Harmonic differential forms for pseudo-reflection groups II. Bi-degree bounds

This paper studies three results that describe the structure of the super-coinvariant algebra of pseudo-reflection groups over a field of characteristic $0$. Our most general result determines the top component in total degree, which we prove for all Shephard--Todd groups $G(m, p, n)$ with $m \neq p$ or $m=1$. Our strongest result gives tight bi-degree bounds and is proven for all $G(m, 1, n)$, which includes the Weyl groups of types $A$ and $B$/$C$. For symmetric groups (i.e. type $A$), this provides new evidence for a recent conjecture of Zabrocki related to the Delta Conjecture of Haglund--Remmel--Wilson. Finally, we examine analogues of a classic theorem of Steinberg and the Operator Theorem of Haiman. Our arguments build on the type-independent classification of semi-invariant harmonic differential forms carried out in the first part of this series. In this paper we use concrete constructions including Gr\"{o}bner and Artin bases for the classical coinvariant algebras of the pseudo-reflection groups $G(m, p, n)$, which we describe in detail. We also prove that exterior differentiation is exact on the super-coinvariant algebra of a general pseudo-reflection group. Finally, we discuss related conjectures and enumerative consequences.

Here S n is the symmetric group on n elements.A classic result of Steinberg [30] describes the harmonic polynomials as precisely those of the form ∂ g ∆ n where ∆ n := 1≤i<j≤n (x j − x i ) is the Vandermonde determinant and ∂ g is a polynomial in the partial derivatives ∂ ∂x i .In particular, the top-degree harmonic polynomial is ∆ n , which transforms by the sgn representation of S n .
Steinberg's result extends uniformly to an arbitrary pseudo-reflection group G.We more generally consider the problem of determining the harmonic differential forms of a pseudo-reflection group G.We have been motivated by Steinberg's result, Haiman's Operator Theorem [18,19], and a recent series of conjectures of Zabrocki [33] and Haglund-Remmel-Wilson [16] related to higher coinvariant algebras described in more detail below.
The first part of this series [31] provides a starting point for such a description by giving a complete, type-independent construction of the det-isotypic harmonic differential forms.In analogy with Steinberg's result, one may expect the det-isotypic forms to be the "top" harmonics in some precise sense.Our main results are as follows: (1) The top total-degree forms are det-isotypic for "almost all" irreducible pseudo-reflection groups G = G(m, p, n), namely those with p = m or m = 1 (Theorem 1.15).(2) The top bi-degree forms are det-isotypic for the groups G(m, 1, n) (Theorem 1.12), which includes S n and the Coxeter groups of type B. (3) All harmonic forms are obtained by applying partial derivatives to det-isotypic forms when the rank is ≤ 2 and either G = G(m, 1, n) or G is real (Theorem 1.6) and also for multiples of the volume form when G = G(m, 1, n) (Theorem 1.7).(4) The t = 0, z = −q specialization of the Hilbert series of the super-diagonal harmonics of the symmetric group agrees with Zabrocki's conjecture (see Section 1.10).
We also show that Theorem 1.6, Theorem 1.7, and Theorem 1.12 fail for certain G; see Remark 1.8 and Remark 1.13.We furthermore provide conjectures describing when they hold more broadly; see Conjecture 1.9 and Conjecture 1. 16.
Our arguments rely on the following tools which may be of independent interest.
(5) The exterior derivative cochain complex on super-harmonic differential forms (or super coinvariants) is exact for all pseudoreflection groups G (Theorem 1.10).( 6) The Artin and Gröbner bases for the coinvariant ideal for G(m, p, n) (Section 5.1, Theorem 5.5, Theorem 5.6).In the following subsections, we summarize classical properties of coinvariant algebras and harmonics for pseudo-reflection groups, introduce super analogues of these constructions, and state our main results and conjectures.See Section 2 and [31, §2] for additional detailed background.
1.2.Coinvariant algebras and harmonics in type A. Let S n be the symmetric group on n elements.The classical coinvariant algebra of S n is the quotient where I n := Q[x 1 , . . ., x n ] Sn + is the coinvariant ideal generated by all homogeneous symmetric polynomials of positive degree.A great deal is known about the structure of R n as a graded S n -module [28].The top-degree component of R n is the full subspace of elements which transform by σ • x = sgn(σ)x and is spanned by the image of the Vandermonde determinant The harmonics of I n are Here ∂ g is the partial differential operator defined by extending x i → ∂ x i = ∂ ∂x i multiplicatively and Q-linearly.The harmonics H n are the orthogonal complement of I n under the natural positive-definite Hermitian form (1) (f, g) := degree zero component of ∂ g f, so the natural projection H n → R n is a graded S n -module isomorphism.
1.3.Coinvariant algebras and harmonics of pseudo-reflection groups.More generally, suppose K ⊂ C is a subfield closed under complex conjugation, V = K n , and G ⊂ GL(V ) is a pseudo-reflection group.That is, G is a finite group of unitary matrices generated by pseudo-reflections, which are non-identity transformations of finite order which fix a hyperplane pointwise.
