Harmonic differential forms for pseudo-reflection groups I. Semi-invariants

We give a type-independent construction of an explicit basis for the semi-invariant harmonic differential forms of an arbitrary pseudo-reflection group in characteristic zero. Our"top-down"approach uses the methods of Cartan's exterior calculus and is in some sense dual to related work of Solomon, Orlik--Solomon, and Shepler describing (semi-)invariant differential forms. We apply our results to a recent conjecture of Zabrocki which provides a representation theoretic-model for the Delta conjecture of Haglund--Remmel--Wilson in terms of a certain non-commutative coinvariant algebra for the symmetric group. In particular, we verify the alternating component of a specialization of Zabrocki's conjecture.


Introduction
Recently there has been a great deal of research activity in algebraic combinatorics studying diagonal actions of the symmetric group S n on k sets of n commuting indeterminants and ℓ sets of n anti-commuting indeterminants. Orellana-Zabrocki [13] describe the S n -invariants of these polynomial rings combinatorially and summarize some of their history. The k = 2, ℓ = 0 case has received a very large amount of attention through the study of the diagonal coinvariants Q[x n , y n ]/D n where D n is the diagonal coinvariant ideal generated by all homogeneous S ninvariants of positive degree and x n is shorthand for x 1 , . . . , x n [9,10]. Zabrocki [29] has recently given a conjectured description of the trigraded S n -module isomorphism type of the coinvariant algebra when m = 2, ℓ = 1: (1) GrFrob(Q[x n , y n , θ n ]/SD n ; q, t, z) = where GrFrob is the tri-graded Frobenius series, SD n is the super diagonal coinvariant ideal generated by homogeneous S n -invariants of positive degree, e n is an elementary symmetric function, and ∆ ′ f is a certain modified Macdonald eigenoperator. Equation (1) may be interpreted as a conjectural representation-theoretic model for the Delta conjecture of Haglund-Remmel-Wilson [7], and both conjectures remain open. See [7,29] for details and further references.
The special case t = 0 of these conjectures involving one set of commuting and one set of anti-commuting variables has received special attention. Haglund-Rhoades-Shimozono [8] had earlier given a different representation-theoretic model for this specialization of the Delta conjecture. Motivated by Zabrocki's conjecture, the second author [28] gave a conjectural description of the harmonics of Q[x n , θ n ]/J n where J n is the super coinvariant ideal generated by homogeneous S n -invariants of positive degree. This description further motivated Rhoades-Wilson [18] to recently construct another representation-theoretic model for the t = 0 specialization of the Delta conjecture arising from the leading terms of those harmonics. In this paper, we restrict our attention to natural pseudo-reflection group generalizations of the t = 0 case of Zabrocki's coinvariant algebra model.
In [28], several of the implications of Zabrocki's conjecture at t = 0 were proven, including a determination of the bi-graded Hilbert series of the alternants in the S n -coinvariants and certain vanishing bounds.
The key to the results in [28] is the fact that one is looking at differential forms with polynomial coefficients and thus one has at one's disposal the full power of Cartan's exterior calculus. The idea of adding anticommuting variables to prove theorems about the commuting variables appeared in the work of Solomon [22] and, later, in the paper of Orlik-Solomon [14] in the case of a finite unitary subgroup G ≤ GL n (C) generated by pseudo-reflections. Differential forms and derivations have also been exploited in this context more recently [16,17]. The purpose of the present paper is to use the methods of differential forms to give a uniform generalization of the type A description in [28] of the alternant polynomial differential forms and coinvariants for an arbitrary pseudo-reflection group.
Our main results are as follows. See the subsequent sections for missing definitions and Section 4 and Section 5 for proofs. Let V be an n-dimensional vector space over an arbitrary field F of characteristic 0, let G ≤ GL(V ) be a pseudo-reflection group, let M be an r-dimensional G-module, and let χ be a one-dimensional character of G.
We consider the semi-invariant differential forms (2) (S(V * ) ⊗ ∧M * ) χ where S(V * ) is the symmetric algebra on V * , ∧M * is the exterior algebra on M * , and W χ := {w ∈ W : σ · w = χ(σ)w, ∀σ ∈ G}. In certain circumstances, such as when M = V , we give two explicit bases for (2). The following result may be thought of as an analogue of Solomon's classic result [22] describing (S(V * ) ⊗ ∧V * ) G as a Grassmann algebra over S(V * ) G . Here ∆ χ is the minimal-degree element in the χ-isotypic component of S(V * ), the d * i are certain differential operators depending on M (see Definition 4.4), the f i are basic invariants of G, and J M * is the Jacobian of M * (see Definition 3.3).
We have the following enumerative corollary. Let e M * i denote the exponents of M * (see Definition 3.1) and let d i denote the degrees of G.
. Remark 1.1. Orlik-Solomon [14] described (S(V * ) ⊗ ∧M * ) G when J M = ∆ M , up to a non-zero scalar, as an exterior algebra over S(V * ) G , and Shepler [20,21] gave an analogous result for (S(V * ) ⊗ ∧V * ) χ using "χ-wedging." These exterior algebra structures may be considered "bottom-up" descriptions. Theorem 4.10 involves the operators d * i , which decrease S(V * )-degree, applied to ∆ χ , so this and related work in [21, §6] can be considered "top-down." The top-down description turns out to be more useful in our analysis of the coinvariant algebra.
Let J * M denote the ideal in S(V * ) ⊗ ∧M * generated by all homogeneous G-invariants of positive degree. Our overarching motivation has been to describe the semi-invariant elements of the coinvariant algebra, We give the following explicit basis for (3) using the harmonics of S(V * ) ⊗ ∧M * (see Definition 4.1).
The hypotheses of Theorem 5.7 are satisfied whenever χ = det M * , M G = 0, and the pseudo-reflections of G act by pseudo-reflections or the identity on M. In this case, the right-hand side of (4) is The alternating component of the t = 0 specialization of Zabrocki's conjecture then follows from Corollary 5.8 when G consists of n × n permutation matrices and M is the (n − 1)-dimensional standard representation.
The classical coinvariant algebra of G is S(V * )/I * where I * is the ideal generated by all non-constant homogeneous G-invariants. It is well-known that the top-degree component of S(V * )/I * is the image of S(V * ) det V , which has motivated much of our work. We will explore which bidegrees of S(V * ) ⊗ ∧M * /J * M are non-zero in a future article [27].
The rest of the paper is organized as follows. In Section 2, we review background material on differential forms and pseudo-reflection groups. Section 3 concerns exponents and basic derivations of pseudo-reflection groups. Section 4 introduces G-harmonics for the polynomial algebra and constructs explicit bases for certain pieces of the differential algebras; see Theorem 4.5 and Theorem 4.10. In Section 5, we prove our main result, Theorem 5.7, which gives a basis for the harmonics and the coinvariants. In Section 6, we discuss the technical condition ∆ M = J M , up to a non-zero scalar, which appears in some of our results. A more explicit but less general version of many of these results that may be more palatable to algebraic combinatorialists can be found in [26].

