaluation duality} \cite{Mac}:
\[
\tilde{P}_\lambda(q^{\mu_1}t^{N-1},q^{\mu_2}t^{N-2},\ldots, q^{\mu_N})=\tilde{P}_\mu(q^{\lambda_1}t^{N-1},q^{\lambda_2}t^{N-2},\ldots, q^{\lambda_N}).
\]

For the wreath case $r>1$, the spectral variables should also have color. 
We assign $s^{(i)}_{l}$ to some $b$ such that $b-\lambda_b\equiv i+1$ mod $r$:
$
s^{(i)}_{l}:=q^{\lambda_b}t^{|N_\bullet|-b}.
$
Here, we point out a natural motivation for imposing our compatibility condition between the $r$-core of $\lambda$ and $N_\bullet$---it forces there to also be $N_i$ spectral variables of color $i$. 
The eigenfunction equation (\ref{WreathEigen}) then describes multiplication by $e_n(s^{(p)}_{1},\ldots, s^{(p)}_{N_p})$.
Note that adding a box to a row will not only contribute a $q$-shift but also change the color, and that is precisely what the cyclic-shift operators $T_{\Ju_a}$ do.
Work of the third author \cite{Wen} provides one constraint on the support of the wreath Pieri rules.
Namely, for a box $(a,b)$, if we call the class of $b-a$ mod $r$ its color, then $P_\mu[X_{N_\bullet};q,t]$ appears as a summand of
$
e_n(x_{p,1},\ldots, x_{p,N_p})P_\lambda[X_{N_\bullet};q,t]
$
only if $\mu\backslash\lambda$ consists of $n$ boxes of each color such that no boxes of color $p$ and $p+1$ are horizontally adjacent. 
One can check that the combinations of $T_{\Ju_a}$ appearing in (\ref{DpnDef}) enforce this condition after swapping $x^{(i)}_{l}\leftrightarrow s^{(i)}_{l}$.
Computer calculations done by the second author also confirm a wreath analogue of evaluation duality.
While we are still a long way from establishing a wreath analogue of the $X_N\leftrightarrow S_N$ symmetry, our strange operators seem to go out of their way to say it must be true.
%Generalizing any of the aforementioned perspectives for understanding this symmetry must surely lead to interesting mathematics.

\subsection{Outline}
\cref{S:comb} discusses the core-quotient combinatorics in detail.
\cref{S:sym} defines wreath Macdonald symmetric functions and their finitizations. 
%into  actual color-symmetric polynomials in finitely many variables. 
\cref{S:operators} defines the ingredients of the formula \eqref{DpnDef} of our wreath Macdonald eigenoperators.

%\subsection{Acknowledgements}

%D.~O. gratefully acknowledges support from the Simons Foundation (Collaboration Grant for Mathematicians, 638577) and the Max Planck Institute for Mathematics (MPIM Bonn). 
%J.~J.~W. was supported by NSF-RTG grant ``Algebraic Geometry and Representation Theory at Northeastern University'' (DMS-1645877).

\section{Abaci, cores, quotients, and root lattice combinatorics}\label{S:comb}
Fix a positive integer $r$ and let $I=\Z/r\Z$.
%\textbf{Warning:} we use nonstandard definitions of the $r$-core and $r$-quotient.

\subsection{Partitions and cores}\label{SS:partitions}
Let $\Y$ be the set of all integer partitions. The diagram of a partition $\mu=(\mu_1,\mu_2,\dotsc)\in \Y$ is by definition the French-style diagram $D(\mu)=\{(a,b)\in\left(\Z_{\ge0}\right)^2 : 0 \le a < \mu_{b+1} \}$ which uses $0$-based first quadrant  Cartesian $(x,y)$-coordinates for the elements of $D(\mu)$. The \emph{residue} or \emph{color} of $(a,b)\in\Z^2$ is the element $b-a\in \Z/r\Z$. 
This is the residue mod $r$ of the \textit{negative} of Macdonald's content of a cell \cite[Ex. I.1.3]{Mac}.

%We often identify a partition with its diagram. Thus we write $(a,b)\in \mu$ to mean $(a,b)\in D(\mu)$, and $\mu \subset \la$ to mean $D(\mu)\subset D(\la)$.

%Let $A(\mu),R(\mu)\subset \Z_{\ge0}^2$ be the sets of $\mu$-addable and $\mu$-removable elements of $\Z_{\ge0}^2$; see the following example.

%\begin{ex} For $\mu=(3,2)$, we have:
%\begin{equation*}
%\ytableausetup{aligntableaux=bottom}
%\begin{ytableau} 01&11 \\ 00 & 10 & 20 \end{ytableau}
%	\qquad 	
%\begin{ytableau}
%+\\
%{}&{}&+\\
%{}&{}&{}&+
%\end{ytableau}
%\qquad
%\begin{ytableau}
%{}& - \\ {} &{} & -
%\end{ytableau}
%\end{equation*}
%$D(\mu)=\{(0,0),(1,0),(2,0),(0,1),(1,1)\}$,
%$A(\mu)=\{(0,2),(2,1),(3,0)\}$ and $R(\mu)=\{(1,1),(2,0)\}$.
%\end{ex}

For $(a,b)\in D(\mu)$ the hook length $h_\mu(a,b)$ is the number of elements $(a',b')\in D(\mu)$ 
that are in the same row as $(a,b)$ and weakly to the right (that is, $(a',b')=(a',b)$ with $a' \ge a$) 
or in the same column as $(a,b)$ and weakly above (that is, $(a',b')=(a,b')$ with $b'\ge b$).
A partition $\mu\in\Y$ is an $r$-core if $h_\mu(a,b)\ne r$ for all $(a,b)\in D(\mu)$. We denote by $\cC=\cC_r$ the set of $r$-cores.

%Let $A_i(\mu)\subset A(\mu)$ and $R_i(\mu)\subset R(\mu)$ denote the subsets of \emph{$\mu$-addable} and \emph{$\mu$-removable} boxes having residue $i$.

\subsection{Edge sequences and partitions}

%\begin{ex} An edge sequence $b$ and its charge and shape are pictured in \cref{F:edge sequence}.
\ytableausetup{boxsize=7pt,aligntableaux=center}
\begin{figure}
\[
\begin{tikzpicture}[>={Stealth[inset=0pt,length=6pt,angle'=30]}]
\node at (0,6) {$\bullet$};
\node at (0,5) {$\bullet$};
\node at (0,4) {$\bullet$};
\node at (1,4) {$\bullet$};
\node at (2,4) {$\bullet$};
\node at (2,3) {$\bullet$};
\node at (2,2) {$\bullet$};
\node at (3,2) {$\bullet$};
\node at (3,1) {$\bullet$};
\node at (4,1) {$\bullet$};
\node at (4,0) {$\bullet$};
\node at (5,0) {$\bullet$};
\node at (6,0) {$\bullet$};

\draw (0,6) -- (0,0) -- (6,0);
\coordinate (zf) at (0,6);
\coordinate (ze) at (0,5);
\coordinate (zd) at (0,4); 
\coordinate (ad) at (1,4);
\coordinate (bd) at (2,4);
\coordinate (bc) at (2,3);
\coordinate (bb) at (2,2);
\coordinate (cb) at (3,2);
\coordinate (ca) at (3,1);
\coordinate (da) at (4,1);
\coordinate (dz) at (4,0);
\coordinate (ez) at (5,0);
\coordinate (fz) at (6,0);

\draw[dotted] (0,0) -- (3,3);