The pseudo-reflection groups were famously classified by Shephard-Todd [26] into an infinite family G(m, p, n) where m, p, n ∈ Z ≥1 and p | m together with 34 exceptional groups.The group G(1, 1, n) = S n consists of n × n permutation matrices.The group G(m, 1, n) consists of n × n pseudo-permutation matrices, namely matrices such that each row and column has one non-zero entry which is an mth complex root of unity.In particular, G(2, 1, n) = B n is the Weyl group of signed permutations.Finally, the group G(m, p, n) is the index-p normal subgroup of G(m, 1, n) consisting of pseudo-permutation matrices where the product of the non-zero elements taken to the m p th power is 1.In particular, G(2, 2, n) = D n is the Weyl group of signed permutations with an even number of signs.
The constructions in Section 1.2 generalize from S n to G for any pseudo-reflection group G (see Section 2).The coinvariant algebra of G and the G-harmonics are given as follows: R G := K[x 1 , . . ., x n ]/I G where where ∂ g is defined by extending x → ∂ x i = ∂ ∂x i multiplicatively and conjugate-linearly in K.For G = G(m, p, n), the G-action is given by taking x i to be the ith coordinate function on K n .As before, H G → R G is an isomorphism of graded G-modules.
Chevalley [6] showed that R G , and hence H G , carries the regular representation.Furthermore, the top-degree component of the harmonics H G is spanned by an element which is unique up to non-zero scalar multiples and which transforms according to g • ∆ G = det(g)∆ G .We call ∆ G the Vandermondian of G, since ∆ Sn = ∆ n .
It is clear from the definition that if f is harmonic, then so is ∂ g f .Steinberg showed that every G-harmonic can be obtained "from the top down" starting with ∆ G as follows.
Theorem 1.1 (Steinberg,[30,Thm. 1.3(c)]).For any pseudo-reflection group G, 1.4.Haiman's Operator Theorem.A famous extension of the classical coinvariant algebra for G = S n was introduced by Garsia and Haiman [13].It involves two sets of commuting variables x 1 , . . ., x n and y 1 , . . ., y n with diagonal S n -action given by σ(x i ) := x σ(i) , σ(y i ) := y σ(i) .The diagonal coinvariants and diagonal harmonics are S n -modules bigraded by x-and y-degree and are defined by where As usual, DH n ∼ = DR n as bi-graded S n -modules.Haiman conjectured and later proved the following description of the diagonal harmonics.Let 18,Thm. 4.2]).We have Haiman's proof of this theorem, which was originally conjectured in [19,Conj. 5.1.1],involves the deep use of algebraic geometry.An elementary proof of the y-degree 1 component of Theorem 1.2 was given by Alfano [1].
A recent conjecture of Zabrocki [33] introduced the super-diagonal coinvariant algebra as a potential representation-theoretic model for the Delta Conjecture of Haglund-Remmel-Wilson [16].The t = 0 case of Zabrocki's conjecture sparked significant interest in the following extension of the classical coinvariant algebras, which is our main object of interest.See [31, §1.2] for an overview of this and related work and further references.This paper studies the extension of the polynomial ring K[x 1 , . . ., x n ] by adjoining anti-commuting variables θ 1 , . . ., θ n where θ i θ j = −θ j θ i and x i θ j = θ j x i .Here θ i is conceptually the differential 1-form dx i , so the products We use the θ variables for consistency with existing literature.Since θ 2 i = 0, we often take be the ring of differential forms with polynomial coefficients.The G-action on the θ i is the same as on the x i and is extended multiplicatively and K-linearly to K[x 1 , . . ., x n , θ 1 , . . ., θ n ], which is a G-module bi-graded by x-degree and θ-degree.Abstractly, this is the ring S(V * ) ⊗ ∧V * ; see [31, §2] for details.
Definition 1.3.The super coinvariant algebra of a pseudo-reflection group G is the quotient + is the super coinvariant ideal generated by bi-homogeneous G-invariant differential forms of positive total degree.We write SR k G for the component of SR G consisting of images of k-forms, i.e. the component of θ-degree k.Note that SR 0 G = R G , so the super coinvariant algebra of G contains the classical coinvariant algebra of G.
To define the harmonics in this context requires the extension of the partial differential operators ∂ g to differential forms.Let ∂ θ i be the usual interior product defined by Here ∂ x i is θ-linear and ∂ θ i is x-linear.Note that ∂ x i and ∂ θ j satisfy the same (anti-)commutation relations as the x i and θ j .For ω ∈ K[x 1 , . . ., x n , θ 1 , . . ., θ n ], let ∂ ω be obtained by replacing each x i with ∂ x i , replacing each θ j with ∂ θ j , reversing the order of the θ's, and taking the conjugate of the coefficients.These twists ensure the extension of (1) remains positive-definite.
Definition 1.4.The super harmonics of a pseudo-reflection group G are The natural projection SH G → SR G is again an isomorphism of bigraded G-modules, so SR G and SH G are frequently interchangeable.
1.6.The det-isotypic harmonic differential forms.From Steinberg's Theorem 1.1, the det-isotypic component of the harmonics, , plays a special role.In [31], the authors gave a "top-down" construction of SH det G in the spirit of Steinberg's Theorem 1.1 and Haiman's Theorem 1.2 using certain differential operators d 1 , . . ., d r ∈ End K (SH G ).
Here r := dim(V /V G ).We call these operators generalized exterior derivatives since in general we may take to be the exterior derivative, where θ i here denotes left multiplication by θ i .If e * 1 , . . ., e * r are the positive co-exponents of G, then d i lowers x-degree by e * i and raises θ-degree by 1. See Section 2.3 for details and The main result of [31] gives a basis of 2 r elements for the det-isotypic component of the super harmonics.