Polynomial differential forms
We now describe several actions and pairings involving polynomial differential forms and related objects. All of the constructions in this section use standard ideas from differential geometry.
Let F be a field of characteristic 0 and let V be an F -vector space of dimension n < ∞. We identify V * * = V . Let G be an arbitrary subgroup of GL(V ) and let M be an F [G]-module of dimension r < ∞. The tensor products S(V ) ⊗ ∧M and S(V * ) ⊗ ∧M * are algebras of differential forms with polynomial coefficients. Each of S(V ), S(V * ), ∧M, and ∧M * is naturally graded, so we have four non-commutative, bigraded F -algebras The G-action on V extends multiplicatively to yield natural G-actions on S(V ) and ∧M. As usual, G acts on V * contragrediently via g * λ := λ • g −1 . Thus S(V ), S(V * ), ∧M, and ∧M * are all naturally graded Gmodules, so the algebras in (5) are bigraded G-modules via the diagonal actions of g ⊗ g, g ⊗ g * , g * ⊗ g, g * ⊗ g * .

2.2.
Pairings and differential operators on S(V * ) ⊗ ∧M * . Our next goal is to describe natural actions of S(V ) ⊗ ∧M and S(V ) ⊗ ∧M * on S(V * ) ⊗ ∧M * . These actions will be fundamental in later sections. The following observation is essentially trivial. By "perfect", we mean that λ, v = 0 for all v ∈ V implies λ = 0, and λ, v = 0 for all λ ∈ V * implies v = 0. We may naturally extend such pairings to symmetric, exterior, and tensor products as follows.
is G-invariant and perfect, where perm denotes the matrix permanent.
Proof. In each case, G-invariance is immediate and non-degeneracy can be checked quickly using dual bases.  Definition 2.6. For s ∈ S(V ) and f ∈ S(V * ), we have adjoint operators defined by For m ∈ ∧M and µ ∈ ∧M * , we have adjoint operators defined by Combining these operators yields actions of S(V ) ⊗ ∧M and S(V ) ⊗ ∧M * on S(V * ) ⊗ ∧M * as follows. These actions will be used prominently in later sections.
Proof. The multiplication operators taken together yield F -algebra morphisms, and the same is true of their adjoints. By definition, g ∈ G acts on ψ ∈ End F (S(V * ) ⊗ ∧M * ) by (gψ)(ω) := gψ(g −1 ω). For Gequivariance, we see immediately that It follows from this and the G-invariance of −, − that The claimed G-equivariance follows.
Example 2.8. A special case of the preceding construction gives a distinguished and familiar G-invariant endomorphism of S(V * ) ⊗ ∧V * .
Let v 1 , . . . , v n ∈ V be a basis and let λ 1 , . . . , λ n ∈ V * be its dual basis. Then is independent of the choice of basis and is hence G-invariant. The action of this element is the exterior derivative, namely The operators above acting on S(V * ) ⊗ ∧M * generate the following super Weyl algebra. All of the operators that we will be using in this paper come from the action of this algebra on S(V * ) ⊗ ∧M * .
where D(V ) is the algebra generated by ∂ v , m λ which is the Weyl algebra on dim(V ) variables, and Cliff(M ⊕ M * ) is the Clifford algebra of the split form m + µ, ℓ + ν cl := µ(ℓ) + ν(m) which is generated by ι m , ǫ µ .