\path[every node/.style={font=\sffamily\small},->]
     (zf)  edge node[midway,right] {5} node[midway,left] {0} (ze);
\path[every node/.style={font=\sffamily\small},->]
     (ze)  edge node[midway,right] {4} node[midway,left] {0} (zd);
\path[every node/.style={font=\sffamily\small},->]
     (zd)  edge node[midway,above] {3} node[midway,below] {1} (ad);
\path[every node/.style={font=\sffamily\small},->]
     (ad)  edge node[midway,above] {2} node[midway,below] {1} (bd);
\path[every node/.style={font=\sffamily\small},->]
     (bd)  edge node[midway,right] {1} node[midway,left] {0} (bc);
 \path[every node/.style={font=\sffamily\small},->]
     (bc)  edge node[midway,right] {{\color{red}{0}}} node[midway,left] {0} (bb);
 \path[every node/.style={font=\sffamily\small},->]
     (bb)  edge node[midway,above] {-1} node[midway,below] {1} (cb);
 \path[every node/.style={font=\sffamily\small},->]
     (cb)  edge node[midway,right] {-2} node[midway,left] {0} (ca);
 \path[every node/.style={font=\sffamily\small},->]
     (ca)  edge node[midway,above] {-3} node[midway,below] {1} (da);
 \path[every node/.style={font=\sffamily\small},->]
     (da)  edge node[midway,right] {-4} node[midway,left] {0} (dz);
 \path[every node/.style={font=\sffamily\small},->]
     (dz)  edge node[midway,above] {-5} node[midway,below] {1} (ez);   
 \path[every node/.style={font=\sffamily\small},->]
     (ez)  edge node[midway,above] {-6} node[midway,below] {1} (fz);
\end{tikzpicture}
\]
\begin{align*}
&\begin{array} {|c||c|c|c|c|c|c|c||c|c|c|c|c|c|c|c|c|c|} \hline
i   &\dotsm  &5&4&3&2&1&{\color{red}{0}}&-1&-2&-3&-4&-5&-6&\dotsm \\ \hline
b_i &0       &0 &0 & 1& 1& 0&0&1&0&1&0&1&1&1\\ \hline
\end{array} \\[2mm]
&\charge(b)={\color{red}{0}} \qquad \shape(b) = \ydiagram{2,2,3,4}
\end{align*}
\caption{The shape and charge of an edge sequence}
\label{F:edge sequence}
\end{figure}
\ytableausetup{boxsize=normal,aligntableaux=bottom}
%\end{ex}

A function $b:\Z\to\{0,1\}$ can be viewed as an infinite indexed binary word 
\[\dotsm b(1) b(0) b(-1) \dotsm;\] 
notice that in writing such a word we index the positions in reverse order. An \emph{inversion} of $b$ is a pair of integers $i>j$ such that $b(i)>b(j)$, a $1$ to the left of a $0$. An \emph{edge sequence} (or Maya diagram) is a function $b:\Z\to\{0,1\}$ such that $b(i)=0$ for $i\gg0$ and $b(i)=1$ for $i\ll0$, that is, $b$ has finitely many inversions. Let $\ES$ denote the set of edge sequences.
The \emph{shape} of $b\in\ES$ is the partition whose French partition diagram has boundary traced out by the 
values of $b$ from top left to bottom right, where $0$ (resp. $1$) indicates a vertical downward (resp. horizontal rightward) unit vector. See \cref{F:edge sequence}, in which the $0$s and $1$s of the edge sequence are written to the left and below their corresponding unit vectors, and the index $i$ of the bit $b(i)$ is written to the right or above the unit vector. 
%The parts of the partition $\shape(b)$ 
%are given by the number of $1$'s to the left of each $0$ in the edge sequence. 

The \emph{charge} of $b\in\ES$ is the index of the unit vector that touches the main diagonal from above and left, or equivalently the index of the rightmost $0$ in the edge sequence of the form $\dotsm0011\dotsm$ obtained from $b$ by repeatedly swapping adjacent pairs $10$ to $01$ until none remain.
There is a bijection 
\begin{equation}
\label{E:edge sequence of partition}
\ES \to \Z \times \Y,\qquad 
b \mapsto (\charge(b),\shape(b)).
\end{equation}

\subsection{Cores and quotients}
Our goal is to define the bijection 
\begin{equation}
 \Y\cong \cC_r \times \Y^r,\qquad
 \la\mapsto (\core_r(\la),\quot_r(\la))
\end{equation}
where $\core_r$ is the \emph{$r$-core} and $\quot_r$ is the \emph{$r$-quotient} map.

In the following diagram all horizontal maps are bijections and vertical maps are inclusions.
\[
\begin{tikzcd}[column sep=huge]
\Z\times \Y & \ES \arrow[l,swap,"\charge\times \shape"] \arrow[r] & \ES^r \arrow[r,"\cd\times \quot_r"] & \Z^r \times \Y^r \\
\{0\} \times \Y \arrow[u] & \ES_0 \arrow[u] \arrow[l]\arrow[u] \arrow[r] & (\ES^r)_0 \arrow[r] \arrow[u] & Q\times \Y^r \arrow[u] \arrow[r,"\kb^{-1}\times\id"] & \cC_r \times \Y^r \\ 
\{0\} \times \cC_r \arrow[u] \arrow[rrr,"\kb"] & & & Q \times \varnothing^r\arrow[u]
\end{tikzcd}
\]
Elements $b^\bullet=(b^0,b^1,\dotsc,b^{r-1})\in \ES^r$ are called \emph{abaci}. We may write them as \linebreak $\{0,1,\dotsc,r-1\}\times \Z$ matrices with entries in $\{0,1\}$ where a $0$ is a bead and a $1$ is a hole (position with no bead) and the $i$-th row represents the edge sequence $b^i$ and is the $i$-th runner in the abacus.

There is a bijection $\ES\to \ES^r$ sending $b$ to $(b^0,b^1,\dotsc,b^{r-1})$ by letting $b^i$ select the bits in $b$ indexed by integers congruent to $i$ mod $r$: $b^i(j) = b(rj+i)$ for $0\le i<r$ and $j\in\Z$. The inverse map is given by interleaving the sequences $b^0,b^1,\dotsc,b^{r-1}$. 
This bijection is charge-additive: $\charge(b) = \sum_{j=0}^{r-1} \charge(b^j)$.
The $r$-fold product of the bijection \eqref{E:edge sequence of partition} yields the 
bijection $\ES^r \cong \Z^r \times \Y^r $. Denote this by $\bd=(b^0,\dotsc,b^{r-1})\mapsto ((c_0,\dotsc,c_{r-1}), \lad)$. We write $\lad = \quot_r(\bd)$; this is the $r$-quotient. Call $(c_0,\dotsc,c_{r-1})=\cd(\bd)$ the charge vector. This indicates the position on each runner where the beads end after pushing all beads to the left within each runner.
This defines the bijections going across the top row of the diagram.

We now restrict all these bijections. Let $\ES_0 = \{b\in \ES\mid \charge(b)=0\}$ and 
$(\ES^r)_0 = \{\bd\in\ES^r\mid \sum_{i=0}^{r-1} c_i(\bd) =0 \}$. 
Then $\cd(\bd)$ can be viewed as an element of the $\mathfrak{sl}_r$ root lattice $Q$ (and belongs to the zero lattice $Q=0$ when $r=1$).
The second row of the diagram (save the last map) 
is given by suitable restrictions of the top row of bijections.

Let $\gamma\in\Y$ and $b\in \ES$ with $\gamma=\shape(b)$. 
We see that the following are equivalent: $\gamma$ is an $r$-core; $h_\gamma(i,j)\ne r$ for all $(i,j)\in \gamma$;
there is no index $k$ such that $b(k)=1$ and $b(k+r)=0$; $\quot_r(\gamma)$ is the empty multipartition $(\varnothing^I)$. 
Therefore the bijection $\{0\} \times\Y \cong Q\times\Y^r$ restricts to the bijection
$\{0\} \times \cC_r \cong Q \times (\varnothing^r)$, that is, $\cC_r\cong Q$. We call this bijection $\kb$.

\begin{ex}\label{X:core-quot}
Let $b\in \ES_0$ be as in the previous example. We have $\la=\shape(b)=(4,3,2,2)$. Set $r=3$. 
We map $b\mapsto (b^0,b^1,b^2)$ which are pictured in the matrix below.
Reading up the columns of the $\{0,1,2\}\times \Z$ matrix we recover $b$. 
Each runner of the abacus is an edge sequence; the corresponding shapes give the $3$-quotient of $(4,3,2,2)$, which is
$((1),\varnothing,(2))$.

To get the $3$-core of $\la$ we move all beads to the left in each runner. This produces the second abacus. 
Reading up columns we obtain the edge sequence $\dotsm 0001|1011\dotsm$. Therefore $\core_3(4,3,2,2)=(2)$.
The charge sequence is $(1,-1,0)\in Q$. 

