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\title[Higher Specht polynomials for the diagonal action]{Higher Specht polynomials for the diagonal action}

\author[Maria Gillespie]{\texorpdfstring{Maria M.~Gillespie\thanks{\href{mailto:maria.gillespie@colostate.edu}{maria.gillespie@colostate.edu}. Partially supported by NSF DMS award number 2054391.}\addressmark{1}}{Maria M.~Gillespie}}

\address{\addressmark{1}Department of Mathematics, Colorado State University}

\received{\today}

\abstract{We introduce higher Specht polynomials - analogs of Specht polynomials in higher degrees - in two sets of variables $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$ under the diagonal action of the symmetric group $S_n$.  This generalizes the classical Specht polynomial construction in one set of variables, as well as the higher Specht basis for the coinvariant ring $R_n$ due to Ariki, Terasoma, and Yamada, which has the advantage of respecting the decomposition into irreducibles.
 
 As our main application, we provide a higher Specht basis for the hook shape Garsia--Haiman modules.  In the process, we obtain a new formula for their doubly graded Frobenius series in terms of new generalized cocharge statistics on tableaux.}

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\keywords{Diagonal coinvariants, Shuffle theorem, Schur positivity, Young tableaux, Garsia--Haiman modules, representation theory of the symmetric group}

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\begin{document}

\maketitle


\section{Introduction and notation}

The polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ comes with a natural action of the symmetric group $S_n$ by permuting the variables.  This gives the polynomial ring the structure of a graded $S_n$-representation, which naturally decomposes into irreducible representations.   In \cite{ATY}, Ariki, Terasoma, and Yamada defined \textbf{higher Specht polyomials} that give a basis for $\mathbb{C}[x_1,\ldots,x_n]$ that respects the decomposition into irreducibles and that generalizes the ordinary Specht polynomial construction for the lowest-degree copy of each irreducible representation. In order to do so, they first found such a basis for the \textbf{coinvariant ring} $R_n=\mathbb{C}[x_1,\ldots,x_n]/(e_1,\ldots,e_n)$  where $e_i$ is the $i$th \textit{elementary symmetric polynomial} consisting of the sum of all square-free monomials of degree $i$ in $x_1,\ldots,x_n$. 

In this extended abstract, we extend the theory of higher Specht polynomials to the \textbf{diagonal action} of the symmetric group $S_n$ on the polynomial ring $$\mathbb{C}[\mathbf{x},\mathbf{y}]:=\mathbb{C}[x_1,\ldots,x_n,y_1,\ldots,y_n],$$ defined by $\pi\cdot  f(x_1,\ldots,x_n,y_1,\ldots,y_n)=f(x_{\pi(1)},\ldots,x_{\pi(n)},y_{\pi(1)},\ldots,y_{\pi(n)})$.
Full details and proofs on the results outlined here can be found in \cite{GillespieFull}.

The coinvariant ring construction can be generalized to two variables as follows.

\begin{definition}
  The ring of \textbf{diagonal coinvariants} is given by $$\DR_n=\CC[{\bf x}, {\bf y}]/I_n$$ where $I_n$ is generated by the positive-degree $S_n$-invariants under the diagonal action.
\end{definition}



 The ring $\DR_n$ arises naturally in the geometry of the Hilbert scheme of $n$ points in the plane $\CC^2$.  Haiman  \cite{HaimanDR} used this connection to prove the famous \textit{$(n+1)^{n-1}$ Conjecture}, which states that $\dim_\CC(\DR_n)=(n+1)^{n-1}$ as a complex vector space.  

Haiman used similar methods to prove the \textit{$n!$ Conjecture} \cite{HaimanGeometry}.  This states that the \textit{Garsia--Haiman modules} $\DR_\mu$ (which are also quotients of $\mathbb{C}[\mathbf{x},\mathbf{y}]$, describing local information of a limit as the $n$ points all approach $0$ in the plane) have dimension $n!$ for any partition $\mu$ of size $n$.  The proof of the $n!$ conjecture was the crucial step in the proof of the Macdonald Positivity Conjecture, which states that the transformed \textit{Macdonald polynomials} \cite{Macdonald} have a positive expansion in terms of the Schur symmetric functions. 

