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\title{Ordered set partitions and the 0-Hecke algebra}

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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Jia Huang\thanks{\href{mailto:huangj2@unk.edu}{huangj2@unk.edu}}\addressmark{1}\and 
Brendon
Rhoades\thanks{\href{mailto:bprhoades@math.ucsd.edu}{bprhoades@math.ucsd.edu}.
Brendon Rhoades was partially supported by NSF Grant DMS-1500838.}\addressmark{2}}

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\address{\addressmark{1}Department of Mathematics and Statistics, University of Nebraska, Kearney, NE 68849, USA
\\ \addressmark{2}Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA}

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\abstract{
Haglund, Rhoades, and Shimozono recently introduced a quotient $R_{n,k}$ of the polynomial ring $\QQ[x_1, \dots, x_n]$ depending on two positive integers $k \leq n$, which reduces to the classical coinvariant algebra of the symmetric group $\symm_n$ if $k = n$.
They determined the graded $\symm_n$-module structure of $R_{n,k}$ and related it to the Delta Conjecture in the theory of Macdonald polynomials.  
We introduce an analogous quotient $S_{n,k}$ and determine its structure as a graded module over the (type A) $0$-Hecke algebra $H_n(0)$, a deformation of the group algebra of $\symm_n$.
When $k=n$ we recover earlier results of the first author regarding the $H_n(0)$-action on the coinvariant algebra.}

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%\resume{\lipsum[2]}

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\keywords{Hecke algebra, set partition, coinvariant algebra}

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\begin{document}

\maketitle


\section{Introduction}
\label{Introduction}
The symmetric group $\symm_n$ acts on the polynomial ring $\QQ[\xx_n] := \QQ[x_1, \dots, x_n]$ by variable permutation.
The corresponding {\em invariant subring} %$\QQ[\xx_n]^{\symm_n}$ consists of all $f \in \QQ[\xx_n]$ with $w(f) = f$ for all $w \in \symm_n$, 
is generated by the \emph{elementary symmetric functions} $e_1(\xx_n), \dots, e_n(\xx_n)$.
%\[ e_d(\xx_n) := e_d(x_1, \dots, x_n) := \sum_{1 \leq i_1 < \cdots < i_d \leq n} x_{i_1} \cdots x_{i_d}. \]
%The {\em invariant ideal} $I_n \subseteq \QQ[\xx_n]$ is the ideal generated by those invariants $\QQ[\xx_n]^{\symm_n}_+$ with vanishing constant term, or equivalently, $I_n := \langle e_1(\xx_n), \dots, e_n(\xx_n) \rangle$.
%I_n := \langle \QQ[\xx_n]^{\symm_n}_+ \rangle = \langle e_1(\xx_n), \dots, e_n(\xx_n) \rangle.
The {\em coinvariant algebra} $R_n := {\QQ[\xx_n]} / {I_n}$, where $I_n:= \langle e_1(\xx_n),\ldots,e_n(\xx_n) \rangle$, plays an important role in algebraic and geometric combinatorics, with properties closely tied to the combinatorics of permutations.
It has $\QQ$-dimension $n!$ and has various $\QQ$-bases constructed by Artin~\cite{Artin}, Garsia--Stanton~\cite{GS}, and others.
Chevalley~\cite{C} proved that $R_n$ is isomorphic to the regular representation $\QQ[\symm_n]$ of $\symm_n$.
Lusztig (unpublished) and Stanley~\cite{Stanley} described the {\em graded} 
$\symm_n$-module structure of $R_n$ using the major index statistic on standard Young tableaux.


Let $k \leq n$ be two positive integers. 
Haglund, Rhoades, and Shimozono~\cite{HRS} introduced a homogeneous ideal $I_{n,k}:= \langle x_1^k, x_2^k, \dots, x_n^k, e_n(\xx_n), e_{n-1}(\xx_n), \dots, e_{n-k+1}(\xx_n)\rangle$ of the polynomial ring $\QQ[\xx_n]$ which is stable under the $\symm_n$-action.
They studied the quotient $R_{n,k} := {\QQ[\xx_n]} / {I_{n,k}}$ which is a graded $\symm_n$-module reducing to the coinvariant algebra $R_n$ when $k=n$.
%Note that the $\symm_n$-action on $x_1^k,x_2^k,\ldots,x_n^k$ is isomorphic to the \emph{defining representation} of $\symm_n$ on the set $[n]:=\{1,2,\ldots,n\}$.
%One has $I_{n,n} = I_n$ and $R_{n,n} = R_n$.
They generalized the Artin basis and the Garsia--Stanton basis of $R_n$ to $R_{n,k}$.
They showed that $R_{n,k}$ is isomorphic to $\QQ[\OP_{n,k}]$ as an
ungraded $\symm_n$-module, where $\QQ[\OP_{n,k}]$ has a basis $\OP_{n,k}$
consisting of {\em ordered set partitions} of $[n]:=\{1,2,\ldots,n\}$
with $k$ blocks and admits an $\symm_n$-action by permuting $1,\ldots,n$;
consequently the dimension of $R_{n,k}$ is $|\OP_{n,k}| = k! \cdot
\Stir(n,k)$, where $\Stir(n,k)$ is the \emph{(signless) Stirling number
of the second kind} counting $k$-block set partitions of $[n]$.
They provided explicit descriptions of the {\em graded} $\symm_n$-module structure of $R_{n,k}$, generalizing the work of Lusztig--Stanley.

The symmetric group $\symm_n$ has an interesting deformation called the \emph{(type $A$) 0-Hecke algebra} and denoted by $H_n(0)$. 
Norton~\cite{Norton} studied the representation theory of $H_n(0)$ over an arbitrary field $\FF$.
Krob and Thibon~\cite{KT} introduced two characteristic maps from representations of $H_n(0)$ to quasisymmetric functions and noncommutative symmetric functions, which are similar to the Frobenius correspondence from representations of symmetric groups to symmetric functions.
%The 0-Hecke algebra $H_n(0)$ acts on the polynomial ring $\FF[\xx_n] := \FF[x_1, \dots, x_n]$ over a field $\FF$ by the {\em isobaric Demazure operators}.

Finding 0-Hecke analogs of results on $\symm_n$-representations has received a great deal of recent attention in algebraic combinatorics \cite{BBSSZ, Huang, HuangTab, TV}.
In particular, the first author~\cite{Huang} showed that the coinvariant algebra $R_n$ is a graded $H_n(0)$-module isomorphic to the regular representation of $H_n(0)$ and has bigraded quasisymmetric characteristic given by a generating function for the pair of Mahonian statistics ($\inv$,  $\maj$) on $\symm_n$. %weighted by Gessel's fundamental quasisymmetric functions $F_{\Des(w^{-1})}$. %$F_{\iDes(w)}$ corresponding to the inverse descent set of $\iDes(w)$ of $w\in\symm_n$ \cite[Cor. 4.9 (i)]{Huang}.

There is an $H_n(0)$-action on the polynomial ring $\FF[\xx_n] :=
\FF[x_1,\ldots,x_n]$ by the \emph{isobaric Demazure operators}; see
Equation~\eqref{Demazure}.
However, the ideal $I_{n,k}$ is not closed under this action, so that $R_{n,k}$ does not have an $H_n(0)$-module structure.  
We remedy this situation as follows.

\begin{defn}
Given positive integers $k \leq n$, define $S_{n,k} := \FF[\xx_n]/J_{n,k}$, where $J_{n,k}$ is the ideal of $\FF[\xx_n]$ generated by $e_n(\xx_n), e_{n-1}(\xx_n), \dots, e_{n-k+1}(\xx_n)$ together with the \emph{complete homogeneous symmetric functions} $h_k(x_1), h_k(x_1, x_2), \dots, h_k(x_1, x_2, \dots, x_n)$.  
\end{defn}

The ideal $J_{n,k}$ is closed under the action of $H_n(0)$, so that $S_{n,k}$ is a graded
$H_n(0)$-module.
The polynomials $h_k(x_1), h_k(x_1, x_2), \dots, h_k(x_1, x_2, \dots, x_n)$ have span isomorphic to the defining representation of $H_n(0)$;  this is analogous to the generators $x_1^k,x_2^k,\ldots,x_n^k$ of the ideal $I_{n,k}$ under the $\symm_n$-action. 
We show that $S_{n,k}$ has algebraic and combinatorial properties analogous to those
of $R_{n,k}$, including a connection to the {\em Delta Conjecture} of Haglund, Remmel, and Wilson~\cite{HRW} in the theory of 
Macdonald polynomials.

The remainder of the paper is structured as follows.
In Section~\ref{Background} we give background on representations of the symmetric groups and 0-Hecke algebras.
In Section~\ref{Hilbert} we study $S_{n,k}$ as a graded vector space.
In Section~\ref{Module} we study $S_{n,k}$ as a module over $H_n(0)$
(both graded and ungraded).
In Section~\ref{Conclusion} we connect our results to the Delta Conjecture.

\section{Background}
\label{Background}

The symmetric group $\symm_n$ consists of all permutations on the set $[n]$.
It is generated by $s_1, s_2, \dots, s_{n-1}$, where $s_i$ is the adjacent transposition $s_i := (i, i+1)$, subject to the quadratic relations $s_i^2=1$ for all $i\in[n-1]$ and the braid relations $s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1}$ for all $i\in[n-2]$ and $s_is_j = s_js_i$ for all $i,j\in[n-1]$ with $|i - j| > 1$.
A permutation $w\in\symm_n$ can be expressed as $w=s_{i_1}\cdots s_{i_k}$ in terms of the generators $s_1,\ldots,s_{n-1}$; 
such an express is \emph{reduced} if $k$ is as small as possible, and the smallest $k$ is the \emph{length} $\ell(w)$ of $w$.

A permutation $w \in \symm_n$ can be written in one-line notation $w = w(1) \cdots w(n)$. 
Define $\Des(w):=\{i\in[n-1]: w(i)>w(i+1)\}$, $\des(w):=|\Des(w)|$, $\maj(w):=\sum_{i\in\Des(w)} i$, and $\inv(w) := |\{(i,j): 1\le i<j\le n-1,\ w(i)>w(j)\}|$; one has $\inv(w)=\ell(w)$.
%The statistics $\inv$ and $\maj$ are equidistributed on $\symm_n$.
%MacMahon \cite{MacMahon} showed that 
It is well known that $\sum_{w \in \symm_n} q^{\inv(w)} = \sum_{w \in \symm_n} q^{\maj(w)} = [n]!_q$,
%\end{equation}
where $[n]!_q := [n]_q [n-1]_q \cdots [1]_q$ and $[n]_q := 1 + q + \cdots + q^{n-1}$.
%${n \brack a_1, \dots , a_r}_q := \frac{[n]!_q}{[a_1]!_q \cdots [a_r]!_q}$, and ${n \brack a}_q := \frac{[n]!_q}{[a]!_q [n-a]!_q}$.
Any statistic on $\symm_n$ with this distribution is {\em Mahonian}.
%The joint distribution $\sum_{w \in \symm_n} q^{\inv(w)} t^{\maj(w)}$ of $(\inv, \maj)$ is called the {\em biMahonian distribution}.

