\documentclass[ALCO,Unicode]{cedram} \usepackage{wasysym} \usepackage{youngtab} \usepackage{ytableau} \newtheorem{open}{Problem} \DeclareMathOperator{\arm}{arm} \DeclareMathOperator{\cc}{cc} \DeclareMathOperator{\CD}{CD} \DeclareMathOperator{\ch}{ch} \DeclareMathOperator{\col}{col} \DeclareMathOperator{\CStat}{CStat} \DeclareMathOperator{\cyc}{cyc} \DeclareMathOperator{\cw}{cw} \DeclareMathOperator{\flip}{flip} \DeclareMathOperator{\Frob}{Frob} \DeclareMathOperator{\gr}{gr} \DeclareMathOperator{\ID}{ID} \DeclareMathOperator{\invcode}{invcode} \DeclareMathOperator{\InvPlot}{InvPlot} \DeclareMathOperator{\majcode}{majcode} \DeclareMathOperator{\MajPlot}{MajPlot} \DeclareMathOperator{\rev}{rev} \DeclareMathOperator{\SSYT}{SSYT} \newcommand{\SYT}{\mathrm{SYT}} \DeclareMathOperator{\Standardize}{Standardize} \DeclareMathOperator{\stat}{stat} \newcommand{\ccind}{\mathrm{ccTab}} \newcommand{\sh}{\mathrm{sh}} \newcommand{\sgn}{\mathrm{sgn}} \newcommand{\maj}{\mathrm{maj}} \newcommand{\comaj}{\mathrm{comaj}} \newcommand{\tH}{\widetilde{H}} \newcommand{\fancy}[1]{\mathcal{#1}} \newcommand{\DR}{\mathrm{DR}} \newcommand{\CC}{\mathbb{C}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\Tab}{\mathrm{Tab}} \newcommand{\Des}{\mathrm{Des}} \newcommand{\des}{\mathrm{des}} \newcommand{\grFrob}{\mathrm{grFrob}} \newtheorem*{MainTheorem}{Theorem~\ref{thm:main}} \title[Higher Specht polynomials under the diagonal action]{Higher Specht polynomials under the diagonal action} \author[\initial{M.} Gillespie]{\firstname{Maria} \lastname{Gillespie}} \address{Colorado State University \\ Department of Mathematics \\ 1874 Campus Delivery \\ Fort Collins, CO 80523 (USA) } \email{Maria.Gillespie@colostate.edu} \thanks{The author was partially supported by NSF DMS award number 2054391.} \keywords{Diagonal harmonics, diagonal coinvariants, representation theory of the symmetric group, Young tableaux, Young symmetrizers, Garsia-Haiman modules, polynomial rings} \subjclass{05E10, 05E05, 05E40, 20C30} \begin{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 of the general theory, 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. \end{abstract} \begin{document} \maketitle \section{Introduction} \noindent The \textit{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 $$\pi\cdot f(x_1,\ldots,x_n,y_1,\ldots,y_n)=f(x_{\pi(1)},\ldots,x_{\pi(n)},y_{\pi(1)},\ldots,y_{\pi(n)}),$$ has been the subject of much study in modern algebraic combinatorics. The ring of \textbf{diagonal coinvariants}, which we call $\DR_n$ in this paper, is the quotient of $\mathbb{C}[\mathbf{x},\mathbf{y}]$ by the ideal generated by the $S_n$-invariants with no constant term. It 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 some of the 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 advancements, 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 (though it is not a higher Specht basis as defined below). In this paper we explore a new approach to understanding and finding bases for these quotients under the diagonal action via the method of \textit{higher Specht polynomials}. 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} for a thorough introduction to the Young tabloid construction). This construction is equivalent 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 Figure~\ref{fig:Specht}.) \end{definition} \begin{figure}[ht] \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} Writing $\Tab(\lambda)$ for the set of all fillings of the boxes of the Young diagram of $\lambda$ with $1,2,\ldots,n$ in some order, 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}, described in Section~\ref{sec:general} below, 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 $F_T=\varepsilon_T(x_T^r)$ 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. 