Despite these advances, it remains open to understand the $n!$ and $(n+1)^{n-1}$ conjectures from a more combinatorial standpoint, in the following sense.

\begin{open}\label{prob:nfact}
 Find an explicit basis of $n!$ polynomials for $\DR_\mu$, where $\mu$ is a partition of $n$.
\end{open}

\begin{open}\label{prob:basis-DR}
 Find an explicit basis of $(n+1)^{n-1}$ polynomials for $\DR_n$.
\end{open}

Problem \ref{prob:nfact} is open for general partitions $\mu$, while Problem \ref{prob:basis-DR} has very recently been addressed by Carlsson and Oblomkov \cite{CarlssonOblomkov}, who gave a construction of a basis for $\mathrm{DR}_n$ by establishing connections with affine Schubert calculus. 
 However, their basis is not a higher Specht basis in the following sense.

 \begin{definition}
    A \textbf{higher Specht basis} for a (graded) $S_n$-module $M$ is a basis $\mathbb{B}$ that admits a set partition $\bigsqcup \mathbb{B}_{\lambda,i}$ where $\lambda$ and $i$ represent a partition and positive integer respectively, such that:
    \begin{enumerate}
        \item Each $\mathbb{B}_{\lambda,i}$ spans one of the copies of an irreducible representation $V_\lambda$ in the decomposition of $M$ into irreducibles (there may be several such copies, so we distinguish with the subscript $i$, and every copy is one such span),
        \item There is a bijection from $\mathbb{B}_{\lambda,i}$ to the set of ordinary Specht polynomials $F_T$ (defined below) for shape $\lambda$, that preserves the $S_n$-action with respect to each basis.
    \end{enumerate} 
\end{definition}

Starting with the work in \cite{ATY} for the coinvariant ring $R_n$, there have been several higher Specht bases constructed for related modules in recent years.  In \cite{GillespieRhoades}, the author and Rhoades found higher Specht bases for both the modules $R_{n,k}$ (the \textit{Haglund--Rhoades--Shimozono modules} defined in the context of the $t=0$ specialization of the Delta conjecture \cite{HRS}) and $R_\mu$ (the \textit{Garsia--Procesi modules}, which are the cohomology rings of the fibers of the Springer resolution \cite{DeConciniProcesi,GarsiaProcesi,Tanisaki}).  The construction was proven to be a basis in the former case and conjectured for the latter, with proof for $\mu$ having two parts.  
 
 This construction was similar to that of \cite{AllenOneVariable}, in which Allen constructed a basis that respects the decomposition into irreducibles for $R_n$, and for $R_\mu$ for $\mu$ having two parts or being a hook shape.  (Allen's basis is not a higher Specht basis in the strongest sense, since it does not satisfy condition 2 above).  In \cite{AllenDescentMonomials} and \cite{AllenBitableaux1}, Allen also began an exploration the two-variable case, focusing on the diagonally symmetric subring of the polynomial ring in two variables and its quotients.  

Most recently, Salois \cite{Salois} defined a higher Specht basis for the cohomology rings of certain \textit{Hessenberg varieties}.  These rings also appear naturally in symmetric function theory, as they directly relate to the \textit{Stanley--Stembridge Conjecture} \cite{Hikita, StanleyStembridge} and the \textit{Shareshian--Wachs Conjecture} \cite{ShareshianWachs} on chromatic (quasi)symmetric functions.

Our main result is a higher Specht basis for the hook shape Garsia--Haiman modules:

\begin{thm}\label{thm:main}
     Suppose $\mu=(n-k+1,1^k)$ is a hook shape of height $k$.  There is an explicitly constructed set of polynomials $\{F_T^S(\mathbf{x},\mathbf{y})\}$ ranging over all pairs $T,S$ of standard Young tableaux of the same shape that forms a higher Specht basis for $\DR_\mu$.
 \end{thm}

 We achieve this result by first developing a general theory of higher Specht polynomial constructions in two sets of variables (see \Cref{sec:general} below). 

\subsection{Notation on Young tableaux and cocharge}

We refer to \cite{Fulton} for notation on Young tableaux and partitions, with the caveat that we are using the ``French'' convention for Young diagrams in this paper, where the $i$-th row from the bottom of the Young diagram of a partition $\lambda$ has $\lambda_i$ squares.  See \Cref{fig:tableaux}.  We will also generalize the notion of a semistandard Young tableau (SSYT) to include entries from any ordered set, not just the natural numbers.  We write $\SYT(\lambda)$ for the set of standard Young tableaux of shape $\lambda$, and $\SYT(n)$ for the set of Young tableaux of size $n$. We write $\Tab(\lambda)$ for the set of \textbf{standard general tableau} $T$ of shape $\lambda$, which are ways of filling the squares of $\lambda$ with the numbers $1,2,\ldots,n$ each used once, with no restrictions on the rows or columns.

