Tail positive words and generalized coinvariant algebras

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.


Introduction
Consider the action of the symmetric group S n on n letters on the polynomial ring Q[x n ] := Q[x 1 , . . . , x n ] given by variable permutation. The polynomials belonging to the invariant subring 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 i 1 1 · · · x in n : 0 ≤ i j < j} descends to a vector space basis for R n , so that the Hilbert series of R n is given by (1.4) Hilb(R n ; q) = (1 + q)(1 + q + q 2 ) · · · (1 + q + · · · + q n−1 ) = [n]! q , the standard q-analog of n!. Chevalley [4] proved that as an ungraded S n -module, we have R n ∼ = 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 Lusztig (unpublished) and Stanley [16] refined this result to describe the graded isomorphism type of R n in terms of the major index statistic on standard Young tableaux.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 Q[x n ] is given by h d (x n ) := 1≤i 1 ≤···≤i d ≤n x i 1 · · · x i d . Definition 1.1. Let n, k, and r be nonnegative integers with r ≤ n. Let I n,k,r ⊆ Q[x n ] be the ideal I n,k,r := h k+1 (x n ), h k+2 (x n ), . . . , h k+n (x n ), e n (x n ), e n−1 (x n ), . . . , e n−r+1 (x n ) and let R n,k,r := Q[x n ]/I n,k,r 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 ⊆ I n , so that R n,k,r projects onto the classical coinvariant algebra R n . If r ≥ 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 π 1 . . . π n of the set {1, 2, . . . , n}, algebraic properties of R n,k,r will be controlled by the combinatorics of permutations π 1 . . . π n+k of the multiset {0 k , 1, 2, . . . , n} whose last r entries π n+k−r+1 . . . π n+k−1 π 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, . . . , n}. For example, we have By considering the possible locations of the k 0's in an element of S n,k,r , it is immediate that (1.5) |S n,k,r | = n + k − r k · |S n | = n + k − r k · n!.
The basic enumeration of Equation 1.5 will manifest in Hilbert series as (1.6) Hilb(R n,k,r ; q) = n + k − r k q · Hilb(R n ; q) = n + k − r k q · [n]! q , where m i q := [m]!q [i]!q[m−i]!q is the usual q-binomial coefficient. Going even further, we have the following graded Frobenius image (1.7) grFrob(R n,k,r ; q) = n + k − r k q · grFrob(R n ; q) = n + k − r k q · T ∈SYT(n) s shape(T ) , which implies that the quotient R n,k,r consists of n+k−r 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 (x n ), h k+2 (x n ), . . . , h k+n (x n ) and the elementary symmetric functions e n (x n ), e n−1 (x n ), . . . , e n−r+1 (x n ). Several mixed ideals have recently been introduced to give combinatorial generalizations of the coinvariant algebra.
• Let k ≤ n. Haglund, Rhoades, and Shimozono [10] studied the quotient of Q[x n ] by the ideal (1.8) x k 1 , x k 2 , . . . , x k n , e n (x n ), e n−1 (x n ), . . . , e n−k+1 (x n ) . The generators are high degree S n -invariants e n (x n ), . . . , e n−k+1 (x n ) together with a homogeneous system of parameters x k 1 , . . . , x k n 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, . . . , n}.
• Let r ≥ 2 and let Z r ≀ S n be the group of n × n monomial matrices whose nonzero entries are r th complex roots of unity (this is the group of 'r-colored permutations' of {1, 2, . . . , n}). Let k ≤ n be non-negative integers. Chan and Rhoades [3] studied the quotient of C[x n ] by the ideal for any polynomial f . The generators here are high degree Z r ≀ S n -invariants e n (x r n ), . . . , e n−k+1 (x r n ) together with a h.s.o.p. x kr+1 1 , . . . , x kr+1 n of degree kr + 1 carrying the dual of the defining representation of Z r ≀ S n . Algebraic properties of the corresponding quotient are controlled by k-dimensional faces in the Coxeter complex attached to Z r ≀ S n .