We review the (ordinary) representation theory of $\symm_n$.
Irreducible $\QQ[\symm_n]$-modules $S^\lambda$ are indexed by partitions $\lambda\vdash n$ and form a free $\ZZ$-basis for the \emph{Grothendieck group} $G_0(\QQ[\symm_n])$ of $\symm_n$.
As $\QQ[\symm_n]$ is \emph{semisimple}, any finite-dimensional $\QQ[\symm_n]$-module $M$ can be written as a direct sum of irreducible submodules, hence an element of $G_0(\QQ[\symm_n])$; sending each irreducible $S^\lambda$ to the \emph{Schur function} $s_\lambda$ gives the {\em Frobenius character} $\Frob(M)$.
%\footnote{The Frobenius character $\Frob(V)$ is indeed a ``character'' since the Schur functions are characters of irreducible polynomial representations of the general linear groups.}
This is an isomorphism of self-dual graded Hopf algebras between the
\emph{Grothendieck group} $G_0(\QQ[\symm_*]):=\bigoplus_{n\ge0}
G_0(\QQ[\symm_n])$ of the tower $\QQ[\symm_*]:
\QQ[\symm_0]\hookrightarrow
\QQ[\symm_1]\hookrightarrow\QQ[\symm_2]\hookrightarrow\cdots$ of algebras
and the ring $\Sym$ of symmetric functions (see, e.g., Grinberg and
Reiner~\cite[Section 4.4]{GrinbergReiner}). 
Moreover, a graded $\QQ[\symm_n]$-module $V = \bigoplus_{d \geq 0} V_d$ with each component $V_d$ finite-dimensional has {\em graded Frobenius series} $\grFrob(V;q) := \sum_{d \geq 0} \Frob(V_d) \cdot q^d$.



Now let $\FF$ be an arbitrary field.
The {\em (type A) $0$-Hecke algebra} $H_n(0)$ is a unital associative $\FF$-algebra with generators $\pi_1, \dots, \pi_{n-1}$ subject to quadratic relations $\pi_i^2 =  \pi_i$ for all $i\in[n-1]$ and the same braid relations as the generators $s_1,\ldots,s_{n-1}$ of $\symm_n$.
One can realize $\pi_i$ as the \emph{bubble sorting operator} acting on a list of entries $(a_1,\ldots,a_n)$ from a totally ordered alphabet by swapping $a_i$ and $a_{i+1}$ if $a_i>a_{i+1}$ or fixing the list otherwise. 
%Roughly speaking, whereas in a typical $\symm_n$-module the generator $s_i$ swaps $i$ and $i+1$, in a typical $H_n(0)$-module the generator $\pi_i$ acts by `sorting' $i$ and $i+1$.

The algebra $H_n(0)$ is also generated by $\pib_1, \dots, \pib_{n-1}$, where $\pib_i := \pi_i - 1$, with quadratic relations $\pib_i^2 =  -\pib_i$ for all $i$, and the same braid relations as above.
%We will also use the relation $\pi_i\pib_i=0$ for all $i$.  
For each permutation $w\in \symm_n$ with any reduced expression $w = s_{i_1} \cdots s_{i_{\ell}}$, the elements $\pi_w := \pi_{i_1} \cdots \pi_{i_{\ell}}$ and $\pib_w := \pib_{i_1} \cdots \pib_{i_{\ell}}$ are well-defined.
The sets $\{ \pi_w \,:\, w \in \symm_n \}$ and $\{ \pib_w \,:\, w \in \symm_n \}$ are both $\FF$-bases for $H_n(0)$.
In particular, $H_n(0)$ has dimension $n!$. 
%In contrast to the situation with the symmetric group, the representation theory of the 0-Hecke algebra is insensitive to the choice of ground field, which motivates our generalization from $\QQ$ to $\FF$.

We recall some notations before reviewing the representation theory of $H_n(0)$. 
A sequence $\alpha = (\alpha_1, \dots, \alpha_{\ell})$ of positive integers is a {\em composition} of $n=|\alpha|:=\alpha_1 + \cdots + \alpha_{\ell}$; this is denoted by $\alpha \models n$.
We call $\alpha_1,\ldots,\alpha_\ell$ the {\em parts} of $\alpha$ and define the \emph{length} of $\alpha$ to be $\ell(\alpha) := \ell$.
The {\em descent set} of $\alpha$ is $\Des(\alpha) := \{\alpha_1, \alpha_1 + \alpha_2, \dots, \alpha_1 + \alpha_2 + \cdots + \alpha_{\ell - 1} \}$.
The map $\alpha \mapsto \Des(\alpha)$ is a bijection from compositions of $n$ to subsets of $[n-1]$.
The {\em major index} of $\alpha$ is $\maj(\alpha) := \sum_{i\in\Des(\alpha)} i$. % = (\ell - 1) \cdot \alpha_1 + \cdots + 1 \cdot \alpha_{\ell-1} + 0 \cdot \alpha_{\ell}.
Given two compositions $\alpha, \beta \models n$, we write $\alpha \preceq \beta$ if 
$\Des(\alpha) \subseteq \Des(\beta)$, i.e., if $\alpha$ is refined by $\beta$.  
The {\em complement} $\alpha^c$ of $\alpha\models n$ is the unique composition of $n$
which satisfies $\Des(\alpha^c) = [n-1] \setminus \Des(\alpha)$.
For $\alpha = (2,3,1,2) \models 8$ we have $\ell(\alpha) = 4$, $\Des(\alpha) = \{2,5,6\}$, $\maj(\alpha) = 2+5+6 = 13$, $\alpha^c = (1,2,1,3,1) \models 8$, and $\Des(\alpha^c) = \{1,3,4,7\} = [7]\setminus\{2,5,6\}$.  

The \emph{descent class} of a composition $\alpha\models n$ consists of permutations $w\in\symm_n$ with $\Des(w) = \Des(\alpha)$; it is an interval under the left weak order of $\symm_n$ whose unique minimal element $w_0(\alpha)$ is the longest element in the parabolic subgroup of $\symm_n$ generated by $\{s_i:i\in \Des(\alpha)\}$. 
For example, if $\alpha = (2,3,1,2)\models 8$ then $w_0(\alpha) = 13247658 \in \symm_8$.

Norton~\cite{Norton} showed that, for each $\alpha\models n$, the module $P_{\alpha} := H_n(0) \pib_{w_0(\alpha)} \pi_{w_0(\alpha^c)}$ has a basis $\{ \pib_w \pi_{w_0(\alpha^c)} : w\in\symm_n,\ \Des(w) = \Des(\alpha) \}$ and a unique maximal submodule spanned by all elements in this basis except the cyclic generator $\pib_{w_0(\alpha)} \pi_{w_0(\alpha^c)}$.
The quotient of $P_\alpha$ by this maximal submodule, denoted by $C_\alpha$, is one-dimensional and admits an $H_n(0)$-action by $\pib_i = -1$ for all $i\in \Des(\alpha)$ and $\pib_i = 0$ for all $i\in \Des(\alpha^c)$. 

The algebra $H_n(0)$ is \emph{non-semisimple}.
The set $\{ P_\alpha:\alpha\models n\}$ (or $\{ C_\alpha: \alpha\models n\}$, resp.) is a complete list of nonisomorphic projective indecomposable (or irreducible, resp.) $H_n(0)$-modules, and gives a $\ZZ$-basis for the \emph{Grothendieck group} $G_0(H_n(0))$ (or $K_0(H_n(0))$, resp.) of $H_n(0)$.
A finite-dimensional $H_n(0)$-module $M$ is identified with the sum of its composition factors (with multiplicities) in $G_0(H_n(0))$. 
If $M$ is also projective then it is a direct sum of projective indecomposable submodules, hence an element of $K_0(H_n(0))$.
With certain product and coproduct, the \emph{Grothendieck groups} $G_0(H_*(0)):=\bigoplus_{n\ge0} G_0(H_n(0))$ and $K_0(H_*(0)) := \bigoplus_{n\ge0} K_0(H_n(0))$ of the tower $H_*(0): H_0(0)\hookrightarrow H_1(0)\hookrightarrow H_2(0)\hookrightarrow\cdots$ of algebras become graded Hopf algebras dual to each other via the pairing defined by $\langle P_\alpha, C_\alpha \rangle := \delta_{\alpha,\beta}$ (Kronecker delta) for all compositions $\alpha$ and $\beta$.

Recall that the ring $\QSym$ of quasisymmetric functions (or the ring $\NSym$ of noncommutative symmetric functions, resp.) has a basis consisting of the \emph{fundamental quasisymmetric functions} $F_\alpha$ (or the \emph{noncommutative ribbon Schur functions} $\sss_\alpha$, resp.) for all compositions $\alpha$.
Krob and Thibon~\cite{KT} defined two isomorphisms of graded Hopf algebras, the {\em quasisymmetric characteristic} $\Ch: G_0(H_*) \to \QSym$ and the {\em noncommutative characteristic} $\ch: K_0(H_*) \to \NSym$ by $\Ch(C_\alpha) := F_\alpha$ and $\ch(P_\alpha) := \sss_\alpha$, respectively. %, for all compositions $\alpha$. 

Let $V = \bigoplus_{d \geq 0} V_d$ be a graded $H_n(0)$-module with finite-dimensional components $V_d$.
It has {\em graded quasisymmetric characteristic} $\Ch_t(V) := \sum_{d \geq 0} \Ch(V_d) \cdot t^d$.
If $V$ is projective, then it has a {\em graded noncommutative characteristic} $\ch_t(V) := \sum_{d \geq 0} \ch(V_d) \cdot t^d$.
Moreover, the {\em length filtration} $H_n(0)^{(0)} \supseteq H_n(0)^{(1)} \supseteq H_n(0)^{(2)} \supseteq \cdots$, where $H_n(0)^{(\ell)}$ is the span of $\{ \pi_w: w \in \symm_n,\ \ell(w) \ge \ell \}$ for all $\ell\ge0$, induces a filtration for any cyclic $H_n(0)$-module $H_n(0)v$. 
Thus if $V$ is a direct sum of cyclic $H_n(0)$-modules, then its has a bi-filtration by $V^{(\ell)} \cap V_d$ for $\ell,d \geq 0$.
Following earlier work~\cite{Huang}, we define the {\em (length-degree-)bigraded quasisymmetric characteristic} of $V$ below, which specializes to $\Ch_{1,t}(V) = \Ch_t(V)$: % and $\Ch_{q,1}(V) = \Ch_q(V)$:
\begin{equation}
\Ch_{q,t}(V) := \sum_{\ell, d \geq 0} \Ch \left( (V^{(\ell)} \cap V_d) \Big/ (V^{(\ell+1)} \cap V_d) \right) \cdot q^{\ell} t^d.
\end{equation} 


The algebra $H_n(0)$ acts on the polynomial ring $\FF[\xx_n]$ by the {\em isobaric Demazure operators}:
\begin{equation}\label{Demazure}
\pi_i(f) := \frac{x_i f - x_{i+1} (s_i(f))}{x_i - x_{i+1}}, \quad \forall f\in \FF[\xx_n],\quad 1\le i\le n-1.
\end{equation}
The quotient algebra $S_{n,k}:=\FF[\xx_n]/J_{n,k}$ defined in Section~\ref{Introduction} is a graded $H_n(0)$-module as one can verify that the ideal $J_{n,k}$ is homogeneous and stable under the $H_n(0)$-action on $\FF[\xx_n]$ using the `Leibniz Rule' $\pib_i(fg) = \pib_i(f) g + s_i(f) \pib_i(g)$ and the observation that $\pi_i(h_k(x_1, \dots, x_i)) = h_k(x_1, \dots, x_i, x_{i+1})$ for all $i\in[n-1]$.
This observation also implies that the span of $\{h_k(x_1, \dots, x_i):i\in[n]\}$ is isomorphic to the \emph{defining representation} of $H_n(0)$ on the span of $[n]$ by $\pi_i(i) = i+1$ and $\pi_i(j) = j$ for all $i\in[n-1]$ and $j\in[n]\setminus\{i\}$.