Theorem 1.6.Let G ⊂ GL(V ) be a pseudo-reflection group with rank Then the θ-degree r component of (5) holds.
Remark 1.8.In Lemma 3.8, we show that in fact the SH r G component of (5) Hence (5) cannot possibly hold for any groups in the infinite family G(m, p, n) beyond these.However, computational data shows that (5) fails for G = D 4 and D 5 .It appears likely to fail for D n with n ≥ 4. On the other hand, we have verified that (5) holds for S n with n ≤ 6, B n with n ≤ 4, G(3, 1, 4), and G(5, 1, 3), among others.See Table 2 for additional data.We also note that the SH 1 Sn case of ( 5) is equivalent to the y-degree 1 case of Haiman's Theorem 1.2.Consequently, we are lead to the following conjecture.Conjecture 1.9 (Differential Operator Conjecture).If G = G(m, 1, n), then (5) holds.
Theorem 1.6 includes the dihedral groups G(m, m, 2) = Dih 2m (m ≥ 1) and 6 exceptional groups.We do not have a complete determination or conjecture for the exceptional groups for which (5) holds.It does hold for H 3 , though perhaps surprisingly it fails for F 4 .
Our proof of Theorem 1.6 is mostly uniform and relies on the following result which may be of independent interest.See Section 1.10 for additional consequences.
Theorem 1.10.For any pseudo-reflection group G ⊂ GL(V ) with r = dim(V /V G ), the exterior derivative cochain complex The complex in Theorem 1.10 is a finite-dimensional, algebraic analogue of the de Rham complex of a smooth manifold.Exactness is proved with an analogue of Hodge theory using total Laplacians.See Section 4 for the proofs of Theorem 1.6 and Theorem 1.10.
1.8.Bi-degree bound results.When (5) holds, the harmonics are the "top-most" elements of SH G .In particular, it implies that for each k, the top x-degree elements of SH G belong to SH det G and fully describes the non-zero bi-degree components of SH G as follows.Write Our strongest result is to show that the following consequence of Conjecture 1.9 is true unconditionally.In particular, it verifies the predicted bi-degree support of Zabrocki's super coinvariant algebra conjecture when t = 0, providing additional evidence for that conjecture.
In contrast to Conjecture 1.9, our data do not rule out the possibility that (7) holds for D n .
Our proof of Theorem 1.12 uses Gröbner and Artin bases of R G(m,1,n) developed in Section 5. See Section 6. 1.9.Total degree bound results.The bi-degree support and top components from Lemma 1.11 imply the following total degree support and top components of SH G , or equivalently SR G .
Lemma 1.14.Let G be a pseudo-reflection group.Suppose Lemma 1.11 holds.Then We show that this weaker description is true even in many cases where Lemma 1.11 fails to hold.Our most general result is the following.
Our proof of Theorem 1.15 again uses Gröbner and Artin bases of R G(m,1,n) developed in Section 5. See Section 7.
Our results and computations have uncovered no cases in which Theorem 1.15 fails to hold.Indeed, our argument shows that it must hold for all G = G(m, p, n) except possibly when m = p > 1 and k ∈ {n − 1, n − 2}.Consequently, we conjecture the following.Conjecture 1.16 (Total Degree Bounds Conjecture).Let G be a pseudoreflection group.Then 1.10.Hilbert series considerations.We finish this introduction with some additional enumerative consequences of the preceding results which support some further conjectures.The Hilbert series of SR k G and SR G are Hilb(SR k G ; q) := where q tracks x-degree and z tracks θ-degree.An immediate consequence of Theorem 1.10 is the following enumerative corollary.
Corollary 1.17.For any pseudo-reflection group G, Our overarching goal has been to provide evidence for Zabrocki's conjecture [33] for the tri-graded Frobenius series of the type A superdiagonal coinvariant algebra.In particular, we may check Zabrocki's conjecture against Corollary 1.17.
Based on computational evidence including Table 2, the first named author has introduced a type B analogue of Conjecture 1.18.Let Stir B q (n, k) be a type B q-Stirling number of the second kind, defined recursively by Stir The type B analogue of Conjecture 1.18 is as follows.
Conjecture 1.19.For 0 ≤ k ≤ n, One may again check that n k=0 19 is also consistent with Corollary 1.17.An investigation of the combinatorial properties of these and other q-analogues is in progress [25].
One may obtain a different complex from ( 6) by replacing the differentials d with d i for 1 ≤ i ≤ r, though the result is typically not exact.The graded Euler characteristic of the complex is A potential approach to Conjecture 1.18 and Conjecture 1.19 is to find homotopic complexes with the correct Euler characteristic.More concretely, Conjecture 1.18 is equivalent to the following variation on Corollary 1.17.
1.11.Paper organization.Section 2 gives background on polynomial differential forms, generalized exterior derivatives, and the invariant theory of pseudo-reflection groups.Section 3 analyzes the structure of the top θ-degree component of SH G and proves Theorem 1.7.Section 4 set up our algebraic Hodge theory argument proving exactness of exterior differentiation on the harmonics, Theorem 1.10, as well as Theorem 1.6.In Section 5 we describe the Artin and Gröbner bases for G(m, p, n); see Theorem 5.5 and Theorem 5.6.Section 6 proves the bi-degree bounds in Theorem 1.12.Section 7 considers the top total degree and proves Theorem 1.15.

