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\DeclareMathOperator{\charge}{charge}
\DeclareMathOperator{\cocharge}{cocharge}
\DeclareMathOperator{\RSK}{\stackrel{RSK}{\sim}}
\DeclareMathOperator{\knuth}{\stackrel{\sim}{Knuth}}
\DeclareMathOperator{\syt}{SYT}
\DeclareMathOperator{\ssyt}{SSYT}
\DeclareMathOperator{\ctype}{ctype}
\DeclareMathOperator{\Frob}{Frob}
\DeclareMathOperator{\Hilb}{Hilb}
\DeclareMathOperator{\sh}{Sh}
\DeclareMathOperator{\shape}{shape}
\DeclareMathOperator{\rev}{rev}
\DeclareMathOperator{\cc}{cc}
\DeclareMathOperator{\cw}{c}
\DeclareMathOperator{\osp}{OSP}
\DeclareMathOperator{\des}{Des}
\DeclareMathOperator{\ch}{ch}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\rw}{rw}
\def\Frobq{\Frob_q}
\def\Hilbq{\Hilb_q}
\def\mHL{\widetilde{H}}
\newcommand{\kat}{\text{kat}}

\usepackage{lipsum}

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\title[A charge monomial basis of $R_\mu$]{A charge monomial basis of the Garsia--Procesi ring}

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\author[Mitsuki Hanada]{\texorpdfstring{Mitsuki Hanada \thanks{\href{mhanada@berkeley.edu}{mhanada@berkeley.edu}.}\addressmark{1}}{Mitsuki Hanada }}

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\address{\addressmark{1}Department of Mathematics, University of California, Berkeley, USA}

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\received{\today}

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\abstract{We construct a basis of the Garsia--Procesi ring using the catabolizability type of standard Young tableaux and the charge statistic, providing the first direct connection between the structure of the Garsia--Procesi ring and the catabolizability formula for the modified Hall--Littlewood polynomial that gives its graded Frobenius character. }

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\keywords{Garsia--Procesi ring, modified Hall--Littlewood polynomials, charge, descent monomials, catabolizability type}

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\begin{document}

\maketitle
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\section{Introduction}\label{section:intro}
The polynomial ring $\mathbb{C}[\mathbf{x}] = \mathbb{C}[x_1,\dots, x_n]$ has a natural $S_n$ action permuting the variables. 
Taking the quotient of $\mathbb{C}[\mathbf{x}]$ by the ideal generated by $S_n$-invariant polynomials with no constant term defines the classical \textbf{coinvariant ring} $R_{1^n}$.
As graded algebras with $S_n$ actions, the coinvariant ring is isomorphic to the cohomology ring of the complex flag variety $\mathcal{F}_n$. The Hilbert series $\Hilbq(R_{1^n})$ is equal to $[n]_q!$.

The \textbf{Garsia--Stanton descent basis} of $R_{1^n}$ is given by the following set of monomials indexed by permutations:
\begin{align}\label{def: descent monomials}
    \{g_w (\mathbf{x}) = \prod_{i\in \des(w)} x_{w_1} \cdots x_{w_i} \ | \ w \in S_n\}.
\end{align}
where $\des (w) : = \{i \ | \ w_i > w_{i+1}\}$ denotes the descent set of $w$. 
We refer to these monomials $g_w(\mathbf{x})$ as \textbf{descent monomials}.
The descent monomials correspond to the permutation statistic $\text{maj}$, defined to be $\text{maj}(w) = \sum_{i \in \des(w)} i$; in particular, we can see that $\text{deg}(g_w(\mathbf{x})) = \text{maj}(w)$.
Using the fact that maj is Mahonian, we have the following equality:
 \[\sum\limits_{w\in S_n} q^{\text{deg}(g_{w}(\mathbf{x}))} = \sum\limits_{w\in S_n} q^{\text{maj}(w)} = [n]_q!.\] 
From this, it is evident that the number of descent monomials of degree $d$ matches the coefficient of $q^d$ in $[n]_q!$.
This gives a combinatorial explanation of $\Hilbq(R_{1^n}) = [n]_q!$.

For $\mu\vdash n$, the \textbf{Garsia--Procesi ring} $R_\mu$ is a quotient of $R_{1^n}$ that corresponds to the cohomology ring of the Springer fiber $\mathcal{F}_\mu \subset \mathcal{F}_n$. The concrete presentation of $R_\mu$ as a quotient of $\mathbb{C}[\mathbf{x}]$ is due to De Concini--Procesi \cite{deCon-Proc} and Tanisaki \cite{Tanisaki}. This quotient is also a graded $S_n$ representation under the same action, where the graded Frobenius character of $R_\mu$ is given by the \textbf{modified Hall--Littlewood polynomial} $\mHL_{\mu}[\mathbf{x};q]$ \cite{HottaSpringer}. One combinatorial formula for $\mHL_\mu[X;q]$, due to 
Lascoux \cite{Lascoux} and Butler \cite{butler}, is:
\begin{align}\label{eq:catabolizable HL formula}
        \mHL_\mu[X;q] &=  \sum\limits_{\substack{T\in \syt_n\\  \ctype(T)\unrhd\mu}}  q^{\cocharge(T)} s_{\shape(T)}(X).
\end{align}
The sum is over all standard Young tableaux $T$ satisfying $\ctype(T) \unrhd \mu$, where $\unrhd$ denotes the dominance order on partitions and $\ctype(T)$ is a partition associated to $T$ called its catabolizability type \cite{Lascoux}.

The question of finding a subset of the Garsia--Stanton descent monomials that is a monomial basis of $R_\mu$ was unresolved until recently, when Carlsson--Chou gave a construction using shuffles of descent compositions \cite{CC}. However, this construction does not give a direct combinatorial explanation for the Hilbert series of $R_\mu$ analogous to how the descent monomials do for the coinvariant ring.