\ytableausetup{boxsize=4pt}

\[
\begin{array} {|c||c|c|c|c|c|c|c||c|c|c|c|c|c|c|c|c|c|} \hline
i   &\dotsm  &5&4&3&2&1&{\color{red}{0}}&-1&-2&-3&-4&-5&-6&\dotsm \\ \hline
b_i &\dotsm       &0 &0 & 1& 1& 0&0&1&0&1&0&1&1&\dotsm\\ \hline
\end{array}
\]

\[
\begin{array}{c||p{10pt}p{10pt}p{10pt}|p{10pt}p{10pt}p{10pt}}
   &2&1&0&-1&-2&-3 \\ \hline \hline
b^0& 0&1& 0&1&1&1 \\
b^1& 0&0& 0&0&1&1 \\
b^2& 0&0& 1&1&0&1 \\ \hline
\end{array}
\]

%\[
%\begin{array}{c||p{10pt}p{10pt}p{10pt}|p{10pt}p{10pt}p{10pt}}
%   &-2&-1&0&1&2&3 \\ \hline \hline
%b^0& 0&0& 1&1&0&1 \\
%b^1& 0&0& 0&0&1&1 \\
%b^2& 0&1& 0&1&1&1 \\ \hline
%\end{array}
%\]
\begin{align*}
\qquad\qquad&
\begin{tikzpicture}[scale=.70]
\node at (-3,2) (aa) {$\dotsm$};
\node at (-3,1) (ab) {$\dotsm$};
\node at (-3,0) (ac) {$\dotsm$};
\node at (-2,2) (ba) {$\bullet$};
\node at (-2,1) (bb) {$\bullet$};
\node at (-2,0) (bc) {$\bullet$};
\node at (-1,2) (ca) {$\circ$};
\node at (-1,1) (cb) {$\bullet$};
\node at (-1,0) (cc) {$\bullet$};
\node at (0,2) (da) {$\bullet$};
\node at (0,1) (db) {$\bullet$};
\node at (0,0) (dc) {$\circ$};
\node at (1,2) (ea) {$\circ$};
\node at (1,1) (eb) {$\bullet$};
\node at (1,0) (ec) {$\circ$};
\node at (2,2) (fa) {$\circ$};
\node at (2,1) (fb) {$\circ$};
\node at (2,0) (fc) {$\bullet$};
\node at (3,2) (ga) {$\circ$};
\node at (3,1) (gb) {$\circ$};
\node at (3,0) (gc) {$\circ$};
\node at (4,2) (ha) {$\dotsm$};
\node at (4,1) (hb) {$\dotsm$};
\node at (4,0) (hc) {$\dotsm$};
\draw[dotted] (.5,-.5) -- (.5,2.5);
\draw (aa) -- (ha);
\draw (ab) -- (hb);
\draw (ac) -- (hc);
\node at (5,2) (ia) {$\ydiagram{1}$};
\node at (5,1) (ib) {$\varnothing$};
\node at (5,0) (ic) {$\ydiagram{2}$};
\end{tikzpicture}
\\
&
\begin{tikzpicture}[scale=.70]
\node at (-3,2) (aa) {$\dotsm$};
\node at (-3,1) (ab) {$\dotsm$};
\node at (-3,0) (ac) {$\dotsm$};
\node at (-2,2) (ba) {$\bullet$};
\node at (-2,1) (bb) {$\bullet$};
\node at (-2,0) (bc) {$\bullet$};
\node at (-1,2) (ca) {$\bullet$};
\node at (-1,1) (cb) {$\bullet$};
\node at (-1,0) (cc) {$\bullet$};
\node at (0,2) (da) {$\circ$};
\node at (0,1) (db) {$\bullet$};
\node at (0,0) (dc) {$\bullet$};
\node at (1,2) (ea) {$\circ$};
\node at (1,1) (eb) {$\bullet$};
\node at (1,0) (ec) {$\circ$};
\node at (2,2) (fa) {$\circ$};
\node at (2,1) (fb) {$\circ$};
\node at (2,0) (fc) {$\circ$};
\node at (3,2) (ga) {$\circ$};
\node at (3,1) (gb) {$\circ$};
\node at (3,0) (gc) {$\circ$};
\node at (4,2) (ha) {$\dotsm$};
\node at (4,1) (hb) {$\dotsm$};
\node at (4,0) (hc) {$\dotsm$};
\draw[dotted] (.5,-.5) -- (.5,2.5);
\draw (aa) -- (ha);
\draw (ab) -- (hb);
\draw (ac) -- (hc);
\node at (5,2) (ia) {$1$};
\node at (5,1) (ib) {$-1$};
\node at (5,0) (ic) {$0$};
\end{tikzpicture}
\end{align*}
\begin{align*}
\text{core} \qquad\begin{array} {|c||c|c|c|c|c|c|c||c|c|c|c|c|c|c|} \hline
i &\dotsm  &5&4&3&2&1&0&-1&-2&-3&-4&-5&-6&\dotsm \\ \hline
a_i &0&0&0 & 0&0&0&1&1&0&1&1&1&1&1\\ \hline
\end{array}\qquad \ydiagram{2}
\end{align*}
\end{ex}

%\begin{rem}
%Our map $\quot_r$ and our definition of charge are the same as in \cite{Wen}, except that we interchange the roles of black and white dots in our Maya diagrams.
%\end{rem}

When considering a fixed $r$, we simply write $\core=\core_r$ and $\quot=\quot_r$.

%\subsection{Cores and ribbons}
%Let $\cC=\cC_r$ be the set of $r$-core partitions.

%Consider $\mu,\la\in\Y$ such that $\mu\subset \la$. The skew shape $\la/\mu:=D(\la)-D(\mu)$ is a %$\mu$-addable and $\la$-removable $r$-ribbon if $|\la|-|\mu|=r$ and the set of boxes $\la/\mu$ is rookwise %connected with at most one element on each southwest-northeast diagonal. Then an $r$-core is precisely a %partition that has no removable $r$-ribbon.
%One way to obtain $\core(\mu)$ is to repeatedly remove (removable) $r$-ribbons starting with $\mu$ until an %$r$-core is reached; by definition this is $\core(\mu)$. This is well-defined: one obtains the same $r$-core  %independently of the order of removal of $r$-ribbons. It is the same as moving the beads in the abacus to the left.

\subsection{Cores to root lattice}
Recall that $Q:=\left\{(c_0,\ldots, c_{r-1})\in\Z^I\, \middle|\, \sum_{i\in I}c_i=0\right\}$
denotes the $\mathfrak{sl}_r$ root lattice (or $Q=0$ in the case $r=1$), realized as the zero sum elements in the lattice $\Z^I$.
Let $\epsilon_i\in\Z^I$ be the $i$-th coordinate vector.
Then $Q$ is the spanned by the elements 
$\{\alpha_i:=\epsilon_{i-1}-\epsilon_i\mid i\in I\}$.
We realize the simple roots of $\mathfrak{sl}_r$ as the $\alpha_i$ for $i\neq 0$.

%Another way to compute the bijection $\kb: \cC \to Q$ is as follows.
Define the map $\kb:\Y\to Q$ by
$
  \kb(\mu) = -\sum_{(p,q)\in\mu} \alpha_{q-p}.
$
%It can be shown that
Then the restriction of $\kb$ to $\cC$ is the same as the bijection $\cC\cong Q$ constructed above.

%Let $Q$ be the root lattice for $\mathfrak{sl}_r$ for $r>1$, and set $Q=0$ if $r=1$. 
%Another way to compute the bijection $\kb: \cC \to Q$ is as follows. If $r=1$, both sets have one element.
%For $r>1$, let $Q_\af$ be the root lattice for $\hat{\mathfrak{sl}}_r$ and $\{\alpha_i\mid i\in I\}$ the simple affine roots. Define the map $\kappa:\Y\to Q_\af$ by
%\begin{align*}
%  \kappa(\mu) = -\sum_{(p,q)\in\mu} \alpha_{q-p}.
%\end{align*}
%
%Let $\cl: Q_\af\to Q$ be the restriction map. Its kernel is $\Z\delta$ where $\delta=\sum_{i\in I} \alpha_i\in Q_\af$ is the null root. We define
%$\kb: \Y \to Q$ by $\kb(\mu) = \cl(\kappa(\mu))$.
%It is not difficult to show that the restriction of $\kb$ to $\cC$ is the same as the bijection $\cC\cong Q$ constructed above, as long as we adopt the convention that 
%\begin{align}\label{albDef}
%\alb_i:=\cl(\alpha_i)=\epsilon_{i-1}-\epsilon_i
%\end{align}
%for all $i\in I$ (with indices understood modulo $r$), where $\{\epsilon_i\}_{i\in I}$ is the standard basis of $\Z^I$.