Background
2.1.Polynomial differentials.We first briefly introduce the standard G-module structures and differential operators underlying our results.See [31, §2] for a general, abstract version.The following concrete version is included in the spirit of much of the combinatorics literature and is intended to make these developments more accessible.
Let K ⊂ C be a subfield closed under complex conjugation.Let x n ] consists of the polynomial functions f : V → K.The group G acts naturally on polynomial functions via the contragredient action (σ To simplify the exposition, we transfer these derivatives to V * by defining operators ∂x i is the usual partial derivative.More generally, we extend these partial derivatives to polynomials in K[x 1 , . . ., x n ] multiplicatively and conjugate-linearly.
We use two fundamental properties of We may define a G-invariant positive-definite Hermitian form on This form is linear in f and conjugate-linear in g.Under this form, the coinvariant ideals and harmonics from Section 1.2 are orthogonal complements, 2.2.Differential forms.In place of polynomial functions f (v) on V , we may consider alternating multilinear k-forms η : We again have the natural contragredient action (σ Following a standard convention in this area of algebraic combinatorics, we abbreviate θ i := dx i and typically suppress the ∧ symbol.The alternating multilinear forms on V under the wedge product together form the K-algebra K[θ 1 , . . ., θ n ] generated by θ 1 , . . ., θ n subject to the relations θ i θ j = −θ j θ i , and in particular θ 2 i = 0.The G-action is given by σ The analogue of the directional derivative ∂ v for alternating forms is given by the interior product.
where θ i ℓ means θ i ℓ is omitted.
Definition 2.2.Let K[x 1 , . . ., x n , θ 1 , . . ., θ n ] be the ring of differential forms on V with coefficients which are polynomial functions on V .This is the K-algebra generated by x 1 , . . ., x n , θ 1 , . . ., θ n subject to the relations x i x j = x j x i , θ i θ j = −θ j θ i , and As before, we extend the interior product operators multiplicatively and conjugate-linearly to ).Moreover, one may check that if ω is bi-homogeneous and non-zero, then ∂ ω ω = α,I |c α,I | 2 α! > 0. Hence we may again define a G-invariant positive-definite Hermitian form on This form is linear in f and conjugate-linear in ω.
Under this form, the super coinvariant ideals and super harmonics from Section 1.5 are orthogonal complements,  3) is consistent with our usage of dx i in Section 2.2.We now describe the operators d i generalizing (3) and underlying our construction of SH det G in [31].We have σ The Gequivariance of the exterior derivative conceptually arises from the Ginvariance of the form More generally, every G-invariant element of K[x 1 , . . ., x n , θ 1 , . . ., θ n ] gives rise to a corresponding G-equivariant operator in obtained by replacing x i with ∂ x i .We now summarize the structure of these operators and define the operators d i used in Section 1.6.
When G is a pseudo-reflection group, Shephard-Todd [26] and later Chevalley [6] The basic invariants are not unique, though the multiset {d 1 , . . ., d n } of their degrees is uniquely determined and is called the multiset of degrees of G.The exponents of G are the multiset {d i − 1, . . ., d n − 1}, which is the multiset of the x-degrees of df i .
Solomon [27] described the G-invariants K[x 1 , . . ., x n , θ 1 , . . ., θ n ] G .He showed that they have a K-basis given by Orlik-Solomon [22] generalized Solomon's exterior algebra construction of the invariants to certain Galois conjugates.Translated to the present language of differential operators, we have the following.
Theorem 2.4 ( [22]; see [31, §3.1]).There are bi-homogeneous, Gequivariant operators d 1 , . . ., d n which raise θ-degree by 1 such that a K-basis for the G-equivariant operators in See Table 1 for an explicit description of the generalized exterior derivatives for G(m, p, n).For example, when G = S n , one may use where 0 ≤ i ≤ n − 1; see Section 2.6 for further details.When K ⊂ R, so G is a (real) reflection group, then by (12) we may take Since the generalized exterior derivatives raise θ-degree by 1, they satisfy d i d j = −d j d i , so in particular d 2 i = 0.The d i are not unique, though the multiset of degrees {e * 1 , . . ., e * n } by which they lower xdegree is uniquely determined.This multiset by definition consists of the co-exponents of G.