We recall the \textbf{RSK bijection}, which takes permutations of $1,\ldots,n$ to pairs $(P,Q)$ of SYT's with the same shape as each other.  Thus there are $n!$ such pairs. 


\begin{figure}
    \centering
    \young(5,34,22,1113) \hspace{1cm} \young(7,48,25,1369) \hspace{1cm} \young(7,14,62,3985)
    \caption{From left to right: A semistandard Young tableau, a standard Young tableau, and a standard general tableau $T$ of shape $\sh(T)=\lambda=(4,2,2,1)$.  The tableau at left has content $(3,2,2,1,1)$, and the reading word of the tableau at middle is $748251369$.} 
    \label{fig:tableaux}
\end{figure}


\begin{definition}
    The \textbf{cocharge labels} of a permutation $\pi$ are defined by labeling the $1$ with a subscript $0$, then searching leftwards cyclically for the $2,3,4,\ldots$, each time incrementing the subscript label unless the search wraps around the word.  The \textbf{cocharge} of $\pi$, written $\cc(\pi)$, is the sum of its cocharge labels.
\end{definition}

\begin{example}
    The permutation $25314$ has cocharge labels $2_15_23_11_04_1$, so $$\cc(25314)=1+2+1+0+1=5.$$

\end{example}


Lusztig and Stanley \cite{Stanley} showed (in a different notation, that is equivalent to the cocharge statistic defined by Lascoux and Sch\"utzenberger \cite{LascouxSchutzenberger}) that the number of copies of the irreducible representation $V_\lambda$ of $S_n$ in the coinvariant ring $R_n$ in degree $d$ is equal to the number of standard Young tableaux of shape $\lambda$ with cocharge $d$.  We will express this fact in terms of the graded Frobenius character in the next section. 



\subsection{Symmetric functions and Frobenius series}
The two bases of the ring of symmetric functions $\Lambda_{\mathbb{C}}(x_1,x_2,\ldots)$ that will be used are the \textbf{elementary} symmetric functions, defined by $e_d(x_1,x_2,\ldots)=\sum_{i_1<i_2<\cdots < i_d} x_{i_1}\cdots x_{i_d}$ and $e_\lambda=\prod e_{\lambda_i}$, and the \textbf{Schur functions}, defined by $s_\lambda=\sum_{T\in \SSYT(\lambda)}\prod_{c \in \lambda} x_{T(c)}$.  The \textbf{Frobenius map} $\Frob$  is the additive map that sends the irreducible $S_n$-module $V_\nu$ to the Schur function $s_\nu$.  For instance, $\Frob(V_{(2)}\oplus V_{(1,1)}\oplus V_{(1,1)})=s_{(2)}+2s_{(1,1)}$.

For $S_n$-modules with a grading (such as the coinvariant ring, which is graded by degree), its \textbf{graded Frobenius} is the generating function $\grFrob_q(R):=\sum_d \Frob(R_d)q^d$ where $R_d$ is the $d$-th graded piece and $q$ is a formal variable.  For doubly graded $S_n$-modules (by the $x$-degree and $y$-degree here) the \textbf{bi-graded Frobenius map}  is $$\grFrob_{q,t}(R):=\sum_d \Frob(R_{d_1,d_2})q^{d_1}t^{d_2}.$$
The theorem of Lusztig and Stanley \cite{Stanley} mentioned above on the decomposition of $R_n$ into irreducibles can therefore be stated as $$\grFrob_q(R_\mu)=\sum_{T\in \SSYT_\mu} q^{\cc(T)}s_{\sh(T)}$$ where $\SSYT_\mu$ is the set of SSYT's of content $\mu$.  We will not define Garsia--Haiman modules in full in this extended abstract, but we discuss their properties when $\mu$ is a \textbf{hook shape} of the form $(n-k+1,1^{k-1})$.   Several monomial bases for $\DR_\mu$ for hook shapes $\mu$ were established in \cite{HookBases}.  The graded Frobenius was derived by Stembridge \cite{Stembridge}, and makes use of ``maj'' and ``comaj'' statistics:
\begin{equation}\label{eq:stembridge}\grFrob_{q,t}(\DR_\mu)=\sum_{T\in \SYT(n)} q^{\maj_{1,n-k+1}(T)}t^{\comaj_{n-k+1,n}(T)}s_{\sh(T)}.\end{equation}