\ytableausetup{boxsize=2mm}
\begin{ex} \label{X:roots and cores} Let $r=3$ and consider the $3$-core $\gamma=\ydiagram{2}$\,. 
%We put $\alpha_{q-p}$ into the box $(p,q)$:
%\ytableausetup{boxsize=normal}
%\[
%\begin{ytableau} \alpha_0 & \alpha_2 \end{ytableau}
%\]
%\ytableausetup{boxsize=2mm}
We have $\kb(\ydiagram{2})=-(\alpha_0+\alpha_2)=\alpha_1$, which agrees with the charge sequence $(1,-1,0)\in Q$ computed above.

We list (in reverse lex order, which refines dominance order) the partitions with core $\gamma=\ydiagram{2}$ and 3-quotient of size $2$. 
\ytableausetup{boxsize=3pt,aligntableaux=bottom}
\begin{equation*}
\begin{array}{|c||c|c|c|c|c|}\hline
 & \ydiagram{0,1,9} & \ydiagram{0,4,6} & \ydiagram{0,2,2,6} & \ydiagram{0,1,1,1,1,6} &\ydiagram{0,2,4,4}   \\ \hline
\text{quot} &  \begin{matrix} \ydiagram{2} &\dt&\dt\end{matrix} &  \begin{matrix}
\ydiagram{1,1} & \dt & \dt
\end{matrix} & \begin{matrix}
\ydiagram{1} & \dt & \ydiagram{1}
\end{matrix} & \begin{matrix}
\ydiagram{1} & \ydiagram{1}&\dt
\end{matrix} & \begin{matrix}
\dt & \dt & \ydiagram{2} 
\end{matrix}  \\ \hline
\end{array}
\end{equation*}

\begin{equation*}
\begin{array}{|c||c|c|c|c|c|}\hline
 & \ydiagram{0,1,1,2,3,3} &  \ydiagram{0,1,2,2,2,3} &\ydiagram{0,1,1,1,2,2,3}  &  \ydiagram{0,1,1,1,1,1,1,1,3} \\ \hline	
\quot &   \begin{matrix}
\dt & \ydiagram{2} & \dt 
\end{matrix} & \begin{matrix}
\dt & \ydiagram{1} & \ydiagram{1} 
\end{matrix} & \begin{matrix}
\dt & \dt & \ydiagram{1,1} 
\end{matrix}& \begin{matrix}
\dt  & \ydiagram{1,1} &\dt
\end{matrix}  \\ \hline
\end{array}
\end{equation*}



\ytableausetup{boxsize=normal}
\end{ex}

%\begin{lem} \label{L:order W equivariant} For any $u\in \Sym_I$ and $w\in \Symaf_I$,
%$\lad\dom_w \mud$ if and only if	$u\lad \dom_{uw} u\mud$.
%\end{lem}


\section{Wreath Macdonald symmetric functions and finitization}
\label{S:sym}

\subsection{Tensor symmetric functions}
Let $\La$ be the algebra of symmetric functions over $\K=\Q(q,t)$ \cite[\S I.2]{Mac}.
Denote by $\La^{I}=\La^{\otimes I}$ the $I$-fold tensor power of $\La$ over $\K$. 
For $f\in \La$, we write $f[X^{(i)}]$ to indicate the element of $\La^I$ with $1$ in tensor factors $j\ne i$ and $f$ in factor $i$. $\Lambda^I$ is a polynomial ring over $\K$ with polynomial generators given by the power sums $p_k[X^{(i)}]$ for $i\in I$ and $k>0$. We write $\Xd$ for the $I$-tuple of alphabets $(X^{(0)},\dotsc,X^{(r-1)})$ and often denote by $f[\Xd]$ a generic element of $\Lambda^I$.
For an $I$-tuple of partitions $\lad=(\la^{(0)},\la^{(1)},\dotsc,\la^{(r-1)})\in \Y^I$, define the tensor Schur function $s_\lad=\bigotimes_{i\in I} s_{\la^{(i)}}=\prod_{i\in I} s_{\la^{(i)}}[X^{(i)}]$. 
The tensor Hall pairing on $\Lambda^I$ is defined by the orthonormality of the tensor Schur basis: $\langle s_\lad, s_\mud \rangle = \delta_{\lad,\mud}$ for $\lad,\mud\in \Y^I$. 
%For $f\in\Lambda^I$, we denote by $f^\perp$ be the adjoint under the tensor Hall pairing to the operator of multiplication by $f$.
%Explicitly, we have
%\[
%p_n^\perp[X^{(i)}](p_m[X^{(j)}])=n\delta_{n,m}\delta_{i,j}.
%\]
%This determines the operator on all of $\La^I$ using that the tensor Hall pairing is a Hopf pairing.

For any $a=a(q,t)\in\K$, define the $\K$-algebra automorphism $\cP_{\id-a\chi^{-1}}$ of $\Lambda^I$ by
%\begin{align}
$\cP_{\id-a\chi^{-1}}(p_k[X^{(i)}]) = p_k[X^{(i)}]-a(q^k,t^k) p_k[X^{(i-1)}]$
%\end{align}
for all $i\in I$ and $k>0$. %(The notation $\cP_{\id-a\chi^{-1}}$ arises from more general matrix plethysms $\cP_A\in\End_\K(\La^I)$ for $A\in\mathrm{Mat}_{I\times I}(\K)$ defined in \cite{OS2}.) 
%We shall only use this for $a=q$ and $a=t^{-1}$.