• Let F be any field and let H n (0) be the 0-Hecke algebra over F of rank n; the algebra H n (0) acts on the polynomial ring F[x n ] by isobaric divided difference operators. Let k ≤ n be positive integers. Huang and Rhoades [13] studied the quotient of F[x n ] by the ideal Once again, the generators consist of high degree H n (0)-invariants e n (x n ), . . . , e n−k+1 (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, . . . , 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 Q[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.

Background
2.1. Words, partitions, and tableaux. Let w = w 1 . . . w n be a word in the alphabet of nonnegative integers. An index 1 ≤ i ≤ n − 1 is a descent of w if w i > w i+1 . The descent set of w is Des(w) := {1 ≤ i ≤ n − 1 : w i > w i+1 } and the major index of w is maj(w) := i∈Des(w) i. A pair of indices 1 ≤ i < j ≤ n is called an inversion of w if w i > w j ; the inversion number inv(w) counts the inversions of w. The word w is called r-tail positive if its last r letters w n−r+1 . . . w n−1 w n are positive.
Let n ∈ Z ≥0 . A partition λ of n is a weakly decreasing sequence λ = (λ 1 ≥ · · · ≥ λ k ) of positive integers with λ 1 + · · · + λ k = n. We write λ ⊢ n or |λ| = n to indicate that λ is a partition of n. The Ferrers diagram of λ (in English notation) consists of λ i left-justified boxes in row i. The Ferrers diagram of (4, 2, 2) ⊢ 8 is shown below on the left. 1 1 2 5  2 3  3 5   1 2 5 8  3 4  6 7 Let λ ⊢ n. A tableau T of shape λ is a filling of the Ferrers diagram of λ 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, . . . , n. The tableau in the center above is semistandard and the tableau on the right above is standard. We let shape(T ) = λ denote the shape of T and let SYT(n) denote the collection of all standard Young tableaux with n boxes.
Given a standard tableau T ∈ SYT(n), an index 1 ≤ i ≤ n − 1 is a descent of T if i + 1 appears in a lower row of T than i. Let Des(T ) denote the set of descents of T and let maj(T ) := i∈Des(T ) i be the major index of T . If T is the standard tableau above we have Des(T ) = {2, 5} so that maj(T ) = 2 + 5 = 7.
2.2. Symmetric functions. Let Λ denote the ring of symmetric functions in an infinite variable set x = (x 1 , x 2 , . . . ) over the ground field Q(q, t). The algebra Λ is graded by degree: Λ = n≥0 Λ n . The graded piece Λ n has dimension equal to the number of partitions λ ⊢ n.
The vector space Λ n has many interesting bases, all indexed by partitions of n. Given λ ⊢ n, let m λ , e λ , h λ , p λ , s λ , H λ denote the associated monomial, elementary, homogeneous, power sum, Schur, and modified Macdonald symmetric function (respectively). As λ ranges over the collection of partitions of n, all of these form bases for the vector space Λ n . Let f (x) ∈ Λ be a symmetric function. We define an eigenoperator ∆ f : Λ → Λ for the modified Macdonald basis of Λ as follows. Given a partition λ, we set where (i, j) ranges over all matrix coordinates of cells in the Ferrers diagram of λ. The reader familiar with plethysm will recognize this formula as For example, if λ = (3, 2) ⊢ 5, we fill the boxes of λ with monomials 1 q q 2 t qt and see that ∆ f ( H (3,2) ) = f (1, q, q 2 , t, qt) · H (3,2) .
When f = e n , the restriction of the delta operator ∆ en to the space Λ n of homogeneous degree n symmetric functions is more commonly denoted ∇: In particular, we have ∆ en e n = ∇e n . Given a partition λ ⊢ n, let S λ 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 λ such that V ∼ = Sn λ⊢n c λ S λ . The Frobenius character of V is the symmetric function If V = d≥0 V d is a graded vector space, the Hilbert series of V is the power series Similarly, if V = d≥0 V d is a graded S n -module, the graded Frobenius character of V is
Let S ⊆ [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 The space QSym of quasisymmetric functions is the Q(q, t)-algebra of formal power series with basis given by {F S,n : n ≥ 0, S ⊆ [n − 1]}. If a subset S ⊆ [n − 1] corresponds to a composition α, we set F α := F S,n .