Our main contribution is a construction of a monomial basis of $R_{\mu}$, consisting of descent monomials, using a catabolizability type condition on  standard Young tableaux (\Cref{thm: desc basis are the same}.)  Our basis coincides with that of Carlsson--Chou, though the two different constructions are independent. We make use of this equality to show that our set is indeed a basis.
 Our construction provides a natural way of describing this set of monomials in light of \eqref{eq:catabolizable HL formula}.
 % The set we construct is the natural subset of the descent basis to consider in light of the combinatorial formula \eqref{eq:catabolizable HL formula} for the modified Hall--Littlewood polynomial.
The description of the basis is compatible with the Hilbert series $\Hilbq(R_{\mu})$ in the same way that the descent basis is compatible with $\Hilbq(R_{1^n})$.
We further highlight the connection between our basis and \eqref{eq:catabolizable HL formula} by using our construction to give an elementary proof of the fact that $\Frobq(R_\mu) = \mHL_\mu[X;q]$ (\Cref{prop:prop2}, \Cref{main cor}) directly from \eqref{eq:catabolizable HL formula} using just the ungraded Frobenius character of $R_\mu$, without relying on other identities regarding $\mHL_\mu[X;q]$.
 Full details of this work are in \cite{hanada}.
 
\section{Background}\label{section:background}
For any word $w =  w_1w_2\dots w_n$, we write $\rev(w) := w_n w_{n-1}\dots w_1$ for the reverse word. For two words $z^{(1)},z^{(2)}$ of length $l_1, l_2$ we define $\text{Sh}(z^{(1)},z^{(2)})$ to be the collection of all words $u$ of length $(l_1+l_2)$ such that $z^{(1)}$ and $z^{(2)}$ are two disjoint subwords of $u$.
 An element $u$ of $\text{Sh}( z^{(1)},z^{(2)})$ is a \textbf{shuffle} of $z^{(1)}$ and $z^{(2)}$. We can extend this definition and let $\text{Sh}(z^{(1)},\dots , z^{(l)})$ denote the set of all shuffles of $z^{(1)},\dots , z^{(l)}$.


Throughout, we use French notation for Young diagrams, meaning that the first part corresponds to the bottom row. We denote the transpose of a partition $\lambda$ by $\lambda^t$.

The \textbf{dominance} ordering on partitions, denoted $\unrhd$, is defined by $\mu\unrhd \lambda  \Leftrightarrow   \mu_1 + \cdots \mu_k \geq \lambda_1+\cdots  \lambda_k \text{ for all } k.$
It is well known that $\mu \unrhd \lambda \ \  \Leftrightarrow \ \   \mu^t \unlhd \lambda^t$. If we move a box of $\lambda$ to a lower row so that the resulting shape $\mu$ is a partition, we have $\mu \unrhd \lambda.$

For any partition $\lambda, $ we denote the set of \textbf{standard Young tableaux} of shape $\lambda$ by $\syt(\lambda)$. We denote the set of standard Young tableaux of all shapes of size $n$ by $\syt_n$. For $T\in \syt(\lambda)$, we let $T^t\in \syt(\lambda^t)$ denote the filling of $\lambda^t$ we get by swapping the rows and columns of $T$. 
We define the descent set of $T$ to be  $\des(T) = \{i  \ | \ i \text{ appears below } i+1 \text{ in } T\}.$

For a \textbf{semistandard Young tableau} $T$ of shape $\lambda$, we say the \textbf{weight} of $T$ is the tuple $(m_1, m_2,\dots)$  where $m_i$ is the number of times $i$ appears in $T$.
For any tableau $T$, we denote the shape of $T$ by $\shape(T)$ and the row reading word of a tableau $T$ by $\rw(T)$. 

The \textbf{RSK (Robinson--Schensted--Knuth) correspondence} gives a bijection from permutations $w\in S_n$ to pairs $(P(w),Q(w))$ of standard Young tableau of size $n$ of the same shape. We say $P(w)$ is the \textbf{insertion tableau} and $Q(w)$ is the \textbf{recording tableau} of $w$.
We note that for any given standard tableau $T$ of shape $\lambda$, the number of permutations $w$ with $P(w) = T$ is $|\syt(\shape(T))|$, since we have one such permutation for each possible choice of $Q(w)$. We recall the following proprety of RSK that we use throughout this paper: for a more detailed exposition of RSK, see \cite[Chapter 7]{EC2}. 

\begin{prop}[{\cite[Corollary A1.2.11]{EC2}}]\label{thm:RSK rev}
For any $w\in S_n$, we have $P(\rev(w))= (P(w))^t.$
\end{prop}

\subsection{Symmetric functions and representation theory of \texorpdfstring{$S_n$}{S\_n}}\label{subsec: symm functions}
We refer to \cite[Chapter 7]{EC2} or \cite{macdonald} for background on symmetric functions or the representation theory of the symmetric group.
We follow Macdonald's notation \cite{macdonald} for the monomial symmetric functions $m_\lambda$, the elementary symmetric functions $e_\lambda$, the complete (homogeneous) symmetric functions $h_\lambda$, and the Schur functions $s_\lambda$.