2.3.
Derivations and anti-derivations. The differential operators ∂ s and ∂ f are the usual polynomial differential operators acting on the polynomial ring or its dual. The operators ǫ m and ǫ µ are called exterior products and the operators ι m and ι µ are called interior products. For completeness and concreteness, we briefly summarize some of their wellknown properties.
If v ∈ V , then ∂ v ∈ End F (S(V * )) is given as usual by The operators ∂ v satisfy the classical Leibniz rule for all v ∈ V and f, h ∈ S(V * ), and hence are derivations. If s ∈ S(V ), we have Combining these last two observations, for all v ∈ V and λ ∈ V * , we have It follows from the perm description in Lemma 2.3 that if v 1 , . . . , v n ∈ V is a basis with dual basis λ 1 , . . . , λ n ∈ V * , then where we have used multi-index notation and all operations are componentwise.
Analogously, one may check that if m 1 , . . . , m r ∈ M is a basis with dual basis µ 1 , . . . , µ r ∈ M * , then where we have used the natural analogue of multi-index notation in this setting, e.g.
The sign may be determined explicitly by iterating the well-known identity More generally, the operators ι m for m ∈ M are anti-derivations in the sense that for all µ ∈ ∧ k M * and ν ∈ ∧M * . In particular, for all m ∈ M and ξ ∈ M * , we have In this subsection, let F be a subfield of C closed under complex conjugation and let G ≤ GL(V ) be an arbitrary finite subgroup.
A Hermitian form on an F -vector space W is a map . That is, a Hermitian form is linear in the first argument and conjugate-linear in the second argument. Note that (w, . By the rank-nullity theorem, dim X + dim X ⊥ ≥ dim W , and the result follows.
Since F is closed under complex conjugation, every finite-dimensional F -vector space W has a positive-definite Hermitian form given by choosing a basis ω 1 , . . . , ω k of W * and using (17) ( for all u, v ∈ W . We may strengthen this construction and relate it to the canonical pairing from Section 2.2 as follows. η, ω = (η, τ (ω)).
Proof. Let W be a finite-dimensional G-module. Since F is assumed closed under complex conjugation, there is a positive-definite Hermitian form on W . Since G is assumed finite, the Hermitian form may be taken to be G-invariant by Weyl's unitarian trick. We may extend Ginvariant positive-definite Hermitian forms in three ways analogous to Lemma 2.3.
The conjugate-linearity and G-equivariance of τ follow quickly from (18) and the corresponding properties of (−, −) and −, − . It remains to show that τ is multiplicative. On ∧M * , we have . The full calculation is exactly analogous.
2.5. Pseudo-reflection groups and coinvariants. We now recall some basic facts concerning pseudo-reflection groups and establish some further notation. We again let F be an arbitrary field of characteristic 0.
Definition 2.12. A pseudo-reflection is an element g ∈ GL(V ) such that dim ker(g − I) = n − 1.
A pseudo-reflection of order two is called a reflection. A (pseudo) reflection group is a finite subgroup of GL(V ) generated by (pseudo) reflections.
For the rest of this subsection, let G denote a pseudo-reflection group. Shephard-Todd [19] and Chevalley [4] showed that the G-invariants S(V * ) G are generated by n algebraically independent, homogeneous elements f 1 , . . . , f n ∈ S(V * ) G of positive degrees d 1 , . . . , d n . The f i are called basic invariants of G and the d i are called the degrees of G.
