Tail positive words and generalized coinvariant algebras

TL;DR

This paper introduces a new family of quotients of polynomial rings, called $R_{n,k,r}$, which generalize classical coinvariant algebras and are governed by combinatorics of tail positive words, with detailed algebraic and representation-theoretic analysis.

Contribution

It defines and analyzes the algebraic structure of $R_{n,k,r}$, including bases, module properties, and connections to 0-Hecke algebras and Macdonald polynomials.

Findings

Calculated the standard monomial basis of $R_{n,k,r}$.

Established $R_{n,k,r}$ as a projective 0-Hecke module.

Connected $R_{n,k,r}$ to tail positive words and conjectured links to Macdonald polynomial operators.

Abstract

Let $n,k,$ and $r$ be nonnegative integers and let $S_n$ be the symmetric group. We introduce a quotient $R_{n,k,r}$ of the polynomial ring $\mathbb{Q}[x_1, \dots, x_n]$ in $n$ variables which carries the structure of a graded $S_n$-module. When $r \geq n$ or $k = 0$ the quotient $R_{n,k,r}$ reduces to the classical coinvariant algebra $R_n$ attached to the symmetric group. Just as algebraic properties of $R_n$ are controlled by combinatorial properties of permutations in $S_n$, the algebra of $R_{n,k,r}$ is controlled by the combinatorics of objects called {\em tail positive words}. We calculate the standard monomial basis of $R_{n,k,r}$ and its graded $S_n$-isomorphism type. We also view $R_{n,k,r}$ as a module over the 0-Hecke algebra $H_n(0)$, prove that $R_{n,k,r}$ is a projective 0-Hecke module, and calculate its quasisymmetric and nonsymmetric 0-Hecke characteristics. We conjecture a relationship between our quotient $R_{n,k,r}$ and the delta operators of the theory of Macdonald polynomials.

Equations

$$\mathbb{Q}[{\mathbf{x}}_n{]}^{S_n}:=\{f({\mathbf{x}}_n)\in \mathbb{Q}[{\mathbf{x}}_n]:\pi .f({\mathbf{x}}_n)=f({\mathbf{x}}_n)\text{ for all π∈Sn}\}$$

$$I_n:=⟨\mathbb{Q}[{\mathbf{x}}_n{]}_{+}^{S_n}⟩=⟨e_1({\mathbf{x}}_n),\dots ,e_n({\mathbf{x}}_n)⟩$$

$$R_n:=\mathbb{Q}[{\mathbf{x}}_n]/I_n.$$

$$\mathrm{Hilb}(R_n;q)=(1+q)(1+q+q^2)\cdots (1+q+\cdots +q^{n-1})=[n]{!}_q,$$

$$I_{n,k,r}:=⟨h_{k+1}({\mathbf{x}}_n),h_{k+2}({\mathbf{x}}_n),\dots ,h_{k+n}({\mathbf{x}}_n),e_n({\mathbf{x}}_n),e_{n-1}({\mathbf{x}}_n),\dots ,e_{n-r+1}({\mathbf{x}}_n)⟩$$

$$R_{n,k,r}:=\mathbb{Q}[{\mathbf{x}}_n]/I_{n,k,r}$$

$$S_{2,2,1}=\{0012,0021,0102,0201,1002,2001\}.$$

$$|S_{n,k,r}|=\left(\genfrac{}{}{0ex}{}{n+k-r}{k}\right)\cdot |S_n|=\left(\genfrac{}{}{0ex}{}{n+k-r}{k}\right)\cdot n!.$$

$$\mathrm{Hilb}(R_{n,k,r};q)={\left[\genfrac{}{}{0ex}{}{n+k-r}{k}\right]}_q\cdot \mathrm{Hilb}(R_n;q)={\left[\genfrac{}{}{0ex}{}{n+k-r}{k}\right]}_q\cdot [n]{!}_q,$$

$$\mathrm{grFrob}(R_{n,k,r};q)={\left[\genfrac{}{}{0ex}{}{n+k-r}{k}\right]}_q\cdot \mathrm{grFrob}(R_n;q)={\left[\genfrac{}{}{0ex}{}{n+k-r}{k}\right]}_q\cdot \sum _{T\in \mathrm{SYT}(n)}{s}_{\mathrm{shape}(T)},$$

$$⟨x_1^k,x_2^k,\dots ,x_n^k,e_n({\mathbf{x}}_n),e_{n-1}({\mathbf{x}}_n),\dots ,e_{n-k+1}({\mathbf{x}}_n)⟩.$$

$$⟨x_1^{kr+1},x_2^{kr+1},\dots ,x_n^{kr+1},e_n({\mathbf{x}}_n^r),e_{n-1}({\mathbf{x}}_n^r),\dots ,e_{n-k+1}({\mathbf{x}}_n^r)⟩,$$

$$⟨h_k(x_1),h_k(x_1,x_2),\dots ,h_k(x_1,x_2,\dots ,x_n),e_n({\mathbf{x}}_n),e_{n-1}({\mathbf{x}}_n),\dots ,e_{n-k+1}({\mathbf{x}}_n)⟩.$$

$$m_{\lambda },e_{\lambda },h_{\lambda },p_{\lambda },s_{\lambda },{\tilde{H}}_{\lambda }$$

$${\Delta }_f({\tilde{H}}_{\lambda }):=f(\dots ,q^{i-1}{t}^{j-1},\dots )\cdot {\tilde{H}}_{\lambda },$$

$${\Delta }_f({\tilde{H}}_{\lambda }):=f[B_{\lambda }]\cdot {\tilde{H}}_{\lambda }.$$

$${\Delta }_f({\tilde{H}}_{(3,2)})=f(1,q,q^2,t,qt)\cdot {\tilde{H}}_{(3,2)}.$$

$${\Delta }_{e_n}{\mid }_{{\Lambda }_n}=\nabla .$$

$$\mathrm{Frob}(V):=\sum _{\lambda ⊢n}{c}_{\lambda }\cdot s_{\lambda }$$

$$\mathrm{Hilb}(V;q)=\sum _{d\ge 0}\dim (V_d)\cdot q^d.$$

$$\mathrm{grFrob}(V;q)=\sum _{d\ge 0}\mathrm{Frob}(V_d)\cdot q^d.$$

$$F_S=F_{S,n}:=\sum _{\begin{array}{c}{i}_1\le i_2\le \cdots \le i_n\\ j\in S\Rightarrow i_j<i_{j+1}\end{array}}{x}_{i_1}\cdots x_{i_n}.$$

$$\{\begin{array}{ll}{T}_i^2=T_i& 1\le i\le n-1\\ T_i{T}_j=T_j{T}_i& |i-j|>1\\ T_i{T}_{i+1}{T}_i=T_{i+1}{T}_i{T}_{i+1}& 1\le i\le n-2.\end{array}$$

$$C_{\alpha }:=\mathrm{top}(P_{\alpha })=P_{\alpha }/\mathrm{rad}(P_{\alpha })$$

$$\mathrm{Ch}:C_{\alpha }\mapsto F_{\alpha }.$$

$$\mathbf{ch}:P_{\alpha }\mapsto {\mathbf{s}}_{\alpha },$$

$$H_n(0{)}^{(0)}\subseteq H_n(0{)}^{(1)}\subseteq H_n(0{)}^{(2)}\subseteq \cdots$$

$$V^{(\ell )}:=H_n(0{)}^{(\ell )}v.$$

$${\mathrm{Ch}}_t(V):=\sum _{\ell \ge 0}\mathrm{Ch}(V^{(\ell )}/V^{(\ell -1)})\cdot t^{\ell }.$$

$${\mathrm{Ch}}_{q,t}(V):=\sum _{\ell ,d\ge 0}\mathrm{Ch}(V^{(\ell ,d)}/(V^{(\ell -1,d)}+V^{(\ell ,d+1)}))\cdot q^d{t}^{\ell }.$$

Full text

1. Introduction

Consider the action of the symmetric group $S_{n}$ on $n$ letters on the polynomial ring ${\mathbb{Q}}[{\mathbf{x}}_{n}]:={\mathbb{Q}}[x_{1},\dots,x_{n}]$ given by variable permutation. The polynomials belonging to the invariant subring

[TABLE]

are the symmetric polynomials in the variable set ${\mathbf{x}}_{n}$. Let $e_{d}({\mathbf{x}}_{n})$ be the elementary symmetric function of degree $d$, that is $e_{d}({\mathbf{x}}_{n})=\sum_{1\leq i_{1}<\cdots<i_{d}\leq n}x_{i_{1}}\cdots x_{i_{d}}$. It is well known that the set $\{e_{1}({\mathbf{x}}_{n}),\dots,e_{n}({\mathbf{x}}_{n})\}$ gives an algebraically independent homogeneous collection of generators for the ring ${\mathbb{Q}}[{\mathbf{x}}_{n}]^{S_{n}}$.

Let ${\mathbb{Q}}[{\mathbf{x}}_{n}]^{S_{n}}_{+}\subset{\mathbb{Q}}[{\mathbf{x}}_{n}]^{S_{n}}$ be the subspace of symmetric polynomials with vanishing constant term. The invariant ideal $I_{n}\subseteq{\mathbb{Q}}[{\mathbf{x}}_{n}]$ is the ideal

[TABLE]

generated by this subspace. The coinvariant algebra $R_{n}$ is the corresponding quotient:

[TABLE]

The algebra $R_{n}$ is a graded $S_{n}$-module.

The coinvariant algebra is among the most important representations in algebraic combinatorics; algebraic properties of $R_{n}$ are deeply tied to combinatorial properties of permutations in $S_{n}$. E. Artin proved [2] that the collection of ‘sub-staircase’ monomials $\{x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}\,:\,0\leq i_{j}<j\}$ descends to a vector space basis for $R_{n}$, so that the Hilbert series of $R_{n}$ is given by

[TABLE]

the standard $q$-analog of $n!$. Chevalley [4] proved that as an ungraded $S_{n}$-module, we have $R_{n}\cong{\mathbb{Q}}[S_{n}]$, the regular representation of $S_{n}$. Lusztig (unpublished) and Stanley [16] refined this result to describe the graded isomorphism type of $R_{n}$ in terms of the major index statistic on standard Young tableaux.

In this paper we will study the following generalization of the coinvariant algebra $R_{n}$. Recall that the degree $d$ homogeneous symmetric function in ${\mathbb{Q}}[{\mathbf{x}}_{n}]$ is given by $h_{d}({\mathbf{x}}_{n}):=\sum_{1\leq i_{1}\leq\cdots\leq i_{d}\leq n}x_{i_{1}}\cdots x_{i_{d}}$.