%\subsection{Pairing}
%In what follows, we will work with the \textit{wreath Hall pairing} $\langle -, -\rangle$ on $\Lambda^{\otimes I}$ (cf. \cite{Mac} Chapter I, Appendix B).
%Specifically, we have
%\[
%\langle p_k[X^{(i)}],p_{k'}[X^{(j)}]\rangle=k\delta_{i,j}\delta_{k,k'}
%\]
%Note that the upward shift matrix plethsym $\sigma$ is adjoint to $\sigma^{-1}$ under this pairing.
%We extend this pairing to $\Lambda^{\otimes r}\otimes \K[Q]$ by setting
%\[
%\langle f\otimes e^{\alpha},g\otimes e^{\beta}\rangle=\delta_{\alpha,\beta}\langle f, g\rangle
%\]
%In this way, we have
%\[
%\langle s_\lambda,s_\mu\rangle=\delta_{\lambda,\mu}
%\]
%
%\subsubsection{Transformed Macdonald pairing}
%We define the \textit{transformed wreath Macdonald pairing} by
%\[
%\langle f\otimes e^\alpha,g\otimes e^\beta\rangle_{q,t}':=\left\langle f\otimes e^\alpha, g[-\sigma(1-t^{-1}\sigma^{-1})(1-q\sigma^{-1})X^\bullet]\otimes e^\beta\right\rangle
%\]
%
%\begin{prop}\label{TransPair}
%We have
%\[
%\langle H_\lambda[X^\bullet;q^{-1},t^{-1}],H_\mu[X^\bullet;q,t]\rangle_{q,t}'\not=0
%\]
%if and only if $\lambda=\mu$.
%\end{prop}
%
%\begin{proof}
%Clearly, we only need to consider the case $\mathrm{core}_r(\lambda)=\mathrm{core}_r(\mu)$.
%If $\lambda\not\ge\mu$, then by adjunction,
%\begin{align*}
%&\langle H_\lambda[X^\bullet;q^{-1},t^{-1}],H_\mu[X^\bullet;q,t]\rangle_{q,t}'\\
%&= \langle H_{\lambda}[-\sigma^{-1}(1-t^{-1}\sigma)X^\bullet ;q^{-1},t^{-1}],H_{\mu}[(1-q\sigma^{-1})X^\bullet ;q,t]\rangle\\
%&= \langle H_{\lambda}[t^{-1}(1-t\sigma^{-1})X^\bullet ;q^{-1},t^{-1}],H_{\mu}[(1-q\sigma^{-1})X^\bullet ;q,t]\rangle
%\end{align*}
%The first entry is the span of $s_{\nu}$ with $\nu\le \lambda$ and the second entry is the span of $s_\nu$ with $\nu\ge \mu$, and $\lambda\not\ge \mu$ prevents any of these Schur functions from coinciding.
%For the case $\lambda\not\le\mu$, we use adjunction to move over $-\sigma^{-1}(1-t^{-1}\sigma^{-1})$ instead.
%%When $\lambda=\mu$, we note that the coefficients of $s_\lambda$ must be nonzero when performing these plethysms because $\{H_\nu\}$ form a basis.
%\end{proof}
%
%%\begin{rem}
%%If we remove the extra $-\sigma$ in the definition of $\langle -,-\rangle_{q,t}'$, then $\{H_\lambda\}$ would be orthogonal to a basis defined like $\{H_\lambda\}$ but using the plethysms $(1-q\sigma)$ and $(1-t^{-1}\sigma)$ instead.
%%Thus, the asymmetry still persists.
%%\end{rem}
%%
%\subsubsection{Ordinary Macdonald pairing}
%The \textit{wreath Macdonald pairing} is given by
%\[
%\langle f\otimes e^\alpha, g\otimes e^\beta\rangle_{q,t}:=\left\langle f\otimes e^\alpha,g\left[ \frac{-\sigma(1-q\sigma^{-1})}{(1-t^{-1}\sigma^{-1})}X^\bullet \right]\otimes e^\beta\right\rangle
%\]
%We have already defined the \textit{ordinary} wreath Macdonald function $P_\lambda[X^\bullet; q,t^{-1}]$ as $H_\lambda[(1-t^{-1}\sigma^{-1})X^\bullet; q,t]$ normalized so that the coefficient of $s_\lambda$ is $1$.
%Correspondingly, we define the \textit{dual} wreath Macdonald function $Q_\lambda^*[X^\bullet ;q,t^{-1}]$ as $H_\lambda[(1-t^{-1}\sigma)X^\bullet; q^{-1},t^{-1}]$ normalized so that the coefficient of $s_\lambda$ in $H_\lambda[(q-\sigma)X^\bullet; q^{-1},t^{-1}]$ is $1$.
%
%\begin{prop}\label{DualMac}
%The following hold for $Q^*_\lambda[X^\bullet ;q, t^{-1}]$:
%\begin{enumerate}
%\item $Q_\lambda[X^\bullet; q,t^{-1}]$ is a scalar multiple of $P_\lambda[\sigma X^\bullet;q^{-1},t]$;
%\item The functions $Q_\lambda^*[X^\bullet; q, t^{-1}]$ is the left dual vector of $P_\lambda[X^\bullet; q,t^{-1}]$ under $\langle -, -\rangle_{q,t}$, i.e.
%\[
%\langle Q^*_\lambda[X^\bullet;q,t^{-1}], P_\mu[X^\bullet;q,t^{-1}]\rangle_{q,t}=\delta_{\lambda,\mu}
%\]
%\end{enumerate}
%\end{prop}
%

\subsection{Wreath Macdonald $H$ and $P$ functions}
For $\la\in\Y$ let $H_\lambda[\Xd;q,t]$ be the wreath Macdonald $H$ function \cite{H2}, as defined in \cite{Wen}.
%\footnote{In the more general framework of \cite{OS2} (due to Haiman \cite{H}), there is a wreath Macdonald basis $\{H^w_\mud\mid \mud\in \Y^I\}$ of $\La^I$ for each element $w$ of the affine Weyl group of type $A_{r-1}^{(1)}$. The wreath Macdonald functions used here are those for which $w$ is a translation element, which corresponds to a fixed core.}
These are characterized by the conditions
\begin{align}
\cP_{\id-q\chi^{-1}}H_\lambda[\Xd;q,t] &\in \K^\times s_{\quot(\lambda)} + \bigoplus_{\substack{\nu>\la\\\kb(\nu)=\kb(\la)}} \K s_{\quot(\nu)}\\
\cP_{\id-t^{-1}\chi^{-1}}H_\lambda[\Xd;q,t] &\in \K^\times s_{\quot(\lambda)} + \bigoplus_{\substack{\nu<\la\\\kb(\nu)=\kb(\la)}} \K s_{\quot(\nu)}\\
\langle s_{(n)}[X^{(0)}], H_\lambda[\Xd;q,t]\rangle &= 1.
\end{align}
where $n=|\quot(\lambda)|$ and $<$ is the (strict) dominance order on partitions \cite[\S I.1]{Mac}.

For any $\la\in\Y$  the wreath Macdonald $P$-function $P_\lambda[\Xd;q,t]$ is defined so that $P_\lambda[\Xd;q,t^{-1}]$ is the scalar multiple of $\cP_{\id-t^{-1}\chi^{-1}}(H_\lambda[\Xd;q,t])$ in which the coefficient of $s_{\quot(\lambda)}$ is $1$. In particular, $P_\lambda[\Xd;q,t]$ satisfies the unitriangularity
\begin{align*}
P_\lambda[\Xd;q,t]\in s_{\quot(\lambda)} + \bigoplus_{\substack{\nu<\la\\\kb(\nu)=\kb(\la)}} \K \,s_{\quot(\nu)}
\end{align*}
For any fixed $\alpha\in Q$, $\{ P_\lambda[\Xd;q,t]\mid \text{$\la\in\Y$ such that $\kb(\lambda)=\alpha$} \}$ 
is a homogeneous basis of $\Lambda^I$, with $P_\lambda[\Xd;q,t]$ having degree $|\quot(\lambda)|$. 

%Our notation $P_\lambda[\Xd;q,t^{-1}]$ agrees with the usual conventions in the classical $r=1$ case. For technical reasons, it is often convenient to work with $P_\lambda[\Xd;q,t^{-1}]$ rather than $P_\lambda[\Xd;q,t]$.

\begin{ex} For $r=1$, $P_\lambda[\Xd;q,t]$ is the usual Macdonald $P$-function.
\end{ex}

\begin{ex} \label{X:P core} If $\la$ is an $r$-core than $H_\la[\Xd;q,t]=1$ and $P_\la[\Xd;q,t]=1$.
\end{ex}

\begin{ex} \label{X:P11}
Let $r=2$ and $\la=(1,1)$. We have $H_{11}(\Xd;q,t) = s_1[X^{(0)}] + t s_1[X^{(1)}]$. To verify this we consider the partitions $(2)\dom(1,1)$. We need only check 
\begin{align*} 
	\cP_{\id-t^{-1} \chi^{-1}}(s_1[X^{(0)}] + t s_1[X^{(1)}]) &= s_1[X^{(0)}] - t^{-1} s_1[X^{(1)} + t(s_1[X^{(1)} - t^{-1}s_1[X^{(0)}] )\\
	&= (t-t^{-1}) s_1[X^{(1)}].
\end{align*}
and note that $s_{\quot((1,1))} = s_{(\varnothing, (1))} = s_1[X^{(1)}]$. It follows that $P_{(1,1)}[\Xd;q,t^{-1}]= s_1[X^{(1)}]$.
\end{ex}


\subsection{Symmetric polynomials}
Recall the sets of variables \eqref{E:variables}. There is a  restriction map
\begin{align}\label{E:to pol}
\pi_{N_\bullet} : \Lambda^I \to \Lambda^I_{N_\bullet}:=\bigotimes_{i\in I} \K\left[x_{1}^{(i)},\dotsc,x_{N_i}^{(i)}\right]^{\fS_{N_i}} 
\end{align}
given by the tensor product $\pi_{N_\bullet} = \otimes_{i\in I} \pi_{N_i}$, where $\pi_N : \Lambda\to\K[x_1,\dotsc,x_N]^{\fS_N}$ is the standard projection to symmetric polynomials. 
We also write $\pi_{N_\bullet}(f)=f[X_{N_\bullet}]$.