% \textbf{we only mention properties we will use later}
% The Kostka numbers $K_{\lambda,\gamma}$, which are the coefficients that appear in the the monomial expansion of $s_\lambda$ counts the number of semistandard Young tableaux of shape $\lambda$ and weight $\gamma$.
%  These coefficients can also be expressed in terms of standard Young tableaux (where $\gamma \models n$ is a composition of length $l)$:
%  \begin{align}\label{eq: kostka numb}
%      K_{\lambda,\gamma} = | \{T\in \syt(\lambda)  \ | \ \des(T) \subset \{\gamma_1,\gamma_1 + \gamma_2, \dots, \gamma_1 +\cdots + \gamma_{l-1}\} \} |.
%  \end{align}

We denote the \textbf{Frobenius character} of an  $S_n$ representation $V$ by $\Frob(V)$. Recall that $\Frob(V_\lambda) = s_\lambda$, where $V_\lambda$ is the irreducible $S_n$ representation indexed by $\lambda$.
For a graded vector space $V = \oplus_{d\geq0} V_d$, the \textbf{graded Frobenius character} is $\Frob_q(V) = \sum_{d \geq 0} q^d \Frob(V_d).$

For any $S_n$ representation $V$, we can recover $\Hilbq(V)$ from $\Frobq(V)$ by the relation $\Hilbq(V) = \langle h_{1^n}, \Frobq(V)\rangle$ where $\langle - , - \rangle$ denotes the Hall inner product on symmetric functions. 
We can also determine $\Frob_q(V)$ using the Hilbert series of certain subspaces of $V$.
For the Young subgroup $S_\gamma = S_{\gamma_1} \times \cdots \times S_{\gamma_l}\subset S_n$ indexed by $\gamma\vdash n$, we define $N_{\gamma} = \sum_{\sigma\in S_\gamma} \sgn(\sigma) \sigma$ to be the antisymmetrizer with respect to $S_\gamma$. 
For any $\mathbb{C}S_n$-module $V$, the vector space $N_\gamma V$ is the subspace of elements of $V$ that are antisymmetric with respect to $S_\gamma$.
Let $\mathcal{E}\uparrow^{S_n}_{S_\gamma}$ denote the induction of the sign representation $\mathcal{E}$ of $S_\gamma$ to $S_n$. 
Using Frobenius reciprocity (see \cite[Chapter 3.3]{Fulton-Harris}) and the fact that $\Frob(\mathcal{E}\uparrow^{S_n}_{S_\gamma}) = e_\gamma$, we get the following result:
\begin{prop}\label{prop: hilb is determined by inner product}
    For any graded $\mathbb{C}S_n$-module $V = \oplus_{d\geq0} V_d$ and Young subgroup $S_\gamma \subset S_n$, we have  $\Hilbq(N_\gamma V) = \langle e_\gamma, \Frobq(V)\rangle.$
\end{prop}

From this, we can see that $\Frobq(V)$ is uniquely determined by $\Hilbq(N_\gamma V)$ for all partitions $\gamma\vdash n$. We use this result in \Cref{sec: remarks} in our proof of $\Frobq(R_\mu) = \mHL_\mu[X;q]$.

\subsection{Charge and cocharge}\label{subsection:charge}
Charge and cocharge are statistics on tableaux that arise in the study of Hall--Littlewood polynomials.
The \textbf{modified Hall--Littlewood polynomial} $\mHL_\mu [X;q]$ is given by the formula
    $\mHL_{\mu}[X;q] = \sum\limits_{\lambda}  \widetilde{K}_{\lambda,\mu}(q) s_\lambda(X)$ where the coefficients $\widetilde{K}_{\lambda,\mu}(q)$ can be expressed using the statistic \textbf{cocharge} on SYT:
\begin{align}\label{eq:kostka cat}
    \widetilde{K}_{\lambda,\mu}(q) = \sum\limits_{\substack{T\in \syt(\lambda)\\  \ctype(T)\unrhd\mu}}  q^{\text{cocharge}(T)}.
\end{align}
We define $\ctype(T)$ in \Cref{section:cat}.
Note that \eqref{eq:kostka cat} is equivalent to the more familiar expression for $\widetilde{K}_{\lambda,\mu}(q)$ with respect to semistandard Young tableaux under a cocharge preserving bijection from $\{T\in \syt(\lambda) \ | \ \ctype(T)\unrhd\mu\}$ to  $\{S\in \ssyt(\lambda) \ | \ \text{weight}(S) = \mu\}$, due to Lascoux \cite{Lascoux}. From the description using semistandard Young tableaux, we can see that $\widetilde{K}_{\lambda,\mu}(1) = K_{\lambda,\mu}$, the Kostka number counting semistandard tableaux of shape $\lambda$, weight $\mu$. 
From this, we can see that $\mHL_\mu[X;1] = h_\mu(X)$.

We first define these statistics on the level of permutations and then extend it to $\syt_n$. For $w\in S_n$, we define $\text{charge}(w) = \text{maj}(\rev(w^{-1}))$ and $\text{cocharge}(w) = \binom{n}{2} - \text{charge}(w).$
Lascoux--Sch\"{u}tzenberger \cite{LS1} proved that charge is the unique statistic on words that satisfies a set of properties, one of them being the following:
\begin{thm}[Lascoux--Sch\"{u}tzenberger \cite{LS1}]
    For $w,w' \in S_n$ such that $P(w) = P(w')$, we have $\text{charge}(w) = \text{charge}(w')$.
\end{thm} 
From this, we can define charge (resp. cocharge) on standard Young tableau $T$ by setting $\text{charge}(T) = \text{charge}(\text{rw}(T))$ (resp. $\text{cocharge}(T) = \text{cocharge}(\text{rw}(T))$) and always have $\charge(P(w)) = \charge(w)$ for any $w\in S_n$.