Definition 1.1**.**

Let $n,k,$ and $r$ be nonnegative integers with $r\leq n$. Let $I_{n,k,r}\subseteq{\mathbb{Q}}[{\mathbf{x}}_{n}]$ be the ideal

[TABLE]

and let

[TABLE]

be the corresponding quotient ring.

The ideal $I_{n,k,r}$ is homogeneous and stable under the action of the symmetric group, so that $R_{n,k,r}$ is a graded $S_{n}$-module. Since the generators of $I_{n,k,r}$ are symmetric polynomials, we have the containment of ideals $I_{n,k,r}\subseteq I_{n}$, so that $R_{n,k,r}$ projects onto the classical coinvariant algebra $R_{n}$. If $r\geq n$ or $k=0$ we have the equality $I_{n,k,r}=I_{n}$, so that $R_{n,k,r}=R_{n}$.

Just as algebraic properties of $R_{n}$ are controlled by combinatorics of permutations $\pi_{1}\dots\pi_{n}$ of the set $\{1,2,\dots,n\}$, algebraic properties of $R_{n,k,r}$ will be controlled by the combinatorics of permutations $\pi_{1}\dots\pi_{n+k}$ of the multiset $\{0^{k},1,2,\dots,n\}$ whose last $r$ entries $\pi_{n+k-r+1}\dots\pi_{n+k-1}\pi_{n+k}$ are all nonzero. Thinking of positive letters as weights, we will call such permutations $r$-tail positive.

Let $S_{n,k,r}$ be the collection of all $r$-tail positive permutations of the multiset $\{0^{k},1,2,\dots,n\}$. For example, we have

[TABLE]

By considering the possible locations of the $k$ [math]’s in an element of $S_{n,k,r}$, it is immediate that

[TABLE]

The basic enumeration of Equation 1.5 will manifest in Hilbert series as

[TABLE]

where ${m\brack i}_{q}:=\frac{[m]!_{q}}{[i]!_{q}[m-i]!_{q}}$ is the usual $q$-binomial coefficient. Going even further, we have the following graded Frobenius image

[TABLE]

which implies that the quotient $R_{n,k,r}$ consists of ${n+k-r\choose k}$ copies of the coinvariant algebra $R_{n}$, with grading shifts given by a $q$-binomial coefficient. The authors know of no direct way to see this from Definition 1.1.

The ideal $I_{n,k,r}$ defining the quotient $R_{n,k,r}$ is of ‘mixed’ type – its generators come in two flavors: the homogeneous symmetric functions $h_{k+1}({\mathbf{x}}_{n}),h_{k+2}({\mathbf{x}}_{n}),\dots,h_{k+n}({\mathbf{x}}_{n})$ and the elementary symmetric functions $e_{n}({\mathbf{x}}_{n}),e_{n-1}({\mathbf{x}}_{n}),\dots,e_{n-r+1}({\mathbf{x}}_{n})$. Several mixed ideals have recently been introduced to give combinatorial generalizations of the coinvariant algebra.

• •

Let $k\leq n$. Haglund, Rhoades, and Shimozono [10] studied the quotient of ${\mathbb{Q}}[{\mathbf{x}}_{n}]$ by the ideal

[TABLE]

The generators are high degree $S_{n}$-invariants $e_{n}({\mathbf{x}}_{n}),\dots,e_{n-k+1}({\mathbf{x}}_{n})$ together with a homogeneous system of parameters $x_{1}^{k},\dots,x_{n}^{k}$ of degree $k$ carrying the defining representation of $S_{n}$. Algebraic properties of the corresponding quotient are controlled by combinatorial properties of $k$-block ordered set partitions of $\{1,2,\dots,n\}$.

• •

Let $r\geq 2$ and let ${\mathbb{Z}}_{r}\wr S_{n}$ be the group of $n\times n$ monomial matrices whose nonzero entries are $r^{th}$ complex roots of unity (this is the group of ‘$r$-colored permutations’ of $\{1,2,\dots,n\}$). Let $k\leq n$ be non-negative integers. Chan and Rhoades [3] studied the quotient of ${\mathbb{C}}[{\mathbf{x}}_{n}]$ by the ideal

[TABLE]

where $f({\mathbf{x}}_{n}^{r})=f(x_{1}^{r},\dots,x_{n}^{r})$ for any polynomial $f$. The generators here are high degree ${\mathbb{Z}}_{r}\wr S_{n}$-invariants $e_{n}({\mathbf{x}}_{n}^{r}),\dots,e_{n-k+1}({\mathbf{x}}_{n}^{r})$ together with a h.s.o.p. $x_{1}^{kr+1},\dots,x_{n}^{kr+1}$ of degree $kr+1$ carrying the dual of the defining representation of ${\mathbb{Z}}_{r}\wr S_{n}$. Algebraic properties of the corresponding quotient are controlled by $k$-dimensional faces in the Coxeter complex attached to ${\mathbb{Z}}_{r}\wr S_{n}$.

• •

Let ${\mathbb{F}}$ be any field and let $H_{n}(0)$ be the 0-Hecke algebra over ${\mathbb{F}}$ of rank $n$; the algebra $H_{n}(0)$ acts on the polynomial ring ${\mathbb{F}}[{\mathbf{x}}_{n}]$ by isobaric divided difference operators. Let $k\leq n$ be positive integers. Huang and Rhoades [13] studied the quotient of ${\mathbb{F}}[{\mathbf{x}}_{n}]$ by the ideal

[TABLE]

Once again, the generators consist of high degree $H_{n}(0)$-invariants $e_{n}({\mathbf{x}}_{n}),\dots,e_{n-k+1}({\mathbf{x}}_{n})$ together with a h.s.o.p. of degree $k$ carrying the defining representation of $H_{n}(0)$. Algebraic properties of the quotient are controlled by 0-Hecke combinatorics of $k$-block ordered set partitions of $\{1,\dots,n\}$.

The novelty of this paper is that our mixed ideals consist of high degree invariants of different kinds: elementary and homogeneous. It would be interesting to develop a more unified picture of the algebraic and combinatorial properties of mixed quotients of polynomial rings.

Our analysis of the rings $R_{n,k,r}$ will share many properties with the analyses of the previously mentioned mixed quotients. Since $I_{n,k,r}$ is not cut out by a regular sequence of homogeneous polynomials in ${\mathbb{Q}}[{\mathbf{x}}_{n}]$, the usual commutative algebra tools (e.g. the Koszul complex) used to study the coinvariant algebra $R_{n}$ are unavailable to us. These will be replaced by combinatorial commutative algebra tools (e.g. Gröbner theory). We will see that the ideal $I_{n,k,r}$ has an explicit minimal Gröbner basis (with respect to the lexicographic term order) in terms of Demazure characters. This Gröbner basis will yield the Hilbert series of $R_{n,k,r}$, as well as an identification of its standard monomial basis. The graded $S_{n}$-isomorphism type of $R_{n,k,r}$ will then be obtainable by constructing an appropriate short exact sequence to serve as a recursion.

The rest of the paper is organized as follows. In Section 2 we give background related to symmetric functions and Gröbner bases. In Section 3 we determine the Hilbert series of $R_{n,k,r}$ and calculate the standard monomial basis for $R_{n,k,r}$ with respect to the lexicographic term order. In Section 4 we determine the graded $S_{n}$-isomorphism type of $R_{n,k,r}$. We also view $R_{n,k,r}$ as a module over the 0-Hecke algebra $H_{n}(0)$ and calculate its graded nonsymmetric and bigraded quasisymmetric 0-Hecke characteristics. We close in Section 5 with some open problems.

2. Background

2.1. Words, partitions, and tableaux

Let $w=w_{1}\dots w_{n}$ be a word in the alphabet of nonnegative integers. An index $1\leq i\leq n-1$ is a descent of $w$ if $w_{i}>w_{i+1}$. The descent set of $w$ is ${\mathrm{Des}}(w):=\{1\leq i\leq n-1\,:\,w_{i}>w_{i+1}\}$ and the major index of $w$ is ${\mathrm{maj}}(w):=\sum_{i\in{\mathrm{Des}}(w)}i$. A pair of indices $1\leq i<j\leq n$ is called an inversion of $w$ if $w_{i}>w_{j}$; the inversion number ${\mathrm{inv}}(w)$ counts the inversions of $w$. The word $w$ is called $r$-tail positive if its last $r$ letters $w_{n-r+1}\dots w_{n-1}w_{n}$ are positive.

Let $n\in{\mathbb{Z}}_{\geq 0}$. A partition $\lambda$ of $n$ is a weakly decreasing sequence $\lambda=(\lambda_{1}\geq\cdots\geq\lambda_{k})$ of positive integers with $\lambda_{1}+\cdots+\lambda_{k}=n$. We write $\lambda\vdash n$ or $|\lambda|=n$ to indicate that $\lambda$ is a partition of $n$. The Ferrers diagram of $\lambda$ (in English notation) consists of $\lambda_{i}$ left-justified boxes in row $i$. The Ferrers diagram of $(4,2,2)\vdash 8$ is shown below on the left.

1

1

2

5

2

3

3

5

              

1

2

5

8

3

4

6

7

Let $\lambda\vdash n$. A tableau $T$ of shape $\lambda$ is a filling of the Ferrers diagram of $\lambda$ with positive integers. The tableau $T$ is called semistandard if its entires increase weakly across rows and strictly down columns. The tableau $T$ is a standard Young tableau if it is semistandard and its entries consist of $1,2,\dots,n$. The tableau in the center above is semistandard and the tableau on the right above is standard. We let $\mathrm{shape}(T)=\lambda$ denote the shape of $T$ and let ${\mathrm{SYT}}(n)$ denote the collection of all standard Young tableaux with $n$ boxes.

Given a standard tableau $T\in{\mathrm{SYT}}(n)$, an index $1\leq i\leq n-1$ is a descent of $T$ if $i+1$ appears in a lower row of $T$ than $i$. Let ${\mathrm{Des}}(T)$ denote the set of descents of $T$ and let ${\mathrm{maj}}(T):=\sum_{i\in{\mathrm{Des}}(T)}i$ be the major index of $T$. If $T$ is the standard tableau above we have ${\mathrm{Des}}(T)=\{2,5\}$ so that ${\mathrm{maj}}(T)=2+5=7$.

2.2. Symmetric functions

Let $\Lambda$ denote the ring of symmetric functions in an infinite variable set ${\mathbf{x}}=(x_{1},x_{2},\dots)$ over the ground field ${\mathbb{Q}}(q,t)$. The algebra $\Lambda$ is graded by degree: $\Lambda=\bigoplus_{n\geq 0}\Lambda_{n}$. The graded piece $\Lambda_{n}$ has dimension equal to the number of partitions $\lambda\vdash n$.