For any composition α |= n, define a symbol s α (the noncommutative ribbon Schur function), formally defined to have homogeneous degree n. Let NSym n be the 2 n−1 -dimensional Q(q, t)-vector space with basis {s α : α |= n} and let NSym be the graded vector space NSym := n≥0 NSym n . The space NSym is the space of noncommutative symmetric functions. Although there is more structure on NSym (and on 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 F be an arbitrary field. The 0-Hecke algebra H n (0) of rank n over F is the unital associative F-algebra with generators T 1 , . . . , T n−1 and relations For all 1 ≤ i ≤ n − 1, let s i := (i, i + 1) ∈ S n be the corresponding adjacent transposition. Given a permutation π ∈ S n , we define T π := T i 1 · · · T i k ∈ H n (0), where π = s i 1 · · · s i k is a reduced (i.e., short as possible) expression for π as a product of adjacent transpositions. It can be shown that {T π : π ∈ S n } is a F-basis for H n (0), so that H n (0) has dimension n! as a F-vector space and may be viewed as a deformation of the group algebra F[S n ]. The algebra H n (0) is not semisimple, even when the field 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 α |= n (see [12,13]). For α |= n, we let P α denote the corresponding indecomposable projective and let 0)) is free with basis given by (isomorphism classes of) the irreducibles {C α : α |= n}. The quasisymmetric characteristic map Ch is defined on G 0 (H n (0)) by 0)) is free with basis given by (isomorphism classes of) the projective indecomposable {P α : α |= n}. The noncommutative characteristic map ch is defined on K 0 (H n (0)) by This extends to give a noncommutative symmetric function ch(P ) for any projective H n (0)-module P . The module P is determined by ch(P ) up to isomorphism.
There are graded refinements of the maps Ch and ch.
The quasisymmetric characteristic Ch admits a bigraded refinement as follows. The 0-Hecke algebra H n (0) has a length filtration The length-graded quasisymmetric characteristic is given by More generally, if V is a direct sum of graded cyclic H n (0)-modules, we define Ch q,t (V ) by applying Ch q,t to its cyclic summands. This may depend on the cyclic decomposition of the module V . 1 1 Our conventions for q and t in the definitions of chq and Chq,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 Sn-modules.

Gröbner theory.
A total order < on the monomials in the polynomial ring Q[x n ] is called a term order if • we have 1 ≤ m for all monomials m, and The term order used in this paper is the lexicographic term order given by If < is any term order any f ∈ Q[x n ] is any nonzero polynomial, let in < (f ) be the leading (i.e., greatest) term of f with respect to the order <. If I ⊆ Q[x n ] is any ideal, the associated initial ideal is A finite collection G = {g 1 , . . . , g r } of nonzero polynomials in an ideal I ⊆ Q[x n ] is called a Gröbner basis of I if we have the equality of monomial ideals (2.17) in If G is a Gröbner basis of I it follows that After fixing a term order, every ideal I ⊆ Q[x n ] has a unique reduced Gröbner basis.
Let I ⊆ Q[x n ] be an ideal and let G be a Gröbner basis for I. The set of monomials in Q[x n ] descends to a vector space basis for the quotient Q[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 Q[x n ]/I is given by where the sum is over all monomials in the standard monomial basis.

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 ⊆ Q n be any finite set of points and consider the ideal We may identify the quotient If Y is stable under the coordinate permutation action of S n , we have the further identification of S n -modules The ideal 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 ∈ I(X), By construction the ideal T(Y ) is homogeneous, so that the quotient Q[x n ]/T(Y ) is graded. Furthermore, we still have the dimension equality and the S n -module isomorphism The symmetric group S n acts on S n,k,r by permuting the positive letters 1, 2, . . . , n. We aim to prove that R n,k,r ∼ = Q[S n,k,r ] as ungraded S n -modules. To do this, our strategy is as follows.
(1) Find a point set Y n,k,r ⊆ 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) Prove that I n,k,r ⊆ T(Y n,k,r ) by showing that the generators of I n,k,r arise as top degree components of polynomials in I(Y n,k,r ).
and use the relation I n,k,r ⊆ T(Y n,k,r ) to conclude that I n,k,r = T(Y n,k,r ). The point set Y n,k,r which accomplishes Step 1 is the following.