Explicitly, we can compute $\text{charge}(w)$ by assigning labels to each letter in $w$.
The \textbf{charge word} of $w$ (denoted $\cw(w)$) is a word of length $n$ consisting of the charge labelling of $w$, which we define in the following way. 
We label $1$ of $w$ with 0. We proceed by labelling the numbers in increasing order: if we label $i$ with a $k$, then we label $(i+1)$ with a $k$ if it is to the left of $i$. We label $(i+1)$ with a $(k+1)$ if it is to the right of $i$.
We compute $\textbf{charge}(w)$ by taking the sum of the letters in $\cw(w)$. 
Using the algorithm for computing charge, we obtain the following lemma about charge words, which we make use of in \Cref{section:descent basis} when proving that the set of monomials we construct is indeed a basis of $R_\mu$.

\begin{lem}\label{lemma: adding a repeated cocharge value is still a cc word}
For $w\in S_n$ and nonnegative integer $x$ where $x= \cw(w)_i$  for some $i$, any word in $\sh(\cw(w),x)$ is a charge word as well. 
\end{lem}



\begin{ex}\label{ex: cocharge word}
    Consider $w =45132$. The corresponding charge word is $\cw (w) = 12011$, thus $\charge(w) = 5$. 
    If $x= 2$, we can see that any word in $\sh(12011,2)$ is a charge word. For example, $120\underline{2}11 = \cw(461532)$. 
\end{ex}

We use these charge words to construct the monomials that appear in our basis. To each $w \in S_n$, we associate a monomial $\mathbf{x}^{\cw(w)}$, where the exponents are given by the charge word of $w$. We refer to this monomial as the \textbf{charge monomial }of $w$. 
\begin{ex}
For $w =45132$, the corresponding charge monomial is $\mathbf{x}^{\cw(w)} = x_1 x_2^2 x_4 x_5.$
\end{ex}

We can see that charge monomials of $S_n$ are exactly the descent monomials given in \eqref{def: descent monomials}. Let $D_n $ denote the set of exponents of descent monomials. We refer to these words as \textbf{descent words.} 

\begin{ex}
Using \eqref{def: descent monomials}, we can compute $g_{321}(\mathbf{x}) = (x_3)(x_3x_2) = x_2x_3^2$. Thus $012 \in D_3$. In this way, we can compute $D_3 = \{012,011, 101, 001,010, 000\}$.
\end{ex}

Since the definition of charge follows from maj, it is clear from definition that $\mathbf{x}^{\cw(w)} = g_\sigma(\mathbf{x})$  for any $w\in S_n$, where $\sigma = \text{rev}(w^{-1})$. From this, we get the following fact:
% \begin{ex}
%     We can see that $g_{321}(\mathbf{x}) = x_2x_3^2 = \mathbf{x}^{\cw(123)}$, since $\cw(123) = 012$.
% \end{ex}


\begin{lem}\label{lemma:charge and majt}
    We have $D_n = \{c(w) \ | \ w \in S_n\}.$
\end{lem}

We define the \textbf{cocharge} word $\cc(w)$ of permutation $w$ by $\cc(w)  = \rev(\cw(\rev(w)))$
We can see that $\cc(w)$ can be computed by the same algorithm we use to compute the charge word, except with left and right exchanged. We compute $\textbf{cocharge}(w)$ by taking the sum of the letters in $\cc(w)$. 
We make the following observation from \Cref{thm:RSK rev} and the relation between cocharge, which we use to switch between the two statistics when describing $\mHL_\mu[X;q]$.

\begin{prop}\label{thm: charge cocharge trabspose}
    If $T$ is a standard Young tableau, then
    $\text{charge}(T) = \text{cocharge}(T^t)$.
\end{prop}


\subsection{Catabolism and catabolizability}\label{section:cat}
To each standard Young tableau $T$, we associate a partition called its \textbf{catabolizability type} $\ctype(T)$. Originally introduced by Lascoux \cite{Lascoux}, this partition keeps track of a sequence of catabolisms to apply to $T$ in order to produce a one row tableau. 
% For a more detailed reference on catabolism and catabolizability type, see \cite{Lascoux} or \cite{SW}.

In general, many things are unknown about catabolism and catabolizability, though the operation of catabolism itself is very easy to compute.
However, there is a concrete algorithm, due to Blasiak \cite{Blasiak}, called the \textbf{catabolism insertion algorithm}, which computes the catabolizability type of a permutation $w$, where we define $\ctype(w) : = \ctype(P(w))$.

We define a modification of the catabolism insertion to use as our main tool in showing that the set we define in \Cref{section:descent basis} is indeed a basis of the Garsia--Procesi rings. For this extended abstract, we reverse our conventions from that used in the full version of this work \cite{hanada} for simplicity. 
We first define the following function:

\begin{defn}
   Consider a tuple $(i,x,T)$ consisting of an integer $i\in [n]$, a word $x$ (with potentially empty spots), and a filling $T$ of partition shape. 
    We construct a new word $x'$ and filling $T'$ from $(x,i,T)$ in the following way:
    \begin{itemize}
        \item If $x_i$ is empty, set $T '= T, x' = x$.
        \item If not, set $T'$ to be the filling we obtain by adding a box containing $i$ to row $(x_i+1)$ of $T$ if the resulting shape is a partition. We create $x'$ by replacing $x_i$ with an empty spot.
        \item If we cannot add a box to row $(x_i+1)$ of $T$ to make a partition shape, we set $T' = T$. We obtain $x'$ from $x$ by replacing $x_i$ with $(x_i+1)$.
    \end{itemize}
   We define the function $f$ on such tuples by $f(i,x,T) = (i+1 ,x',T')$ if $i<n$ and $f(n,x,T) = (1 ,x',T')$.
\end{defn}


\begin{alg}\label{alg: cat insertion}  
   Let $(1, \cw(w),\emptyset)$ be the initial input, where $w\in S_n$. Apply $f$ repeatedly to this tuple until it results in the tuple  $(j, \emptyset, T_w)$ for some index $j$, where $T_w$ is a standard filling (but not necessarily a SYT) of a partition shape. 
\end{alg}


We illustrate this algorithm in \Cref{ex: cat alg}. Note that this algorithm may not terminate if the word in the initial tuple is not a charge word.  For example: if $z = 20$, the second letter will never be deleted, since we cannot add a box to the partition $(1)$ in rows $2$ or higher. The shape of the resulting filling $T_w$ gives us the catabolizability type of a certain permutation.

\begin{prop}[Blasiak{\cite{Blasiak}}]
  We have $\ctype(\rev(w)) = \shape(T_w)$, where $T_w$ is the filling produced by Algorithm \ref{alg: cat insertion}.
\end{prop}

The original algorithm of Blasiak produced a partition shape with no filling. We choose to fill the shape in order to construct disjoint subwords of $\cw(w)$ using the entries in the columns of $T_w$. We make use of these subwords when proving the main result in \Cref{section:descent basis}.