The vector space $\Lambda_{n}$ has many interesting bases, all indexed by partitions of $n$. Given $\lambda\vdash n$, let

[TABLE]

denote the associated monomial, elementary, homogeneous, power sum, Schur, and modified Macdonald symmetric function (respectively). As $\lambda$ ranges over the collection of partitions of $n$, all of these form bases for the vector space $\Lambda_{n}$.

Let $f({\mathbf{x}})\in\Lambda$ be a symmetric function. We define an eigenoperator $\Delta_{f}:\Lambda\rightarrow\Lambda$ for the modified Macdonald basis of $\Lambda$ as follows. Given a partition $\lambda$, we set

[TABLE]

where $(i,j)$ ranges over all matrix coordinates of cells in the Ferrers diagram of $\lambda$. The reader familiar with plethysm will recognize this formula as

[TABLE]

For example, if $\lambda=(3,2)\vdash 5$, we fill the boxes of $\lambda$ with monomials

$\ignorespaces 1$

$\ignorespaces q$

$\ignorespaces q^{2}$

t

$\ignorespaces qt$

and see that

[TABLE]

When $f=e_{n}$, the restriction of the delta operator $\Delta_{e_{n}}$ to the space $\Lambda_{n}$ of homogeneous degree $n$ symmetric functions is more commonly denoted $\nabla$:

[TABLE]

In particular, we have $\Delta_{e_{n}}e_{n}=\nabla e_{n}$.

Given a partition $\lambda\vdash n$, let $S^{\lambda}$ denote the associated irreducible representation of the symmetric group $S_{n}$; for example, we have that $S^{(n)}$ is the trivial representation and $S^{(1^{n})}$ is the sign representation. Given any finite-dimensional $S_{n}$-module $V$, there exist unique integers $c_{\lambda}$ such that $V\cong_{S_{n}}\bigoplus_{\lambda\vdash n}c_{\lambda}S^{\lambda}$. The Frobenius character of $V$ is the symmetric function

[TABLE]

obtained by replacing the irreducible $S^{\lambda}$ with the Schur function $s_{\lambda}$.

If $V=\bigoplus_{d\geq 0}V_{d}$ is a graded vector space, the Hilbert series of $V$ is the power series

[TABLE]

Similarly, if $V=\bigoplus_{d\geq 0}V_{d}$ is a graded $S_{n}$-module, the graded Frobenius character of $V$ is

[TABLE]

2.3. Quasisymmetric and nonsymmetric functions

The space $\Lambda$ of symmetric functions has many generalizations; in this paper we will also use the spaces ${\mathrm{QSym}}$ of quasisymmetric functions and ${\mathbf{NSym}}$ of noncommutative symmetric functions. We briefly review their definition below, as well as their relationship with the 0-Hecke algebra $H_{n}(0)$; for more details see [12, 13].

Let $n$ be a positive integer. A (strong) composition $\alpha$ of $n$ is a sequence $(\alpha_{1},\dots,\alpha_{k})$ of positive integers with $\alpha_{1}+\cdots+\alpha_{k}=n$. We write $\alpha\models n$ or $|\alpha|=n$ to indicate that $\alpha$ is a composition of $n$. The map $\alpha=(\alpha_{1},\dots,\alpha_{k})\mapsto\{\alpha_{1},\alpha_{1}+\alpha_{2},\dots,\alpha_{1}+\cdots+\alpha_{k-1}\}$ provides a bijection between compositions of $n$ and subsets of $[n-1]$; we will find it convenient to identify compositions with subsets.

Let $S\subseteq[n-1]$ be a subset. The Gessel fundamental quasisymmetric function $F_{S}=F_{S,n}$ attached to $S$ is the degree $n$ formal power series

[TABLE]

The space ${\mathrm{QSym}}$ of quasisymmetric functions is the ${\mathbb{Q}}(q,t)$-algebra of formal power series with basis given by $\{F_{S,n}\,:\,n\geq 0,\,S\subseteq[n-1]\}$. If a subset $S\subseteq[n-1]$ corresponds to a composition $\alpha$, we set $F_{\alpha}:=F_{S,n}$.

For any composition $\alpha\models n$, define a symbol ${\mathbf{s}}_{\alpha}$ (the noncommutative ribbon Schur function), formally defined to have homogeneous degree $n$. Let ${\mathbf{NSym}}_{n}$ be the $2^{n-1}$-dimensional ${\mathbb{Q}}(q,t)$-vector space with basis $\{{\mathbf{s}}_{\alpha}\,:\,\alpha\models n\}$ and let ${\mathbf{NSym}}$ be the graded vector space ${\mathbf{NSym}}:=\bigoplus_{n\geq 0}{\mathbf{NSym}}_{n}$. The space ${\mathbf{NSym}}$ is the space of noncommutative symmetric functions. Although there is more structure on ${\mathbf{NSym}}$ (and on ${\mathrm{QSym}}$) than the graded vector space structure (namely, they are dual graded Hopf algebras), only the vector space structure will be relevant in this paper.

Let ${\mathbb{F}}$ be an arbitrary field. The 0-Hecke algebra $H_{n}(0)$ of rank $n$ over ${\mathbb{F}}$ is the unital associative ${\mathbb{F}}$-algebra with generators $T_{1},\dots,T_{n-1}$ and relations

[TABLE]

For all $1\leq i\leq n-1$, let $s_{i}:=(i,i+1)\in S_{n}$ be the corresponding adjacent transposition. Given a permutation $\pi\in S_{n}$, we define $T_{\pi}:=T_{i_{1}}\cdots T_{i_{k}}\in H_{n}(0)$, where $\pi=s_{i_{1}}\cdots s_{i_{k}}$ is a reduced (i.e., short as possible) expression for $\pi$ as a product of adjacent transpositions. It can be shown that $\{T_{\pi}\,:\,\pi\in S_{n}\}$ is a ${\mathbb{F}}$-basis for $H_{n}(0)$, so that $H_{n}(0)$ has dimension $n!$ as a ${\mathbb{F}}$-vector space and may be viewed as a deformation of the group algebra ${\mathbb{F}}[S_{n}]$. The algebra $H_{n}(0)$ is not semisimple, even when the field ${\mathbb{F}}$ has characteristic zero, so its representation theory has a different flavor from that of $S_{n}$.

The indecomposable projective representations of $H_{n}(0)$ are naturally labeled by compositions $\alpha\models n$ (see [12, 13]). For $\alpha\models n$, we let $P_{\alpha}$ denote the corresponding indecomposable projective and let

[TABLE]

be the corresponding irreducible $H_{n}(0)$-module.

The Grothendieck group $G_{0}(H_{n}(0))$ is the ${\mathbb{Z}}$-module generated by all isomorphism classes $[V]$ of finite-dimensional $H_{n}(0)$-modules with a relation $[V]-[U]-[W]=0$ for every short exact sequence $0\rightarrow U\rightarrow V\rightarrow W\rightarrow 0$ of $H_{n}(0)$-modules. The ${\mathbb{Z}}$-module $G_{0}(H_{n}(0))$ is free with basis given by (isomorphism classes of) the irreducibles $\{C_{\alpha}\,:\,\alpha\models n\}$. The quasisymmetric characteristic map ${\mathrm{Ch}}$ is defined on $G_{0}(H_{n}(0))$ by

[TABLE]

If a $H_{n}(0)$-module $V$ has composition factors $C_{\alpha^{(1)}},\dots,C_{\alpha^{(k)}}$, then ${\mathrm{Ch}}(V)=F_{\alpha^{(1)}}+\cdots+F_{\alpha^{(k)}}$. Since $H_{n}(0)$ is not semisimple, the characteristic ${\mathrm{Ch}}(V)$ does not determine $V$ up to isomorphism.

Let $K_{0}(H_{n}(0))$ be the ${\mathbb{Z}}$-module generated by all isomorphism classes $[P]$ of finite-dimensional projective $H_{n}(0)$-modules with a relation $[P]-[Q]-[R]=0$ for every short exact sequence $0\rightarrow Q\rightarrow P\rightarrow R\rightarrow 0$ of projective modules. The ${\mathbb{Z}}$-module $K_{0}(H_{n}(0))$ is free with basis given by (isomorphism classes of) the projective indecomposable $\{P_{\alpha}\,:\,\alpha\models n\}$. The noncommutative characteristic map ${\mathbf{ch}}$ is defined on $K_{0}(H_{n}(0))$ by

[TABLE]

This extends to give a noncommutative symmetric function ${\mathbf{ch}}(P)$ for any projective $H_{n}(0)$-module $P$. The module $P$ is determined by ${\mathbf{ch}}(P)$ up to isomorphism.

There are graded refinements of the maps ${\mathrm{Ch}}$ and ${\mathbf{ch}}$. Let $V=\bigoplus_{d\geq 0}V_{d}$ be a graded $H_{n}(0)$-module with each $V_{d}$ finite-dimensional. The degree-graded quasisymmetric characteristic is ${\mathrm{Ch}}_{q}(V):=\sum_{d\geq 0}{\mathrm{Ch}}(V_{d})\cdot q^{d}$. If each $V_{d}$ is projective, the degree-graded noncommutative characteristic is ${\mathbf{ch}}_{q}(V):=\sum_{d\geq 0}{\mathbf{ch}}(V_{d})\cdot q^{d}$.

The quasisymmetric characteristic ${\mathrm{Ch}}$ admits a bigraded refinement as follows. The 0-Hecke algebra $H_{n}(0)$ has a length filtration

[TABLE]

where $H_{n}(0)^{(\ell)}$ is the subspace of $H_{n}(0)$ with ${\mathbb{F}}$-basis $\{T_{\pi}\,:\,\pi\in S_{n},{\mathrm{inv}}(\pi)\leq\ell\}$. If $V=H_{n}(0)v$ is a cyclic $H_{n}(0)$-module with distinguished generator $v$, we get an induced length filtration of $V$ by

[TABLE]

The length-graded quasisymmetric characteristic is given by

[TABLE]

Now suppose $V=\bigoplus_{d\geq 0}V_{d}$ is a graded $H_{n}(0)$-module which is also cyclic. We get a bifiltration of $V$ consisting of the modules $V^{(\ell,d)}:=V^{(\ell)}\cap V_{d}$ for $\ell,d\geq 0$. The length-degree-bigraded quasisymmetric characteristic is

[TABLE]

More generally, if $V$ is a direct sum of graded cyclic $H_{n}(0)$-modules, we define ${\mathrm{Ch}}_{q,t}(V)$ by applying ${\mathrm{Ch}}_{q,t}$ to its cyclic summands. This may depend on the cyclic decomposition of the module $V$. 111Our conventions for $q$ and $t$ in the definitions of ${\mathbf{ch}}_{q}$ and ${\mathrm{Ch}}_{q,t}$ are reversed with respect to those in [12, 13] and elsewhere. We make these conventions so as to be consistent with the case of the graded Frobenius map on $S_{n}$-modules.