\subsection{Specht polynomial constructions}
The standard construction of the \textit{Specht modules} $V_\lambda$, which are the irreducible representations of $S_n$ where $\lambda$ ranges over all partitions of $n$, is often presented via \textit{Young tabloids} (see \cite{Sagan}).  This construction is equivalent (see \cite[p.\ 125, Exercise 15(c)]{Macdonald}) to defining a submodule of $\mathbb{C}[x_1,\ldots,x_n]$ spanned by \textit{Specht polynomials} as follows.

\begin{definition}
  The \textbf{Specht polynomial} $F_T$ corresponding to a Young tableau $T$ whose entries are $1,2,3,\ldots,n$ is given by $$F_T=\prod_{C\in\mathrm{col}(T)}\prod_{\substack{i,j\in C\\ i\text{ above }j}}(x_i-x_j)$$ where $\mathrm{col}(T)$ is the set of columns of $T$. (See \Cref{fig:Specht}.)
\end{definition} 

\begin{figure}[b]
    \centering
   $T=\raisebox{-0.5cm}{\young(3,716,2845)}$\hspace{2cm} $F_T=(x_3-x_7)(x_7-x_2)(x_3-x_2)\cdot (x_1-x_8)\cdot (x_6-x_4)$
    \caption{A tableau $T\in \Tab(4,3,1)$ in French notation, and the Specht polynomial $F_T$.}
    \label{fig:Specht}
\end{figure}

One can then define the Specht module 
as $$M_\lambda=\mathrm{span}\{F_T: T\in \Tab(\lambda)\}\subseteq \mathbb{C}[x_1,\ldots,x_n],$$ and we have $M_\lambda\cong V_\lambda$ with a basis given by the $F_T$ such that $T$ is a \textbf{standard Young tableau (SYT)}, in which the entries are increasing along rows and up columns.  The \textbf{Garnir relations} (see \cite{Sagan}) give a straightening algorithm for expressing any $F_T$ in terms of the standard Young tableau basis, and thereby gives a rule for computing with the $S_n$-action on the Specht module.

 In \cite{ATY}, Ariki, Terasoma, and Yamada noted that $F_T$ may also be defined (up to a constant) as a Young idempotent operator applied to a monomial.  In particular, define $$\varepsilon_T=\sum_{\tau \in C(T)}\sum_{\sigma \in R(T)}\sgn(\tau) \tau \sigma$$ where $C(T)\subseteq S_n$ is the group of \textit{column permutations} that preserve the columns of $T$, and $R(T)\subseteq S_n$ is the group of \textit{row permutations}. 
 Then it is not hard to check that $F_T$ is a scalar multiple of $\varepsilon_T(x_T^r)$ where $x_T^r=\prod x_i^{\mathrm{row}(i)-1}$ with $\mathrm{row}(i)$ denoting the row that $i$ occurs in in T, indexed from bottom to top.  
 
\begin{example}
    We have   $$\varepsilon_{\text{\tiny$\young(3,12)$}}=\mathrm{id}+(12)-(13)-(123)$$ and $F_{\text{\tiny$\young(3,12)$}}=x_3-x_1$.
\end{example}
 
 Ariki, Terasoma, and Yamada then generalized this construction to define \begin{equation}\label{eq:ATY} F_T^S(\mathbf{x}) = \varepsilon_T(x_T^S)\end{equation} where $T,S$ are a pair of standard Young tableaux of the same shape, and $x_T^S$ is defined as follows.  Let $\ccind(S)$ be the tableau consisting of the cocharge subscripts of the reading word of $S$, written in the corresponding squares of the diagram of $\lambda$.  Define $$x_T^S=\prod_c x_{T(c)}^{\ccind(S)(c)}$$ where the product is over all squares $c$ in the diagram of $\lambda$. 

 \begin{thm}[\cite{ATY}]
     The polynomials $F_T^S(\mathbf{x})$ form a higher Specht basis for the one-variable coinvariant ring $R_n=\mathbb{C}[x_1,\ldots,x_n]/(e_1,\ldots,e_n)$.
 \end{thm}