We now describe these subwords.
 Consider $w\in S_n$ where $\shape(T_w) = \lambda$. For a column $j$ of $\lambda$, define $\cw(w)^{(j)}$ to be the subword of $\cw(w)$ consisting of the entries $\cw(w)_k$ where $k$ appears in column $j$ of $T_w$. We give examples of this construction in \Cref{ex: cat alg}.
 Using \Cref{lemma: adding a repeated cocharge value is still a cc word}, we can prove the following proposition.
\begin{prop}\label{prop: subword of column is a cocharge word}
For $w\in S_n$ where $\shape(T_w) = \lambda$, consider $j$ such that $1\leq j \leq \lambda_1$.
Then $\cw(w)^{(j)} = \cw(\sigma)$ for some $\sigma \in S_r$.
\end{prop}

From this, we can see that each of the subwords of $\cw(w)$ corresponding to columns of $T_w$ are themselves charge words. This observation will be essential when proving the main result in \Cref{section:descent basis}.

\begin{rem}
    The fact that $\cw(w)^{(j)}$ is a charge word is not obvious a priori, since it is not true that any subword of a charge word is a charge word. For example, consider $w = 12 $, which has $\cw(w) =01.$ Though $1$ is a subword of $\cw(w)$, it is not a charge word.
\end{rem}

\begin{ex}\label{ex: cat alg}
    Consider $w = 52143$, with $\cw(w) = 10011$.
    The position of the word that we are considering at each step is underlined. 
    Our initial input is $ (1, \mathbf{\underline{1}}0011,\emptyset)$. We can see that at the first step, the resulting tuple is  $ (2, 20011,\emptyset)$, since we cannot add a box to row 2 of the empty partition. We proceed through the rest of the word, where we underline $x_i$ when considering the tuple $(i,x,T)$:
\begin{align*}  \ytableausetup{smalltableaux, centertableaux}
    & (2 , \  2 \ \mathbf{\underline{0}} \ 0 \ 1 \ 1 \hspace{.1cm} , \hspace{.1cm}\emptyset \hspace{.1cm}) \xrightarrow{f}
     (3, \ 2 \  \text{\textvisiblespace}  \  \mathbf{\underline{0}} \ 1 \ 1 , \hspace{.3cm} \ytableaushort{2} \hspace{.1cm}) 
       \xrightarrow{f} \ 
     (4,\  2 \ \text{\textvisiblespace} \  \text{\textvisiblespace}  \  \mathbf{\underline{1}} \ 1 \hspace{.1cm},\hspace{.1cm} \ytableaushort{23}\hspace{.1cm}) \\
     &\quad \xrightarrow{f} ( 5, \  2 \   \text{\textvisiblespace}  \   \text{\textvisiblespace}  \  \text{\textvisiblespace}  \  \mathbf{\underline{1}}\hspace{.1cm} , \hspace{.3cm}\ytableaushort{4,23}\hspace{.1cm})
     \xrightarrow{f}
     (1, \  \mathbf{\underline{2}} \  \text{\textvisiblespace}  \  \text{\textvisiblespace}   \   \text{\textvisiblespace}  \  \text{\textvisiblespace}  \    \ \ , \hspace{.3cm}\ytableaushort{45,23}\hspace{.1cm})   \xrightarrow{f}
     (2, \ \text{\textvisiblespace}\ \text{\textvisiblespace} \  \text{\textvisiblespace} \  \text{\textvisiblespace} \ \hspace{.2cm} , \hspace{.3cm}\ytableaushort{1,45,23}\hspace{.1cm})
\end{align*}
 Hence we have $T_{52143} = \ytableaushort{1,45,23}\  , \ \ctype(34125) =\ctype(\rev(52143)) = (2,2,1).$
In this case, we see that the subwords corresponding to the columns of $T_w$ are $\cw(w)^{(1)} = \cw(w)_1\cw(w)_2\cw(w)_4 = 101 = \cw(312)$ and $\cw(w)^{(2)} = 01 = \cw(12)$. 
\end{ex}

% In the full version of this work, we use Algorithm \ref{alg: cat insertion}, as well as different modified versions of it, to prove properties about catabolizability type. For more, see Section 5 of \cite{hanada}. We give one result here without proof, which was used \ref{sec: remarks} when proving \Cref{prop:prop2}:
% \begin{prop}\label{prop: swaps raise ctype}
%    Consider $w\in S_n$ and index $i$ such that $w_i  +1 < w_{i+1}$. If $\tilde{w} = w_1\dots w_{i+1}w_i\\\dots w_n$, we have $\ctype(\tilde{w}) \unrhd \ctype(w)$.
% \end{prop}

\section{Descent basis of Carlsson--Chou}
We now review the construction of the Carlsson--Chou descent basis.

For $\mu = (\mu_1,\dots,\mu_l) \vdash n$, define $\mathcal{D}_\mu$ to be $ \mathcal{D}_\mu = \bigcup_{(z^{(1)},\dots , z^{(l)})} \sh (z^{(1)},\dots , z^{(l)})$ where $(z^{(1)},\dots , z^{(l)})$ ranges over all $l$-tuples in $\mathcal{D}_{\mu_1}\times \cdots \times \mathcal{D}_{\mu_l}$.