2.4. Gröbner theory

A total order $<$ on the monomials in the polynomial ring ${\mathbb{Q}}[{\mathbf{x}}_{n}]$ is called a term order if

• •

we have $1\leq m$ for all monomials $m$, and

• •

$m\leq m^{\prime}$ implies $m\cdot m^{\prime\prime}\leq m^{\prime}\cdot m^{\prime\prime}$ for all monomials $m,m^{\prime},m^{\prime\prime}$.

The term order used in this paper is the lexicographic term order given by $x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}<x_{1}^{b_{1}}\cdots x_{n}^{b_{n}}$ if there exists an index $i$ with $a_{1}=b_{1},\dots,a_{i-1}=b_{i-1}$ and $a_{i}<b_{i}$.

If $<$ is any term order any $f\in{\mathbb{Q}}[{\mathbf{x}}_{n}]$ is any nonzero polynomial, let ${\mathrm{in}}_{<}(f)$ be the leading (i.e., greatest) term of $f$ with respect to the order $<$. If $I\subseteq{\mathbb{Q}}[{\mathbf{x}}_{n}]$ is any ideal, the associated initial ideal is

[TABLE]

A finite collection $G=\{g_{1},\dots,g_{r}\}$ of nonzero polynomials in an ideal $I\subseteq{\mathbb{Q}}[{\mathbf{x}}_{n}]$ is called a Gröbner basis of $I$ if we have the equality of monomial ideals

[TABLE]

If $G$ is a Gröbner basis of $I$ it follows that $I=\langle G\rangle$. A Gröbner basis $G=\{g_{1},\dots,g_{r}\}$ is called minimal if the $<$-leading coefficient of each $g_{i}$ is $1$ and ${\mathrm{in}}(g_{i})\nmid{\mathrm{in}}(g_{j})$ for all $i\neq j$. A minimal Gröbner basis $G=\{g_{1},\dots,g_{r}\}$ is called reduced if in addition, for all $i\neq j$, no term of $g_{j}$ is divisible by ${\mathrm{in}}_{<}(g_{i})$. After fixing a term order, every ideal $I\subseteq{\mathbb{Q}}[{\mathbf{x}}_{n}]$ has a unique reduced Gröbner basis.

Let $I\subseteq{\mathbb{Q}}[{\mathbf{x}}_{n}]$ be an ideal and let $G$ be a Gröbner basis for $I$. The set of monomials in ${\mathbb{Q}}[{\mathbf{x}}_{n}]$

[TABLE]

descends to a vector space basis for the quotient ${\mathbb{Q}}[{\mathbf{x}}_{n}]/I$. This is called the standard monomial basis; it is completely determined by the ideal $I$ and the term order $<$. If $I$ is a homogeneous ideal, the Hilbert series of ${\mathbb{Q}}[{\mathbf{x}}_{n}]/I$ is given by

[TABLE]

where the sum is over all monomials in the standard monomial basis.

3. Hilbert series

In this section we will derive the Hilbert series and ungraded isomorphism type of the module $R_{n,k,r}$. The method that we use dates back to Garsia and Procesi in the context of Tanisaki ideals and quotients [6].

Let $Y\subseteq{\mathbb{Q}}^{n}$ be any finite set of points and consider the ideal ${\mathbf{I}}(Y)\subseteq{\mathbb{Q}}[{\mathbf{x}}_{n}]$ of polynomials which vanish on $Y$. That is, we have

[TABLE]

We may identify the quotient ${\mathbb{Q}}[{\mathbf{x}}_{n}]/{\mathbf{I}}(Y)$ with the collection of all (polynomial) functions $Y\rightarrow{\mathbb{Q}}$; since $Y$ is finite we have

[TABLE]

If $Y$ is stable under the coordinate permutation action of $S_{n}$, we have the further identification of $S_{n}$-modules

[TABLE]

The ideal ${\mathbf{I}}(Y)$ is usually not homogeneous; we wish to replace it by a homogeneous ideal so that the associated quotient is graded. For any nonzero polynomial $f\in{\mathbf{I}}(X)$, write $f=f_{d}+\cdots f_{1}+f_{0}$ where $f_{i}$ is homogeneous of degree $i$ and $f_{d}\neq 0$. Let $\tau(f)=f_{d}$ be the top homogeneous component of $f$. The ideal ${\mathbf{T}}(Y)\subseteq{\mathbb{Q}}[{\mathbf{x}}_{n}]$ is given by

[TABLE]

By construction the ideal ${\mathbf{T}}(Y)$ is homogeneous, so that the quotient ${\mathbb{Q}}[{\mathbf{x}}_{n}]/{\mathbf{T}}(Y)$ is graded. Furthermore, we still have the dimension equality

[TABLE]

and the $S_{n}$-module isomorphism

[TABLE]

whenever the point set $Y$ is $S_{n}$-stable.

The symmetric group $S_{n}$ acts on $S_{n,k,r}$ by permuting the positive letters $1,2,\dots,n$. We aim to prove that $R_{n,k,r}\cong{\mathbb{Q}}[S_{n,k,r}]$ as ungraded $S_{n}$-modules. To do this, our strategy is as follows.

1. (1)

Find a point set $Y_{n,k,r}\subseteq{\mathbb{Q}}^{n}$ which is stable under the action of $S_{n}$ such that there is a $S_{n}$-equivariant bijection from $Y_{n,k,r}$ to $S_{n,k,r}$. 2. (2)

Prove that $I_{n,k,r}\subseteq{\mathbf{T}}(Y_{n,k,r})$ by showing that the generators of $I_{n,k,r}$ arise as top degree components of polynomials in ${\mathbf{I}}(Y_{n,k,r})$. 3. (3)

Prove that

[TABLE]

and use the relation $I_{n,k,r}\subseteq{\mathbf{T}}(Y_{n,k,r})$ to conclude that $I_{n,k,r}={\mathbf{T}}(Y_{n,k,r})$.

The point set $Y_{n,k,r}$ which accomplishes Step 1 is the following.

Definition 3.1**.**

Fix $n+k$ distinct rational numbers $\alpha_{1},\alpha_{2},\dots,\alpha_{n+k}\in{\mathbb{Q}}$. Let $Y_{n,k,r}$ be the set of points $(y_{1},y_{2},\dots,y_{n})\in{\mathbb{Q}}^{n}$ such that

• •

the coordinates $y_{1},y_{2},\dots,y_{n}$ are distinct and lie in $\{\alpha_{1},\alpha_{2},\dots,\alpha_{n+k}\}$, and

• •

the numbers $\alpha_{n+k-r+1},\dots,\alpha_{n+k-1},\alpha_{n+k}$ all appear as coordinates $(y_{1},y_{2}\dots,y_{n})$.

It is clear that $Y_{n,k,r}$ is stable under the action of $S_{n}$. We have a natural identification of $Y_{n,k,r}$ with permutations in $S_{n,k,r}$ given by letting a copy of $\alpha_{i}$ in position $j$ of $(y_{1},\dots,y_{n})$ correspond to the letter $j$ in position $i$ of the corresponding permutation in $S_{n,k,r}$. For example, if $(n,k,r)=(4,3,2)$ then

[TABLE]

This bijection $Y_{n,k,r}\leftrightarrow S_{n,k,r}$ is clearly $S_{n}$-equivariant, so Step 1 of our strategy is accomplished. Step 2 of our strategy is achieved in the following lemma.

Lemma 3.2**.**

We have $I_{n,k,r}\subseteq{\mathbf{T}}(Y_{n,k,r})$.

Proof.

We show that every generator of $I_{n,k,r}$ arises as the leading term of a polynomial in ${\mathbf{I}}(Y_{n,k,r})$. We begin with the elementary symmetric function generators $e_{n-r+1}({\mathbf{x}}_{n}),\dots,e_{n-1}({\mathbf{x}}_{n}),e_{n}({\mathbf{x}}_{n})$. Consider the rational function in $t$ given by

[TABLE]

If $(x_{1},\dots,x_{n})\in Y_{n,k,r}$, the $r$ factors in the denominator cancel with $r$ factors in the numerator, so that this rational expression is a polynomial in $t$ of degree $n-r$. In particular, for $n-r+1\leq m\leq r$ taking the coefficient of $t^{m}$ on both sides gives

[TABLE]

so that $e_{m}({\mathbf{x}}_{n})\in{\mathbf{T}}(Y_{n,k,r})$.

A similar trick shows that the homogeneous symmetric functions $h_{k+1}({\mathbf{x}}_{n}),\dots,h_{k+n}({\mathbf{x}}_{n})$ lie in ${\mathbf{T}}(Y_{n,k,r})$. Consider the rational function

[TABLE]

If $(x_{1},\dots,x_{n})\in Y_{n,k,r}$ the $n$ factors in the denominator cancel with $n$ factors in the numerator, giving a polynomial in $t$ of degree $k$. For $m\geq k+1$, taking the coefficient of $t^{m}$ on both sides gives

[TABLE]

so that $h_{m}({\mathbf{x}}_{n})\in{\mathbf{T}}(Y_{n,k,r})$. ∎

Step 3 of our strategy will take more work. To begin, we identify a convenient collection of monomials in the initial ideal ${\mathrm{in}}_{<}(I_{n,k,r})$ with respect to the lexicographic term order. Given a subset $S=\{s_{1}<\cdots<s_{m}\}=\subseteq[n]$ the corresponding skip monomial ${\mathbf{x}}(S)$ is given by

[TABLE]

In particular, if $n=8$ we have ${\mathbf{x}}(2458)=x_{2}^{2}x_{4}^{3}x_{5}^{3}x_{8}^{5}$.

Lemma 3.3**.**

Let $<$ be the lexicographic term order on ${\mathbb{Q}}[{\mathbf{x}}_{n}]$. If $S\subseteq[n]$ satisfies $|S|=n-r+1$ we have ${\mathbf{x}}(S)\in{\mathrm{in}}_{<}(I_{n,k,r})$. Moreover, we have $x_{1}^{k+1},x_{2}^{k+2},\dots,x_{n}^{k+n}\in{\mathrm{in}}_{<}(I_{n,k,r})$.