%$\K\left[\{x_{i,l}\}_{i\in I}^{1\le l\le n_i}\right]^{\prod_{i\in I} \fS_{n_i}}$

%\subsection{Finitization}\label{Finite} 
Let $\gamma$ be an $r$-core and $\kb(\gamma)= (c_0,c_1,\dotsc,c_{r-1})\in Q\subset \Z^I$.
We say that $\Nd\in\Z_{\ge0}^I$ is compatible with $\gamma$ if 
\begin{align}\label{Compatibility}
N_{i}-N_{i-1}=(\alpha_i^\vee,\kb(\gamma))=c_{i-1}-c_{i},\quad \text{for all $i\in I$,}
\end{align}
where $(-,-):Q^\vee\times Q\to \Z$ is the standard pairing between $\mathfrak{sl}_r$ root and coroot lattices. 
%Identifying the lattices $Q^\vee\cong Q$ and realizing $Q$ inside $\Z^I$ as above, $(-,-)$ becomes the dot product on $\Z^I$ and $\alpha_i^\vee=\epsilon_{i-1}-\epsilon_i$ for all $i\in I$. 
All $\Nd$ compatible with a given $\gamma$ differ from each other by a vector in  $\Z(1^r)$.

\begin{ex}
In the setting of \cref{X:roots and cores}, the core is $\gamma=(2)$ and $\kb(\gamma)=(1,-1,0)$. The smallest $\Nd$ compatible with $\gamma$ is $N_\bullet=(0,2,1)=-\kb(\lambda)+(1,1,1)$. To this we can add the vector $(1,1,1)$ any number of times to obtain the other $\Nd$ compatible with $\gamma$.
\end{ex}

\begin{lem}\label{SumR} Let $\gamma$ be an $r$-core and $N_\bullet\in \Z_{\ge0}^I$ be compatible with $\gamma$.
Then
\begin{enumerate}
\item The quantity $N=|N_\bullet|:=\sum_{i\in I}N_i$ is divisible by $r$.
\item For all $\lambda\in\Y_{N,\gamma}$ and all $p\in I$,
\[
N_p=|\left\{ 1\le i\le N :  i-\lambda_i- 1 \equiv p \,\text{mod}\, r \right\}|.
\]
In particular, $\mathrm{quot}(\lambda)=\lambda^\bullet$ satisfies $\ell(\lambda^{(i)})\leq N_i$ for all $i\in I$.
\item For any $\lambda^\bullet\in\Y^I$ satisfying $\ell(\lambda^{(i)})\leq N_i$ for all $i$, the unique partition $\la$
with core $\gamma$ and quotient $\la^\bullet$ satisfies $\la\in \Y_{N,\gamma}$.
\end{enumerate} 
\end{lem}

%\begin{proof}
%(1) This follows from the fact that $N_\bullet$ and $\kb(\lambda)$ are congruent modulo $\Z(1,\dotsc,1)$, and the coordinates of the latter sum %to zero.
%%and the latter belongs to $Q$. %
%
%(2) This follows from \cite[I.1, Ex. 8]{Mac} after taking our labeling conventions into account.
%
%(3) For any edge sequence $b$, the length of $\shape(b)$ is precisely the number of $0$'s positioned to the right of at least one $1$. 
%Given $\alpha\in Q$, our choice of $N_\bullet$ ensures that the number of $0$'s positioned to the right of $1$'s in the interleaved edge sequence %defining $\lambda$ will not exceed $|N_\bullet|$.
%\end{proof}

An immediate consequence of parts (2) and (3) of \cref{SumR} is the following:

\begin{prop}\label{Basis} Let $\gamma$ be an $r$-core and $\Nd$ be compatible with $\gamma$.
Then $\{P_\lambda[X_{N_\bullet};q,t]\mid \la\in\Y_{N,\gamma}\}$ forms a basis of the space of color symmetric polynomials $\Lambda^I_{N_\bullet}$.
\end{prop}

\section{Wreath Macdonald eigenoperators}\label{S:operators}

%\subsection{Cyclic-shift operators}\label{Cyclic}
Fix an $r$-core $\gamma$ and compatible $\Nd\in\Z_{\ge0}^I$. 
%We introduce some difference operators that also permute variables.
As before, let $X_{N_\bullet}$ be the variables as in \eqref{E:variables} satisfying \eqref{Compatibility}. 
Define a \textit{shift pattern of $X_{N_\bullet}$} to be a subset of $X_{N_\bullet}$ that contains no more than one variable of each color.
A shift pattern \textit{contains the color $p\in I$} if it contains a variable of color $p$.
Let $Sh_p(X_{N_\bullet})$ denote the set of all shift patterns containing $p$. 

For a shift pattern $\Ju$, let $J\subset I$ denote the set of colors of the variables in $\Ju$.
We denote the variables in $\Ju$ by $x_{\Ju}^{(i)}$, so $\Ju=\{ x_{\Ju}^{(i)} \}_{i\in J}$.
To $\underline{J}$ we associate the following:
\begin{enumerate}
    \item \textit{Gap labels:} For $i\in I$  %$i^{\triangle}\in J$ be first element greater than \textit{or} equal to $i$ in the cyclic order.
     let $i^{\triangledown}\in J$ be the first element less than or equal to $i$ in the cyclic order. 
We stipulate that %$0\le i^{\triangle}-i,  i-i^{\triangledown}\le r-1$.
$0\le i-i^{\triangledown} \le r-1$. Define
\begin{align*}
%   x_{\Ju^\triangle}^{(i)}&=q^{(i-i^{\triangle})}x_{\Ju}^{(i^\triangle)}\\
   x_{\Ju^\triangledown}^{(i)}&=q^{(i-i^{\triangledown})}x_{\Ju}^{(i^\triangledown)}. 
\end{align*}
In particular  %$x_{\Ju^\triangle}^{(i)}=x_{\Ju^\triangledown}^{(i)}=x_{\Ju}^{(i)}$ if $i\in J$.
$x_{\Ju^\triangledown}^{(i)}=x_{\Ju}^{(i)}$ if $i\in J$.
Thus, while $\Ju$ gives a list of variables colored by $J\subset I$, we `fill in the gaps' for values $i\in I\backslash J$ with $q$-shifts of the elements of $\Ju$.
%Note that the $q$-shifts are %negative for $x_{\Ju^\triangle}^{(i)}$ and 
%positive for $x_{\Ju^\triangledown}^{(i)}$.
    \item \textit{A cyclic-shift operator:} For $i\in J$, let $i^{\blacktriangledown}\in J$ be the first element \textit{strictly} less than $i$ in the cyclic order.
        We set $1\le i-i^\blacktriangledown\le r$, where $r$ occurs if and only if $|J|=\{i\}$. 
        We then define the operator $T_{\Ju}$ on $\K[X_{N_\bullet}]$ as the $\K$-algebra map induced by
        \begin{align*}
            T_{\Ju}(x_{l}^{(i)})&=
            \left\{
                \begin{array}{ll}
                    q^{(i-i^\blacktriangledown)}x_{\Ju}^{(i^\blacktriangledown)} & \text{if $i\in J$ and $x_{l}=x_{\Ju}^{(i)}$} \\
                     x_{l}^{(i)}&\hbox{otherwise.}
                \end{array}
            \right.
        \end{align*}
	%Note that this $q$-shift is positive.
%	If we let $i^{\blacktriangle}\in J$ be the first element strictly greater than $i$ in the cyclic order, then observe that
  %      \begin{align*}
    %        T^{-1}_{\Ju}(x_{l}^{(i)})&=
      %      \left\{
        %        \begin{array}{ll}
          %5          q^{(i-i^\blacktriangle)}x_{\Ju}^{(i^\blacktriangle)} & \text{if $i\in J$ and $x_{l}^{(i)}=x_{\Ju}^{(i)}$} \\
               %      x_{l}^{(i)}&\hbox{otherwise} 
              %  \end{array}
            %\right.
       % \end{align*}
%	where as before, we view $1\le i^{\blacktriangle}-i\le r$.
%	Finally, we note the following: for $i\in J$
%\begin{equation}
%\begin{aligned}
%T_\Ju(x_{\Ju}^{(i)})&= qx_{\Ju^\triangledown}^{(i-1)}. %\\
%T_\Ju^{-1}(x_{\Ju}^{(i)})&= q^{-1}x_{\Ju^\triangle}^{(i+1)}.
%\end{aligned}
%\label{ShiftRename}
%\end{equation}
\end{enumerate}