\section{General constructions in two sets of variables}\label{sec:general}

For a tableau $T\in \Tab(\lambda)$, fix an ordering on the squares of $\lambda$ and write $T_1,T_2,\ldots,T_n$ to denote the values of $T$ in each of those squares.  Also, for any tuples $c=(c_1,\ldots,c_n)$ and $d=(d_1,\ldots,d_n)$ of nonnegative integers, we write $$x_T^c=x_{T_1}^{c_1} x_{T_2}^{c_2}\cdots x_{T_n}^{c_n}\qquad\text{ and }\qquad y_T^d=y_{T_1}^{d_1} y_{T_2}^{d_2}\cdots y_{T_n}^{d_n}.$$

\begin{definition}
We write $$F_T^{c,d}=\varepsilon_T (x_T^cy_T^d).$$
\end{definition}


\begin{remark}
    All proofs in this section  hold for any number of variables; for instance, we could have three sets of variables $\mathbf{x},\mathbf{y},\mathbf{z}$ and three exponent sequences $c,d,e$ and define $F_T^{c,d,e}=\varepsilon_T (\mathbf{x}^c\mathbf{y}^d\mathbf{z}^e)$ and we would obtain analogous results.
\end{remark}

We will first show the general statement that, assuming the polynomials $F_T^{c,d}$ are independent for $T$ standard of shape $\lambda$, the submodule $V^{c,d}\subseteq \mathbb{C}[\mathbf{x},\mathbf{y}]$ spanned by these polynomials is a copy of the irreducible $S_n$-module $V_\lambda$.  We first recall (see, for instance, \cite{Peel,Sagan}) that the Garnir relations govern the $S_n$-module structure of $V_\lambda$ with respect to the standard Specht basis.


Our first preliminary result is that Lemma 3.3 in \cite{Peel} (or equivalently \cite[Lemma 3.13]{GillespieRhoades}) generalizes to this setting (see \cite{GillespieFull} for the proof):

\begin{prop}\label{prop:Garnir}
We have $\pi F_T^{c,d} = F_{\pi T}^{c,d}$ for any $\pi\in S_n$, and the $F_T^{c,d}$ elements satisfy the Garnir relations.
\end{prop}

It follows that the span $M_\lambda$ of the polynomials $F_T^{c,d}$
is an $S_n$-module.  The Garnir straightening algorithm (see \cite{Sagan}) along with \Cref{prop:Garnir} then implies the following.

\begin{prop}\label{prop:irreducible}
 The subspace $M_\lambda=\mathrm{span}(F_T^{c,d}:T\in \Tab(\lambda))\subseteq \mathbb{C}[\mathbf{x},\mathbf{y}]$ is generated by the subset $\mathbb{B}=\{F_T^{c,d}:T\in \SYT(\lambda)\}$.  Moreover, if the polynomials in $\mathbb{B}$ are independent, then $M_\lambda\cong V_\lambda$ as an $S_n$-module, and there is an action-preserving isomorphism induced by the bijection $F_T^{c,d}\to F_T$ that sends the basis $\mathbb{B}$ to the ordinary Specht basis.
\end{prop}


\subsection{Conditions for independence}

\Cref{prop:irreducible} shows that if the basis elements are independent, then $M_\lambda$ is a copy of the irreducible Specht module.  We now state some conditions for determining independence in quotients by $S_n$-invariant ideals or in the full polynomial ring (see \cite{GillespieFull} for proofs). 

\begin{prop}\label{prop:nonzero}
    The polynomials in $\mathbb{B}=\{F_T^{c,d}:T\in \SYT(\lambda)\}$ are independent in $\mathbb{C}[\mathbf{x},\mathbf{y}]/I$ if and only if some particular element $F_T^{c,d}$ is nonzero in the quotient (i.e., not in the ideal $I$).  
\end{prop}


We now state a sufficient condition for one of the $F_T^{c,d}$ elements to be nonzero in the full polynomial ring $\mathbb{C}[\mathbf{x},\mathbf{y}]$ that we will use below.

\begin{prop}\label{prop:SSYT-values}
    Suppose that $c=(c_1,\ldots,c_n)$ and $d=(d_1,\ldots,d_n)$ are tuples of nonnegative integers, with their ordering corresponding to a chosen ordering of the boxes of the Young diagram of a partition $\lambda$ of $n$.  Suppose further that there exists a total ordering on the possible pairs of values $(m,n)$ that arise among the pairs $c_i,d_i$ such that, when the boxes of $\lambda$ are filled in order with $(c_i,d_i)$, it forms a semistandard Young tableau with respect to the ordering on the pairs.