Proof.

The assertion regarding skip monomials comes from combining [10, Lem. 3.4] (and in particular [10, Eqn. 3.5]) and [10, Lem. 3.5]. To prove the second assertion, the identities

[TABLE]

(for $1\leq i\leq n$ and $m\geq 0$) imply that

[TABLE]

so that $x_{1}^{k+1},x_{2}^{k+2},\dots,x_{n}^{k+n}\in{\mathrm{in}}_{<}(I_{n,k,r})$. ∎

The initial terms provided by Lemma 3.3 will be all we need. We name the monomials $m\in{\mathbb{Q}}[{\mathbf{x}}_{n}]$ which are not divisible by any of these initial terms as follows.

Definition 3.4**.**

A monomial $m\in{\mathbb{Q}}[{\mathbf{x}}_{n}]$ is $(n,k,r)$-good if

• •

we have ${\mathbf{x}}(S)\nmid m$ for all $S\subseteq[n]$ with $|S|=n-r+1$, and

• •

we have $x_{i}^{k+i}\nmid m$ for all $1\leq i\leq n$.

Let ${\mathcal{M}}_{n,k,r}$ denote the set of all $(n,k,r)$-good monomials.

By Lemma 3.3, the monomials in ${\mathcal{M}}_{n,k,r}$ contain the standard monomial basis of $R_{n,k,r}$, and so descend to a spanning set of $R_{n,k,r}$. We will see that ${\mathcal{M}}_{n,k,r}$ is in fact that standard monomial basis of $R_{n,k,r}$. We will do this using the following combinatorial result.

Lemma 3.5**.**

There is an injection $\Psi:S_{n,k,r}\rightarrow{\mathcal{M}}_{n,k,r}$ with the property that $\deg(\Psi(\pi))={\mathrm{inv}}(\pi)$ for all $\pi\in S_{n,k,r}$.

It will develop that the map $\Psi$ of Lemma 3.5 is actually a bijection.

Proof.

The map $\Psi$ will essentially be the inversion code. Let $\pi=\pi_{1}\dots\pi_{n+k}\in S_{n,k,r}$ be a $r$-tail positive permutation of the multiset $\{0^{k},1,2,\dots,n\}$. The code of $\pi$ is the sequence $(c_{1},\dots,c_{n})$ where

[TABLE]

For example, if $\pi=40130052$ the code is $(c_{1},c_{2},c_{3},c_{4},c_{5})=(2,0,3,6,1)$. It is clear that the sum of the code of $\pi$ gives the inversion number ${\mathrm{inv}}(\pi)$. If $\pi\in S_{n,k,r}$ has code $(c_{1},\dots,c_{n})$, we define $\Psi(\pi):=x_{1}^{c_{1}}\cdots x_{n}^{c_{n}}$.

We argue that $\Psi$ is a well defined function $S_{n,k,r}\rightarrow{\mathcal{M}}_{n,k,r}$, that is, we have $\Psi(\pi)\in{\mathcal{M}}_{n,k,r}$ for all $\pi\in S_{n,k,r}$. Let $\pi\in S_{n,k,r}$ have code $(c_{1},\dots,c_{n})$. Since $\pi$ contains $k$ copies of [math], it is clear that $c_{i}<k+i$ for all $1\leq i\leq n$, so that $x_{i}^{k+i}\nmid\Psi(\pi)$ for all $1\leq i\leq n$.

Now let $S=\{s_{1}<\cdots<s_{n-r+1}\}\subseteq[n]$ and suppose ${\mathbf{x}}(S)\mid\Psi(\pi)$. This means that $c_{s_{i}}\geq s_{i}-i+1$ for all $1\leq i\leq n-r+1$. Let $T=\{\pi_{n+k-r+1},\dots,\pi_{n+k-1},\pi_{n+k}\}$ be the $r$-tail of $\pi$; since $\pi\in S_{n,k,r}$ the set $T$ consists of $r$ positive numbers. We argue that $S\cap T=\emptyset$ as follows.

• •

If $s_{1}\in T$ we would have $c_{s_{1}}\leq s_{1}-1$ (since $s_{1}$ could form inversions with only $1,2,\dots,s_{1}-1$), contradicting the inequality $c_{s_{1}}\geq s_{1}$. We conclude that $s_{1}\notin T$.

• •

If $s_{1},\dots,s_{i-1}\notin T$ and $s_{i}\in T$, we would have $c_{s_{i}}\leq s_{i}-i$ (since $s_{i}$ can only form inversions with those letters in $1,2,\dots,s_{i}-1$ which lie in $T$), contradicting the inequality $c_{s_{i}}\geq s_{i}-1+1$. We conclude that $s_{i}\notin T$.

Induction gives the result that $S\cap T=\emptyset$. However, this contradicts the facts that $|S|=n-r+1$, $|T|=r$, and that there are a total of $n$ positive letters in $\pi$. This concludes the proof that the map $\Psi:S_{n,k,r}\rightarrow{\mathcal{M}}_{n,k,r}$ is well defined.

The relation $\deg(\Psi(\pi))={\mathrm{inv}}(\pi)$ is clear from construction. The fact that $\Psi$ is an injection is equivalent to the fact that a permutation $\pi=\pi_{1}\dots\pi_{n+k}\in S_{n,k,r}$ is determined by its code $(c_{1},\dots,c_{n})$. This assertion is true more broadly for any permutation of the multiset $\{0^{k},1,2,\dots,n\}$ (whether or not it is $r$-tail positive); we leave the verification to the reader. ∎

We are ready to derive the Hilbert series of $R_{n,k,r}$.

Theorem 3.6**.**

Endow monomials in ${\mathbb{Q}}[{\mathbf{x}}_{n}]$ with the lexicographic term order. The standard monomial basis of $R_{n,k,r}$ is ${\mathcal{M}}_{n,k,r}$. The Hilbert series of $R_{n,k,r}$ is given by

[TABLE]

Proof.

Let ${\mathcal{B}}^{{\mathbf{T}}}$ be the standard monomial basis of ${\mathbb{Q}}[{\mathbf{x}}_{n}]/{\mathbf{T}}(Y_{n,k,r})$ and let ${\mathcal{B}}^{J}$ be the standard monomial basis of $R_{n,k,r}={\mathbb{Q}}[{\mathbf{x}}_{n}]/I_{n,k,r}$. We know that $|S_{n,k,r}|=|{\mathcal{B}}^{{\mathbf{T}}}|$. Lemma 3.2 implies that ${\mathcal{B}}^{{\mathbf{T}}}\subseteq{\mathcal{B}}^{J}$. Lemma 3.3 further implies the containment ${\mathcal{B}}^{J}\subseteq{\mathcal{M}}_{n,k,r}$. Finally, Lemma 3.5 gives the relation $|{\mathcal{M}}_{n,k,r}|\leq|S_{n,k,r}|$. Putting these facts together gives

[TABLE]

and the fact that all of these sets have size $|S_{n,k,r}|$. In particular, the standard monomial basis of $R_{n,k,r}$ is ${\mathcal{M}}_{n,k,r}$.

By the last paragraph, the map $\Psi$ of Lemma 3.5 is a bijection. It follows that

[TABLE]

It is well known that $\sum_{\pi\in S_{n}}q^{{\mathrm{inv}}(\pi)}=[n]!_{q}$. The $q$-binomial coefficient in

[TABLE]

comes from the ways of inserting $k$ copies of [math] among the first $n-r$ letters of a permutation in $S_{n}$. ∎

We can also derive the ungraded $S_{n}$-isomorphism type of the quotient $R_{n,k,r}$.

Corollary 3.7**.**

As an ungraded $S_{n}$-module we have $R_{n,k,r}\cong_{S_{n}}{\mathbb{Q}}[S_{n,k,r}]$.

Proof.

Lemma 3.2 and Theorem 3.6 give the isomorphisms

[TABLE]

of ungraded $S_{n}$-modules. ∎

We describe a minimal Gröbner basis for the ideal $I_{n,k,r}$. Given a subset $S=\{s_{1}<\dots<s_{m}\}\subseteq[n]$, let $\gamma(S)=(\gamma(S)_{1},\dots,\gamma(S)_{n})$ be the length $n$ skip vector of nonnegative integers given by

[TABLE]

Let $\gamma(S)^{*}=(\gamma(S)_{n},\dots,\gamma(S)_{1})$ be the reversal of the vector $\gamma(S)$. If $\gamma=(\gamma_{1},\dots,\gamma_{n})$ is any length $n$ vector of nonnegative integers, let $\kappa_{\gamma}({\mathbf{x}}_{n})\in{\mathbb{Q}}[{\mathbf{x}}_{n}]$ be the associated Demazure character (see [10, Sec. 2] for its definition). Finally, if $f({\mathbf{x}}_{n})\in{\mathbb{Q}}[{\mathbf{x}}_{n}]$ is any polynomial, let $f({\mathbf{x}}_{n}^{*})$ be the polynomial obtained by reversing the variables in $f({\mathbf{x}}_{n})$ so that

[TABLE]

Corollary 3.8**.**

Endow monomials in ${\mathbb{Q}}[{\mathbf{x}}_{n}]$ with the lexicographic term order. A Gröbner basis for the ideal $I_{n,k,r}$ consists of the polynomials

[TABLE]

together with the polynomials

[TABLE]

where $S$ ranges over all $n-r+1$-element subsets of $[n]$. When $r<n$ and $k>0$ this Gröbner basis is minimal.

The Gröbner basis in Corollary 3.8 is typically not reduced.

Proof.

The proof of Lemma 3.3 shows that the polynomial $h_{k+i}(x_{i},x_{i+1},\dots,x_{n})$ lies in the ideal $I_{n,k,r}$. By [10, Lem. 3.4] (and in particular [10, Eqn. 3.5]) shows that the relevant variable reversed Demazure characters lie in $I_{n,k,r}$.