Recall that the Hilbert series of a graded vector space If g ∈ GL(V ) is a (pseudo) reflection, then g * ∈ GL(V * ) is as well, so G * ≤ GL(V * ) is a pseudo-reflection group, and so S(V ) G is similarly generated by algebraically independent, homogeneous elements z 1 , . . . , z n ∈ S(V ) G . Since G acts completely reducibly, dim M G = dim(M * ) G in general, so Hilb(S(V ) G ; q) = Hilb(S(V * ) G ; q), and we may take deg f i = deg z i . Remark 2.14. Chevalley [4] showed that S(V * )/I * , a graded algebra, carries the regular representation of G and that (19) S as a graded S(V * ) G -module.
2.6. Fields of definition. In order to use the machinery of Hermitian forms, we require representations defined over subfields of the complex numbers which are closed under complex conjugation. In practice, pseudo-reflection groups are typically constructed in terms of explicit unitary matrices over cyclotomic fields, in which case these properties are trivial. More care is required to handle the general case and avoid artificial assumptions. Suppose W is an n-dimensional vector space over a field K of characteristic 0. The character field of a representation ρ : G → GL(W ) is the subfield Q(ρ) ⊂ K generated by the set {Tr ρ(g) | g ∈ G}.
Lemma 2.15. If |G| < ∞, then Q(ρ) is isomorphic with a subfield of C closed under complex conjugation.
Proof. Choosing some basis for W , the matrix coefficients of ρ(g) are c g ij ∈ K. Clearly Q(ρ) ⊂ Q(c g ii ). Since {c g ii } is finite, we may identify Q(c g ii ) with a subfield of C, so Q(ρ) may be identified with a subfield of C. Diagonalizing now shows that Tr ρ(g) is of the form i ζ α i for ζ = exp(2πi/|G|) ∈ C. Thus We say ρ : G → GL(W ) is defined over a subfield K ′ of K if there is some basis of W for which ρ(G) ⊂ GL n (K ′ ). If ρ is defined over K ′ , then clearly K ′ ⊃ Q(ρ). In favorable circumstances, the representation is actually defined over its character field.   . Let G be a pseudo-reflection group over a field of characteristic 0. Then the character field K of G is a splitting field of G. That is, every representation of G over a field containing K is defined over K.
Corollary 2.18. If G is a pseudo-reflection group over a field K of characteristic 0 and ρ : G → GL(W ) is a representation of G over K, then ρ is defined over a subfield of K which is isomorphic to a subfield of C closed under complex conjugation.

Exponents, Jacobians, and Vandermondians
We continue to let V be an n-dimensional vector space over a field F of characteristic 0. Let G ≤ GL(V ) be a pseudo-reflection group and suppose M is an r-dimensional G-module. In this section we recall the M-exponents, the basic derivations for M, and related notions.
Since basic invariants f 1 , . . . , f n ∈ S(V * ) G are algebraically independent and char F = 0, the classical Jacobian criterion gives det(∂ v j f i ) = 0 for any basis v 1 , . . . , v n of V . This observation and the well-known fact that e i = d i −1 imply df 1 , . . . , df n ∈ (S(V * )⊗V * ) G form a set of basic derivations for V . That is, we may take where λ 1 , . . . , λ n is the dual basis of v 1 , . . . , v n . Likewise, the basic invariants z 1 , . . . , z n ∈ S(V ) G yield basic deriva- Jacobians and Vandermondians. The preceding example motivates the following general notion. Suppose M * has basis µ 1 , . . . , µ r . We may expand the basic derivations for M over S(V * ) as The Jacobian is non-zero and is uniquely determined up to a nonzero constant [14,Prop. 2.5]. Note that deg J M = e M 1 + · · · + e M r . Example 3.4. We have Steinberg [24] proved that Indeed, e i = d i − 1 follows from (22). For general M, we similarly have some 0 = ξ ∈ ∧ r M such that Consequently, though the containment may be strict, which motivates the following.
By (20), there exists an element