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 a monomial whose subscripts are given by the entries of $T$ and whose exponents are determined by the \textit{cocharge} algorithm on the corresponding boxes in $S$. Their main result was as follows. \begin{thm}[\cite{ATY}] The polynomials $F_T^S(\mathbf{x})$ form a basis for the one-variable coinvariant ring $R_n=\mathbb{C}[x_1,\ldots,x_n]/(e_1,\ldots,e_n)$, where $e_i$ is the $i$-th elementary symmetric polynomial. \end{thm} The importance of this basis is that it is a \textit{higher Specht basis}, which respects the decomposition into irreducible $S_n$ representations, and which we define precisely in this paper as follows. \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 a copy 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 $i$), \item There is a bijection from $\mathbb{B}_{\lambda,i}$ to the set of ordinary Specht polynomials $F_T$ for shape $\lambda$, that preserves the $S_n$ action with respect to each basis. \end{enumerate} \end{definition} The terminology ``higher Specht'' in this setting is also due to the fact that the polynomials $F_T^S(\mathbf{x})$ are in general higher-degree analogs of the ordinary Specht polynomials. 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}). They proved their construction was a basis in the former case and conjectured for the latter, proving it 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 the $S_n$ action on the basis is different.) 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. In this paper, we establish a general theory for constructing higher Specht polynomials in $\mathbb{C}[\mathbf{x},\mathbf{y}]$ under the diagonal action. In particular, we show that for certain conditions on the sequences of exponents $c=(c_1,\ldots,c_n)$ and $d=(d_1,\ldots,d_n)$, the set of polynomials $$F_T^{c,d}:=\varepsilon_T (x_1^{c_1}\cdots x_n^{c_n}y_1^{d_1}\cdots y_n^{d_n}),$$ for all standard Young tableaux $T$ of a fixed shape $\lambda$, form a higher Specht basis for a copy of the irreducible Specht module $V_\lambda$. We then use our theory to develop our main result, a higher Specht basis for the Garsia--Haiman modules for hook shapes: \begin{thm}\label{thm:main} Suppose $\mu=(n-k+1,1^k)$ is a hook shape of height $k$ and $T,S$ denotes a pair of standard Young tableaux of the same shape $\lambda$. Then there are modified cocharge labelings (see Section~\ref{sec:hooks}) $\ccind_\mu(S)$ and $\ccind_\mu'(S)$ such that if $$\mathbf{xy}_T^S=\prod_{b\in \lambda}x^{\ccind_\mu(S)(b)}_{T(b)}y_{T(b)}^{\ccind'_\mu(S)(b)}\hspace{0.5cm}\text{ and }\hspace{0.5cm}F_T^S(\mathbf{x},\mathbf{y})=\varepsilon_T(\mathbf{xy}_T^S),$$ then the set of polynomials $\{F_T^S(\mathbf{x},\mathbf{y})\}$ ranging over all pairs $T,S$ of standard Young tableaux of the same shape forms a higher Specht basis for $\DR_\mu$. \end{thm} We have verified this computationally on Sage up to $n=8$ as well, and we provide the proof in Section~\ref{sec:hooks}. The proof relies on some of the structural results for hook shape Garsia--Haiman modules established in~\cite{HookBases} by Adin, Remmel, and Roichman. In particular, we multiply the $F_T^S(\mathbf{x},\mathbf{y})$ constructions by certain diagonally symmetric polynomials to construct a higher Specht basis for an intermediate quotient $\mathcal{P}_n^{(k)}$ between $\mathbb{C}[\mathbf{x},\mathbf{y}]$ and $\DR_\mu$. The paper is organized as follows. In Section~\ref{sec:background}, we establish notational and mathematical background on higher Specht bases and the Frobenius series of $\DR_\mu$, and also provide a bijective proof of a known equidistribution result on the major index and cocharge. This does not to our knowledge directly