Let $<$ be the lexicographic term order on ${\mathbb{Q}}[{\mathbf{x}}_{n}]$. We have ${\mathrm{in}}_{<}(h_{k+i}(x_{i},x_{i+1},\dots,x_{n}))=x_{i}^{k+i}$ and ${\mathrm{in}}_{<}(\kappa_{\gamma(S)^{*}}({\mathbf{x}}_{n}^{*})={\mathbf{x}}(S)$ (see [10, Lem. 3.5]). We know that these initial terms generate ${\mathrm{in}}_{<}(I_{n,k,r})$, proving the assertion about the claimed collection of polynomials being a Gröbner basis. When $r<n$ and $k>0$, none of the relevant skip monomials ${\mathbf{x}}(S)$ are divisible by any of the variable powers $x_{1}^{k+1},\dots,x_{n}^{k+n}$. This proves the claim about minimality. ∎

For example, consider the case $(n,k,r)=(5,2,3)$. A minimal Gröbner basis for $J_{5,2,3}$ is given by the polynomials

[TABLE]

together with the variable reversed Demazure characters

[TABLE]

Theorem 3.6 describes the standard monomial basis ${\mathcal{M}}_{n,k,r}$ of $R_{n,k,r}$ in terms of divisibility by skip monomials. However, a more direct characterization of this standard monomial basis is available. Let $k\geq 0$ and $r\leq n$. For any $(n-r)$-element subset $T\subseteq[n]$, define a length $n$ sequence $\delta(T):=(\delta(T)_{1},\dots,\delta(T)_{n})$ by the formula

[TABLE]

where $[n]-T=\{s_{1}<\cdots<s_{r}\}$. Any of the ${n\choose r}$ sequences which can be obtained in this way is an $(n,k,r)$-staircase. For example, the $(5,2,3)$-staircases are

[TABLE]

Proposition 3.9**.**

Endow monomials in ${\mathbb{Q}}[{\mathbf{x}}_{n}]$ with the lexicographic term order. The standard monomial basis ${\mathcal{M}}_{n,k,r}$ of $R_{n,k,r}$ consists of those monomials in ${\mathbb{Q}}[{\mathbf{x}}_{n}]$ whose exponent vectors are componentwise $\leq$ at least one $(n,k,r)$-staircase.

Proof.

Let ${\mathcal{N}}_{n,k,r}$ be the collection of monomials in ${\mathbb{Q}}[{\mathbf{x}}_{n}]$ whose exponent vectors are componentwise $\leq$ at least one $(n,k,r)$-staircase. If $\delta_{n}(T)=(a_{1},\dots,a_{n})$ is an $(n,k,r)$-staircase for some $(n-r)$-element set $T\subseteq[n]$ and $m=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ is the corresponding monomial, it is clear that $a_{i}<k+i$ for all $i$, so that $x_{i}^{k+i}\nmid m$. If $S\subseteq[n]$ satisfies $|S|=n-r+1$ then at least one index $i\in S$ satisfies $i\notin T$, which forces ${\mathbf{x}}(S)\nmid m$. It follows that ${\mathcal{N}}_{n,k,r}\subseteq{\mathcal{M}}_{n,k,r}$.

On the other hand, we may construct a map

[TABLE]

by letting $\Phi(\pi)=(c_{1},\dots,c_{n})$ be the code of any $r$-tail positive permutation $\pi\in S_{n,k,r}$. The fact that $\pi$ is $r$-tail positive implies that $\Phi(\pi)\in{\mathcal{N}}_{n,k,r}$, so that $\Phi$ is well defined. It is clear that $\Phi$ is injective, so that

[TABLE]

and we have ${\mathcal{N}}_{n,k,r}={\mathcal{M}}_{n,k,r}$, as desired. ∎

For example, if $(n,k,r)=(2,2,1)$ the $(2,2,1)$-staircases are $(0,3)$ and $(2,0)$ so that

[TABLE]

4. Frobenius series

In this section we derive the Frobenius series of $R_{n,k,r}$. Our first lemma is a short exact sequence which establishes a Pascal-type recursion for ${\mathrm{grFrob}}(R_{n,k,r};q)$.

Lemma 4.1**.**

Suppose $n,k,r\geq 0$ with $r<n$ and $k>0$. There is a short exact sequence of $S_{n}$-modules

[TABLE]

where the first map is homogeneous of degree $n-r$ and the second map is homogeneous of degree [math]. Equivalently, we have the equality of graded Frobenius characters

[TABLE]

Proof.

We have the inclusion of ideals $I_{n,k,r}\subseteq I_{n,k,r+1}$; we let the second map be the canonical projection $\pi:R_{n,k,r}\twoheadrightarrow R_{n,k,r+1}$. We have a homogeneous map $\widetilde{\varphi}:{\mathbb{Q}}[{\mathbf{x}}_{n}]\rightarrow R_{n,k,r}$ of degree $n-r$ given by multiplication by $e_{n-r}({\mathbf{x}}_{n})$, and then projecting onto $R_{n,k,r}$.

We claim that $\widetilde{\varphi}(I_{n,k-1,r})=0$, so that $\widetilde{\varphi}$ induces a well defined map $\varphi:R_{n,k-1,r}\rightarrow R_{n,k,r}$. This is equivalent to showing that $h_{k}({\mathbf{x}}_{n})\cdot e_{n-r}({\mathbf{x}}_{n})\in I_{n,k,r}$. The Pieri Rule implies that

[TABLE]

we will show that both terms on the right hand side lie in $I_{n,k,r}$.

To see that $s_{(k,1^{n-r})}({\mathbf{x}}_{n})\in I_{n,k,r}$, observe that, for $1\leq i\leq r$ we have

[TABLE]

It follows that modulo $I_{n,k,r}$ we have the congruences

[TABLE]

where the last congruence used the fact that $s_{(k+n-r)}({\mathbf{x}}_{n})=h_{k+n-r}({\mathbf{x}}_{n})\in I_{n,k,r}$ since $r<n$. This chain of congruences also shows that $s_{(k+1,1^{n-r-1})}({\mathbf{x}}_{n})\in I_{n,k,r}$.

By the last paragraph, we have a well defined induced map $\varphi:R_{n,k-1,r}\rightarrow R_{n,k,r}$. It is clear that $\mathrm{Im}(\varphi)=\mathrm{Ker}(\pi)$. Moreover, the Pascal relation implies that

[TABLE]

so that by Theorem 3.6 we have

[TABLE]

Since $\pi$ is a surjection, this forces the sequence

[TABLE]

to be exact. To finish the proof, observe that the maps $\varphi$ and $\pi$ commute with the action of $S_{n}$. ∎

We are ready to state the graded Frobenius image of $R_{n,k,r}$. We will give several formulas for this image. For any word $w$ over the nonnegative integers, define the monomial ${\mathbf{x}}^{w}$ to be

[TABLE]

in particular, any copies of [math] in $w$ do not affect ${\mathbf{x}}^{w}$.

Theorem 4.2**.**

The graded Frobenius image of $R_{n,k,r}$ is given by

[TABLE]

The last sum ranges over all length $n+k$ words $w=w_{1}\dots w_{n+k}$ in the alphabet of nonnegative integers which contain precisely $k$ copies of [math] and are $r$-tail positive.

Proof.

By considering the placement of the $k$ copies of [math] in a $r$-tail positive word $w$ appearing in the final sum, we see that

[TABLE]

On the other hand, we have

[TABLE]

where the first equality uses the equidistribution of the statistics ${\mathrm{inv}}$ and ${\mathrm{maj}}$ on permutations of a fixed multiset of positive integer, the second follows from standard properties of the RSK correspondence, and the third is a consequence of the work of Lusztig-Stanley [16].

By the last paragraph, it suffices to prove the first equality asserted in the statement of the theorem. If $r\geq n$ or $k=0$ then $R_{n,k,r}=R_{n}$ and this equality is trivial. Otherwise, we have the $q$-Pascal relation

[TABLE]

so that the theorem follows from Lemma 4.1 and induction. ∎

The short exact sequence in Lemma 4.1 gives a recipe for constructing bases of $R_{n,k,r}$ from bases of the classical coinvariant algebra $R_{n}$. We switch from working over ${\mathbb{Q}}$ to working over an arbitrary field ${\mathbb{F}}$, so that the ideals $I_{n,k,r},I_{n}$ are defined inside the ring ${\mathbb{F}}[{\mathbf{x}}_{n}]:={\mathbb{F}}[x_{1},\dots,x_{n}]$ and we have $R_{n,k,r}:={\mathbb{F}}[{\mathbf{x}}_{n}]/I_{n,k,r},R_{n}:={\mathbb{F}}[{\mathbf{x}}_{n}]/I_{n}$.

Theorem 4.3**.**

Let ${\mathcal{C}}_{n}=\{b_{\pi}({\mathbf{x}}_{n})\,:\,\pi\in S_{n}\}$ be a collection of polynomials in ${\mathbb{F}}[{\mathbf{x}}_{n}]$ indexed by permutations in $S_{n}$ which descends to a basis of $R_{n}$. The collection of polynomials

[TABLE]

in ${\mathbb{F}}[{\mathbf{x}}_{n}]$ descends to a basis of $R_{n,k,r}$.

Proof.

This is trivial when $k=0$ or $r\geq n$, so we assume $k>0$ and $r<n$.

The arguments of Section 3 apply to show that $\dim(R_{n,k,r})=|S_{n,k,r}|$ when working over the arbitrary field ${\mathbb{F}}$. 222If ${\mathbb{F}}$ is a finite field, there might not be enough elements in ${\mathbb{F}}$ for the point set $Y_{n,k,r}$ of Definition 3.1 to make sense. To get around this, we may apply [13, Lem. 3.1] to harmlessly replace ${\mathbb{F}}$ by an extension field $\mathbb{K}$. The proof of Lemma 4.1 then applies over ${\mathbb{F}}$ to give a short exact sequence of graded ${\mathbb{F}}$-vector spaces

[TABLE]

where $\pi$ is the canonical projection. We may inductively assume that ${\mathcal{C}}_{n,k-1,r}$ descends to a ${\mathbb{F}}$-basis of $R_{n,k-1,r}$ and that ${\mathcal{C}}_{n,k,r+1}$ descends to a ${\mathbb{F}}$-basis of $S_{n,k,r+1}$. Exactness implies that

[TABLE]

descends to an ${\mathbb{F}}$-basis of $R_{n,k,r}$. ∎

Theorem 4.3 reinforces the fact that $R_{n,k,r}$ consists of ${n+k-r\choose k}$ copies of $R_{n}$, graded by the $q$-binomial coefficient ${n+k-r\brack k}_{q}$. Interesting bases ${\mathcal{C}}_{n}$ to which Theorem 4.3 can be applied include

• •

the Artin basis [2]

[TABLE]

(which is connected to the ${\mathrm{inv}}$ statistic on permutations in $S_{n}$) and

• •

the Garsia-Stanton basis (or the descent monomial basis [5, 7] ${\mathcal{C}}_{n}=\{gs_{\pi}\,:\,\pi\in S_{n}\}$ where

[TABLE]

(which is connected to the ${\mathrm{maj}}$ statistic on permutations in $S_{n}$).