\begin{thm}[Carlsson--Chou \cite{CC}]
The set $\mathbf{x}^{\mathcal{D}_\mu} = \{\mathbf{x}^\alpha \ | \ \alpha \in  \mathcal{D}_\mu\}$  is a monomial basis of $R_{\mu^t}$. 
\end{thm}   

\begin{ex}\label{ex: 31}
    Let $\mu = (3,1).$ Since 
        $D_3 = \{012,011, 101, 001,010, 000\}$ and $
        D_1 = \{0\}$, we have
  $\mathcal{D}_{3,1}  = \{0012,\ 0102,\ 0120, \ 0011, \ 0101, \ 1001, \ 1010, \ 0110,\  0001,\ 0010,\ 0100, \ 0000\}.$
  The corresponding set of monomials, given below, is a basis of $R_{2,1,1}$ since $(3,1)^t = (2,1,1)$: 
  \begin{align*}
       \{x_3x_4^2,
  \ x_2x_4^2, \ x_2x_3^2, \ x_3x_4 ,\  x_2x_4, \ x_1x_4, \ x_1x_3, \ x_2x_3, \ x_4, \ x_3 , \ x_2, \ 1\}.
  \end{align*}
\end{ex}

Carlsson--Chou show that their basis is a subset of the descent basis. Though the description of the basis is combinatorial, it lacks nice combinatorial properties that the descent basis had. For one, it is not obvious that $|\mathcal{D}_{\mu}| = \dim(R_{\mu^t})$, since multiple shuffles can correspond to the same descent word. For example, $\mathbf{0}101 \in \sh(101,\mathbf{0})$ and $01\mathbf{0}1 \in \sh(011,\mathbf{0})$ both correspond to the same element in $\mathcal{D}_{3,1}$.
It is also not obvious how to see directly for which $w\in S_n$ we have $g_w(\mathbf{x}) \in \mathbf{x}^{\mathcal{D}_{\mu}}$ without computing $\mathcal{D}_\mu$.

This construction also fails to provide direct connections to the $q=0$ specialization of the  modified Macdonald polynomials $\mHL_\mu[X;q,t]$. Since $\mHL_\mu[X;0,t]$ recovers the modified Hall--Littlewood polynomials, one would expect there to be a nice bijection between the monomials $\mathbf{x}^{\mathcal{D}_\mu}$ and the terms that arise in the combinatorial formula due to Haglund--Haiman--Loehr \cite{HHL}.
However, Carlsson--Chou state that there does not seem to be a ``reasonable''  weight-preserving bijection between the indexing sets of the two.

\section{Charge monomial basis of \texorpdfstring{$R_{\mu^t}$}{R\_mu\^t}}\label{section:descent basis}
Our main contribution is that we construct sets of charge monomials that are monomial bases of Garsia--Procesi rings. 
Our construction involves charge words of permutations whose insertion tableaux satisfy a catabolizability condition. 

\begin{thm}
\label{thm: desc basis are the same} 
   Let $ \mathcal{C}_\mu := \{\cw(w)\ | \ w\in S_n, \  \ctype(P(w) ^t)\unrhd\mu^t\}.$
    The set $\mathbf{x}^{\mathcal{C}_\mu} = \{\mathbf{x}^\alpha \ | \ \alpha \in \mathcal{C}_\mu\}$ 
     is a monomial basis of $R_{\mu^t}$. 
       In fact, it coincides with the basis $  \mathbf{x}^{\mathcal{D}_\mu}$ given by Carlsson--Chou \cite{CC}, ie., $\mathcal{C}_\mu = \mathcal{D}_\mu$.
\end{thm}
From \Cref{lemma:charge and majt}, we know $\mathcal{C}_\mu\subset D_n$, thus this set of monomials is indeed a subset of the descent basis of the coinvariant ring.
Though the two sets coincide, the definitions of $\mathcal{C}_\mu$ and $\mathcal{D}_\mu$ are independent, which we can see through the difference in the combinatorics used in the constructions. 

\begin{ex}\label{ex: cat basis (3,1)}
Consider $\mu = (3,1)$. There are five standard tableaux $S$ such that $\ctype(S^t) \unrhd (2,1,1) = \mu^t$. We list them in \Cref{figure: table of SYT} along with all words $w$ such that $P(w) = S$ and their charge monomials.
The resulting set of charge monomials is  \[\{x_3x_4^2,
  \ x_2x_4^2, \ x_2x_3^2, \ x_3x_4 ,\  x_2x_4, \ x_1x_4, \ x_1x_3, \ x_2x_3, \ x_4, \ x_3 , \ x_2, \ 1\},\]
which is the same as in \Cref{ex: 31}.
\end{ex}

\begin{figure}
    \centering
    \begin{center}
     \ytableausetup{smalltableaux,  aligntableaux = center}
   \begin{tabular}{ p{2cm}|p{5cm}  p{5cm}  }
$S$ & $\{w \ | \ P(w) = S\}$ & $\{x^{\cw(w)} \ | \ P(w) = S\}$\\
 \hline
\\[-.5em]
 \ytableaushort{2,134}  &\{2134,
  \ 2314, \ 2341\}& $\{x_3x_4^2,
  \ x_2x_4^2, \ x_2x_3^2\}$ \\[-.5em] \\
\ytableaushort{24,13} &  \{2143 ,\  2413\} &  $\{x_3x_4 ,\  x_2x_4\}$  \\[-.5em] \\
\ytableaushort{4,2,13} & \{4213, \ 4231, \ 2431\} &  $\{x_1x_4, \ x_1x_3, \ x_2x_3\}$\\[-.5em] \\
\ytableaushort{3,2,14} & \{3214, \ 3241 , \ 3421\} & $\{x_4, \ x_3 , \ x_2\}$\\[-.5em] \\
\ytableaushort{4,3,2,1}&  \{4321\}& $\{1\}$
\end{tabular}
\end{center}
    \caption{Charge monomials $\mathbf{x}^{\cw(w)}$ for $w\in S_n$ with $\ctype(P(w)^t) \unrhd (2,1,1)$}
    \label{figure: table of SYT}
\end{figure}