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 α i in position j of (y 1 , . . . , 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 (α 7 , α 2 , α 4 , α 6 ) ↔ 0203041. This bijection Y n,k,r ↔ 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 ⊆ 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 I(Y n,k,r ). We begin with the elementary symmetric function generators e n−r+1 (x n ), . . . , e n−1 (x n ), e n (x n ). Consider the rational function in t given by If (x 1 , . . . , x n ) ∈ 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 ≤ m ≤ r taking the coefficient of t m on both sides gives so that e m (x n ) ∈ T(Y n,k,r ).
A similar trick shows that the homogeneous symmetric functions h k+1 (x n ), . . . , h k+n (x n ) lie in T(Y n,k,r ). Consider the rational function If (x 1 , . . . , x n ) ∈ 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 ≥ k + 1, taking the coefficient of t m on both sides gives Step 3 of our strategy will take more work. To begin, we identify a convenient collection of monomials in the initial ideal in < (I n,k,r ) with respect to the lexicographic term order. Given a subset S = {s 1 < · · · < s m } =⊆ [n] the corresponding skip monomial x(S) is given by In particular, if n = 8 we have x(2458) = x 2 2 x 3 4 x 3 5 x 5 8 . To prove the second assertion, the identities so that x k+1 1 , x k+2 2 , . . . , x k+n n ∈ in < (I n,k,r ).
The initial terms provided by Lemma 3.3 will be all we need. We name the monomials m ∈ Q[x n ] which are not divisible by any of these initial terms as follows. By Lemma 3.3, the monomials in 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 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.
It will develop that the map Ψ of Lemma 3.5 is actually a bijection.
• If s 1 ∈ T we would have c s 1 ≤ s 1 −1 (since s 1 could form inversions with only 1, 2, . . . , s 1 −1), contradicting the inequality c s 1 ≥ s 1 . We conclude that s 1 / ∈ T . We conclude that s i / ∈ T . Induction gives the result that S ∩ T = ∅. However, this contradicts the facts that |S| = n − r + 1, |T | = r, and that there are a total of n positive letters in π. This concludes the proof that the map Ψ : S n,k,r → M n,k,r is well defined.
The relation deg(Ψ(π)) = inv(π) is clear from construction. The fact that Ψ is an injection is equivalent to the fact that a permutation π = π 1 . . . π n+k ∈ S n,k,r is determined by its code (c 1 , . . . , c n ). This assertion is true more broadly for any permutation of the multiset {0 k , 1, 2, . . . , 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 Q[x n ] with the lexicographic term order. The standard monomial basis of R n,k,r is M n,k,r . The Hilbert series of R n,k,r is given by (3.15) Hilb(R n,k,r ; q) = n + k − r k q · [n]! q .
Proof. Let B T be the standard monomial basis of Q[x n ]/T(Y n,k,r ) and let B J be the standard monomial basis of R n,k,r = Q[x n ]/I n,k,r . We know that |S n,k,r | = |B T |. Lemma 3.2 implies that B T ⊆ B J . Lemma 3.3 further implies the containment B J ⊆ M n,k,r . Finally, Lemma 3.5 gives the relation |M n,k,r | ≤ |S n,k,r |. Putting these facts together gives (3.16) B T = B J = M n,k,r , 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 M n,k,r . By the last paragraph, the map Ψ of Lemma 3.5 is a bijection. It follows that It is well known that π∈Sn q inv(π) = [n]! q . The q-binomial coefficient in comes from the ways of inserting k copies of 0 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 . of ungraded S n -modules.