The GS basis above can be deformed somewhat to describe the isomorphism type of $R_{n,k,r}$ as a module over the 0-Hecke algebra. The algebra $H_{n}(0)$ acts on the polynomial ring ${\mathbb{F}}[{\mathbf{x}}_{n}]$ by letting the generator $T_{i}$ act by the Demazure operator $\sigma_{i}$, where

[TABLE]

Here $s_{i}(f)$ is the polynomial obtained by interchanging $x_{i}$ and $x_{i+1}$ in $f({\mathbf{x}}_{n})$. It can be shown that if $f\in{\mathbb{F}}[{\mathbf{x}}_{n}]^{S_{n}}$ is any symmetric polynomial and $g\in{\mathbb{F}}[{\mathbf{x}}_{n}]$ is an arbitrary polynomial then

[TABLE]

Therefore, any ideal $I\subseteq{\mathbb{F}}[{\mathbf{x}}_{n}]$ generated by symmetric polynomials is stable under the action of $H_{n}(0)$. In particular, the ideal $I_{n,k,r}$ is stable under the action of $H_{n}(0)$, and the quotient $R_{n,k,r}={\mathbb{F}}[{\mathbf{x}}_{n}]/I_{n,k,r}$ carries the structure of an $H_{n}(0)$-module.

Huang [12] studied the coinvariant ring $R_{n}$ as a graded module over the 0-Hecke algebra $H_{n}(0)$. We apply Theorem 4.3 to generalize Huang’s results to the quotient $R_{n,k,r}$. If $V$ is any graded $H_{n}(0)$-module, we let $V(i)$ denote the graded $H_{n}(0)$-module with components $V(i)_{j}:=V_{i+j}$.

Theorem 4.4**.**

Let $n,k,$ and $r$ be nonnegative integers with $r\leq n$. We have an isomorphism of graded $H_{n}(0)$-modules

[TABLE]

Here the direct sum is over all partitions $\lambda$ which satisfy $\lambda_{1}\leq n-r$ and have at most $k$ parts. The module $R_{n}={\mathbb{F}}[{\mathbf{x}}_{n}]/I_{n}$ is the coinvariant algebra viewed as a graded $H_{n}(0)$-module.

Proof.

Huang [12] introduced the following modified GS basis of $R_{n}$. For $1\leq i\leq n-1$, define an operator $\bar{\sigma}_{i}$ on ${\mathbb{F}}[{\mathbf{x}}_{n}]$ by the rule $\bar{\sigma}_{i}:=\sigma_{i}-1$. For any permutation $\pi\in S_{n}$, define $\bar{\sigma}_{\pi}:=\bar{\sigma}_{i_{1}}\cdots\bar{\sigma}_{i_{k}}$ where $s_{i_{1}}\cdots s_{i_{k}}$ is any reduced word for $\pi$. Finally, given $\pi\in S_{n}$, let ${\mathbf{x}}_{{\mathrm{Des}}(\pi)}$ be the monomial

[TABLE]

For example, we have ${\mathbf{x}}_{21543}=(x_{1})\cdot(x_{1}x_{2}x_{3})\cdot(x_{1}x_{2}x_{3}x_{4})$. Huang proves [12, Thm. 4.5] that the collection of polynomials

[TABLE]

in ${\mathbb{F}}[{\mathbf{x}}_{n}]$ descends to a ${\mathbb{F}}$-basis for $R_{n}$.

Applying Theorem 4.3 to Huang’s basis of $R_{n}$, we get a collection of polynomials ${\mathcal{C}}_{n,k,r}$ given by

[TABLE]

which descends to a basis of $R_{n,k,r}$. The symmetric polynomial $e_{\lambda}({\mathbf{x}}_{n})$ has degree $|\lambda|$ and the symmetry of $e_{\lambda}({\mathbf{x}}_{n})$ gives

[TABLE]

It follows that, for $\lambda\subseteq(n-r)\times k$ fixed, the collection of polynomials

[TABLE]

descends inside $R_{n,k,r}$ to a ${\mathbb{F}}$-basis of a copy of $R_{n}$ with degree shifted up by $|\lambda|$. ∎

It may be tempting to try to prove Theorem 4.4 in the same fashion as Theorem 4.2 – by applying the short exact sequence of Lemma 4.1 directly and without appealing to Theorem 4.3. However, although the maps in this sequence commute with the action of $H_{n}(0)$, since $H_{n}(0)$ is not semisimple it is not a priori clear that this sequence splits in the category of $H_{n}(0)$-modules. Theorem 4.4 guarantees that this sequence splits; the authors do not know of a more direct way to see this splitting.

Corollary 4.5**.**

Let $n,k,$ and $r$ be nonnegative integers with $r\leq n$.

1. (1)

The length-degree bigraded quasisymmetric characteristic $\mathrm{Ch}_{q,t}(R_{n,k,r})$ is given by

[TABLE]

where $F_{{\mathrm{Des}}(\pi^{-1}),n}$ is the fundamental quasisymmetric function. 2. (2)

The degree graded quasisymmetric characteristic $\mathrm{Ch}_{q}(R_{n,k,r})$ is in fact symmetric and given by

[TABLE] 3. (3)

The $H_{n}(0)$-module $R_{n,k,r}$ is projective. Its degree graded noncommutative characteristic $\mathbf{ch}_{q}(R_{n,k,r})$ is

[TABLE]

where $\alpha$ ranges over all strong compositions of $n$, the major index is ${\mathrm{maj}}(\alpha)=(\alpha_{1})+(\alpha_{1}+\alpha_{2})+\cdots$, and $\mathbf{s}_{\alpha}$ is the noncommutative ribbon Schur function.

Proof.

Parts 1 and 2 follow from the work of Huang [12, Cor. 4.9] and Theorem 4.4. Since $R_{n}$ is a projective $H_{n}(0)$-module (see [12, Thm. 4.5]) and direct sums of projective modules are projective, we can apply [12, Cor. 8.4, $\mu=(1^{n})$] to get Part 3. ∎

Since the characteristics $\mathrm{Ch}_{q,t}$ and $\mathrm{Ch}_{t}$ are defined in terms of the Grothendieck group $G_{0}(H_{n}(0))$ of $H_{n}(0)$, we may apply the short exact sequence of Lemma 4.1 to obtain Parts 1 and 2 of Corollary 4.5 more directly. However, since extensions of projective modules are not in general projective, Lemma 4.1 does not immediately imply that $R_{n,k,r}$ is a projective $H_{n}(0)$-module.

Although Theorem 4.3 gives a collection of polynomials in ${\mathbb{F}}[{\mathbf{x}}_{n}]$ generalizing the GS monomials which descend to a basis of $R_{n,k,r}$, the authors have been unable to find a collection of monomials in ${\mathbb{F}}[{\mathbf{x}}_{n}]$ which generalizes the GS monomials and descends to a basis of $R_{n,k,r}$ (such monomial bases were found for the quotients appearing in the work of Haglund-Rhoades-Shimozono and Huang-Rhoades [10, 13]). Judging from the construction in [10, Sec. 5] and the Hilbert series of $R_{n,k,r}$, one might expect that the set of monomials

[TABLE]

would descend to a basis of $R_{n,k,r}$, but this set of monomials is linearly dependent in the quotient in general. A potential combinatorial obstruction to finding a GS monomial basis for $R_{n,k,r}$ is the fact that the statistics ${\mathrm{inv}}$ and ${\mathrm{maj}}$ do not share the same distribution on $S_{n,k,r}$.

5. Open problems

5.1. Bivariate generalization for $r=1$

We propose a relationship between our quotient ring $R_{n,k,r}$ and the theory of Macdonald polynomials. In particular, consider the ideal $I^{\prime}_{n,k,r}\subseteq{\mathbb{Q}}[{\mathbf{x}}_{n}]$ given by

[TABLE]

and let $R^{\prime}_{n,k,r}:={\mathbb{Q}}[{\mathbf{x}}_{n}]/I_{n,k,r}$ be the corresponding quotient. The ideal $I^{\prime}_{n,k,r}$ is obtained from the ideal $I_{n,k,r}$ by replacing the homogeneous symmetric functions with power sum symmetric functions.

As with the quotient $R_{n,k,r}$, the quotient $R^{\prime}_{n,k,r}$ has the structure of a graded $S_{n}$-module. Although the ideals $I_{n,k,r}$ and $I^{\prime}_{n,k,r}$ are not equal in general, we present

Conjecture 5.1**.**

There is an isomorphism of graded $S_{n}$-modules $R_{n,k,r}\cong R^{\prime}_{n,k,r}$.

The main reason for preferring the quotient rings $R^{\prime}_{n,k,r}$ over the quotient rings $R_{n,k,r}$ is that they generalize more readily to two sets of variables. Let ${\mathbf{x}}_{n}=(x_{1},\dots,x_{n})$ and ${\mathbf{y}}_{n}=(y_{1},\dots,y_{n})$ be two sets of $n$ variables and let ${\mathbb{Q}}[{\mathbf{x}}_{n},{\mathbf{y}}_{n}]$ be the polynomial ring in these variables. The symmetric group $S_{n}$ acts on ${\mathbb{Q}}[{\mathbf{x}}_{n},{\mathbf{y}}_{n}]$ by the diagonal action $\pi.x_{i}=x_{\pi_{i}},\pi.y_{i}=y_{\pi_{i}}$.

For any $a,b\geq 0$, let $p_{a,b}({\mathbf{x}}_{n},{\mathbf{y}}_{n})$ be the polarized power sum

[TABLE]

Moreover, let ${\mathcal{M}}_{n}$ be the set of the $2^{n}$ monomials $z_{1}\dots z_{n}$ in ${\mathbb{Q}}[{\mathbf{x}}_{n},{\mathbf{y}}_{n}]$ where $z_{i}\in\{x_{i},y_{i}\}$ for all $1\leq i\leq n$. For example, we have

[TABLE]

For a nonnegative integer $k$, let $DI_{n,k}\subseteq{\mathbb{Q}}[{\mathbf{x}}_{n},{\mathbf{y}}_{n}]$ be the ideal generated by the polarized power sums $p_{a,b}({\mathbf{x}}_{n},{\mathbf{y}}_{n})$ with $a+b\geq k+1$ together with the monomials in ${\mathcal{M}}_{n}$. Let $DR_{n,k}:={\mathbb{Q}}[{\mathbf{x}}_{n},{\mathbf{y}}_{n}]/DI_{n,k}$ be the corresponding quotient, which is a bigraded $S_{n}$-module.