appear in the literature as a bijective proof, so we include it for completeness, as we will use similar methods to construct our new generalized cocharge statistics. In Section~\ref{sec:general}, we build the theory of higher Specht polynomials in two sets of variables, and show that many of the constructions from one set of variables apply to any number of sets of variables. In Section~\ref{sec:hooks}, we prove Theorem~\ref{thm:main}. Finally, in Section~\ref{sec:future}, we outline several possible directions for exploration that this paper opens up. We also give example higher Specht constructions in special cases for the diagonal coinvariant rings, in particular for $\DR_2$ and $\DR_3$. \section{Definitions and background}\label{sec:background} \subsection{Tableaux and statistics} The \textbf{Young diagram} of a partition $\lambda=(\lambda_1,\ldots,\lambda_k)$, written so that $\lambda_1\ge \lambda_2\ge \cdots$, is the left-justified partial grid of squares having $\lambda_i$ squares in the $i$-th row from the bottom (we use the ``French'' convention for diagrams in this paper). We write $n=|\lambda|$ for the number of squares of $\lambda$, and $\ell(\lambda)=k$. A \textbf{semistandard Young tableau}, or SSYT, of shape $\lambda$ is a way of filling each square of $\lambda$ with a positive integer such that the rows are weakly increasing left to right and the columns are strictly increasing from bottom to top. In general we will also allow entries from any totally ordered set (not necessarily the positive integers), with the same definition. For an SSYT $T$, we write $\sh(T)=\lambda$ if $T$ has shape $\lambda$. The \textbf{reading word} of any tableau is the word formed by concatenating the rows from top to bottom (and this ordering of the squares is called \textbf{reading order}). For instance, the reading word of the tableau at middle in Figure~\ref{fig:tableaux} is $748251369$. The \textbf{content} of a tableau or word is the tuple $(m_1,m_2,\ldots)$ where $m_i$ is the number of times $i$ appears in the tableau or word. For instance, the tableau at left in Figure~\ref{fig:tableaux} has content $(3,2,2,1,1)$. An SSYT is \textbf{standard} if each of the entries $1,2,\ldots,n$ is used exactly once (where~$n=|\lambda|$). 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 $n$. The \textbf{standardization} of an SSYT is the standard Young tableau formed by labeling the boxes $1,2,3,\ldots,n$ in a way that respects the ordering of the labels in the SSYT and breaks ties in reading order. It is straightforward to verify that the standardization of an SSYT is an SYT. We recall the well-known fact that permutations of $1,\ldots,n$ are in bijection with pairs $(P,Q)$ of SYT's with the same shape as each other, via the \textbf{RSK bijection}. Thus there are $n!$ such pairs. In particular, we will need the \textbf{RSK insertion} algorithm: given a word $w$, we define its \textbf{insertion tableau} $T$ by starting with the empty tableau, and inserting $w_1,w_2,\ldots$ inductively in the following manner. Given a semistandard young tableau $S$, we insert the letter $a$ into the bottom row $r$, by either placing it at the end of row $r$ if it is greater than every element of $r$, or replacing entry $b$ of $r$ by $a$ where $b$ is the leftmost entry greater than $a$. Then we insert $b$ into the second row in the same manner, and so on. (See~\cite{Fulton} for details and examples.) A \textbf{standard general tableau} $T$ of shape $\lambda$ is a way 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 write $\Tab(\lambda)$ for the set of all standard general tableaux of shape $\lambda$, and note that $|\Tab(\lambda)|=n!$. (See Figure~\ref{fig:tableaux}.) \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.