%We will also make use of $n$-tuples of shift patterns.
For an $n$-tuple $\mathbf{\underline{J}}=(\underline{J}_1,\ldots, \underline{J}_n)$ of shift patterns and $0\le k\le n$, we denote
%\begin{align*}
%    T_{\mathbf{\underline{J}}}^k&:=T_{\underline{J}_1}\cdots T_{\underline{J}_k}\\
%    T_{\mathbf{\underline{J}}}^{-k}&:=T_{\underline{J}_n}^{-1}\cdots T_{\underline{J}_{n-k+1}}^{-1}
%\end{align*}
%(at $k=0$ this is the identity operator).
%We will also denote
\begin{align*}
    |\mathbf{\underline{J}}|&=\underline{J}_1\cup\cdots\cup\underline{J}_n\subset X_{N_\bullet},\quad
    |\mathbf{\underline{J}}|_{\le k}=\underline{J}_1\cup\cdots\cup\underline{J}_{k}\subset X_{N_\bullet},\quad
    |\mathbf{\underline{J}}|_{\ge k}=\underline{J}_k\cup\cdots\cup\underline{J}_{n}\subset X_{N_\bullet}.
\end{align*}
If $\mathbf{\underline{J}}$ is an $n$-tuple of shift patterns all containing color $p$, we say $\mathbf{\underline{J}}$ is \textit{$p$-distinct} if the $p$-colored variables $x_{\underline{J}_k}^{(p)}$ are all distinct.
Let $Sh_p^{[n]}(X_{N_\bullet})$ denote the set of all $p$-distinct $n$-tuples of shift patterns containing color $p$.
%and as before, we will simply write it as $Sh_p^{[n]}$ when the variable set is clear from context.

Finally, if $a_1,a_2,\dotsc,a_n$ are operators then define 
%\begin{align}
$
\displaystyle\overset{\curvearrowright}{\prod_{j=1}^n} a_j := a_1 \circ a_2 \circ \dotsm \circ a_n.
$
%\end{align}
This completes the definition of the operators $D_{p,n}$ in \eqref{DpnDef}.

%\subsubsection{Degree one wreath Macdonald operators}\label{FirstOp}


\begin{ex}
Let $r=1$, $p=1$, $N_\bullet=(2,1,0)$, and $\lambda=(3,1,1)$.
In this case, $\lambda$ is a $3$-core and by \cref{X:P core} $P_\la[\Xd;q,t]=1$ so that $P_\lambda[X_{N_\bullet};q,t]=1$.
There are three shift patterns containing $1$: $\Ju_1=\{x_1^{(1)}\}$, $ \Ju_2=\{x_1^{(0)},x_1^{(1)}\}$, and
$\Ju_3=\{x_2^{(0)},x_1^{(1)}\}$. %The operator $D_{1,1}(X_{N_\bullet};q,t)$ is then
We have
\begin{align}
    \label{Ex11}
    D_{1,1}(X_{N_\bullet};q,t)&=q\left(\frac{qtx_1^{(1)}-x_1^{(0)}}{q^2x_1^{(1)}-x_1^{(0)}}\right)\left(\frac{qtx_1^{(1)}-x_2^{(0)}}{q^2x_1^{(1)}-x_2^{(0)}}\right)T_{\Ju_1}\\
    \label{Ex12}
    &+(1-qt^{-1})q\left(\frac{qtx_1^{(1)}-x_2^{(0)}}{x_1^{(0)}-x_2^{(0)}}\right)\left(\frac{qtx_1^{(1)}}{x_1^{(0)}-q^2x_1^{(1)}}\right)T_{\Ju_2}\\
    \label{Ex13}
    &+(1-qt^{-1})q\left(\frac{qtx_1^{(1)}-x_1^{(0)}}{x_2^{(0)}-x_1^{(0)}}\right)\left(\frac{qtx_1^{(1)}}{x_2^{(0)}-q^2x_1^{(1)}}\right)T_{\Ju_3}.
\end{align}
The cyclic-shift operators act trivially on $P_\lambda(X_{N_\bullet};q,t)=1$. After a miraculous simplification one obtains
%Consolidating \eqref{Ex12} and \eqref{Ex13} gets us
%\begin{align}
%    \nonumber
%    &(1-qt^{-1})q\left\{\left(\frac{qtx_1^{(1)}-x_2^{(0)}}{x_1^{(0)}-x_2^{(0)}}\right)\left(\frac{qtx_1^{(1)}}{x_1^{(0)}-q^2x_1^{(1)}}\right)+
%    \left(\frac{qtx_1^{(1)}-x_1^{(0)}}{x_2^{(0)}-x_1^{(0)}}\right)\left(\frac{qtx_1^{(1)}}{x_2^{(0)}-q^2x_1^{(1)}}\right)\right\}\\
%    \nonumber
%    &=(1-qt^{-1})q\left\{
%   \frac{qtx_1^{(1)}\left(
%   qtx^{(1)}{\color{red}x_2^{(0)}}-qtx_1^{(1)}{\color{red}x_1^{(0)}}-{\color{red}x_2^{(0)}x_2^{(0)}+x_1^{(0)}x_1^{(0)}}+q^2{\color{red}x_2^{(0)}}x_1^{(1)}-q^2{\color{red}x_1^{(0)}}x_1^{(1)}
%   \right)}
%   {{\color{red}(x_1^{(0)}-x_2^{(0)})}(x_1^{(0)}-q^2x_1^{(1)})(x_2^{(0)}-q^2x_1^{(1)})}
%    \right\}\\
%    \nonumber
%    &=(1-qt^{-1})q\left\{
%   \frac{qtx_1^{(1)}\left(
%   -qtx_1^{(1)}+x_1^{(0)}+x_2^{(0)}-q^2x_1^{(1)}
%   \right)}
%   {(x_1^{(0)}-q^2x_1^{(1)})(x_2^{(0)}-q^2x_1^{(1)})} 
%    \right\}\\
%    \nonumber
%    &=(1-qt^{-1})q\left\{
%   \frac{
%   -(q^2t^2+q^3t)x_1^{(1)}x_1^{(1)}+qtx_1^{(0)}x_1^{(1)}+qtx_2^{(0)}x_1^{(1)}
%   }
%   {(x_1^{(0)}-q^2x_1^{(1)})(x_2^{(0)}-q^2x_1^{(1)})}
%    \right\}\\
%   \label{Ex14} 
%    &=q\left\{
%   \frac{
%   (-q^2t^2+q^4)x_1^{(1)}x_1^{(1)}+(qt-q^2)x_1^{(0)}x_1^{(1)}+(qt-q^2)x_2^{(0)}x_1^{(1)}
%   }
%   {(x_1^{(0)}-q^2x_1^{(1)})(x_2^{(0)}-q^2x_1^{(1)})}
%    \right\}.
%\end{align}
%On the other hand, \eqref{Ex11} becomes
%\begin{align}\label{Ex15}
% q\left(\frac{qtx_1^{(1)}-x_1^{(0)}}{q^2x_1^{(1)}-x_1^{(0)}}\right)\left(\frac{qtx_1^{(1)}-x_2^{(0)}}{q^2x_1^{(1)}-x_2^{(0)}}\right)
%    =q\left\{
%   \frac{q^2t^2x_1^{(1)}x_1^{(1)}-qtx_1^{(1)}x_2^{(0)}-qtx_1^{(0)}x_1^{(1)}+x_1^{(0)}x_2^{(0)}}
%   {(x_1^{(0)}-q^2x_1^{(1)})(x_2^{(0)}-q^2x_1^{(1)})} 
%    \right\}.
%\end{align}
%Combining \eqref{Ex14} and \eqref{Ex15} gets us
%\begin{align*}
$    D_{1,1}(X_{N_\bullet};q,t)P_\lambda[X_{N_\bullet};q,t]
    %&=
%    q\left\{
%    \frac{q^4x_1^{(1)}x_1^{(1)}-q^2x_1^{(1)}x_2^{(0)}-q^2x_1^{(0)}x_1^{(1)}+x_1^{(0)}x_2^{(0)}}
%    {(x_1^{(0)}-q^2x_1^{(1)})(x_2^{(0)}-q^2x_1^{(1)})}
%    \right\}\\
%    &=q\frac{(x_1^{(0)}-q^2x_1^{(1)})(x_2^{(0)}-q^2x_1^{(1)})}{(x_1^{(0)}-q^2x_1^{(1)})(x_2^{(0)}-q^2x_1^{(1)})}\\
    =qP_\lambda[X_{N_\bullet};q,t].
$
%\end{align*}
\end{ex}