Conjecture 5.2**.**

The bigraded Frobenius image of $DR_{n,k}$ is given by the delta operator image

[TABLE]

The latter three quantities in the conjecture are trivially equal by the definition of the delta operator. When $k=0$, the ring $DR_{n,0}$ is the classical diagonal coinvariant ring $DR_{n}$, so that Conjecture 5.2 reduces to Haiman’s celebrated result [11] that ${\mathrm{grFrob}}(DR_{n})=\Delta_{e_{n}}e_{n}$. Setting the ${\mathbf{y}}_{n}$ variables equal to zero in the quotient $DR_{n,k}$ yields the ring $R^{\prime}_{n,k,1}$, so that the ring $R^{\prime}_{n,k,1}$ conjecturally gives the analog of the coinvariant ring (for one set of variables) attached to the operator $\Delta_{h_{k}e_{n}}$.

The following proposition states that our module $R_{n,k,1}$ has graded Frobenius series which agrees with any of the delta operator expressions in Conjecture 5.2 upon setting $q=0$ and $t=q$.

Proposition 5.3**.**

We have

[TABLE]

Proof.

In this proof we will use the notation of plethysm; we refer the reader to [8] for the relevant details on plethysm and symmetric functions.

Let ${\mathrm{rev}}_{t}$ be the operator which reverses the coefficient sequences of polynomials with respect to the variable $t$. For a partition $\lambda\vdash n$, let $Q^{\prime}_{\lambda}=Q^{\prime}_{\lambda}({\mathbf{x}};t)$ be the corresponding Hall-Littlewood symmetric function. It is well known that the modified Macdonald polynomial $\widetilde{H}_{\lambda}=\widetilde{H}_{\lambda}({\mathbf{x}};q,t)$ satisfies

[TABLE]

This means that, for any symmetric function $f$ and any partition $\lambda\vdash n$, we have

[TABLE]

where $\ell(\lambda)$ is the number of parts of $\lambda$.

In order to exploit Equation 5.4, we need to express $e_{n}$ in terms of the modified Macdonald basis. This expansion is found in [8, Eqn. 2.72]: we have

[TABLE]

where

• •

$M=(1-q)(1-t)$,

• •

$B_{\lambda}=\sum_{c=(i,j)\in\lambda}q^{i-1}t^{j-i}$, where the sum is over all cells $c$ with matrix coordinates $(i,j)$ in the Ferrers diagram of $\lambda$,

• •

$\Pi_{\lambda}=\prod_{c=(i,j)\neq(0,0)}(1-q^{i-1}t^{j-1})$, where $c=(i,j)$ is a cell in $\lambda$ other than the corner $(0,0)$,

• •

$w_{\lambda}=\prod_{c\in\lambda}(q^{a(c)}-t^{l(c)+1})(t^{l(c)}-q^{a(c)+1})$, where the product is over all cells $c$ in the Ferrers diagram of $\lambda$ and $a(c),l(c)$ denote the arm and leg lengths of $\lambda$ at $c$.

We apply the operator $\Delta_{h_{k}e_{n}}=\Delta_{h_{k}}\Delta_{e_{n}}$ to both sides of Equation 5.5 to get

[TABLE]

Setting $q=0$ on both sides of Equation 5.6 gives

[TABLE]

For any $\lambda\vdash n$ and any symmetric function $f$, we have $f[B_{\lambda}]\mid_{q=0}=f(1,t,t^{2},\dots,t^{\ell(\lambda)-1})$. In particular, we have $e_{n}[B_{\lambda}]=0$ unless $\lambda=(1^{n})$ and Equation 5.7 reduces to

[TABLE]

The right hand side of Equation 5.8 simplifies to

[TABLE]

where we used Theorem 4.2 at $r=1$ and the well known fact that the graded Frobenius image of the classical coinvariant algebra $R_{n}$ is ${\mathrm{grFrob}}(R_{n};t)={\mathrm{rev}}_{t}(Q^{\prime}_{(1^{n})})$. ∎

5.2. Other bivariate generalizations

One may wonder if there is a bivariate generalization of the entire ring $R_{n,k,r}$, as we have only discussed the $r=1$ case so far. While we have not been able to find a full generalization, there is some progress in the Hilbert series case. The skewing operator acts on a symmetric function $f$ of degree $d$ uniquely so that

[TABLE]

for all symmetric functions $g$ of degree $d-1$, where the inner product is the usual Hall inner product on symmetric functions.

Given a vector $\alpha=(\alpha_{1},\dots,\alpha_{n})$ of $n$ positive integers, an $\alpha$-Tesler matrix $U=(u_{i,j})_{1\leq i,j\leq n}$ is an $n\times n$ upper triangular matrix with nonnegative integer entries such that, for $i=1$ to $n$,

[TABLE]

We write $U\in\mathcal{T}(\alpha)$. For example, the matrix

[TABLE]

satisfies $U\in\mathcal{T}(3,2,2,3)$. The weight of an $n\times n$ $\alpha$-Tesler matrix $U$ is equal to

[TABLE]

where $\operatorname{pos}(U)$ is the number of positive entries in $U$ and $[k]_{q,t}$ is the usual $q,t$-integer, i.e. $[k]_{q,t}=\frac{q^{k}-t^{k}}{q-t}$. For example, if $U$ is the Tesler matrix shown above, we have $\operatorname{pos}(U)=7$ and

[TABLE]

Finally, the $\alpha$-Tesler polynomial is

[TABLE]

This corollary follows from work in [1, 9, 14].

Corollary 5.4**.**

[TABLE]

It would be interesting to find an extension of this corollary to the entire graded Frobenius series of $R_{n,k,r}$ for general $r$.

5.3. A Schubert basis

There is also a basis for $R_{n,k,r}$ given by certain Schubert polynomials. We let $\Pi_{n,k,r}$ be all the permutations $\pi$ of $\{1,2,\ldots,n+k\}$ that satisfy

• •

all descents in $\pi$ occur weakly left of position $n$, and

• •

$1,2,\ldots,r$ all appear in $\pi_{1}\pi_{2}\ldots\pi_{n}$.

If $\pi\in\Pi_{n,k,r}$ is a permutation, let ${\mathfrak{S}}_{\pi}({\mathbf{x}}_{n})$ be the Schubert polynomial attached to $\pi$. Note that, since each $\pi$ has no descents after position $n$, there are at most $n$ variables that appear in the Schubert polynomial associated to $\pi$, so we have not truncated the variable set in any meaningful way. We will show that $\{{\mathfrak{S}}_{\pi}({\mathbf{x}}_{n}^{*}):\pi\in\Pi_{n,k,r}\}$ is a basis for $R_{n,k,r}$, where the asterisk represents the reversal of the vector of variables. This will follow from the fact that the leading terms are all $(n,k,r)$-good monomials.

Proposition 5.5**.**

Let $<$ be the lexicographic monomial order and let

[TABLE]

Then $\mathcal{LT}_{n,k,r}={\mathcal{M}}_{n,k,r}$.

Proof.

We will construct a bijection $\Phi:\Pi_{n,k,r}\to{\mathcal{M}}_{n,k,r}$ that satisfies $\Phi(\pi)={\mathrm{in}}_{<}({\mathfrak{S}}_{\pi}({\mathbf{x}}_{n}^{*}))$. The bijection itself is

[TABLE]

where $d_{i}$ counts the number of $j>i$ such that $\pi_{i}>\pi_{j}$. The fact that $\Phi(\pi)={\mathrm{in}}_{<}({\mathfrak{S}}_{\pi}({\mathbf{x}}_{n}^{*}))$ follows directly from the definition of the Schubert polynomial. We need to show that $m=\Phi(\pi)\in{\mathcal{M}}_{n,k,r}$ and to construct its inverse. Our proof will be similar to that of Lemma 3.5. First, we check that

• •

we have ${\mathbf{x}}(S)\nmid m$ for all $S\subseteq[n]$ with $|S|=n-r+1$, and

• •

we have $x_{i}^{k+i}\nmid m$ for all $1\leq i\leq n$.

To check the first condition, we recall that ${\mathbf{x}}(S)=x_{s_{1}}^{s_{1}}x_{s_{2}}^{s_{2}-1}\ldots x_{s_{n-r+1}}^{s_{n-r+1}-n+r}$ if $S=\{s_{1}<s_{2}<\ldots<s_{n-r+1}\}$. Since $S\subseteq[n]$ and the entries $1$ through $r$ all appear in $\pi_{1}$ through $\pi_{n}$, there is some $s_{i}$ such that $\pi_{s_{i}}\leq r$. Choose $i$ as large as possible such that $\pi_{s_{i}}\leq r$. Since $j>n$ implies $\pi_{j}>r$, $\pi_{s_{i}}$ can only be greater than at most $n-s_{i}$ entries to its right, i.e. $d_{s_{i}}\leq n-s_{i}$. Hence the power of $x_{n-s_{i}+1}$ in $m$ is at most $n-s_{i}$, which means ${\mathbf{x}}(S)\nmid m$. The second condition follows from the definition of $m$.

Given a monomial $m\in{\mathcal{M}}_{n,k,r}$, we would like to construct $\Phi^{-1}(m)$. This can be done using the usual bijection from codes $(d_{1},d_{2},\ldots,d_{n})$ to permutations. For $i=1$ to $n$, we choose $\pi_{i}$ such that it is greater than exactly $d_{i}$ of the entries in $[n+k]$ that have not already been placed to the left of position $i$ in $\pi$. The second condition for $(n,k,r)$-good monomials implies that the result is an honest permutation, and the first condition implies that $1,2,\ldots,r$ all appear in the first $n$ entries. ∎

Corollary 5.6**.**

$\{{\mathfrak{S}}_{\pi}({\mathbf{x}}_{n}^{*}):\pi\in\Pi_{n,k,r}\}$* descends to a basis for $R_{n,k,r}$.*

It would be interesting to explore if this Schubert basis maintains many of the properties of the Schubert basis for the usual ring of coinvariants. For example, the following suggests that the structure constants of this Schubert basis are positive modulo $R_{n,k,r}$.

Question 5.7**.**

For two permutations $\pi,\pi^{\prime}\in\Pi_{n,k,r}$, is it always true that the product

[TABLE]

has positive integer coefficients when expanded in the basis $\{{\mathfrak{S}}_{\pi}({\mathbf{x}}_{n}^{*}):\pi\in\Pi_{n,k,r}\}$ modulo $I_{n,k,r}$? Using Sage, we have checked that this is true for $1\leq n,k\leq 4$ and $0\leq r\leq n$. If so, do these coefficients count intersections in some family of varieties?

6. Acknowledgements

B. Rhoades was partially supported by the NSF Grant DMS-1500838. A. T. Wilson was partially supported by a NSF Graduate Sciences Postdoctoral Research Fellowship.