Before we prove \Cref{thm: desc basis are the same}, we point out some key connections between our construction of $\mathbf{x}^{\mathcal{C}_\mu}$ and the combinatorics of $\mHL_\mu[X;q]$.
Using the combinatorial formula \eqref{eq:catabolizable HL formula} for the modified Hall--Littlewood polynomials and \Cref{thm: charge cocharge trabspose}, we obtain the following expression for $\mHL_{\mu^t}[X;q]$ with respect to charge:
\begin{align}\label{eq: cat HL charge}
    \mHL_{\mu^t} [X;q] &=  \sum\limits_{\substack{T\in \syt_n\\  \ctype(T^t)\unrhd\mu^t}}  q^{\charge(T)} s_{\shape(T^t)}(X).
\end{align}
Using \eqref{eq: cat HL charge} we can express $\Hilbq(R_{\mu^t})$ as a sum over certain permutations:
\begin{align}\label{eqref: hilb gp}
     \Hilbq(R_{\mu^t}) &=  \sum\limits_{\substack{w\in S_n\\  \ctype(P(w)^t)\unrhd\mu^t}}  q^{\text{charge}(w)} .
\end{align}

By our definition of $\mathcal{C}_\mu$, the number of monomials in $\mathbf{x}^{\mathcal{C}_\mu}$ of degree $d$ is exactly the number of permutations that satisfy $\ctype(P(w) ^t)\unrhd\mu^t$ and $\charge(w) = d$. Comparing this observation with \eqref{eqref: hilb gp}, it is evident $\mathbf{x}^{\mathcal{C}_\mu}$ has the correct cardinality in each degree to be a basis of $R_{\mu^t}$. This is analogous to the result that we had for the descent basis and the coinvariant ring that we discussed in \Cref{section:intro}.
This is also in contrast to the Carlsson--Chou construction where it was nontrivial to show the ungraded version of this equality ($|\mathcal{D}_\mu| = \dim(R_{\mu^t})$).
Our construction also gives a direct way to determine for which $w\in S_n$ we have $g_w(\mathbf{x})\in \mathbf{x}^{\mathcal{C}_\mu}$.
In particular, we have $g_w(\mathbf{x})\in \mathbf{x}^{\mathcal{C}_\mu}$ for $w = \text{rev}(\sigma^{-1})$ where $\ctype(P(\sigma)^t)\unrhd \mu^t$.

This construction also highlights a natural connection to the $q=0$ specialization of the modified Macdonald polynomials $\mHL_\mu[X;q;t]$ that was difficult to see from the Carlsson--Chou construction. In \cite{HHL}, the authors give a bijection between the fillings $\tau$ of $\mu$ that satisfy $\text{inv}(\tau)= 0$, which index the terms in the combinatorial formula of $\mHL_\mu[X;0;t]$ and pairs $(S,T)$ where $S\in \ssyt, T\in \syt$ are tableaux of the same shape. We can easily extend this bijection to $\mathcal{C}_\mu$ using Lascoux's cocharge preserving standardization.

We prove \Cref{thm: desc basis are the same} by showing that $\mathcal{C}_{\mu} = \mathcal{D}_\mu$, which implies that the set $\mathbf{x}^{\mathcal{C}_\mu}$ is a basis of $R_{\mu^t}$.
Since we know $|\mathcal{C}_\mu | =  \dim(R_{\mu^t})$ from the observation above and $\dim(R_{\mu^t}) = |\mathcal{D}_\mu|$ from the results of Carlsson--Chou, it suffices to show that $\mathcal{C}_\mu\subset \mathcal{D}_\mu$. This containment is almost immediate using the algorithm we defined in \Cref{section:cat}.
\begin{prop}\label{prop: ctype implies shuffle}
    Let $\mu$ be a partition of $n$ of length $l$ and $w$ be a permutation of $n$ such that $\ctype(\rev(w)) \unrhd \mu^t$. 
    Then $\cw(w) \in \mathcal{D}_\mu$.
\end{prop}

If $\ctype(\rev(w)) = \mu^t$, this follows from \Cref{prop: subword of column is a cocharge word}, which gives a decomposition of $\cw(w)$ into $l$ disjoint subwords $\cw(w)^{(1)}, \dots, \cw(w)^{(l)}$. Note that $\cw(w)^{(j)} = \cw(u) $ for some $u\in S_{\mu_j}$, hence $\cw(w)^{(j)}\in D_{\mu_j}$.

\begin{ex}\label{ex: cat ex}
    Let $\mu = (3,2)$. Consider $w = 52143$ from \Cref{ex: cat alg} with $\cw(w) =  10011$ and $\ctype(\rev(w)) = (2,2,1) = \mu^t$.
    Using the subwords coming from the columns of $T_w$, we can write $\cw(w) = 10\mathbf{0}1\mathbf{1}$ as a shuffle of $\cw(w)^{(1)} = 101$ and $\cw(w)^{(2)}=01$ where the bold letters indicate those in $\cw(w)^{(2)}$.
    Thus $\cw(w) \in \sh(101,01)$ where $(101,01) \in D_{3}\times D_{2}$ since $101 = \cw(312)$ and $01 = \cw(12)$.
\end{ex}

It remains to consider $\ctype(\rev(w)) \rhd \mu^t$. 
Recall that for any partition $\lambda \rhd \mu^t$, we can go from shape $\lambda$ to $\mu^t$ by moving boxes to higher rows, which results in lowering the shape in the dominance ordering.  
Using this idea and \Cref{lemma: adding a repeated cocharge value is still a cc word}, we define an algorithm that takes the filling $T_w$ of shape $\lambda$ and constructs a filling $T_{w}^{\mu^t}$ of shape $\mu^t$ by moving certain boxes to higher rows. By definition of the algorithm, the resulting columns of $T_w^{\mu^t}$ correspond to disjoint subwords of $\cw(w)$ that are each charge words themselves as well. For details, see Section 4 of \cite{hanada}. 