We have $J_{n,1} = \langle x_1, x_2, \dots, x_n \rangle$, so that $S_{n,1} \cong \FF$ is the trivial $H_n(0)$-module in degree $0$.
It can be shown that $J_{n,n} = I_n$, so that $S_{n,n} = R_n$ (over $\FF=\QQ$) is the classical coinvariant algebra.  
The first author~\cite{Huang} proved that $R_n$ is isomorphic to the regular representation of $H_n(0)$ and obtained its length-degree-bigraded quasisymmetric characteristic (with $F_{\Des(w^{-1})}:=F_\alpha$ for the composition $\alpha\models n$ satisfying $\Des(\alpha) = \Des(w^{-1})$)
\[ \Ch_{q,t} (R_n) = \sum_{w\in\symm_n} q^{\inv(w)} t^{\maj(w)} F_{\Des(w^{-1})}. \]


To study $R_{n,k}$ we need to use ordered set partitions.
An {\em ordered set partition} $\sigma$ of size $n$ is a set partition of $[n]$ with a total order on its blocks.
Let $\OP_{n,k}$ denote the collection of ordered set 
partitions of size $n$ with $k$ blocks.  
In particular, we may identify $\OP_{n,n}$ with $\symm_n$.
We can write $\sigma\in\OP_{n,k}$ as a permutation in $\symm_n$ with $k$ blocks separated by $k-1$ bars such that letters within each block are increasing and blocks are ordered from left to right.
For example, we have $\sigma = (245 | 6 | 13) \in \OP_{6,3}$. 
The {\em shape} of an ordered set partition $\sigma = (B_1 | \cdots | B_k)\in\OP_{n,k}$ is
the composition $\alpha = (|B_1|, \dots, |B_k|) \models n$.  For example, $\sigma = (245 | 6 | 13)$ has shape $(3,1,2) \models 6$.
Given $\alpha \models n$, let $\OP_{\alpha}$ denote the collection of ordered set partitions of size $n$ with shape $\alpha$.  
We can represent $\sigma \in \OP_{\alpha}$ as the pair $(w, \alpha)$, where $w = w(1) \cdots w(n)\in \symm_n$ is obtained by erasing the bars in $\sigma$.  For example, $\sigma = (245 | 6 | 13) = (245613, (3,1,2))$.
This notation establishes a bijection between $\OP_{n,k}$ and pairs $(w, \alpha)$ where $\alpha \models n$, $\ell(\alpha) = k$, $w \in \symm_n$, $\Des(w) \subseteq \Des(\alpha)$.

The algebra $H_n(0)$ acts on the $\FF$-vector space $\FF[\OP_{n,k}]$ with basis $\OP_{n,k}$ by the rule
\begin{equation}
\label{pi-defining-action}
\pib_i . \sigma := \begin{cases}
-\sigma, & \text{if $i+1$ appears in a block to the left of $i$ in $\sigma$,} \\
s_i(\sigma) & \text{if $i+1$ appears in a block to the right of $i$ in $\sigma$.} \\
0, & \text{if $i+1$ appears in the same block as $i$ in $\sigma$,} 
%\pi_i . \sigma := \begin{cases}
%0 & \text{if $i+1$ appears in a block to the left of $i$ in $\sigma$,} \\
%\sigma + s_i(\sigma), & \text{if $i+1$ appears in a block to the right of $i$ in $\sigma$.} \\
%\sigma & \text{if $i+1$ appears in the same block as $i$ in $\sigma$,} 
\end{cases}
\end{equation}
% where $s_i.\sigma$ is the ordered set partition $\sigma$ with $i$ and $i+1$ interchanged.
% This is already defined.
For example, if $\sigma =  (25 | 6 | 134)$ then $\pib_1 (\sigma) = -\sigma$, $\pib_2 (\sigma) = (35 | 6 | 124)$, and $\pib_3 (\sigma) = 0$.
This $H_n(0)$-action preserves $\FF[\OP_\alpha]$ for each $\alpha \models n$ and turns out to be a special case of an $H_n(0)$-action on \emph{generalized ribbon tableaux} introduced by the first author~\cite{HuangTab}.

We next extend the major index from permutations to ordered set partitions in a `reverse' way to some other work~\cite{HRS,RW}.  
%We caution the reader that our definition of $\maj$ is {\bf not} equivalent to, or even equidistributed with, the corresponding statistics for ordered set partitions in other work~\cite{RW, HRS}.  
%However, the distribution of our $\maj$ on $\OP_{n,k}$ is the reversal of the distribution of their $\maj$.
For $\sigma = (B_1 | \cdots | B_k) = (w,\alpha) \in\OP_{n,k}$, define
\begin{equation}
\maj(\sigma) = \maj(w, \alpha) := \maj(w) + \sum_{i \,  : \,  \max(B_i) < \min(B_{i+1})} (\alpha_1 + \cdots + \alpha_i - i).
\end{equation}
For example, $\maj ( 2 4 | 5 7 | 1 3 6 | 8) = \maj(24571368) + (2 - 1) +(2 + 2 + 3 - 3) = 4 + 5 = 9$.


%The generating function for $\maj$ on $\OP_{n,k}$ may be described as follows.  
Let $\rev_q$ be the operator on polynomials in $q$ that reverses coefficient sequences.
For example, $\rev_q(3q^3 + 2q^2 + 1) = q^3 + 2q + 3$.
The {\em $q$-Stirling number} $\Stir_q(n,k)$ is defined by $\Stir_q(0,k) := \delta_{0,k}$ and $\Stir_q(n,k) := \Stir_q(n-1,k-1) + [k]_q \cdot \Stir_q(n-1,k)$ for $n\ge1$.

\begin{proposition}
For $n\ge k\ge1$ we have $\sum_{\sigma \in \OP_{n,k}} q^{\maj(\sigma)} = \rev_q( [k]!_q \cdot \Stir_q(n,k))$. %where $\Stir_q(n,k)$ is defined by the recursion $\Stir_q(n,k) = \Stir_q(n-1,k-1) + [k]_q \cdot \Stir_q(n-1,k)$ and the initial condition $\Stir_q(0,k) = \delta_{0,k}$ (the Kronecker delta) and $\rev_q$ be the operator on polynomials in the variable $q$ which reverses coefficient sequences.
\end{proposition}




\section{Graded vector space structure}
\label{Hilbert}

In this section we give the Hilbert series and describe the standard monomial basis for $S_{n,k} := \FF[\xx_n]/J_{n,k}$ as a graded vector space, where $k\le n$ are two positive integers. 

%We use tools from elementary algebraic geometry similarly as Haglund, Rhoades, and Shimozono~\cite{HRS} in their study of the $\symm_n$-module $R_{n,k}$.
%This method dates back to the work of Garsia and Procesi on Springer
%fibers and Tanisaki quotients \cite{GP}.

%Given a finite point set $Z \subseteq \FF^n$, There is a natural identification of the collection of
%polynomial functions $Z \rightarrow \FF$ with the quotient of $\FF[\xx_n]$ by the ideal $\II(Z)$ of all polynomials vanishing on $Z$.
%\begin{equation}
%\II(Z) := \{ f \in \FF[\xx_n] \,:\, f(\zz) = 0 \text{ for all $\zz \in Z$} \}.
%\end{equation} 
%In particular, the dimension of this quotient as an $\FF$-vector space is
%\begin{equation}
%\dim \left( {\FF[\xx_n]} / {\II(Z)} \right) = |Z|.
%\end{equation}
%The ideal $\II(Z)$ is almost never homogeneous, but the ideal
%For any nonzero polynomial $f \in \FF[\xx_n]$, let $\tau(f)$ be the top degree component of $f$.  
%That is, if $f = f_d + f_{d-1} + \cdots + f_0$ where $f_i$ has homogeneous degree $i$ for all $i$ and $f_d \neq 0$, then $\tau(f) = f_d$.  
%$\TT(Z)$ generated by the top degree components of all nonzero polynomials in $\II(Z)$ is homogeneous, so that ${\FF[\xx_n]} / {\TT(Z)}$ is a graded $\FF$-vector space.
%Moreover, it is well known that $\dim \left( {\FF[\xx_n]} / {\TT(Z)} \right) = \dim \left( {\FF[\xx_n]} / {\II(Z)} \right) = |Z|$.
%Thus if we can find a point set $Z_{n,k} \subseteq \FF^n$ which is in bijective correspondence with $\OP_{n,k}$ such that the generators of $J_{n,k}$ arise as top degree components of polynomials in $\II(Z_{n,k})$, then we have $\dim(S_{n,k}) = |\OP_{n,k}|$. %using Gr\"obner theory 

%By field extension we may assume $|\FF|\ge n+k-1$, without loss of generality.
%Let $\alpha_1, \alpha_2, \dots, \alpha_{n+k-1} \in \FF$ be distinct elements.  Define $Z_{n,k} \subseteq \FF^n$ to be the collection of points $(z_1, z_2, \dots, z_n)$ such that $z_1, z_2, \dots, z_n$ are distinct, $z_i \in \{\alpha_1, \alpha_2, \dots, \alpha_{k+i-1} \}$ for all $i\in[n]$, and $\{\alpha_1, \alpha_2, \dots, \alpha_k\} \subseteq \{z_1, z_2, \dots, z_n\}$.
%We define a bijection $\varphi: \OP_{n,k} \rightarrow Z_{n,k}$ by sending each $\sigma = (B_1 | \cdots | B_k) \in \OP_{n,k}$ to $\varphi(\sigma) = (z_1, \dots, z_n) \in Z_{n,k}$ as follows.
%For each $i\in[k]$, let $z_{j_i} := \alpha_i$, where $j_i := \min(B_i)$.  
%Write the set of unassigned indices of $(z_1, \dots, z_n)$ as  $S = [n] - \{\min(B_1), \dots, \min(B_k)\} = \{s_1 < \cdots < s_{n-k} \}$.
%For each $s \in [n] \setminus \{j_1,\ldots,j_k\} = \{s_1 < \cdots < s_{n-k} \} $, let $\ell_s$ be the number of blocks $B$ weakly to the left of $s$ in $\sigma$ with $\min(B) < s$.
%Let $z_{s_1} := \alpha_{k + \ell_{s_1}}$.  Assuming $z_{s_1}, z_{s_2}, \dots, z_{s_{j-1}}$ have already 
%been defined, let $z_{s_j}$ be the $\ell_{s_j}$-th term in the sequence formed by deleting
%$z_{s_1}, z_{s_2}, \dots, z_{s_{j-1}}$ from $(\alpha_{k+1}, \alpha_{k+2}, \dots, \alpha_{n+k-1})$.
%For example, $\varphi$ sends $( 7   | 2 4 8  | 1 3  | 5 9)$ to $(\alpha_3, \alpha_2,  \alpha_6,  \alpha_5 , \alpha_4, \alpha_9, \alpha_1,  \alpha_8, \alpha_{12})$.