2.4.
Removing invariant vectors.Our description of the semi-invariant differential forms in Theorem 1.5 from Section 1.6 involves r operators d 1 , . . ., d r where r = dim(V /V G ) ≤ n, rather than n operators.This arises from a slight mismatch between Theorem 1.5 and the result [31, Thm.5.7] underlying it.That result involves semi-invariant harmonic forms in S(V * ) ⊗ ∧M * , where M is a finite-dimensional G-module with M G = 0.The generalization of the basis (4) from [31] involves dim M differential operators in general.When G = S n acts on Q n by permutation matrices, V G = Span Q {(1, . . ., 1)} = 0, so the result does not directly apply.However, we may use M = V /V G in [31, Thm.5.7], which in the case of G = S n is the standard representation.
The relationship between the coinvariants and harmonics of S(

and SR ′
G for the super coinvariant ideal, super harmonics, and super coinvariant algebra of S(V * ) ⊗ ∧(V /V G ) * .Lemma 2.6.The super harmonics SH G and SH ′ G are naturally isomorphic.
2.5.Vandermondians and Jacobians.We now briefly give an explicit construction of the key element ∆ G from Section 1.2, which we call the Vandermondian of G, along with a related element ∆ * G , which we call the co-Vandermondian of G. See [31, §3] for a more complete summary and references to the literature.We continue the notation from Section 2.3, so G is a pseudo-reflection group and f 1 , . . ., f n are basic invariants of G.By Steinberg's Theorem 1.1, the top-degree component of H G ⊂ K[x 1 , . . ., x n ] is spanned by an element ∆ G ∈ H G , uniquely defined up to non-zero scalar multiples, which transforms according to g • ∆ G = det(g)∆ G .Similarly, there is an element ∆ * G ∈ H G , uniquely defined up to non-zero scalar multiples, which transforms according to g In fact, ∆ * G | ∆ G and equality holds if and only if G is generated by reflections (that is, order 2 pseudo-reflections).These facts may be read off from a formula of Gutkin [15], which expresses ∆ G and ∆ * G explicitly in terms of the reflecting hyperplanes of G as follows.
Let A(G) be the set of reflecting hyperplanes of G, i.e. the fixed spaces of pseudo-reflections of G.For each H ∈ A(G), fix some α H ∈ V * with ker α H = H.Let G H denote the subgroup of G fixing H pointwise.The Vandermondian is defined uniquely up to a non-zero scalar by The co-Vandermondian is defined uniquely up to a non-zero scalar by is the sum of the exponents of G, which is the x-degree of df 1 • • • df n .Similarly, the sum of the co-exponents is the number of reflecting hyperplanes, Given a set of homogeneous G-invariants f 1 , . . ., f n , one may verify that they are indeed basic invariants using Saito's criterion [23,Thm. 4.19] or an appropriate generalization [22,Thm. 3.1], which says that it suffices to check that the Jacobian determinant of f 1 , . . ., f n agrees with the Vandermondian of G.That is, we require Likewise, for the generalized exterior derivatives Given a proposed set of bihomogeneous G-equivariant operators, one may verify that they are indeed generalized exterior derivatives by checking that J * G agrees with the co-Vandermondian of G, det(f ij ) 1≤i,j≤n = ∆ * G , up to a non-zero constant.Note that d 6. Explicit formulas.We now give explicit descriptions for the basic invariants, generalized exterior derivatives, Vandermondians, co-Vandermondians, and co-exponents of the pseudo-reflection groups G = G(m, p, n) described in Section 1.3.See Table 1 at the end of the paper for a quick summary.The special cases of real reflection groups and G(m, m, 2) = Dih m are also written out in Table 1.
When G = G(1, 1, n) = S n is the symmetric group, we may use power-sums for the basic invariants, namely f i = n j=1 x i j for 1 ≤ i ≤ n.The reflections are the n 2 transpositions with reflecting hyperplanes x j − x i = 0, so ∆ Sn = 1≤i<j≤n (x j − x i ) is the classical Vandermonde determinant of degree n 2 .Since G is a real reflection group, the co-Vandermondian is equal to the Vandermondian, and we may use the exterior derivatives of the f i to construct the generalized exterior derivatives d i .It is most convenient to use In particular, d 1 = d is the exterior derivative and Note that d 0 acts as 0 on SH Sn , so only d 1 , . . ., d n−1 are of interest, and r = n − 1.The co-exponents are e * i = i and their sum is n 2 .When G = G(m, 1, n) is the group of pseudo-permutation matrices whose non-zero entries belong to the group µ m of mth complex roots of unity, we may use basic invariants f i = n j=1 x mi j for 1 ≤ i ≤ n.The pseudo-reflections come in two types.First, the m n 2 generalized transpositions indexed by 1 ≤ i < j ≤ n and a ∈ µ m where σ( ), which has degree m n 2 + n.In this case, the generalized exterior derivatives are where the product of the non-zero entries raised to the (m/p)th power is 1, we may use basic invariants . The pseudo-reflections come in the same types as for G(m, 1, n), except that there are n(m/p − 1) "rotations" which additionally require a ∈ µ m/p − {1}, and ), which has degree m n 2 + n.In this case, the generalized exterior derivatives are d i = n j=1 ∂ x (i−1)m+1 j θ j , and the co-exponents are e * i = (i−1)m+1.When p = m, the only pseudo-reflections are the generalized transpositions and the co-Vandermondian is 1≤i<j≤m (x m j − x m i ), which has degree m n 2 .In this case, the generalized exterior derivatives are

The structure of SH r G
We begin by considering the top θ-degree component of the key differential operator equation (5).Let G be a pseudo-reflection group with basic invariants f 1 , . . ., f n .By Lemma 2.6, we may streamline our exposition by supposing without loss of generality throughout this subsection that V G = 0. Hence r = n and the basic invariants f 1 , . . ., f n all have degree at least 2.
Recall that ∆ G , ∆ * G ∈ H G ⊂ K[x 1 , . . ., x n ] are the unique elements up to non-zero scalar multiples in H det G and H det G , respectively.Since H G is isomorphic to the regular representation of G, there is also a non-zero element unique up to non-zero scalar multiples.We may describe Γ G in terms of ∆ G and ∆ * G as follows.Lemma 3.1.We have Proof.The n-forms ω ∈ SH n G are of the form g θ 1 • • • θ n for some g ∈ K[x 1 , . . ., x n ].We have ω ∈ SH n G if and only if ∂ f i ω = 0 and ∂ df i ω = 0 for all 1 ≤ i ≤ n.The first condition occurs if and only if g ∈ H G .Noting that df i = n j=1 ∂ x j f i θ j , the second condition implies that Hence ∂ df i ω = 0 if and only if ∂ ∂x j f i g = 0 for all j, so the second condition is equivalent to g ∈ H ′ G .The result follows since We now consider Ann Γ G .
We will now show that Ann Γ G = I ′ G is equivalent to the k = n case of (5) when V G = 0. We then restate and prove Theorem 1.7.Proposition 3.7.Suppose that G is a pseudo-reflection group and Ann Γ G = I ′ G .Proof.By (4) and Lemma 3.1, By Lemma 3.4, equation ( 15) is hence equivalent to ( 17) given by ψ(g) := ∂ g Γ G , so ker ψ = Ann Γ G .Equation ( 17 G .This is a restatement of ( 16).
Proof.First suppose G is real.Then det 2 = 1, so Γ G = 1 and Ann Γ G = x 1 , . . ., x n .Furthermore, x n as well.The result follows from Proposition 3.7.
If G = G(m, 1, n) for m > 1, then r = n.Table 1 gives Finally, the annihilator of ( The assertion is false for G(m, p, n) not covered by the above theorem.Here we consider the cyclic groups G(m, p, 1) = G(m/p, 1, 1) as part of the family G(m, 1, n).The simplest example with strict containment is G(4, 2, 2), which is generated by reflections but is not real.
Hence (16) does not hold, so the r-form component of (5) does not hold.
Moreover, in these cases, the top x-degree component of SH r G is strictly higher than the top x-degree component of (SH r G ) det , so the r-form component of (7) does not hold.