Proof. Let N n,k,r be the collection of monomials in Q[x n ] whose exponent vectors are componentwise ≤ at least one (n, k, r)-staircase. If δ n (T ) = (a 1 , . . . , a n ) is an (n, k, r)-staircase for some (n − r)element set T ⊆ [n] and m = x a 1 1 · · · x an n is the corresponding monomial, it is clear that a i < k + i for all i, so that x k+i i ∤ m. If S ⊆ [n] satisfies |S| = n − r + 1 then at least one index i ∈ S satisfies i / ∈ T , which forces x(S) ∤ m. It follows that N n,k,r ⊆ M n,k,r . On the other hand, we may construct a map (3.23) Φ : S n,k,r → N n,k,r by letting Φ(π) = (c 1 , . . . , c n ) be the code of any r-tail positive permutation π ∈ S n,k,r . The fact that π is r-tail positive implies that Φ(π) ∈ N n,k,r , so that Φ is well defined. It is clear that Φ is injective, so that (3.24) |S n,k,r | ≤ |N n,k,r | ≤ |M n,k,r | = |S n,k,r | and we have N n,k,r = M n,k,r , as desired.

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 grFrob(R n,k,r ; q). Lemma 4.1. Suppose n, k, r ≥ 0 with r < n and k > 0. There is a short exact sequence of S n -modules where the first map is homogeneous of degree n − r and the second map is homogeneous of degree 0. Equivalently, we have the equality of graded Frobenius characters (4.2) grFrob(R n,k,r ; q) = grFrob(R n,k,r+1 ; q) + q n−r · grFrob(R n,k−1,r ; q).
Proof. We have the inclusion of ideals I n,k,r ⊆ I n,k,r+1 ; we let the second map be the canonical projection π : R n,k,r ։ R n,k,r+1 . We have a homogeneous map ϕ : Q[x n ] → R n,k,r of degree n − r given by multiplication by e n−r (x n ), and then projecting onto R n,k,r . We claim that ϕ(I n,k−1,r ) = 0, so that ϕ induces a well defined map ϕ : R n,k−1,r → R n,k,r . This is equivalent to showing that h k (x n ) · e n−r (x n ) ∈ I n,k,r . The Pieri Rule implies that (4.3) h k (x n ) · e n−r (x n ) = s (k,1 n−r ) (x n ) + s (k+1,1 n−r−1 ) (x n ); 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 ) (x n ) ∈ I n,k,r , observe that, for 1 ≤ i ≤ r we have It follows that modulo I n,k,r we have the congruences where the last congruence used the fact that s (k+n−r) (x n ) = h k+n−r (x n ) ∈ I n,k,r since r < n. This chain of congruences also shows that s (k+1,1 n−r−1 ) (x n ) ∈ I n,k,r . By the last paragraph, we have a well defined induced map ϕ : R n,k−1,r → R n,k,r . It is clear that Im(ϕ) = Ker(π). Moreover, the Pascal relation implies that (4.6) |S n,k−1,r | + |S n,k,r+1 | = |S n,k,r |, so that by Theorem 3.6 we have (4.7) dim(R n,k−1,r ) + dim(R n,k,r+1 ) = dim(R n,k,r ).
Since π is a surjection, this forces the sequence to be exact. To finish the proof, observe that the maps ϕ and π 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 x w to be (4.9) x in particular, any copies of 0 in w do not affect x w .
Theorem 4.2. The graded Frobenius image of R n,k,r is given by The last sum ranges over all length n + k words w = w 1 . . . w n+k in the alphabet of nonnegative integers which contain precisely k copies of 0 and are r-tail positive.
Proof. By considering the placement of the k copies of 0 in a r-tail positive word w appearing in the final sum, we see that On the other hand, we have where the first equality uses the equidistribution of the statistics inv and 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 ≥ n or k = 0 then R n,k,r = R n and this equality is trivial. Otherwise, we have the q-Pascal relation (4.14) n + k − r k 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 Q to working over an arbitrary field F, so that the ideals I n,k,r , I n are defined inside the ring F[x n ] := F[x 1 , . . . , x n ] and we have R n,k,r := F[x n ]/I n,k,r , R n := F[x n ]/I n .
Theorem 4.3. Let C n = {b π (x n ) : π ∈ S n } be a collection of polynomials in F[x n ] indexed by permutations in S n which descends to a basis of R n . The collection of polynomials (4.15) C n,k,r := {b π (x n ) · e λ (x n ) : π ∈ S n , λ 1 ≤ n − r, and λ has ≤ k parts} in F[x n ] descends to a basis of R n,k,r .