%As an example of the map $\varphi$, let $\sigma = (7 | 248 | 13 | 569) \in \OP_{9,4}$.
%The following table computes the image $\varphi(\sigma) = (z_1, \dots, z_9)$.
%We start by assigning the coordinates $(z_7, z_2, z_1, z_5) = (\alpha_1, \alpha_2, \alpha_3, \alpha_4)$ of the letters in the minimal blocks of $\sigma$.
%t the top row of the table, the coordinates corresponding to the letters $S = \{3,4,6,8,9\}$ which are not minimal in their blocks of $\sigma$ are unassigned and we have the sequence of possible values  $(\alpha_{k+1}, \dots, \alpha_{n+k-1}) = (\alpha_5, \dots, \alpha_{12})$.
%We add the elements of $S$ to the blocks of $\sigma$ one at a time, from smallest to largest.
%At each stage, we record the letter $s$ added together with the number $\ell_s$ of blocks $B$ weakly to the left of $s$ in $\sigma$ which satisfy $\min(B) < s$.
%We assign the coordinate $z_s$ the value of the $\ell_s^{th}$ term in the list of unassigned values, and then erase the value from the list.


%\begin{lemma}
%\label{j-contained-in-t}
%Assume $|\FF|\ge n+k-1$.  We have $J_{n,k} \subseteq \TT(Z_{n,k})$.
%\end{lemma}


We endow monomials in $\FF[\xx_n]$ with the {\em negative lexicographical term order} $<$
defined by 
$x_1^{a_1} \cdots x_n^{a_n} < x_1^{b_1}  \cdots x_n^{b_n}$ if and only if there exists $j\in[n]$ such that $a_{j+1} = b_{j+1}, \dots, a_n = b_n$, and $a_j < b_j$.
Following the notation of $\textsc{sage}$, we denote this term order by {\tt neglex}.
For any nonzero $f \in \FF[\xx_n]$, let $\initial_<(f)$ be its leading (i.e., largest) term with respect to $<$.  
The {\em initial ideal} of an ideal $I$ of $\FF[\xx_n]$ is the monomial ideal $\initial_<(I) := \langle \initial_<(f) \,:\, f \in I \setminus \{0\} \rangle$.
The set of all monomials $m \in \FF[\xx_n]$ with $m \notin \initial_<(I)$
descends to an $\FF$-basis for the quotient  ${\FF[\xx_n]} / {I}$; this
basis is called the {\em standard monomial basis}~\cite[Proposition 1, pp. 230]{CLO}.
 
%We describe the standard monomial basis of the quotient $S_{n,k}$ with respect to the $\neglex$ term order on $\FF[\xx_n]$.  
Following the notion of \emph{skip monomials} in %earlier work
\cite[Definition 3.2]{HRS}, we define the {\em reverse skip
monomial} of $S = \{s_1 < \cdots < s_m \} \subseteq [n]$ as
$\xx(S)^* :=
x_{n-s_1+1}^{s_1} x_{n-s_2+1}^{s_2-1} \cdots x_{n-s_m+1}^{s_m-m+1}$.
For example, $\xx(2578)^* = x_8^2 x_5^4 x_3^5 x_2^5$ if $n = 9$. 
%The following definition is the reverse of \cite[Defn. 4.4]{HRS}.
A monomial $m \in \FF[\xx_n]$ is {\em $(n,k)$-reverse nonskip} if $x_i^k \nmid m$ for all $i\in [n]$ and $\xx(S)^* \nmid m$ for all $S \subseteq [n]$ with $|S| = n+k-1$.
Let $\CCC_{n,k}$ be the set of all $(n,k)$-reverse nonskip monomials in $\FF[\xx_n]$.
%There is some redundancy in this definition. 
%In particular, if $n \in S$, the power of $x_1$ in $\xx(S)^*$  where $|S| = n-k+1$ is $x_1^k$, so that we need only consider those sets $S$ with $n \notin S$.

\begin{theorem}
\label{standard-monomial-basis}
For any field $\FF$, the dimension of  $S_{n,k} = {\FF[\xx_n]} / {J_{n,k}}$ is $|\OP_{n,k}|$ and the set $\CCC_{n,k}$ is the standard monomial basis of $S_{n,k}$ with respect to the $\neglex$ term order on $\FF[\xx_n]$.
\end{theorem}


If $k = n$ then $\CCC_{n,n}$ consists of `sub-staircase' monomials $x_1^{a_1} \cdots x_n^{a_n}$ with $0 \leq a_i \leq n-i$ for all $i\in[n]$; this is the basis for the coinvariant algebra $R_n$ obtained by E. Artin~\cite{Artin} using Galois theory.
To generalize this `staircase' characterization to $\CCC_{n,k}$, we define 
an {\em $(n,k)$-staircase} to be a shuffle of $(k-1, k-2, \dots, 1, 0)$ and
$(k-1, k-1, \dots, k-1)$, where the second sequence
has $n-k$ copies of $k-1$.  For example, the $(5,3)$-staircases are
\begin{center}
$(2,1,0,2,2)$, $(2,1,2,0,2)$, $(2,2,1,0,2)$, $(2,1,2,2,0)$, $(2,2,1,2,0)$, and
$(2,2,2,1,0)$.
\end{center}
The following result follows from Theorem~\ref{standard-monomial-basis}
and a previous result~\cite[Theorem 4.13]{HRS}.

\begin{corollary}
\label{staircase-shuffle}
The standard monomial basis $\CCC_{n,k}$ of $S_{n,k}$ consists of those monomials
$x_1^{a_1} \cdots x_n^{a_n}$ whose exponent sequences $(a_1, \dots, a_n)$ 
are componentwise $\leq$ {\em some} $(n,k)$-staircase.
\end{corollary}
%Let us mention an analogous characterization of $\CCC_{n,k}$ derived from earlier work~\cite{HRS}.
%The reversal of a previous result~\cite[Thm. 4.13]{HRS} shows that $\CCC_{n,k}$ consists of all monomials $x_1^{a_1} \cdots x_n^{a_n}$ with exponent sequence $(a_1, \dots, a_n)$ componentwise $\leq$ {\em some} $(n,k)$-staircase.
%Recall that a {\em shuffle} of two sequences $(a_1, \dots, a_r)$ and $(b_1, \dots, b_s)$ is an interleaving $(c_1, \dots, c_{r+s})$ of these sequences which preserves the relative order of the $a$'s and the $b$'s.  
%Here a {\em $(n,k)$-staircase} is a shuffle of $(k-1, k-2, \dots, 1, 0)$ and $(k-1, k-1, \dots, k-1)$; the latter sequence has $n-k$ copies of $k-1$.
%For example, the $(5,3)$-staircases are the shuffles of $(2,1,0)$ and $(2,2)$, i.e., $(2,1,0,2,2)$, $(2,1,2,0,2)$, $(2,2,1,0,2)$, $(2,1,2,2,0)$, $(2,2,1,2,0)$, and $(2,2,2,1,0)$.
%The following result is just the reversal of a previous result~\cite[Thm. 4.13]{HRS}.
For example, $(4,2)$-staircases are shuffles of $(1,0)$ and $(1,1)$, i.e., $(1,0,1,1), (1,1,0,1),$ and $(1,1,1,0)$, so 
\begin{equation*}
\CCC_{4,2} = \{1, x_1, x_2, x_3, x_4, x_1 x_2, x_1 x_3, x_1 x_4, x_2 x_3, x_2 x_4, x_3 x_4, x_1 x_3 x_4, x_1 x_2 x_4, x_1 x_2 x_3 \}.
\end{equation*}
%Consequently, we have the Hilbert series $\Hilb(S_{4,2};q) = 1 + 4q + 6q^2 + 3q^3$.

%We next describe a Gr\"obner basis of the ideal $J_{n,k}$, which is analogous to a result on the ideal $I_{n,k}$~\cite[Thm. 4.14]{HRS}, although the Gr\"obner basis here is not reduced. 
Our next result gives a Gr\"obner basis of $J_{n,k}$.
Recall that a finite set $G = \{g_1, \dots, g_r\}$ of nonzero polynomials in an ideal $I$ of the polynomial ring $\FF[\xx_n]$ is a {\em Gr\"obner basis} of $I$ if $\initial_<(I) = \langle \initial_<(g_1), \dots , \initial_<(g_r) \rangle$; this implies $I = \langle G \rangle$. % \cite[Cor. 6, p. 77]{CLO}.
%The Gr\"obner basis $G$ is called {\em minimal} if for any $g \in G$, the leading coefficient of $g$ with respect to $\leq$ is $1$, and for any $g \neq g'$ in $G$, the leading monomial of $g$ does not divide the leading monomial of $g'$.
%A minimal Gr\"obner basis $G$ is called {\em reduced} if in addition for any $g \neq g'$ in $G$, the leading monomial of $g$ does not divide any term in the polynomial $g'$.
%Up to a choice of term order, every ideal $I$ has a unique reduced Gr\"obner basis \cite[Prop. 6, p. 92]{CLO}.
The reader is referred to Cox, Little, and O'Shea~\cite{CLO} for an introduction to Gr\"obner theory.
Given a {\em weak composition} (i.e., a sequence of nonnegative integers) $\gamma$ of length $n$, let $\kappa_{\gamma}(\xx_n) \in \FF[\xx_n]$ be the associated {\em Demazure character} (or {\em key polynomial});
see e.g. \cite[Section 2.4]{HRS}.  
For any $S\subseteq [n]$, let $\gamma(S)^*$ be the exponent sequence of the reverse skip monomial $\xx(S)^*$.


\begin{theorem}
\label{groebner-basis}
Let $k \leq n$ be positive integers and endow monomials in $\FF[\xx_n]$ with the $\neglex$ term order.
Then the ideal $J_{n,k}$ has a Gr\"obner basis consisting of $h_k(x_1,\ldots,x_i)$ for all $i\in[n]$ and $\kappa_{\gamma(S)^*}(\xx_n)$ for all $S \subseteq [n-1]$ with $|S| = n-k+1$.
This Gr\"obner basis is minimal when $k < n$.
\end{theorem}

For example, if $(n,k) = (6,4)$, a (minimal) Gr\"obner basis of $J_{6,4} \subseteq \FF[\xx_6]$ consists of the polynomials $h_4(x_1,x_2,\ldots,x_i)$ for $i=1,2,\ldots,6$ and the Demazure characters
\begin{center} 
$\kappa_{(0,0,0,1,1,1)}(\xx_6)$, $\kappa_{(0,0,2,0,1,1)}(\xx_6)$, $\kappa_{(0,3,0,0,1,1)}(\xx_6)$, $\kappa_{(0,0,2,2,0,1)}(\xx_6)$, $\kappa_{(0,3,0,2,0,1)}(\xx_6)$, $\kappa_{(0,3,3,0,0,1)}(\xx_6)$, $\kappa_{(0,0,2,2,2,0)}(\xx_6)$, $\kappa_{(0,3,0,2,2,0)}(\xx_6)$, $\kappa_{(0,3,3,0,2,0)}(\xx_6)$, $\kappa_{(0,3,3,3,0,0)}(\xx_6)$.
 \end{center}

%The authors do not have a conjecture for the reduced Gr\"obner basis for the ideal $J_{n,k}$.