Proof. By Table 1, we may use basic invariants
) is typically not real), so Γ G = 1 and Ann Γ G = x 1 , . . ., x n .The top x-degree component of (SH n G ) det is hence degree 0. By assumption, m ≥ 3, so the generators x m−1 i of I ′ G are at least quadratic.Also by assumption, n ≥ 3, so the generators x −1 i (x 1 • • • x n ) are also at least quadratic, so Ann Γ G I ′ G .Moreover, I ′ G does not contain the linear polynomials, so H ′ G contains all linear polynomials, and the top x-degree component of . The generators of I ′ G have strictly larger degree since n ≥ 2, so the containment is strict.Indeed, it is easy to see that G is at least 1 higher than that of (SH n G ) det .Example 3.9.When G = G(4, 2, 2), we have On the other hand, (SH 2 G ) det = K θ 1 θ 2 .Hence ( 5) and ( 7) are false in this case.We may also use the explicit description of H ′ G to determine the highest degree of multiples of the volume form in SH G .This will be used below in Section 7.
x s = 0, or if and only if s ≤ m/p − 2. Now take n ≥ 2. As above, we have , so the top-degree monomial not in x n and again SH r G = K.Now take n ≥ 2, m ≥ 3, and p ≥ 2.Here m/p ≤ m/2 < m − 1. Suppose x α ∈ I ′ G for |α| maximal.By symmetry, we may suppose The degree of the latter minus the degree of the former is m − 2 − m/p ≥ 0, so the latter is the top-degree element.

Exactness of exterior differentiation on SH G
In this section we introduce a generalization of the complex ( 6) for a pseudo-reflection group G.We then summarize an algebraic analogue of well-known results in Hodge theory.Finally we prove Theorem 1.10 and deduce Theorem 1.6.Our argument follows an approach to proving exactness of the Koszul complex; see for example [14,Prop. 7.2.11].
4.1.Super harmonic cochain complexes.We begin by showing that two "dual" types of operators preserve the super harmonics.The first of these appeared in [31].
Lemma 4.1.Let d = n j=1 ∂ g j θ j with g j ∈ K[x 1 , . . ., x n ] be a Gequivariant operator which strictly lowers x-degree.Then d : SH G → SH G preserves the super harmonics of G.In particular, we have a cochain complex (18) 0 [31,Cor. 5.6].The argument in the next proof is very similar.
In particular, the generalized exterior derivatives d 1 , . . ., d r preserve the super harmonics SH G .Moreover, we have the following "dual" result.
Proof.Suppose η ∈ SH G , so ∂ ω η = 0 for all ω ∈ J G .We must show We may suppose δ and ω are bihomogeneous, so |J| and |K| are constant in the expansions, and that

It is easily seen that
and g is linear.Indeed, it may be reduced to the identity ∂ a Since δ strictly lowers θ-degree, λ has positive θ-degree or is 0, so λ ∈ J G , and we find ∂ ω δη = 0.
In particular, d † = n j=1 x j ∂ θ j satisfies the hypotheses of Lemma 4.2 and hence preserves the harmonics SH G .However, the adjoints d † i of the generalized exterior derivatives do not in general preserve the harmonics.

Hodge theory and Laplacians. A classic technique due to Hodge
[20, Ch.III] for analyzing cohomology on Riemannian manifolds involves replacing the cohomology groups with kernels of Laplacians.An algebraic version of this decomposition for cell complexes was introduced by Eckmann [10] (see also [11] and [9, §3] for further references).We state the version of this decomposition appropriate to our context and include a standard proof sketch using elementary linear algebra for the benefit of the reader.

Corollary 4.4. The homology of the sequence
In particular, the sequence is exact at B if and only if L is invertible.
When d = d is the exterior derivative, the adjoint operator is d † = n j=1 x j ∂ θ j , and Thus L acts by multiplying by the total degree, so it is invertible except on constants.Many of the operators d i in Table 1 are of the form d = n j=1 ∂ x N j θ j for some N ∈ Z ≥0 , which remain "largely" invertible.
Proof.We first show that L acts diagonally on the monomial basis.Recall that θ j ∂ θ ℓ + ∂ θ ℓ θ j = δ j=ℓ .We compute for the falling factorial.Now If I = ∅, the coefficient is positive.If I = ∅, the coefficient is zero if and only if α j < N for all N.