} \label{fig:tableaux} \end{figure} \begin{definition} A \textbf{descent} of a standard Young tableau $T$ is an entry $i$ such that $i+1$ occurs weakly to the left of (and necessarily above) $i$ in $T$. We write $\Des(T)$ for the set of all descents of $T$ and $\des(T)$ for the number of descents. The \textbf{major index} of a standard Young tableau $T$ is defined as the sum of the descents: $$\maj(T)=\sum_{i\in \Des(T)} i.$$ In a permutation (for instance a reading word of a tableau), we say a \textbf{descent} is an entry $i$ occurring to the right of $i+1$ in the word. Note that this is the inverse of the usual notion of descents for words, but matches our notation for tableaux here. \end{definition} \begin{example} The middle tableau in Figure~\ref{fig:tableaux} has descents $1$, $3$, and $6$. Therefore, its major index is $1+3+6=10$. \end{example} It is well-known (due to Lusztig and Stanley~\cite{Stanley}) 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 major index $d$. We will express this fact in terms of the graded Frobenius character in Section~\ref{sec:Frobenius}. Lascoux and Sch\"utzenberger, in~\cite{LascouxSchutzenberger} generalized the major index to a statistic on semistandard Young tableaux called \textbf{cocharge}. They used it to obtain the graded decomposition of the Garsia--Procesi modules $R_\mu$ into irreducible $S_n$-modules. We recall its definition here. \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} It is known~\cite{LascouxSchutzenberger} that the RSK insertion tableau of $\pi$ has the same cocharge as~$\pi$. In particular, cocharge is preserved under \textbf{Knuth equivalence} of words (which is also preserved by RSK), defined as the reachability of one word from another via \textit{elementary Knuth moves} of the form $$bac \leftrightarrow bca \hspace{1cm} \text{or} \hspace{1cm} acb\leftrightarrow cab$$ where $a< b< c$ and the three entries are consecutive in the word. \subsection{A weight-preserving bijection} We now give a bijection $\phi:\SYT(\lambda)\to \SYT(\lambda)$ (for any shape $\lambda$) that sends $\cc$ to $\maj$ and preserves the number of descents, and that will come up in our work on hook shapes in Section~\ref{sec:hooks}. We will require the following two operations on words. \begin{definition} The \textbf{flip} of a permutation $\pi\in S_n$, written $\flip(\pi)$, is the permutation formed by replacing $\pi_i$ with $n+1-\pi_i$ for each entry $\pi_i$. The \textbf{reverse} of $\pi$, written $\rev(\pi)$, is formed by writing $\pi$ backwards: $\pi_n\pi_{n-1}\cdots \pi_1$. \end{definition} \begin{example} The flip of $836791245$ is $274319865$. The reverse of the latter is $568913472$. \end{example} \begin{definition} Given an SYT $T$, let $w$ be its reading word. Apply the RSK insertion algorithm to $\rev(\flip(w))$. We define $\phi(T)$ to be the resulting tableau. \end{definition} \begin{example} Let $T$ be the following tableau: $$\young(8,3679,1245)$$ Its reading word is $836791245$. Applying flip and reverse yields $568913472$, whose RSK insertion tableau is $\phi(T)$, shown below at left with its tableau of cocharge labels shown at right: $$\young(5,3689,1247) \hspace{2cm} \young(2,1233,0012)$$ Notice that the descents of $T$ are $2,5,7$ and the major index is $2+5+7=14$. The descents of $\phi(T)$ are $2,4,7$, and its cocharge is the sum of the cocharge labels, which is $14$. \end{example} \begin{lemma} The map $\phi$ is shape-preserving: $\sh(T)=\sh(\phi(T))$. \end{lemma} \begin{proof} The shape of an RSK insertion is determined by the length of the longest increasing subsequence~\cite{Fulton}, and then the longest pair of disjoint increasing subsequences, and so on. These statistics are preserved by performing $\rev$ and $\flip$ in succession (each of which simply switch increasing to decreasing). \end{proof} \begin{lemma} The map $\phi$ is its own inverse (and is hence a bijection). \end{lemma} \begin{proof} Let $T$ be a standard Young tableau with reading word $w$. Let $\pi=\rev(\flip(w))$, and let $P$ be the RSK insertion tableau of $\pi$. Let $v$ be the reading word of $P$. Then~$v$ is Knuth equivalent to $\pi$ since the Knuth equivalence class is determined by its insertion tableau~\cite{Fulton}, and the insertion tableau of $v$ is $P$ as well. Thus $\rev(\flip(v))$ is Knuth equivalent to $w$, since we may simply flip and reverse the Knuth moves used to form an equivalence. Thus the RSK insertion tableau of $\rev(\flip(v))$ is equal to $T$. Hence~$\phi(\phi(T))=T$. \end{proof} \begin{prop} We have $\cc(\phi(T))=\maj(T)$ for any standard Young tableau $T$. \end{prop} \begin{proof} Let $w$ be the reverse of the flip of the reading word of $T$. Then $\cc(\phi(T))=\cc(w)$ since RSK insertion preserves cocharge. The cocharge subscripts on $w$ may be alternatively defined by labeling the $1$ with~$0$, and then looking for $2,3,4,\ldots$ and whenever the next number is to the left of the previous, we increment the subscript, and if it is to the right, we do not increment, for instance: $$5_2 6_2 8_3 9_3 1_0 3_1 4_1 7_2 2_0$$ This is the same as the \textit{charge} of the reversed word of $w$ (which is the flip of the reading word of $T$), in which we increment if we move to the right and do not otherwise: $$2_0 7_2 4_1 3_1 1_0 9_3 8_3 6_2 5_2$$ The sum of the charge labels can be computed as follows: for all numbers after the first rightward move, we add $1$. Then we add $1$ for all numbers after the next rightward move, and so on. If we moved rightwards from $j$ to $j+1$, this means we added $n-(j-1)=n+1-j$ to the total charge for each such move. The rightward moves in the flipped word correspond to the descents in the original word (in the running example, $836791245$), and those descents are precisely the values $n+1-j$ by the definition of the flip bijection. So we have shown that $\cc(\rev(\flip(w))=\maj(w)$. This completes the proof. \end{proof} \begin{lemma} The map $\phi$ preserves the number of descents of the tableau. \end{lemma} \begin{proof} First, note that the number of descents is preserved under RSK because $i$ and $i+1$ cannot switch positions under any Knuth move in a permutation (see the definition of Knuth equivalence above, or from~\cite{Fulton}). Then, the descent pairs become ascents under $\flip$, which then become descents again under $\rev$. This completes the proof. \end{proof} \subsection{Symmetric functions and graded Frobenius series}\label{sec:Frobenius} The two bases of the ring of symmetric functions $$\Lambda_{\mathbb{C}}(x_1,x_2,\ldots)=\lim_{\leftarrow} \Lambda_{\mathbb{C}}(x_1,x_2,\ldots,x_n)=\lim_{\leftarrow}\mathbb{C}[x_1,\ldots,x_n]^{S_n}$$ that will be used in this paper are the elementary and Schur symmetric functions. The elementary are defined as $e_d(x_1,x_2,\ldots)=\sum_{i_1(x,y)$ if and only if either $m>x$ or $m=x$ and $n>y$. Consider the following tableau, where pairs $(m,n)$ are written as $mn$: $$\begin{ytableau} 22 & 22 \\ 21 & 21 & 22 & 22 \\ 01 & 10 & 10 & 13 \\ 00 & 00 & 00 & 01 & 10 \end{ytableau}$$ Also, for simplicity let $T$ be the tableau that standardizes the above (the choice of $T$ will not matter in the proof): $$\begin{ytableau} 12 & 13 \\ 10 & 11 & 14 & 15 \\ 4 & 6 & 7 & 9 \\ 1 & 2 & 3 & 5 & 8 \end{ytableau}$$ 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 below, we will show that this leading term does not cancel after applying $\varepsilon_T$. \end{example} \begin{proof} We will show that any term $\tau\sigma$ arising in $\varepsilon_T$ that fixes the leading term $x_T^c y_T^d$ actually fully lies in $R(T)$, so that $\tau$ must be the identity permutation and the sign is positive. This shows that the leading term has a nonzero coefficient, so the polynomial is nonzero. We first make some notational simplifications; the proof will not depend on the choice of $T$, so we may assume $T$ is the standardization of the tableau labeled by the $(c_i,d_i)$ pairs and that this standardization order is the ordering on the boxes in $\lambda$ as well. This means that $(c_1,d_1)\le (c_2,d_2)\le \cdots \le (c_n,d_n)$ with respect to the total ordering on pairs, and for the purposes of showing that a coefficient is nonzero, we can change the pairs $(c_i,d_i)$ to numbers $b_1\le \ldots\le b_n$ such that $b_i=b_{i+1}$ if and only if $(c_i,d_i)=(c_{i+1},d_{i+1})$. We can then replace the monomial $x_T^cy_T^d$ with $x_T^b$ and reduce to the standard action of $S_n$ rather than the diagonal action. For instance, in Example~\ref{ex:horz-strips}, the tableau of pairs can be replaced with the SSYT below: $$\begin{ytableau} 6 & 6 \\ 5 & 5 & 6 & 6 \\ 2 & 3 & 3 & 4 \\ 1 & 1 & 1 & 2 & 3 \end{ytableau}$$ and the monomial $x_T^b$ is $$x_{15}^6x_{14}^6x_{13}^6x_{12}^6x_{11}^5x_{10}^5x_{9}^4x_{8}^3x_7^3x_6^3x_5^2x_4^2x_3x_2x_1.$$ We first show that if $\tau\in C(T)$ and $\sigma \in R(T)$ and $\tau\sigma(x_T^b)=x_T^b$, then if $n-r,n-r+1,\ldots,n$ are the entries in the row of $n$ such that $b_{n-r}=b_{n-r+1}=\cdots = b_n$, then $\tau\sigma(n)$ is among $n-r,\ldots,n$. (In the example above, $b_{14}=b_{15}=6$ are the entries in the row of $15$ with the largest exponent, and we want to show that $\tau\sigma(15)$ is either~$14$ or $15$.) Indeed, in the example, since it fixes $x_T^b$, $\tau\sigma$ must send $15$ to either $12, 13, 14$, or~$15$ so that it retains its exponent. But $\sigma$ is row preserving, and so it cannot send any element in a row below the $15$, like $9$ or $5$, to $12, 13, 14,$ or $15$. Since $\tau$ is column preserving, it cannot send $15$ to the horizontal strip of elements with exponent $6$ without $\sigma$ already sending it there. In general, this reasoning means $\tau \sigma$ must permute the entries $n-r,\ldots,n$. By a similar argument, $\tau\sigma$ must permute the entries in the next row up that also have exponent $b_n$, and so on. Therefore, all of the entries for which $b_i=b_n$ are sent to other entries in their row with the same $b$ value. Since these entries are fixed amongst themselves, we then consider the second highest value of $b_i$ (in this example, $5$), and repeat the argument, and so on. Thus $\tau\sigma\in R(T)$ as desired. \end{proof} \section{A higher Specht basis for the 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 the hook shape Garsia--Haiman modules. \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}, written $\ccind'_\mu(S)$, 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 $$\young(57,368,124),$$ and $k=5$. Then $\ccind_\mu(S)$ is the tableau $$\young(12,012,000),$$ and $\ccind'_\mu(S)$ is the tableau $$\young(00,000,110).$$ Thus we have $\cc_\mu'(S)=1+1=2$ and $\cc_\mu(S)=1+1+2+2=6$. Notice also that $n-k+1=4$ and $\maj_{1,4}(S)=2$ and $\comaj_{4,7}(S)=(8-4)+(8-6)=6$ (see Definition~\ref{def:maj-comaj}). \end{example} \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{proof} To compute $\cc_\mu(S)$, notice that we begin at $n-k+1$ and label with a subscript of $0$, and then continue to find $n-k+2,n-k+3,\ldots,n$. If we are at a descent, we increment the subscript label at the next step. Thus each descent $d$ we encounter contributes $n-d$ to the total cocharge, and so the total is equal to $\comaj_{n-k+1,n}(S)=\sum(n-d)$ where the sum is over descents $d$ such that $n-k+1\le d