Recall that the {\em Hilbert series} of a graded vector space $V = \bigoplus_{d \geq 0} V_d$ with each component $V_d$ finite-dimensional is $\Hilb(V;q) := \sum_{d \geq 0} \dim(V_d) \cdot q^d$. 
%Combining our results with previous work~\cite{HRS} gives the Hilbert series of $S_{n,k}$.

\begin{theorem}
\label{hilbert-series}
Let $k \leq n$ be positive integers.  We have
$\Hilb(S_{n,k};q) = \rev_q([k]!_q \cdot \Stir_q(n,k))$.
\end{theorem}

The Garsia--Stanton basis is another important basis of the coinvariant algebra $R_n$.
For any composition $\alpha \models n$, let $\xx_{\alpha} := \prod_{j \in \Des(\alpha)} (x_1 x_2 \dots x_j)$.  
The {\em Garsia--Stanton monomial} (or {\em descent monomial}) of a permutation $w \in \symm_n$ is 
$gs_w := w(\xx_{\alpha})$, where $\alpha \models n$ is characterized by 
$\Des(\alpha) = \Des(w)$.  
By construction the degree of $gs_w$ is $\maj(w)$.
Garsia~\cite{Garsia} proved that the set 
$\GS_n := \{ gs_w \,:\, w \in \symm_n \}$ of all GS monomials descends to a 
basis of $R_n$.
Garsia and Stanton \cite{GS} later studied $\GS_n$ in the context
of Stanley--Reisner theory.


The $(n,k)$-generalization of the GS monomials is as follows.
If $\ii = (i_1, \dots, i_n)$ is a sequence of nonnegative integers and $\alpha \models n$ then define $\xx_{\alpha,\ii} := \xx_\alpha\cdot x_1^{i_1} x_2^{i_2} \cdots x_{n}^{i_{n}}$.
For any $w\in\symm_n$, let $\alpha$ be the composition of $n$ with $Des(\alpha) = \Des(w)$ and define  a monomial $gs_{w,\ii} := w(\xx_{\alpha,\ii})\in \FF[\xx_n]$ of degree $\maj(w) + |\ii|$, where $|\ii| := i_1 + \cdots  + i_n$.
%For example, let $(n,k) = (9,5)$, $\ii = (2,2,1,1,0,0,0,0,0)$, and $w = 254689137 \in \symm_9$. 
%Then $\alpha = (2,4,3)\models 9$ satisfies $\Des(\alpha) = \{2,6\} = \Des(w)$.
%It follows that $\xx_{\alpha,\ii} = (x_1 x_2) (x_1 x_2 x_3 x_4 x_5 x_6) (x_1^2 x_2^2 x_3^1 x_4^1)$ and $gs_{w,\ii} = (x_2 x_5) (x_2 x_5 x_4 x_6 x_8 x_9) (x_2^2 x_5^2 x_4^1 x_6^1)$.
%Haglund, Rhoades, and Shimozono introduced \cite[Defn. 5.2]{HRS} introduced the set 
Let 
\begin{equation}
\GS_{n,k} := \{ gs_{w,\ii} \,:\, w \in \symm_n, \, k - \des(w) > i_1 \geq \cdots \geq i_{n-k} \geq 0 = i_{n-k+1} = \cdots = i_n\}.
\end{equation}
%For example, $\GS_{7,5}$ consists of $(x_6)(x_6 x_2 x_1 x_3 x_7)$, $(x_6)(x_6 x_2 x_1 x_3 x_7)(x_6)$, $(x_6)(x_6 x_2 x_1 x_3 x_7)(x_6^2)$, $(x_6)(x_6 x_2 x_1 x_3 x_7) (x_6 x_2)$, $(x_6)(x_6 x_2 x_1 x_3 x_7) (x_6^2 x_2)$, and $(x_6)(x_6 x_2 x_1 x_3 x_7) (x_6^2 x_2^2)$.
%When $k = n$, we have $gs_{w,\ii}\in\GS_{n,n}$ if and only if $w\in\symm_n$ and $\ii=0^n$.
%Garsia~\cite{Garsia} proved that $\GS_{n,n}$ descends to a basis of $R_n$.
%Garsia and Stanton \cite{GS} later studied $\GS_{n,n}$ in the context of Stanley-Reisner theory. 


Haglund, Rhoades, and Shimozono~\cite[Theorem 5.3]{HRS} proved that $\GS_{n,k}$ descends to a basis of $R_{n,k}$.
We generalize this result to $S_{n,k}$.
%Using work of Adin, Brenti, and Roichman~\cite{ABR} and a similar construction as in work of the first author~\cite{Huang}, we prove that that $\GS_{n,k}$ is among a family of bases of $S_{n,k}$.
%To state this result, we need some notation.
Given a monomial $m = x_1^{a_1} \cdots x_n^{a_n}$, let $\lambda(m)$ be the sequence obtained by sorting the exponent sequence $(a_1,\ldots,a_n)$ of $m$ into weakly decreasing order.
Let $\prec$ be the partial order on monomials in $\FF[\xx_n]$ defined by $m \prec m'$ if and only if $\lambda(m) < \lambda(m')$ in lexicographical order.
The next result describes a family of sets of polynomials, including $\GS_{n,k}$,
which all descend to bases of $S_{n,k}$.

%Following Adin, Brenti, and Roichman \cite{ABR}, we associate a collection of objects to any monomial $m = x_1^{a_1} \cdots x_n^{a_n}$ in $\FF[\xx_n]$ as follows.
%Let $\sigma(m)=\sigma_1\cdots \sigma_n\in\symm_n$ be the permutation (in one-line notation) obtained by listing the indices of variables in weakly decreasing order of the exponents in $m$, breaking ties by listing smaller indexed variables first.
%Let $d(m) = (d_1, \dots, d_n)$ be the integer sequence given by $d_j = |\Des(\sigma(m)) \cap \{j, j+1, \dots, n\} |$.  
%Adin, Brenti, and Roichman~\cite{ABR} showed that the componentwise difference $\lambda(m) - d(m)$ is an integer partition (i.e., has weakly decreasing components).
%Let $\mu(m)$ be the {\em conjugate} of this integer partition.
%For example, if $m = x_1^3 x_2^4 x_3^0 x_4^2 x_5^2 x_6^0 x_7^0$, then we have $\lambda(m) = (4,3,2,2,0,0,0)$ and $\sigma(m) = 2145367$.  
%It follows that $d(m) = (2,1,1,1,0,0,0), \lambda(m) - d(m) = (2,2,1,1,0,0,0),$ and $\mu(m) = (4,2)$.


\begin{theorem}
\label{garsia-stanton-type-bases}
%Let $\leq$ be the $\neglex$ term order on $\FF[\xx_n]$.
A set $\BBB_{n,k} = \{b_{w,\ii}\}$ indexed by pairs $(w, \ii)$ with $w \in \symm_n$, $\ii = (i_1, \dots, i_n)\in\ZZ^n$, and $k - \des(w) > i_1 \geq \cdots \geq i_{n-k} \geq 0 = i_{n-k+1} = \cdots = i_n$ descends to a basis of $S_{n,k}$ if each $b_{w,\ii} \in \BBB_{n,k}$ satisfies $b_{w,\ii} = gs_{w,\ii} + \sum_{m \prec gs_{w,\ii}} c_m \cdot m$ where $c_m \in \FF$ could depend on $(w,\ii)$.

In particular, the set $\GS_{n,k}$ descends to a basis of $S_{n,k}$.
\end{theorem}

The bases given by Theorem~\ref{garsia-stanton-type-bases} are important for studying $S_{n,k}$ as an $H_n(0)$-module.

%\begin{corollary}\label{garsia-stanton-basis}
%The set $\GS_{n,k}$ descends to a basis of $S_{n,k}$.
%\end{corollary}

%For example, let $(n,k) = (4,2)$.  The permutations in $\symm_4$ with $< 2$ descents are
%\begin{center}
%$1234, 2134, 3124, 4123, 1324, 1423, 2314, 2413, 3412, 1243, 1342,$ and $2341.$
%\end{center}
%The corresponding GS monomials are 
%\begin{equation*}
%1, x_2, x_3, x_4, x_1 x_3, x_1 x_4, x_2 x_3, x_2 x_4, x_3 x_4, x_1 x_2 x_4, x_1 x_3 x_4, \text{ and } x_2 x_3 x_4.
%\end{equation*}
%To obtain $\GS_{4,2}$, we add to this set $x_1$ and $x_1 x_2$ (coming from the permutation $1234$) to obtain
%\begin{equation*}
%\GS_{4,2} = \{1, x_1, x_1 x_2, 
%x_2, x_3, x_4, x_1 x_3, x_1 x_4, x_2 x_3, x_2 x_4, x_3 x_4, x_1 x_2 x_4, x_1 x_3 x_4, x_2 x_3 x_4\}.
%\end{equation*}
%Corollary~\ref{garsia-stanton-basis} implies that this set descends to a basis of $S_{4,2}$.
%\edit{(Is it a coincidence that $\GS_{4,2} = \CCC_{4,2}$?)}


\section{Module structure over the 0-Hecke algebra}
\label{Module}

%In this section we study $S_{n,k}$ as a (graded) $H_n(0)$-module for $n\ge k\ge1$.
%We first study the (ungraded) $H_n(0)$-module $\FF[\OP_{n,k}]$, which is the direct sum of (ungraded) $H_n(0)$-module $\FF[\OP_\alpha]$ for all $\alpha\models n$ with $\ell(\alpha)=k$.
%For $\alpha, \beta \models n$ with $\alpha \preceq \beta$, we define $P_{\alpha,\beta} := H_n(0) \pib_{w_0(\alpha)} \pi_{w_0(\beta^c)}$, whose structure is given by the next lemma.
%As an example, %the module $P_{(4,1),(1,2,1,1)}$ breaks up into projective indecomposable submodules as
%we have $P_{(4,1),(1,2,1,1)} \cong P_{(4,1)} \oplus P_{(1,3,1)} \oplus P_{(3,1,1)} \oplus %P_{(1,2,1,1)}$.
%In particular, $P_{\alpha,\alpha} = P_{\alpha}$.