    Then for any standard tableau $T$ of shape $\lambda$, the polynomial $F_{T}^{c,d}$ is nonzero in $\mathbb{C}[x,y]$.
\end{prop}

\begin{example}\label{ex:horz-strips}
    Suppose we choose the ordering on pairs $(m,n)$ by $(m,n)>(x,y)$ if and only if either $m>x$ or $m=x$ and $n>y$.  Consider the following tableau below at left, where pairs $(m,n)$ are written as $mn$:
    $$\begin{ytableau}
        22 & 22 \\
        21 & 21 & 22 & 22 \\
        01 & 10 & 10 & 13 \\
        00 & 00 & 00 & 01 & 10
    \end{ytableau}\hspace{2cm} \begin{ytableau}
        12 & 13 \\ 
        10 & 11 & 14 & 15 \\
        4 &  6 & 7  & 9 \\
        1 & 2 & 3 & 5 & 8
    \end{ytableau}$$
    Also, for simplicity let $T$ be the tableau above at right that standardizes the above (the choice of $T$ does not matter in the proof). 
    Then $x_T^c y_T^d=x_{15}^2 y_{15}^2 \cdot x_{14}^2y_{14}^2\cdot x_{13}^2y_{13}^2\cdot x_{12}^2y_{12}^2\cdot x_{11}^2y_{11}\cdots $.  In the proof \cite{GillespieFull}, this leading term does not cancel after applying $\varepsilon_T$.
\end{example}



\section{New basis for hook-shape Garsia--Haiman modules}\label{sec:hooks}

Throughout this subsection, we set $\mu=(n-k+1,1^{k-1})$ to be the \textit{hook shape} of height $k$ and size $n$.  We now define our higher Specht polynomials for $\DR_\mu$.

\begin{definition}
  For $S\in \SYT(n)$, define the \textbf{$\mu$-cocharge tableau} of $S$, written $\ccind_\mu(S)$, to be the tableau of shape $\sh(S)$ that has $0$ in the squares occupied by $1,2,\ldots,n-k+1$ in $S$, and where for $n-k+1,n-k+2,\ldots,n$ we apply the cocharge algorithm in reverse reading order, incrementing the cocharge label whenever we reach an entry $i+1$ that is above $i$. The \textbf{$\mu$-cocharge} of $S$, written $\cc_\mu(S)$, is the sum of the entries of $\ccind_\mu(S)$.
\end{definition}

\begin{definition}
    For $S\in \SYT(n)$, define the \textbf{reverse $\mu$-cocharge tableau}, which we write $\ccind'_\mu(S)$, to be the tableau of shape $\sh(S)$ that has $0$ in the squares occupied by $n-k+1,\ldots,n$ in $S$, and where for $n-k+1,n-k,n-k-1,\ldots,1$ we calculate cocharge in \textit{reverse}, that is, in forward reading order and labeling the numbers from biggest to smallest, incrementing the label on $i$ when it is below $i+1$.     The \textbf{$\mu$-reverse cocharge}, written $\cc'_\mu(S)$, is the sum of the entries of the reverse $\ccind'_\mu(S)$.
\end{definition}


\begin{example}\label{ex:cctab}
  Suppose $n=8$, $S$ is the tableau below at left,  and $k=5$.  Then $\ccind_\mu(S)$ is the tableau below at middle and $\ccind'_\mu(S)$ is at right: $$\young(57,368,124) \hspace{2cm} \young(12,012,000) \hspace{2cm}\young(00,000,110)$$ Thus we have $\cc_\mu'(S)=1+1=2$ and $\cc_\mu(S)=1+1+2+2=6$.
\end{example}

In \cite{GillespieFull}, we show that these statistics agree with maj and comaj from Equation \eqref{eq:stembridge}:

\begin{lemma}\label{lem:degrees}
    We have that $\cc_\mu(S)=\comaj_{n-k+1,n}(S)$ and $\cc_\mu'(S)=\maj_{1,n-k+1}(S)$ for any standard Young tableau $S$.
\end{lemma}

\begin{definition}
  Let $(T,S)\in \mathrm{Tab}(\lambda)\times \SYT(\lambda)$. The $\mu$-\textbf{monomial} of $(T,S)$ is $$\mathbf{xy}_T^S:=\prod_{b\in \lambda} x_{T(b)}^{\ccind_\mu(S)(b)} y_{T(b)}^{\ccind'_\mu(S)(b)}.$$  
Then we define $F_{T}^S(x,y)=\varepsilon_T \mathbf{xy}_T^S.$
\end{definition}

\begin{example}
    Keeping $S$ as in \Cref{ex:cctab} and $k=5$, we set $$T=\young(68,247,135)\hspace{1cm}\text{and} \hspace{1cm}S=\young(57,368,124)$$ and we find $\mathbf{xy}^S_T=x_{4}x_6x_7^2x_8^2\cdot y_1y_3.$
\end{example}