\Cref{thm: desc basis are the same} follows from Propositions \ref{thm:RSK rev} and  \ref{prop: ctype implies shuffle}. Hence we have that our set $\mathbf{x}^{\mathcal{C}_\mu}$ is indeed a basis of $R_{\mu^t}$.
Note that this proof does not depend on the fact that $\Frobq(R_\mu) = \mHL_\mu[X;q]$. It suffices to know that the ungraded Frobenius character of $R_\mu$ is given by $h_\mu$, which can easily be identified directly from the structure of $R_\mu$. 


\section{Antisymmetric part}\label{sec: remarks}
We now use the construction of our basis to give a direct, elementary proof of the fact that $\Frobq (R_{\mu^t}) = \mHL_{\mu^t}[X;q].$
We do this by showing that for any $\gamma = (\gamma_1,\dots, \gamma_l)\vdash n$, we have $\Hilbq(N_\gamma R_{\mu^t}) = \langle e_\gamma , \mHL_{\mu^t}[X;q]\rangle$.

Using \eqref{eq: cat HL charge}, we can explicitly write out $\langle e_{\gamma},\  \mHL_{\mu^t} \rangle$ as the following:
\begin{align}\label{eq:Hilb using words}
    \langle e_{\gamma},\  \mHL_{\mu^t} \rangle  &= \sum\limits_{\substack{T\in \syt\\  \ctype(T^t)\unrhd\mu^t}}  q^{\charge(T)}   K_{\shape(T),\gamma} \\ &= \sum\limits_{\substack{w\in S_n\\  \ctype(P(w)^t)\unrhd\mu^t \\ \des(Q(w)) \subset \{\gamma_1,\gamma_1 + \gamma_2, \dots, \gamma_1 +\cdots + \gamma_{l-1}\} }}  q^{\charge(w)} \label{eq: Hilb on words}.
\end{align}
Note that we use the interpretation of $K_{\lambda,\gamma}$ with respect to standard tableaux:
\[K_{\lambda,\gamma} =  | \{w \in S_n  \ | \ \shape(P(w)) = \lambda, \des(Q(w)) \subset \{\gamma_1,\gamma_1 + \gamma_2, \dots, \gamma_1 +\cdots + \gamma_{l-1}\} \}|.\]


From this, proving $\Frob_q(R_{\mu^t}) = \mHL_{\mu^t}[X;q]$ is equivalent to showing that for any $\gamma\vdash n$, the Hilbert series $\Hilbq(N_\gamma R_{\mu^t})$ is equal to \eqref{eq: Hilb on words}.
We show this by proving that the natural subset of charge monomials to expect from Equation \eqref{eq: Hilb on words} gives a basis of $N_\gamma R_{\mu^t}$ by antisymmetrization. 
\begin{prop}
\label{prop:prop2}
    Let $\mu \vdash n$  and $\gamma = (\gamma_1,\dots, \gamma_l)\vdash n$. 
    The set
    \begin{align*}\label{eq:antisym basis}
    \{ N_\gamma \mathbf{x}^{\cw(w)} \ | \  w \in S_n, \ \ctype(P(w)^t) \unrhd\mu^t, \ \des(Q(w)) \subset \{\gamma_1,\gamma_1 + \gamma_2, \dots, \gamma_1 + \cdots + \gamma_{l-1}\}\}.
    \end{align*}
    is a basis of $N_\gamma R_{\mu^t}$.
\end{prop}

Using the construction of our basis, we can translate questions about the structure of $R_{\mu^t}$ into questions about conditions on tableaux. In this case, determining the elements that are a basis of $N_\gamma R_{\mu^t}$ is equivalent to looking at pairs of standard tableaux $(P,Q)$ of the same shape with conditions on both $P$ and $Q$. 
We illustrate this with an example:
\begin{ex}
Consider $\mu= (3,1)$ and $\gamma = (2,2)$. All standard tableaux $P$ that satisfy $\ctype(P^t) \unrhd (2,1,1) = (3,1)^t$ are listed below:
   \ytableausetup{mathmode,smalltableaux,  aligntableaux = bottom}
\[\ytableaushort{2,134} \ , \  \ytableaushort{24,13} \ , \ \ytableaushort{4,2,13} \ , \  \ytableaushort{3,2,14}\ , \ \ytableaushort{4,3,2,1}. \]
We also list all standard tableaux $Q$ such that $\des(Q) \subset \{2\}$:
\[\ytableaushort{1234} \ , \  \ytableaushort{3,124}. \]
Thus the only pair $(P,Q)$ of standard tableaux of the same shape that satisfy $\ctype(P^t) \unrhd (2,1,1)$ and $\des(Q) = (2,2)$ is the pair $ \ytableausetup{smalltableaux,  aligntableaux = top}\left(\ytableaushort{2,134} \ , \ \ytableaushort{3,124}\right)$, which corresponds to the permutation $w = 2314$. Since $\cw(w) = 0102$,  we conclude the basis of $N_\gamma R_{\mu^t}$ is given by the polynomial
$N_\gamma(x_2x_4^2) = x_2x_4^2 - x_1x_4^2 - x_2x_3^2 + x_1x_3^2.$
\end{ex}


\begin{cor}\label{main cor}
For any partition $\mu$ of $n$ we have $\Frobq(R_{\mu^t}) =\mHL_{\mu^t}[X;q].$
\end{cor}

Though Garsia--Procesi \cite{GP} provide the first combinatorial proof of this result, their proof requires deriving recursive properties of $\mHL_\mu[X;q]$ that match the recursion occurring in the construction of their monomial basis of $R_\mu$. Not only is our proof independent from theirs, it also relies solely on the combinatorial formula \eqref{eq: cat HL charge} for $\mHL_{\mu}[X;q]$, as well as some basic symmetric function theory, providing a direct connection between the Schur expansion of $\mHL_\mu[X;q]$ and $\Frobq(R_\mu)$.

\acknowledgements{The author thanks Mark Haiman, Sylvie Corteel, and Raymond Chou for helpful conversations.}

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