%\begin{lemma}[{Huang~\cite[Thm. 3.2]{HuangTab}}]
%\label{alpha-beta-structure}
%Let $\alpha, \beta \models n$ and assume $\alpha \preceq \beta$.  
%Then $P_{\alpha,\beta}$ has basis $\{ \pib_w \pi_{w_0(\beta^c)} \,:\, w \in \symm_n, \, \Des(\alpha) \subseteq \Des(w) \subseteq \Des(\beta) \}$ and is the direct sum of projective indecomposable modules $P_{\gamma}$ for all compositions $\gamma\models n$ satisfying $\alpha\preceq \gamma\preceq \beta$.
%\end{lemma}


%Recall that, for each composition $\alpha\models n$, we denote by $\OP_{\alpha}$ the collection of ordered set partitions of shape $\alpha$, i.e., pairs $(w,\alpha)$ for all $w\in\symm_n$ with $\Des(w)\subseteq \Des(\alpha)$.

In this section we study $S_{n,k}$ as a module (ungraded and graded) over the 0-Hecke algebra
$H_n(0)$.  
The ungraded $H_n(0)$-module structure of $S_{n,k}$ is given by the next result.


%\begin{lemma}
%\label{osp-substructure}
%If $\alpha\models n$ then $\FF[\OP_\alpha] = H_n(0)(12\cdots n,\alpha)$ and is isomorphic to $P_{(n),\alpha}$ via the map defined by sending $(w,\alpha)$ to $\pib_w \pi_{w_0(\alpha^c)}$ for all $w\in\symm_n$ with $\Des(w)\subseteq \Des(\alpha)$.
%\[ \FF[\OP_{\beta}] \cong \bigoplus_{\beta \preceq \alpha} P_{\beta}.\]
%\end{lemma}


%\begin{proposition}
%\label{osp-structure}
%We have an $H_n(0)$-module isomorphism $\FF[\OP_{n,k}] \cong \bigoplus_{\beta \models n} P_{\beta}^{\oplus{n - \ell(\beta) \choose k - \ell(\beta)}}$.
%\end{proposition}

%For example, we have
%$\FF[\OP_{(1,3)}] \cong P_{(1,3)} \oplus P_{(4)}$, $\FF[\OP_{(2,2)}]\cong P_{(2,2)} \oplus P_{(4)}$, $\FF[\OP_{(3,1)}] \cong P_{(3,1)}\oplus P_{(4)}$, and summing these gives
%$\FF[\OP_{4,2}] \cong P_{(1,3)}\oplus P_{(2,2)} \oplus P_{(3,1)} \oplus P_{(4)}^{\oplus 3}$; see Figure~\ref{OP42}.




%\subsection{0-Hecke action on polynomials}
%Our next task is to give an ungraded $H_n(0)$-module isomorphism $S_{n,k} \cong \FF[\OP_{n,k}]$.

%\begin{lemma}
%\label{demazure-leading-term}
%Let $\dd := (d_1,\ldots,d_n)$, $d_1,\ldots,d_n\in\ZZ$, $d_1 \geq \dots \geq d_n\ge0$, and $\xx^{\dd} := x_1^{d_1} \cdots x_n^{d_n}$. 
%If $w \in \symm_n$ and $\Des(w) \subseteq \Des(\dd)$ then
%the polynomial $\pib_w (\xx^{\dd})$ has the form
%$\pib_w(\xx^{\dd}) = w(\xx^{\dd}) + \sum_{m \prec \xx^{\dd}} c_m \cdot m$, where $c_m \in \FF$.
%Then the leading term of the polynomial
%$\pib_w (\xx^{\dd})$ under {\tt neglex} is the monomial $w(\xx^{\dd})$.
%\end{lemma}

%We will decompose the quotient $S_{n,k}$ into a direct sum of projective modules of the form $P_{\alpha,\beta}$ defined in \eqref{P-alpha-beta}.
%This will ultimately rest on the following lemma.

%\begin{lemma}
%\label{alpha-beta-monomial}
%Let $\alpha,\beta \models n$, $\alpha\preceq\beta$, $\dd := (d_1,\ldots,d_n)$, $d_1,\ldots,d_n\in\ZZ$, $d_1 \geq \dots \geq d_n\ge0$, and $\Des(\dd) = \Des(\beta)$.
%Then $H_n(0) \pib_{w_0(\alpha)} \xx^{\dd}$ has basis
%$\left\{ \pib_w(\xx^{\dd}) \,:\, \Des(\alpha) \subseteq \Des(w) \subseteq \Des(\beta) \right\}$ and
%sending $\pib_w(\xx^\dd)$ to $\pib_w\pi_{w_0(\beta^c)}$ gives an $H_n(0)$-module isomorphism $H_n(0) \pib_{w_0(\alpha)} \xx^{\dd} \cong P_{\alpha,\beta}$.
%\end{lemma}



%\subsection{Decomposition of $S_{n,k}$}
%We begin by introducing a family of $H_n(0)$-submodules of $S_{n,k}$.

%Let $A_{n,k}$ be the set of all pairs $(\alpha, \ii)$, where $\alpha=(\alpha_1,\ldots,\alpha_\ell) \models n$ and $\ii = (i_1, \dots, i_n)$ satisfy $\alpha_1> n-k$, $i_1,\ldots,i_n\in\ZZ$, and $k - \ell \geq i_1 \geq \cdots \geq i_{n-k} \geq 0 = i_{n-k+1} = \cdots = i_n$.
%For example, let $(n,k) = (6,3)$. 
%Omitting the $n-k = 3$ trailing zeros from $\ii$ and parentheses and commas from $\alpha$ and $\ii$, we have the set $A_{6,3}$ consists of the following elements.
%\begin{small} \begin{equation*}
%\begin{array}{c}
%(411, 000), (42, 111), (42, 110), (42, 100), (42, 000), (51, 111), (51, 110), (51, 100), (51,000), \\
%(6, 222), (6,221), (6,220), (6,211), (6,210), (6,200), (6,111), (6,110), (6,100), (6,000)
%\end{array}
%\end{equation*}
%\end{small}
%Given $(\alpha, \ii) \in A_{n,k}$, let $N_{\alpha,\ii}$ be the $H_n(0)$-module generated by the image of $\pib_{w_0(\alpha)}(\xx_{\alpha,\ii})$ in the quotient $S_{n,k}$.
%To study this $H_n(0)$-module, we define
%We show that $N_{\alpha,\ii} \cong P_{\alpha,\alpha \cup \ii}$, where 
%a composition $\alpha \cup \ii \models n$ by the {\em disjoint} union decomposition $\Des(\alpha \cup \ii) = \Des(\alpha) \sqcup \{1 \leq j \leq n-1 \,:\, i_j > i_{j+1} \}$. 
%In fact, each element of $\Des(\ii)$ lies in the interval $1 \leq j \leq n-k$ whereas each element of $\Des(\alpha)$ lies in the interval $n-k+1 \leq j \leq n-1$.

%\begin{lemma}
%\label{gs-n-lemma}  
%Let $k \leq n$ be positive integers. %and endow monomials in $\FF[\xx_n]$ with the partial order $\prec$.
%Let $(\alpha, \ii) \in A_{n,k}$ and $w \in \symm_n$ be such that $\Des(\alpha) \subseteq \Des(w) \subseteq \Des(\alpha \cup \ii)$. 
%Then the unique $\prec$-leading term of $\pib_w(\xx_{\alpha,\ii})$ is $w(\xx_{\alpha,\ii}) = gs_{w,\ii'} \in \GS_{n,k}$, where $\ii' = (i'_1, \dots, i'_n)$ is related to $\ii = (i_1, \dots, i_n)$ by
%$i'_j = i_j - | \{r \in \Des(w)\cap[n-k] \,:\, r \geq j \} |$.

%Conversely, let $gs_{w,\ii'} \in \GS_{n,k}$.  
%Then $gs_{w,\ii'}$ is the unique $\prec$-leading term of $\pib_w(\xx_{\alpha,\ii})$ 
%for some $w\in\symm_n$ and some $(\alpha,\ii) \in A_{n,k}$ satisfying
%$\Des(\alpha) \subseteq \Des(w) \subseteq \Des(\alpha \cup \ii)$ if and only if $\alpha \models n$ is characterized by $\Des(\alpha) = \Des(w) \setminus [n-k]$ and the sequence $\ii = (i_1, \dots, i_n)$ is related to the sequence $\ii' = (i'_1, \dots, i'_n)$ in the same way as above.
%by Equation~\eqref{i2i'}.
%\end{lemma}


%Lemma~\ref{gs-n-lemma} can be used to derive a new basis for the quotient  $S_{n,k}$.
%This basis will be helpful in decomposing $S_{n,k}$ into a direct sum of $H_n(0)$-modules of the form $N_{\alpha,\ii}$.

%\begin{lemma}
%\label{new-basis-lemma}
%Let $k \leq n$ be positive integers.
%The set $\{ \pib_w(\xx_{\alpha,\ii}) \,:\, (\alpha,\ii) \in A_{n,k}, \, w \in \symm_n, \, \Des(\alpha) \subseteq \Des(w) \subseteq \Des(\alpha \cup \ii) \}$ descends to an $\FF$-basis of the quotient ring $S_{n,k}$. 
%For any $(\alpha, \ii) \in A_{n,k}$ and any $w \in \symm_n$
%with $\Des(\alpha) \subseteq \Des(w) \subseteq \Des(\alpha \cup \ii)$ we have $\deg( \pib_w(\xx_{\alpha,\ii}) ) = \deg( \xx_{\alpha,\ii}) = \maj(\alpha) + |\ii|$.
%\end{lemma} 


%In the coinvariant algebra case $k = n$, the basis of Lemma~\ref{new-basis-lemma} appeared in \cite{Huang}.
%As in \cite{Huang}, this modified GS-basis will facilitate analysis of the $H_n(0)$-structure of $S_{n,k}$.



\begin{theorem}
\label{s-decomposition-theorem}
Let $k \leq n$ be positive integers. 
%If $(\alpha,\ii) \in A_{n,k}$ then $\{ \pib_w(\xx_{\alpha,\ii}) \,:\, w \in \symm_n, \, \Des(\alpha) \subseteq \Des(w) \subseteq \Des(\alpha \cup \ii) \}$ descends to a basis for $N_{\alpha,\ii}$ and we have an isomorphism $N_{\alpha,\ii} \cong P_{\alpha,\alpha \cup \ii}$ of $H_n(0)$-modules by $\pib_w(\xx_{\alpha,\ii}) \mapsto \pib_w\pi_{w_0((\alpha\cup\ii)^c)}$.
As an ungraded $H_n(0)$-module, $S_{n,k}$ is projective and isomorphic to $\FF[\OP_{n,k}]$ with the following direct sum decomposition into indecomposables:
\begin{equation}
S_{n,k} 
\cong \bigoplus_{\beta \models n} P_{\beta}^{\oplus\binom{n - \ell(\beta)}{k - \ell(\beta)}}.
\end{equation}
%In particular, $S_{n,k}$ is a projective $H_n(0)$-module.
\end{theorem}