In \cite{GillespieFull}, we show that the values of $c:=\ccind_\mu(S)$ and $d:=\ccind'_\mu(S)$ form an SSYT according to a particular ordering on the pairs of values, so that it satisfies the conditions of \Cref{prop:SSYT-values}.  Then, we obtain the following by \Cref{prop:irreducible}.

\begin{prop}\label{prop:if-basis-then-specht}
The polynomials $F_T^S(\mathbf{x},\mathbf{y})$ for $T\in \SYT(\lambda)$ span a copy of $V_\lambda$ in $\mathbb{C}[\mathbf{x},\mathbf{y}]$.
\end{prop}


We now note that we can recover the one-variable higher Specht polynomials from our construction by a substitution and a degree shift.

\begin{lemma}\label{lem:x-inverse}
    Consider the map to the Laurent polynomial ring (also defined in \cite{HookBases}) $$\psi:\mathbb{C}[x_1,\ldots,x_n,y_1,\ldots,y_n]\to \mathbb{C}[x_1,\ldots,x_n,x_1^{-1},\ldots,x_n^{-1}]$$ that sends $y_i\mapsto x_i^{-1}$.   Then if $q$ is the largest entry in $\ccind'_\mu(S)$, we have $$\psi(F_T^S(\mathbf{x},\mathbf{y}))\cdot x_1^qx_2^q\cdots x_n^q=F_T^S(\mathbf{x})$$ where $F_T^S(\mathbf{x})$ is the one-variable higher Specht polynomial  defined in \cite{ATY} (see Equation \eqref{eq:ATY}).
\end{lemma}


  For notational brevity, we define $M_\lambda^S$ to be the copy of $V_\lambda$ inside $\mathbb{C}[\mathbf{x},\mathbf{y}]$ generated by the polynomials $F_T^S$ over all $T\in\SYT(\sh(S))$.  We can use \Cref{lem:x-inverse} and the results of \cite{ATY} to show these copies are distinct for different $S$.

\begin{lemma}\label{lem:disjoint}
    If $S_1$ and $S_2$ are two distinct standard Young tableaux of size $n$ and shapes $\lambda$ and $\rho$ respectively, then $M_\lambda^{S_1}$ and $M_{\rho}^{S_2}$ are disjoint submodules of $\mathbb{C}[\mathbf{x},\mathbf{y}]$.
\end{lemma}

We now recall a generating set for the ideal $I_\mu$ defining $\DR_\mu$ in the hook shape case, and a sub-ideal defined in \cite{HookBases}.

\begin{prop}[\cite{HookBases}] \label{prop:ideal}
    If $\mu=(n-k+1,1^{k-1})$, the ideal $I_\mu$ is generated by:
    \begin{enumerate}
        \item The elementary symmetric functions $e_1(\mathbf{x}),\ldots,e_n(\mathbf{x})$,
        \item The elementary symmetric functions 
        $e_1(\mathbf{y}),\ldots,e_n(\mathbf{y})$,
        \item All products $x_{i_1}\cdots x_{i_k}$ of $k$ distinct $x$ variables,
        \item All products $y_{j_1}\cdots y_{j_{n-k+1}}$ of $n-k+1$ distinct $y$ variables.
        \item The products $x_iy_i$ for $i=1,\ldots,n$.
    \end{enumerate}
\end{prop}

\begin{definition}[\cite{HookBases}]
    Define $\mathcal{I}_k$ to be the ideal generated by  the last three bullet points in the definition above, that is, just the products $x_{i_1}\cdots x_{i_k}$, $y_{j_1}\cdots y_{j_{n-k+1}}$, and $x_iy_i$.  Define $$\mathcal{P}_{n}^{(k)}=\mathbb{C}[\mathbf{x},\mathbf{y}]/\mathcal{I}_k.$$
\end{definition}

It is not hard to see that the elements $F_T^S(\mathbf{x},\mathbf{y})$ are nonzero in $\mathcal{P}_n^{(k)}$. To show they descend to a basis of $\DR_\mu$, we recall  the $S_n$-invariant basis $e_\nu^{(k)}$ of $(\mathcal{P}_n^{(k)})^{S_n}$ from \cite{HookBases}.

\begin{definition}
    For $d=1,\ldots,n$, define $$e_d^{(k)}=\begin{cases}
        e_d(x_1,\ldots,x_n) & d\le k-1 \\
        e_{n-d}(y_1,\ldots,y_n) & d\ge k
    \end{cases}$$ and $e_\nu^{(k)}=\prod_i e_{\nu_i}^{(k)}$.
\end{definition}