Proof. This is trivial when k = 0 or r ≥ 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 F. 2 The proof of Lemma 4.1 then applies over F to give a short exact sequence of graded F-vector spaces where π is the canonical projection. We may inductively assume that C n,k−1,r descends to a F-basis of R n,k−1,r and that C n,k,r+1 descends to a F-basis of S n,k,r+1 . Exactness implies that descends to an F-basis of R n,k,r . Theorem 4.3 reinforces the fact that R n,k,r consists of n+k−r k copies of R n , graded by the q-binomial coefficient n+k−r k q . Interesting bases C n to which Theorem 4.3 can be applied include • the Artin basis [2] (4.18) C n = {x i 1 1 · · · x in n : 0 ≤ i j < j} (which is connected to the inv statistic on permutations in S n ) and • the Garsia-Stanton basis (or the descent monomial basis [5,7] C n = {gs π : π ∈ S n } where (4.19) gs π = π i >π i+1 x π 1 · · · x π i (which is connected to the 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 F[x n ] by letting the generator T i act by the Demazure operator σ i , where (4.20) σ Here s i (f ) is the polynomial obtained by interchanging x i and x i+1 in f (x n ). It can be shown that if f ∈ F[x n ] Sn is any symmetric polynomial and g ∈ F[x n ] is an arbitrary polynomial then Therefore, any ideal I ⊆ F[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 = F[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 ≤ n. We have an isomorphism of graded H n (0)-modules (4.22) R n,k,r ∼ = λ⊆(n−r)×k R n (−|λ|).
Here the direct sum is over all partitions λ which satisfy λ 1 ≤ n − r and have at most k parts. The module R n = F[x n ]/I n is the coinvariant algebra viewed as a graded H n (0)-module.
It follows that, for λ ⊆ (n − r) × k fixed, the collection of polynomials (4.27) C n,k,r (λ) := {e λ (x n ) ·σ π .x Des(π) : π ∈ S n } descends inside R n,k,r to a F-basis of a copy of R n with degree shifted up by |λ|.
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 ≤ n.
(1) The length-degree bigraded quasisymmetric characteristic Ch q,t (R n,k,r ) is given by where F Des(π −1 ),n is the fundamental quasisymmetric function. (2) The degree graded quasisymmetric characteristic Ch q (R n,k,r ) is in fact symmetric and given by (4.29) Ch q (R n,k,r ) = n + k − r k q · Ch q (R n ) = n + k − r k q · T ∈SYT(n) q maj(T ) s shape(T ) .
Since the characteristics Ch q,t and 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 F[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 F[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 (4.31) {gs π · x i 1 π 1 · · · x i n−r π n−r : π ∈ S n and k ≥ i 1 ≥ · · · ≥ i n−r ≥ 0} 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 inv and maj do not share the same distribution on S n,k,r .

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 ′ n,k,r ⊆ Q[x n ] given by (5.1) I ′ n,k,r := p k+1 (x n ), p k+2 (x n ), . . . , p k+n (x n ), e n (x n ), e n−1 (x n ), . . . , e n−r+1 (x n ) and let R ′ n,k,r := Q[x n ]/I n,k,r be the corresponding quotient. The ideal I ′ 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 ′ n,k,r has the structure of a graded S n -module. Although the ideals I n,k,r and I ′ 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 ∼ = R ′ n,k,r .
The main reason for preferring the quotient rings R ′ n,k,r over the quotient rings R n,k,r is that they generalize more readily to two sets of variables. Let x n = (x 1 , . . . , x n ) and y n = (y 1 , . . . , y n ) be two sets of n variables and let Q[x n , y n ] be the polynomial ring in these variables. The symmetric group S n acts on Q[x n , y n ] by the diagonal action π.x i = x π i , π.y i = y π i .
For any a, b ≥ 0, let p a,b (x n , y n ) be the polarized power sum Moreover, let M n be the set of the 2 n monomials z 1 . . . z n in Q[x n , y n ] where z i ∈ {x i , y i } for all 1 ≤ i ≤ n. For example, we have For a nonnegative integer k, let DI n,k ⊆ Q[x n , y n ] be the ideal generated by the polarized power sums p a,b (x n , y n ) with a + b ≥ k + 1 together with the monomials in M n . Let DR n,k := Q[x n , 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 grFrob(DR n,k ; q, t) = ∆ h k en e n = ∆ s k+1,1 n−1 e n = ∆ h k ∇e n .