For example, $S_{4,2} \cong P_{(2,2)} \oplus P_{(1,3)} \oplus P_{(3,1)} \oplus P_{(4)}^{\oplus 3} \cong \FF[\OP_{4,2}]$.
See Figure~\ref{OP42} and \ref{S42}.
% See Figure~\ref{S42}.
%Figure~\ref{S42} illustrates this isomorphism via the action of $H_4(0)$ on the basis of $S_{4,2}$ given by Lemma~\ref{new-basis-lemma}, whose elements are polynomials in general.
%, although they happen to be monomials in this example.

\begin{figure}[h]
\begin{footnotesize}
\[ \xymatrix{
1|234 \ar@(ur,rd)[]^{\pib_2 = \pib_3 = 0 } \ar[d]_{\pib_1} \\
2|134 \ar@(ur,rd)[]^{\substack{\pib_1 = -1 \\ \pib_3 = 0 }} \ar[d]_{\pib_2} \\
3|124 \ar@(ur,rd)[]^{\substack{\pib_1 = 0 \\ \pib_2 = -1 }} \ar[d]_{\pib_3} \\
4|123 \ar@(ur,rd)[]^{\substack{\pib_1 = \pib_2 = 0 \\ \pib_3 = -1 }} } \\ \\
\quad \xymatrix{
& 12|34 \ar@(ur,rd)[]^{\pib_1 = \pib_3 = 0 } \ar[d]_{\pib_2} \\
& 13|24 \ar@(ur,r)[]^{\pib_2 = -1} \ar[ld]^{\pib_1} \ar[rd]_{\pib_3} \\
23|14 \ar@(ul,dl)[]_{\substack{\pib_1 = -1 \\ \pib_2 = 0 }} \ar[rd]^{\pib_3} &&
14|23 \ar@(ur,rd)[]^{\substack{\pib_2 = 0 \\ \pib_3 = -1 }} \ar[ld]_{\pib_1} \\
& 24|13 \ar@(r,rd)[]^{\pib_1 = \pib_3 = -1} \ar[d]_{\pib_2} \\
& 34|12 \ar@(ur,rd)[]^{ \substack{ \pib_1 = \pib_3 = 0 \\ \pib_2 = -1 } } } 
\qquad \xymatrix{
123|4 \ar@(ur,rd)[]^{\pib_1 = \pib_2 = 0} \ar[d]_{\pib_3} \\
124|3 \ar@(ur,rd)[]^{\substack{\pib_1 = 0 \\ \pib_3 = -1 }} \ar[d]_{\pib_2} \\
134|2 \ar@(ur,rd)[]^{\substack{\pib_2 = -1 \\ \pib_3 = 0 }} \ar[d]_{\pib_1} \\
234|1 \ar@(ur,rd)[]^{\substack{\pib_1 = -1 \\ \pib_2 = \pib_3 = 0 }} 
} \]
\[ \FF[\OP _{(1,3)}] \cong P_{(4)} \oplus P_{(1,3)} \hspace{.7in} 
\FF[\OP _{(2,2)}] \cong P_{(4)} \oplus P_{(2,2)} \hspace{.7in} 
\FF[\OP _{(3,1)}] \cong P_{(4)} \oplus P_{(3,1)} \]
\caption{A decomposition of $\FF[\OP_{4,2}]$} 
\label{OP42}
\end{footnotesize}
\end{figure}

\begin{proof} (Sketch.)
It suffices to show that $\FF[\OP_{n,k}]$ and $S_{n,k}$ both have the claimed direct sum decomposition
into projective indecomposable modules.  

We have a disjoint union decomposition $\OP_{n,k} = \bigsqcup_{\beta\models n,\ell(\beta)=k} \OP_{\beta}$,
%where $\beta$ ranges over all compositions of $n$ with $k$ parts.  
giving a direct sum decomposition $\FF[\OP_{n,k}] = \bigoplus_{\beta\models n,\ell(\beta)=k} \FF[\OP_{\beta}]$; 
%a picture of this decomposition when $n = 4$ and $k = 2$ is shown in
see Figure~\ref{OP42} when $(n,k)=(4,2)$.
One shows that $\FF[\OP_{\beta}] \cong \bigoplus_{\beta \preceq \alpha} P_{\alpha}$ for
any fixed composition $\beta$, which leads to the desired decomposition of $\FF[\OP_{n,k}]$ into  
projective indecomposables.

To analyze the 0-Hecke structure of $S_{n,k}$, we use a strategically chosen
member of the family of bases of $S_{n,k}$ afforded by
Theorem~\ref{garsia-stanton-type-bases}.
Let $A_{n,k}$ be the set of all pairs $(\alpha, \ii)$, where $\alpha=(\alpha_1,\ldots,\alpha_\ell) \models n$ and $\ii = (i_1, \dots, i_n)$ satisfy $\alpha_1> n-k$, $i_1,\ldots,i_n\in\ZZ$, and $k - \ell \geq i_1 \geq \cdots \geq i_{n-k} \geq 0 = i_{n-k+1} = \cdots = i_n$.
For each pair $(\alpha, \ii)\in A_{n,k}$, the sets $\Des(\alpha)$ and $\Des(\ii) := \{1 \leq j \leq n-1 \,:\, i_j > i_{j+1} \}$ are disjoint.
The set 
\begin{equation}
\{ \pib_w(\xx_{\alpha,\ii}) \,:\, (\alpha,\ii) \in A_{n,k}, \, w \in \symm_n, \, \Des(\alpha) \subseteq \Des(w) \subseteq \Des(\alpha) \sqcup \Des(\ii) \} 
\end{equation}
satisfies the conditions of Theorem~\ref{garsia-stanton-type-bases}, and hence
descends to an $\FF$-basis of $S_{n,k}$.
The $H_n(0)$-action on this basis is easy to describe; see Figure~\ref{S42} for the case $(n,k) = (4,2)$.

The above basis of $S_{n,k}$ gives a direct sum decomposition $S_{n,k} \cong \bigoplus_{(\alpha, \ii) \in A_{n,k}} N_{\alpha,\ii}$ of $S_{n,k}$
into certain direct summands $N_{\alpha,\ii}$ indexed by $(\alpha,\ii) \in A_{n,k}$;
see Figure~\ref{S42}.  
The modules $N_{\alpha,\ii}$ have explicit decompositions into projective 
indecomposables; one shows that summing over $(\alpha,\ii) \in A_{n,k}$ gives the 
claimed
decomposition of $S_{n,k}$. 
\end{proof}


\begin{figure}[h]
\begin{footnotesize}
\[ \xymatrix{
x_1 \ar@(ur,rd)[]^{\pib_2 = \pib_3 = 0 } \ar[d]_{\pib_1} \\
x_2 \ar@(ur,rd)[]^{\substack{\pib_1 = -1 \\ \pib_3 = 0 }} \ar[d]_{\pib_2} \\
x_3 \ar@(ur,rd)[]^{\substack{\pib_1 = 0 \\ \pib_2 = -1 }} \ar[d]_{\pib_3} \\
x_4 \ar@(ur,rd)[]^{\substack{\pib_1 = \pib_2 = 0 \\ \pib_3 = -1 }} \\ N_{4,1000}  \cong P_{(4)} \oplus P_{(1,3)} }
\quad \xymatrix{
& x_1x_2 \ar@(ur,rd)[]^{\pib_1 = \pib_3 = 0 } \ar[d]_{\pib_2} \\
& x_1x_3 \ar@(ur,r)[]^{\pib_2 = -1} \ar[ld]^{\pib_1} \ar[rd]_{\pib_3} \\
x_2x_3 \ar@(ul,dl)[]_{\substack{\pib_1 = -1 \\ \pib_2 = 0 }} \ar[rd]^{\pib_3} &&
x_1x_4 \ar@(ur,rd)[]^{\substack{\pib_2 = 0 \\ \pib_3 = -1 }} \ar[ld]_{\pib_1} \\
& x_2x_4 \ar@(r,rd)[]^{\pib_1 = \pib_3 = -1} \ar[d]_{\pib_2} \\
& x_3x_4 \ar@(ur,rd)[]^{ \substack{ \pib_1 = \pib_3 = 0 \\ \pib_2 = -1 } }   }
\qquad \xymatrix{
1 \ar@(ur,rd)[]^{\pib_1 = \pib_2 = \pib_3 = 0 } \\  N_{4,0000} \cong P_{(4)} \\
x_1x_2x_4 \ar@(r,rd)[]^{\substack{\pib_1 = 0 \\ \pib_3 = -1 }} \ar[d]_{\pib_2} \\
x_1x_3x_4 \ar@(r,rd)[]^{\substack{\pib_2 = -1 \\ \pib_3 = 0 }} \ar[d]_{\pib_1} \\
x_2x_3x_4 \ar@(r,rd)[]^{\substack{\pib_1 = -1 \\ \pib_2 = \pib_3 = 0 }} 
} \]
\[   \hspace{1.8in}
N_{4,1100}  \cong P_{(4)} \oplus P_{(2,2)} \hspace{1in}
N_{31,0000} \cong P_{(3,1)} \]
\caption{A decomposition of $S_{4,2}$} \label{S42}
\end{footnotesize}
\end{figure}

We next give the graded noncommutative characteristic and bigraded quasisymmetric characteristic of the projective $H_n(0)$-module $S_{n,k}$.

\begin{theorem}
\label{s-characteristic-theorem}
Let $k \leq n$ be positive integers.   We have
\begin{align}
\ch_t(S_{n,k}) &= \sum_{\alpha \models n} t^{\maj(\alpha)} {n - \ell(\alpha) \brack k - \ell(\alpha)}_t \sss_{\alpha} \quad\text{and} \\
\Ch_{q,t}(S_{n,k}) &
= \sum_{w \in \symm_n} q^{\inv(w)} t^{\maj(w)} {n - \des(w) - 1 \brack k - \des(w) - 1}_t F_{\Des(w^{-1})} \\
&= \sum_{(w,\alpha) \in \OP_{n,k}} q^{\inv(w)} t^{\maj(w,\alpha)} F_{\Des(w^{-1})}.
\end{align}
\end{theorem}

We define the \emph{length} of $\sigma=(w,\alpha) \in \OP_{n,k}$ to be $\ell(\sigma):=\inv(w)$, since $w$ is the minimal representative of the parabolic coset $w \symm_{\alpha} = w (\symm_{\alpha_1} \times \cdots \times \symm_{\alpha_k})$ corresponding to $\sigma$.
We have the distributions
\begin{equation}
\label{length-equation}
\sum_{\sigma \in \OP_{\alpha}} q^{\ell(\sigma)} = {n \brack \alpha_1, \dots, \alpha_k}_q
\quad\text{and}\quad
%Summing Equation~\eqref{length-equation} over all $\alpha \models n$ with $\ell(\alpha) = k$ gives a {\bf different} distribution than the generating function of $\maj$:
\sum_{\sigma \in \OP_{n,k}} q^{\maj(\sigma)} = \rev_q([k]!_q \cdot \Stir_q(n,k)).
\end{equation}
Both distributions equal $[n]!_q$ in the case $k = n$.
There is a different extension of the inversion/length statistic on $\symm_n$ to $\OP_{n,k}$ whose distribution is $[k]!_q \cdot \Stir_q(n,k)$~\cite{HRW, HRS, RW, Rhoades, WMultiset}.
By Theorem~\ref{s-characteristic-theorem}, $\Ch_{q,t}(S_{n,k})$ is the generating function
for the `biMahonian pair' $(\ell, \maj)$ on $\OP_{n,k}$ with quasisymmetric function weights. %$F_{\iDes(\sigma)}$.