4.3.
Proving exactness and the rank 2 case.We may now restate and prove Theorem 1.10 from the introduction.
Theorem 1.10.For any pseudo-reflection group G ⊂ GL(V ) with r = dim(V /V G ), the exterior derivative cochain complex Proof.By Lemma 4.1, the exterior derivative d preserves SH G and the complex is well-defined.By Lemma 4.2, the adjoint d † also preserves SH G .Thus by Corollary 4.4, exactness at SH k G for k > 0 is equivalent to invertibility of the total Laplacian L = d † d + dd † acting on SH k G .In this case, L is invertible by Lemma 4.5, in either K[x 1 , . . ., x n , θ 1 , . . ., θ n ] or SH k G .Finally, ker d| SH 0 G is the set of all harmonics f ∈ K[x 1 , . . ., x n ] where ∂ x j f = 0 for all j, which is precisely K.
Remark 4.7.For i > 1, the adjoints d † i do not typically preserve SH G .Hence we cannot simply use Lemma 4.5 to analyze the homology of the complex (SH • G , d i ).A possible approach to Conjecture 1.20 would be to quantify this failure.
Before restating and proving Theorem 1.6, we give a criterion for the second component in (21) to satisfy (5).Here we use the notation Theorem 1.6.Let G ⊂ GL(V ) be a pseudo-reflection group with rank G from Corollary 4.6 satisfies (5) as well.The result follows for r ≤ 1, so take r = 2.
First suppose G is real.By Theorem 1.7, SH 2 G = Span K {vol} where vol is the volume form on V /V G , which transforms by det.
From Table 1, we have We may obtain x m−2 x m−2 2 θ 1 symmetrically, which completes the proof.

Gröbner and Artin bases for G(m, p, n)
Our proofs of Theorem 1.12 and Theorem 1.15 will use explicit Gröbner and monomial bases of the coinvariant algebras R G(m,p,n) with respect to the lexicographic order on monomials.Artin [2, p.41] implicitly gave the first monomial basis in type A, which corresponds in a natural way to the inversion statistic on permutations.Garsia [12] gave a separate descent basis in type A which corresponds to the major index statistic.The descent basis was subsequently generalized to Weyl groups and G(m, p, n) by a variety of authors; see [3, p.324] for details and further references.
The Artin bases are very well-known for S n , they appear to be folklore for G(m, 1, n), and we have been unable to locate a description of them for G(m, p, n) with p > 1.We give Artin bases A(m, p, n) for general G(m, p, n) in Section 5.1; see Definition 5.3.In Section 5.2, we give some combinatorial properties of A(m, p, n).In Section 5.3, we use Gröbner bases to prove the main results of this section, Theorem 5.5 and Theorem 5.6.Our arguments are elementary and self-contained, except for an appeal to Chevalley's result that dim K R G = |G| when G is a (pseudo-)reflection group.A }.These monomials may be visualized as "sub-staircase diagrams"; see Figure 1.A(m, 1, n) := {x a 1 1 • • • x an n : 0 ≤ a i < im, ∀i ∈ [n]}.These too may be visualized as sub-staircase diagrams; see Figure 2. We typically draw these sub-staircase diagrams horizontally rather than vertically for convenience.The Artin basis for general G(m, p, n) is an index p subset of A(m, 1, n) which we may describe using the following operation on sub-staircase diagrams.
Definition 5.2.Suppose m, p, n ∈ Z ≥1 with p | m.Let A be a substaircase diagram for G(m, 1, n).Define the p-contraction of A as follows.Let i be the largest index such that the ith row of A has fewer than m cells.Take the lower-left rectangle of width m using rows i, i + 1, . . ., n and shrink this rectangle horizontally by a factor of p.If this creates a partial cell, delete it.The result is the p-contraction of A.
Note that such an i exists since the first row has length < m.See Figure 3 below for an example.The Gröbner basis for the type A coinvariant ideal relative to lexicographic order (actually relative to any linear order withx 1 > • • • > x n ) is well-known and can be extracted from Artin's original argument in [2] with some effort.Let h j (x 1 , . . ., x n ) denote the complete homogeneous symmetric polynomial of degree j in n variables.The general Gröbner basis is as follows, which is proved in Section 5.3.
Theorem 5.6.The reduced Gröbner basis of I G(m,p,n) with respect to the lexicographic term order with Recall that a Gröbner basis G ⊂ F [x 1 , . . ., x n ] is reduced if for all g ∈ G, the leading coefficient of g is 1 and for all monomials m in all h ∈ G − {g}, the leading monomial of g does not divide m.See [8, §2.4,2.7] for details.

Combinatorial properties of the Artin bases for G(m, p, n).
We now give some combinatorial properties of the Artin bases A(m, p, n).Lemma 5.8.There are m n n!/p sub-staircase diagrams of type G(m, p, n).Moreover, the Hilbert series of the Artin basis of type G(m, p, n) is (25) Hilb(A(m, p, n) Proof.Note that p-contraction results in some fraction k/p of a cell, k ∈ {0, . . ., p − 1}, which is then discarded.It follows that the fibers of the p-contraction map are all of cardinality p. Hence there are m n n!/p sub-staircase diagrams of type G(m, p, n).
For the Hilbert series, condition on which row j is the lowest with length < m.The resulting Hilbert series is easily seen to be