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 grFrob(DR n ) = ∆ en e n . Setting the y n variables equal to zero in the quotient DR n,k yields the ring R ′ n,k,1 , so that the ring R ′ n,k,1 conjecturally gives the analog of the coinvariant ring (for one set of variables) attached to the operator ∆ h k en .
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 grFrob(R n,k,1 ; t) = ∆ h k en e n | q=0 = ∆ s k−1,1 n−1 e n | q=0 = ∆ h k ∇e n | q=0 .
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 rev t be the operator which reverses the coefficient sequences of polynomials with respect to the variable t. For a partition λ ⊢ n, let Q ′ λ = Q ′ λ (x; t) be the corresponding Hall-Littlewood symmetric function. It is well known that the modified Macdonald polynomial H λ = H λ (x; q, t) satisfies . This means that, for any symmetric function f and any partition λ ⊢ n, we have , where ℓ(λ) is the number of parts of λ.
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 where the sum is over all cells c with matrix coordinates (i, j) in the Ferrers diagram of λ, ) is a cell in λ other than the corner (0, 0), • w λ = c∈λ (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 λ and a(c), l(c) denote the arm and leg lengths of λ at c. We apply the operator ∆ h k en = ∆ h k ∆ en to both sides of Equation .

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 (5.10) ∂f, g = f, p 1 g 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 α = (α 1 , . . . , α n ) of n positive integers, an α-Tesler matrix U = (u i,j ) 1≤i,j≤n is an n × n upper triangular matrix with nonnegative integer entries such that, for i = 1 to n, This corollary follows from work in [1,9,14].

A Schubert basis.
There is also a basis for R n,k,r given by certain Schubert polynomials. We let Π n,k,r be all the permutations π of {1, 2, . . . , n + k} that satisfy • all descents in π occur weakly left of position n, and • 1, 2, . . . , r all appear in π 1 π 2 . . . π n . If π ∈ Π n,k,r is a permutation, let S π (x n ) be the Schubert polynomial attached to π. Note that, since each π has no descents after position n, there are at most n variables that appear in the Schubert polynomial associated to π, so we have not truncated the variable set in any meaningful way. We will show that {S π (x * n ) : π ∈ Π 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.
Proof. We will construct a bijection Φ : Π n,k,r → M n,k,r that satisfies Φ(π) = in < (S π (x * n )). The bijection itself is (5.17) where d i counts the number of j > i such that π i > π j . The fact that Φ(π) = in < (S π (x * n )) follows directly from the definition of the Schubert polynomial. We need to show that m = Φ(π) ∈ M n,k,r and to construct its inverse. Our proof will be similar to that of Lemma 3.5. First, we check that if S = {s 1 < s 2 < . . . < s n−r+1 }. Since S ⊆ [n] and the entries 1 through r all appear in π 1 through π n , there is some s i such that π s i ≤ r. Choose i as large as possible such that π s i ≤ r. Since j > n implies π j > r, π s i can only be greater than at most n − s i entries to its right, i.e. d s i ≤ n − s i . Hence the power of x n−s i +1 in m is at most n − s i , which means x(S) ∤ m. The second condition follows from the definition of m.
Given a monomial m ∈ M n,k,r , we would like to construct Φ −1 (m). This can be done using the usual bijection from codes (d 1 , d 2 , . . . , d n ) to permutations. For i = 1 to n, we choose π 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 π. 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, . . . , r all appear in the first n entries.
Corollary 5.6. {S π (x * n ) : π ∈ Π 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 π, π ′ ∈ Π n,k,r , is it always true that the product S π (x * n ) × S π ′ (x * n ) (5.18) has positive integer coefficients when expanded in the basis {S π (x * n ) : π ∈ Π n,k,r } modulo I n,k,r ? Using Sage, we have checked that this is true for 1 ≤ n, k ≤ 4 and 0 ≤ r ≤ n. If so, do these coefficients count intersections in some family of varieties?