\begin{ex}
Let $r=2$, $p=0$, $N_\bullet=(1,1)$, and $\lambda=(1,1)$. We have $P_{(1,1)}[\Xd;q,t]=s_1[X^{(1)}]$ from \cref{X:P11}.
Finitizing we obtain $P_\lambda[X_{N_\bullet};q,t]=x_1^{(1)}$.
There are two shift patterns containing $0$:
$ \Ju_1=\{x_1^{(0)}\}$, $ \Ju_2=\{x_1^{(0)},x_1^{(1)}\}$.
We then have
\begin{align*}
    D_{0,1}(X_{N_\bullet};q,t)&=
    q\left(
   \frac{tx_1^{(0)}-x_1^{(1)}}{qx_1^{(0)}-x_1^{(1)}} 
    \right)T_{\Ju_1}
    +(1-qt^{-1})
    \frac{x_1^{(1)}}{x_1^{(0)}}
    \left(
    \frac{tx_1^{(0)}}{x_1^{(1)}-qx_1^{(0)}}
    \right)T_{\Ju_2}.
\end{align*}
Observe that $ T_{\Ju_1}x_1^{(1)}=x_1^{(1)}$ and $T_{\Ju_2}x_1^{(1)}=qx_1^{(0)}$.
Altogether then,
\begin{align*}
    D_{0,1}(X_{N_\bullet};q,t)P_\lambda[X_{N_\bullet};q,t]
    &=q\left(
   \frac{tx_1^{(0)}-x_1^{(1)}}{qx_1^{(0)}-x_1^{(1)}} 
    \right)x_1^{(1)}
    +(1-qt^{-1})
    \frac{x_1^{(1)}}{x_1^{(0)}}
    \left(
    \frac{tx_1^{(0)}}{x_1^{(1)}-qx_1^{(0)}}
    \right)qx_1^{(0)}\\
    &=qx_1^{(1)}\left(\frac{tx_1^{(0)}-x_1^{(1)}-(t-q)x_1^{(0)}}{qx_1^{(0)}-x_1^{(1)}}\right)
    %&=qx_1^{(1)}\left(\frac{qx_1^{(0)}-x_1^{(1)}}{qx_1^{(0)}-x_1^{(1)}}\right)\\
    =qP_\lambda[X_{N_\bullet};q,t].
\end{align*}
\end{ex}



%\subsection{Spectral variables}
%Letting $\lambda$ vary over partitions with $\core(\lambda)$ compatible with $N_\bullet$ and $\ell(\lambda)\le |N_\bullet|$, we note that by %\cref{SumR}, the stabilized eigenvalues
%\[
%e_n\left[\sum_{\substack{b=1\\b-\lambda_b\equiv p+1}}^\infty q^{\lambda_b} t^{b}\right]
%\]
%depend only on the $N_p$ values of $b$ where $1\le b\le |N_\bullet|$ and $b-\lambda_b=p+1$.
%We define the \textit{color $p$ spectral variables} $\{ s^{(p)}_{ a} \}_{a=1}^{N_p}$ by setting
%\[
%s^{(p)}_{a}=q^{\lambda_{b_a}}t^{b_a}
%\]
%where $1\le b_a\le|N_\bullet|$ is the $a$th number where $b_a-\lambda_{b_a}\equiv p+1$.
%Using these variables, we can rewrite
%\[
%e_n\left[\sum_{\substack{b=1\\b-\lambda_b\equiv p+1}}^\infty q^{\lambda_b} t^{b}\right]
%=e_n\left[ \frac{t^{|N_\bullet|+p+1}}{1-t^{nr}}+\sum_{a=1}^{N_p}s^{(p)}_{a} \right]
%\]
%where $t\ll 1$.
%The following is but a slight alteration of Lemma 3.2 from \cite{FHHSY}:
%\begin{lem}\label{EigenStab}
%	For $|t|\ll 1$, we have
%\begin{equation}
%e_n\left[\frac{t^{|N_\bullet|+p+1}}{1-t^{nr}}+\sum_{a=1}^{N_p}s^{(p)}_{a} %\right]=\sum_{k=0}^n\frac{t^{n|N_\bullet|+(n-k)(p+1)+r\binom{n-k}{2}}}{\displaystyle\prod_{l=1}^{n-k}(1-t^{rl})}
%e_k\left[ \sum_{a=1}^{N_p}t^{-|N_\bullet|}s^{(p)}_{a} \right]
%\label{PlusSpec}
%\end{equation}
%\end{lem}
%
%\begin{proof}
%As in \textit{loc. cit.}, it is an easy application of the quantum binomial theorem.
%\end{proof}


%\subsection{Eigenfunction equation}
%
%\begin{thm}\label{ThmEigenHigher}
%For $\lambda$ with core $\kb(\lambda)$ compatible with $N_\bullet$ according to \eqref{Compatibility} and $\ell(\lambda)\le|N_\bullet|$, %the wreath Macdonald polynomial $P_\lambda[X_{N_\bullet};q,t]$ satisfies the equations:
%\begin{align*}
%D_{p,n}^*(X_{N_\bullet};q,t)P_\lambda[X_{N_\bullet};q,t] &= e_n\left[ \sum_{\substack{b=1\\ b-\lambda_b\equiv %p+1}}^{|N_\bullet|}q^{-\lambda_b}t^{-|N_\bullet|+b} \right]
%P_\lambda[X_{N_\bullet};q,t]\\
%D_{p,n}(X_{N_\bullet};q,t)P_\lambda[X_{N_\bullet};q,t] &= e_n\left[ \sum_{\substack{b=1\\ b-\lambda_b\equiv %p+1}}^{|N_\bullet|}q^{\lambda_b}t^{|N_\bullet|-b} \right]
%P_\lambda[X_{N_\bullet};q,t].
%\end{align*}
%Here, $x_{i,l}$, $q$, and $t$ take generic values.
%
%\end{thm}

%\begin{rmk}\label{Rmk2Colors}
%Even though $r\ge 3$ was assumed throughout, we have verified experimentally that \cref{ThmEigenHigher} continues to hold as 
%stated for $r=2$. 
%The $r=1$ case is discussed in Remark~\ref{MacCompare} above.
%\end{rmk}

\begin{ex}
Let $r=2$, $p=1$, $N_\bullet=(0,2)$, and $\lambda=(1)$.
Because $\lambda$ is a $2$-core, %we have
$P_\lambda[X_{N_\bullet};q,t]=1$.
There are only two shift patterns containing $1$: $ \Ju_1=\{x_1^{(1)}\}$, 
$ \Ju_2=\{x_2^{(1)}\}$. Note that
$T_{\Ju_1}x_1^{(1)}=q^2x_1^{(1)}, T_{\Ju_1}x_2^{(1)}=x_1^{(1)}, T_{\Ju_2}x_1^{(1)}=x_1^{(1)},
T_{\Ju_2}x_2^{(1)}=q^2x_2^{(1)}.$
Therefore,
\begin{align*}
    D_{1,1}(X_{N_\bullet};q,t)P_\lambda[X_{\bullet};q,t]&=
    \frac{(-1)(1-qt^{-1})}{1-q^2t^{-2}}
    \left\{
    \frac{qtx_2^{(1)}-q^2x_1^{(1)}}{x_1^{(1)}-x_2^{(1)}}+\frac{qtx_1^{(1)}-q^2x_2^{(1)}}{x_2^{(1)}-x_1^{(1)}}
    \right\}\\
    &=\frac{(-1)(1-qt^{-1})(-qt-q^2)}{1-q^2t^{-2}}
    =qtP_\lambda[X_{N_\bullet};q,t].
\end{align*}
\end{ex}

\acknowledgements{We thank Mark Haiman, Andrei Negu\cb{t}, and Alexander Tsymbaliuk for helpful conversations.
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