The result follows by observing
Lemma 5.9 Conversely, suppose we have a sub-staircase diagram A = (a 1 , . . ., a n ) of type G(m, 1, n) which does not contain the above hooks.Since A does not contain the first hook, an m/p by n rectangle, there is some row in A of length < m/p.Thus we may apply the reverse pcontraction process described above to A starting at some row j.We must only show the result, call it B = (b 1 , . . ., b n ), remains a substaircase diagram of type G(m, 1, n), i.e. b i < im for all i ∈ [n].We have b 1 = a 1 , . . ., b j−1 = a j−1 , so the necessary constraint is satisfied for i < j.When i = j, we have b j = a j p < (m/p)p = m ≤ jm as required.Finally, since rows i > j of A have length ≥ m/p and yet the ith hook is not contained in the diagram, we must have a i < (m/p) + (i − 1)m for i > j.Thus We also have the following recursive description of the Artin bases.The Hilbert series formula (25) also follows easily from this description and induction.
Lemma 5.10.The Artin bases of types G(m, p, n) are defined recursively by for n ≥ 2, with base cases A(m, p, 1) = {x a 1 1 : 0 ≤ a 1 < m/p}.Proof.The base cases are immediate.For the recursive formula, the first term consists of elements where j = n when performing p-contraction.When j < n, we may keep or remove the final row without affecting the procedure materially, and that row has length satisfying 0 ≤ a n − m/p < (n − 1)m.
The standard monomial basis of K[x 1 , . . ., x n ]/ LT(G) consists of all monomials not divisible by any of the leading terms of G. Encoding monomials in diagrams as in Section 5.1, Lemma 5.9 and Lemma 5.12 together show that the Artin basis A(m, p, n) is precisely the standard monomial basis for K[x 1 , . . ., x n ]/ LT(G).By Lemma 5.8, this basis has size m n n!/p.
As for the right-hand side of ( 27), a set of representatives of a basis of K[x 1 , . . ., x n ]/ LT(I m,p,n ) descends to a basis of K[x 1 , . . ., x n ]/I m,p,n as usual, and in particular their dimensions agree.Chevalley [6] Thus equality must hold in (27), so LT(G) = LT(I m,p,n ) and the result follows.
Proof of Theorem 5.6.By the preceding argument, LT(G) = LT(I m,p,n ), so the proposed Gröbner basis is in fact a Gröbner basis.All that is left is to check that G is reduced.The leading coefficients are all 1, so we must only verify the divisibility condition.
We begin with the p = 1 case.Consider a general term Then a k m ≥ km, so a k ≥ 1, and hence j ≤ k.Now h j (x m j , . . ., x m n ) has degree jm while x km k has degree km, so km ≤ jm, forcing k = j, a contradiction.
We now turn to the case p > 1. Again consider a general term x ; where the exponents in the numerator arise from the degrees of the elements in the regular sequence.See [29, I.5, p.39-42] for details and [7] for results similar to Lemma 6.2.The right-hand side has degree (mj − 2)(n − j + 1).
The final ingredient needed for Theorem 1.12 is the following observation.Afterwards we restate and prove Theorem 1.12.Lemma 6.5.If j ∈ [n] and i ∈ [n], then Proof.The result is trivial if i < j, so take i ≥ j.Since the exterior derivative d = n ℓ=1 ∂ x ℓ θ ℓ is G-equivariant and an anti-derivation, it preserves J G(m,p,n) .By Theorem 5.6, The degree of x   (5).The q = 1 specialization has been taken to make the output manageable.(5) holds for a given G if and only if the two columns agree.In each case, ( 7) and ( 9) indeed hold, except (7) fails for G(4, 2, 4) and G(4, 4, 4).Calculations were done using SageMath [32].The exceptional group calculations were done with [21].

Lemma 3 . 10 .
Let G = G(m, p, n).Then the top x-degree component of SH r G has degree

1 ,
then G is real and SH r G = K has top degree 0 by Theorem 1.7.If p = 1 and m ≥ 2, we have I ′ G = x m−1 . . ., x m−1 n

Definition 5 . 1 .
A sub-staircase diagram of type G(m, 1, n) is a leftjustified arrangement of n rows consisting of a 1 , . . ., a n square cells from top to bottom satisfying 0 ≤ a i < im for all i ∈ [n].

x 4 2 x 3 3 x 15 4 x 6 5 .
A sub-staircase diagram for G(4, 1, 5) representing the monomial x 1 The bottom-most row with < m cells is row i = 3.The lower-left rectangle involved in p-contraction is highlighted.

1 x 4 2 x 3 x 13 4 x 4 5
The result of 2-contracting the previous sub-staircase diagram.The halfcell arising from contracting row 3's three cells by a factor of two has been removed to create a sub-staircase diagram of type G(4, 2, 5) representing the monomial x .

Table 1
for explicit descriptions of d 1 , ..., d r for G(m, p, n).Table1lists the real cases G = S n , B n , D n , Dih 2n explicitly. 1)n.
31, Lemmas 2.7, 2.11, 5.4] for more information.2.3.Generalized exterior derivatives.Recall the exterior derivative (3), which is defined by d = n i=1 ∂ x i θ i .Here and elsewhere, θ i refers to left multiplication by θ i .Equation ( ) is equivalent to im ψ = H ′ G .By Lemma 3.6, we have ψ : K[x 1 , . . ., x n ] → H ′ G .Again by Lemma 3.6, ker ψ ⊃ I ′ G .Thus by Lemma 3.3, im ψ = im ψ| H ′ G and ker ψ = ker ψ| H ′ G ⊕I ′ G .Since H ′ G is finite-dimensional, ψ| H ′ G is surjective if and only if it is injective, which occurs if and only if ker ψ = I ′ We may reverse the p-contraction process by horizontally expanding the width m/p rectangle in A involving rows i, i + 1, . . ., n where i is the lowest row of length < m/p.The result, call it B, necessarily expands the rectangle of width m/p involving rows j, j + 1, . . ., n in the hook diagram.After expansion, row j of B has length ≥ m + (j − 1)m = jm, contradicting the fact that B must be a sub-staircase diagram of type G(m, 1, n).Thus sub-staircase diagrams of type G(m, p, n) do not contain the above hooks.
. A sub-staircase diagram of type G(m, 1, n) is a .Proof.First suppose to the contrary that there exists a sub-staircase diagram A of type G(m, p, n) containing the jth hook diagram.