%Taking the commutative image of $\ch_t(S_{n,k})$ or setting $q=1$ in $\Ch_{q,t}(S_{n,k})$ gives the following expression for $\Ch_t(S_{n,k})$.
Since $S_{n,k}$ is projective, the graded noncommutative characteristic $\displaystyle \Ch_t(S_{n,k})$ is symmetric. 
We will expand it in Schur functions.
Let $\SYT(n)$ be the set of standard Young tableaux with $n$ boxes.
For each $Q\in\SYT(n)$, its \emph{shape} is the corresponding partition $\shape(Q)\vdash n$, its \emph{descent set} $\Des(Q)$ consists of all $i\in[n-1]$ appearing in a row above  $i+1$ in $Q$, and its \emph{major index} is $\maj(Q):=\sum_{i\in\Des(Q)} i$.
We also let $\des(Q) := |\Des(Q)|$.
The next result follows from Theorem~\ref{s-characteristic-theorem} and the Robinson--Schensted correspondence.


\begin{corollary}
\label{s-characteristic-corollary}
$\displaystyle \Ch_t(S_{n,k}) = \sum_{Q \in \SYT(n)} t^{\maj(Q)} {n  - \des(Q) - 1 \brack k - \des(Q) - 1}_t\, s_{\shape(Q)}$.
%\begin{align}
%\Ch_t(S_{n,k}) &= \sum_{(w,\alpha) \in \OP_{n,k}} t^{\maj(w,\alpha)} F_{\Des(w^{-1})}  = \sum_{w \in \symm_n}  t^{\maj(w)} {n - \des(w) - 1 \brack k - \des(w) - 1}_t F_{\Des(w^{-1})}  \\
%&= \sum_{\alpha \models n} t^{\maj(\alpha)} {n - \ell(\alpha) \brack k - \ell(\alpha)}_t s_{\alpha} = \sum_{Q \in \SYT(n)} t^{\maj(Q)} {n  - \des(Q) - 1 \brack k - \des(Q) - 1}_t s_{\shape(Q)}.
%\end{align}
\end{corollary}

For example, each projective indecomposable $P_\alpha$ in Figure~\ref{S42} is graded by polynomial degree and corresponds to $\sss_\alpha$ and $s_\alpha$.
By Corollary~\ref{s-characteristic-corollary}, the characteristic
$\Ch_{t}(S_{n,k})$ coincides with the Frobenius image of the graded
$\symm_n$-module $R_{n,k}$~\cite[Corollary 6.13]{HRS}:
\begin{equation}
\Ch_t(S_{n,k}) = \grFrob(R_{n,k}; t).
\end{equation}
It would be interesting to have a conceptual explanation of this coincidence.
One can always use the positive expansion $s_\lambda = \sum_{ Q\in \SYT(\lambda) } F_{\Des(Q)}$ due to Gessel~\cite{Gessel} to define a $H_n(0)$-module structure on every $\symm_n$-module, but this is ad-hoc and does not explain such a coincidence for the pair of modules $S_{n,k}$ and $R_{n,k}$.

%Krob and Thibon~\eqref{ChP} showed that $\Ch(P_\alpha) = s_\alpha \in\Sym$ for any composition $\alpha$.
%Hence $\Ch_t(S_{n,k}) \in Sym$.
%Krob and Thibon also showed \cite{KT} that for any composition $\alpha$, the characteristic $\Ch(P_\alpha)$ equals the corresponding ribbon Schur function $s_{\alpha} \in \Sym$:
%\begin{equation}\label{ChP}
%\Ch(P_{\alpha}) =  \sum_{w \in \symm_n \,:\, \Des(w) = \Des(\alpha)} F_{\iDes(w)} = s_{\alpha}.
%\end{equation}
%We give an explicit expansion of $\Ch_t(S_{n,k})$ in the Schur basis.





\section{Connection with Macdonald Theory}
\label{Conclusion}

Our results have the following connection to the theory of Macdonald polynomials.
Given a partition $\mu$, let $\widetilde{H}_{\mu}$ be the corresponding {\em modified
Macdonald symmetric function} and let $B_{\mu} := \sum  q^i t^j$, where $(i,j)$ ranges over the coordinates of the cells of $\mu$. 
%For example, if $\mu = (3,2)$ then $B_{\mu} = 1 + q + q^2 + t + q*t$.
If $F \in \Sym$ is any symmetric function, then the 
{\em delta operator} $\Delta'_F: \Sym \rightarrow \Sym$ is the Macdonald operator
defined (using plethystic notation) by 
$\Delta'_F: \widetilde{H}_{\mu} \mapsto F[B_{\mu}(q,t) - 1] \cdot \widetilde{H}_{\mu}$.

When $F = e_{k-1}$ is an elementary symmetric function, the {\em Delta Conjecture}
of Haglund, Remmel, and Wilson \cite{HRW} predicts the monomial expansion
of $\Delta'_{e_{k-1}} e_n$: it has the form
\begin{equation}
\Delta'_{e_{k-1}} e_n = \Rise_{n,k}(\xx;q,t) = \Val_{n,k}(\xx;q,t),
\end{equation}
where $\Rise(\xx;q,t)$ and $\Val(\xx;q,t)$ are certain combinatorially defined formal power series 
in the variable set $\xx = (x_1, x_2, \dots )$ involving the parameters $q$ and $t$.
%see \cite{HRW} for details.

When $k = n$, the Delta Conjecture reduces to the Shuffle Theorem proved recently by Carlsson and Mellit~\cite{CM}.
Although the Delta Conjecture is open for general $k \leq n$, its truth has been 
established when one of the variables $q,t$ is set to zero by Garsia--Haglund--Remmel--Yoo~\cite{GHRY}.
Combining this with results of the second author~\cite{Rhoades} and Wilson~\cite{WMultiset},
we have
\begin{equation}\label{delta-zero}
\begin{aligned}
& \Delta'_{e_{k-1}} e_n|_{q=0} = \Delta'_{e_{k-1}} e_n|_{t=0,\,q=t}  \\
=& \Rise_{n,k}(\xx;0,t) = \Rise_{n,k}(\xx;t,0) = \Val_{n,k}(\xx;0,t) = \Val_{n,k}(\xx;t,0).
\end{aligned}
\end{equation}
If we let $C_{n,k}(\xx;t)$ be the common symmetric function in
Equation~\eqref{delta-zero}, then Corollary~\ref{s-characteristic-corollary} and results from earlier work~\cite{HRW} show that 
\begin{equation}
\Ch_t(S_{n,k}) = (\rev_t \circ \omega) C_{n,k}(\xx;t)
\end{equation}
where $\omega$ is the involution on $\Sym$ which interchanges $e_n$ and $h_n$.



%Corollary~\ref{s-characteristic-corollary} shows that $Ch_t(S_{n,k})$ coincides with $\grFrob(R_{n,k},t)$, and hence is related to the {\em Delta Conjecture} of Haglund, Remmel, and Wilson~\cite{HRW} by work of Haglund--Rhoades--Shimozono~\cite[Thm. 6.11]{HRS}, Rhoades~\cite{Rhoades}, and Wilson~\cite{WMultiset}.
%The derivation of $\grFrob(R_{n,k};t)$ in \cite{HRS} has a different flavor from our derivation of 
%$\Ch_t(S_{n,k})$.
%We do not have a conceptual explanation for the coincidence $Ch_t(S_{n,k})=\grFrob(R_{n,k},t)$.

%The coinvariant algebra $R_n$ can be generalized to a quotient $R_\lambda:=\QQ[\xx_n]/I_\lambda$ for any $\lambda\vdash n$, where $I_{\lambda}$ is the {\em Tanisaki ideal} indexed by $\lambda$. 
%It is known that $R_\lambda$ is isomorphic to the cohomology ring of the Springer fiber indexed by $\lambda$, and is a graded $\symm_n$-module with $\grFrob(R_{\lambda};q) = \rev_q( Q'_{\lambda}(\xx;q))$, where $Q'_{\lambda}(\xx;q)$ is the dual Hall-Littlewood polynomial. 
%See, e.g., Garsia and Procesi~\cite{GP}.
%The first author~\cite{Huang} showed that $I_{\lambda}$ is closed under the $H_n(0)$-action if and only if $\lambda$ is a hook, and in that case $R_{\lambda}$ is a graded 0-Hecke module with
%$\Ch_t(R_{\lambda}) = \grFrob(R_{\lambda}; t) = \rev_t (Q'_{\lambda}(\xx;t))$.
%Now we have ideals $I_{n,k}$ and $J_{n,k}$ stable under the actions of $\symm_n$ and $H_n(0)$, respectively, such that $\Ch_t \left( {\FF[\xx_n]} / {J_{n,k}} \right) = \grFrob \left( {\QQ[\xx_n]} / {I_{n,k}}; t \right)$.
%Can one define, for any $\lambda\vdash n$, a homogeneous ideal $J_{\lambda}$ of $\FF[\xx_n]$ stable under the $H_n(0)$-action such that $\Ch_t \left( {\FF[\xx_n]} / {J_{\lambda}} \right) = \grFrob(R_{\lambda};t) = \rev_t(Q'_{\lambda}(\xx;t))$?

%The $0$-Hecke algebra of a Weyl group $W$ acts on the Laurent ring of the weight lattice of $W$; this Laurent ring is isomorphic to $\FF[x_1, \dots, x_r, x_1^{-1}, \dots, x_r^{-1}]$ if the rank of $W$ is $r$.
%The first author~\cite{Huang} described the 0-Hecke structure of the corresponding
%coinvariant algebra.
%On the other hand, Chan and Rhoades~\cite{CR} generalized the ideal $I_{n,k}$ from $\symm_n$ to the complex reflection groups $G(r,1,n) \cong \ZZ_r \wr \symm_n$.
%It would be interesting to give an analog of the work in this paper for a wider class of reflection groups.



\acknowledgements{The authors are grateful to Jim Haglund, Adriano Garsia, Jeff Remmel, Mark Shimozono, and Andy Wilson for helpful conversations, and to the anonymous referees for valuable comments.
}



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