As a stepping stone to proving our main result, we show the following \cite{GillespieFull}.

\begin{prop}\label{prop:intermediate-basis}
    The set of polynomials $\{F_T^S(\mathbf{x},\mathbf{y})\cdot e_\nu^{(k)}\}$, ranging over all pairs $(T,S)$ of SYT's of the same shape and all (possibly empty) $\nu$ with $\nu_1\le n$, forms a higher Specht basis of $\mathcal{P}_n^{(k)}$.
\end{prop}


We finally conclude our main result. 

\begin{MainTheorem}
    The set $\{F_T^S(\mathbf{x},\mathbf{y})\}$ is a higher Specht basis for $\DR_\mu$.
\end{MainTheorem}



\section{Future directions and observations}\label{sec:future}

One corollary of \Cref{thm:main} is that we can express the graded Frobenius series for hook shapes $\mu=(n-k+1,1^{k-1})$ as
$$\grFrob_{q,t}(\DR_\mu)=\sum_{S\in \SYT(n)} q^{\cc_\mu(S)}t^{\cc'_\mu(S)}s_{\sh(T)}.$$
It is a more general open problem than Problem \ref{prob:nfact} to find a combinatorial Schur expansion for the Frobenius series of $\DR_\mu$, and one possible route would be to generalize these new $\cc_\mu$ and $\cc'_\mu$ statistics to more general shapes $\mu$. 


For the diagonal coinvariant ring (Problem \ref{prob:basis-DR}), it would be interesting to see if a higher Specht basis could be constructed by combining the general tools from \Cref{sec:general} with the new basis of Carlsson and Oblomkov \cite{CarlssonOblomkov}, or by interpreting parking functions as certain pairs of tableaux.  

To conclude, we provide higher Specht bases for $\DR_2$ and $\DR_3$.  The Shuffle theorem (conjectured in \cite{HHLRU}, proven in \cite{shuffle}) gives a combinatorial formula for the graded Frobenius series of $\DR_n$, which is also equal to a Macdonald eigenoperator $\nabla$ applied to the elementary symmetric function $e_n$.  
We used Sage \cite{sage} to expand $\nabla e_2$ and $\nabla e_3$ in terms of Schur functions, giving us the decompositions of $\DR_2$ and $\DR_3$ into irreducibles, and used that data to guess and verify higher Specht bases.

\begin{example}
For $\DR_2$, a higher Specht basis is: $1, x_2-x_1, y_2-y_1$.
\end{example}

\begin{example}
    For $\DR_3$, its graded Frobenius series from the Shuffle theorem is $$\nabla e_3 = s_{(3)}+(q+t)s_{(2,1)}+(q^2+qt+t^2)s_{(2,1)}+qt s_{(1,1,1)}+(q^3+t^3+q^2t+t^2q)s_{(1,1,1)}.$$  The following set of polynomials, corresponding to each term in this expansion left to right, form a higher Specht basis for $\DR_3$:
    $$1,\hspace{0.5cm} x_2-x_1,\,x_3-x_1, \hspace{0.5cm}y_2-y_1,\,y_3-y_1, \hspace{0.5cm} \varepsilon_{\text{\tiny$ \young(3,12)$}}x_3x_2,\,\varepsilon_{\text{\tiny$ \young(2,13)$}}x_2x_3, \hspace{0.5cm} \varepsilon_{\text{\tiny$ \young(3,12)$}}x_3y_2,\,\varepsilon_{\text{\tiny$ \young(2,13)$}}x_2y_3$$
    $$ \varepsilon_{\text{\tiny$\young(3,12)$}}y_3y_2,\,\varepsilon_{\text{\tiny$\young(2,13)$}}y_2y_3, \hspace{0.5cm} \varepsilon_{\text{\tiny$ \young(3,2,1)$}}x_3y_1, \hspace{0.5cm} (x_3-x_2)(x_3-x_1)(x_2-x_1),$$
    $$(y_3-y_2)(y_3-y_1)(y_2-y_1), \hspace{0.5cm} \varepsilon_{\text{\tiny$ \young(3,2,1)$}}x_3^2y_1, \hspace{0.5cm}\varepsilon_{\text{\tiny$\young(3,2,1)$}}x_3y_1^2$$
\end{example}


\acknowledgements{
The author thanks Kelvin Chan, Bryan Gillespie, Sean Griffin, Jake Levinson, and Brendon Rhoades for helpful mathematical conversations pertaining to this project.  Special thanks to Jake Levinson for his support and encouragement that made it possible.}

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