3.3.
Gutkin's formula. Stanley [23] expressed ∆ M and Gutkin [6] expressed J M as a product of linear forms vanishing on the reflecting hyperplanes of G, which we now summarize. See [16, §4] for historical discussion and [3, §4.5.2] for a proof using Molien's theorem.
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. Since σ ∈ G H acts trivially on a codimension 1 subspace, σ → det V * σ is a faithful representation G H ֒→ F × . Since finite subgroups of F × are cyclic, G H is cyclic, G H is generated by a pseudo-reflection, and the irreducible G H -representations are powers of det V * . Consequently,  Recall that ∆ M = J det M up to a non-zero scalar, so Theorem 3.7 gives product formulas for ∆ M as well. We have the following well-known special cases. See [3,Prop. 4.34(2)] and [14].

Harmonics and semi-invariant bases
Our first goal in this section is to define the well-known G-harmonic polynomials using the constructions in Section 2. We prove their basic properties by constructing a G-invariant Hermitian form, and then use the harmonics to construct explicit bases for semi-invariant differential forms. The main results of this section are Theorem 4.5 and Theorem 4.10. F remains a field of characteristic 0, V is an n-dimensional F -vector space, G ≤ GL(V ) is a pseudo-reflection group, and M is an r-dimensional G-module. That is, the harmonics are the orthogonal complements of the coinvariant ideals I and I * with respect to the natural perfect pairing −, − from (6). The harmonics have the following basic properties.

Lemma 4.2. We have
and similarly with H(V ).
As for (27) For the general case, by Corollary 2.18, G is defined over a subfield K of F which may be identified with a subfield of C closed under complex conjugation. Let v 1 , . . . , v n be a basis for V over F such that the matrix of each σ ∈ G is in GL n (K). Set from Lemma 2.7.
Proof. We suppose throughout that F is a subfield of C closed under complex conjugation. The general case follows by arguing as in the proof of Lemma 4.2 using extension of scalars.
..,i k ∈ F . By homogeneity, we may suppose the subsets are each of size k. For a fixed choice of {i 1 , . . . , i k }, let {j 1 , . . . , j r−k } be its complement in [r].
Remark 4.6. The d * i essentially appear in [21,§6,Prop. 18] in slightly less generality. The preceding proof exploits their exterior algebra structure in a fundamental way. The condition J M * | ∆ χ is a generalization of the condition from [21] that χ is wholly-nontrivial, meaning that ∆ det V * | ∆ χ .
Proof. The d * i alter bidegree by (−e M * i , 1). We additionally have the following result. Alternatively, it is a straightforward consequence of Orlik-Solomon's generalization [14,Thm. 3.1] of Solomon's description [22] of (S(V * ) ⊗ ∧V * ) G as the exterior algebra over S(V * ) G generated by df i . By   (1 + zq e M i ).
This follows from the well-known fact that for each 0 ≤ k ≤ r, there is a (non-canonical) isomorphism of GL(V )-modules Remark 4.9. By Benard's Theorem 2.17, every irreducible representation of a pseudo-reflection group G is absolutely irreducible over the character field of G. Steinberg noted that if G is generated by n = dim(V ) pseudo-reflections and V is irreducible, then ∧ k V is (absolutely) irreducible for all 0 ≤ k ≤ n (see [2, Ch. V, §2, Exercise 3(d)], [11, p. 250, Thm. A]).
We now restate and prove Theorem 4.10 and Corollary 4.11 from the Introduction.
Proof. The fact that the first set is a basis follows from Theorem 4.5 and (29). For the second set, it suffices to verify linear independence. To do so, we sketch how to modify the proof of Theorem 4.5. Suppose F is a subfield of C closed under complex conjugation, pick a homogeneous, orthogonal basis {g α } of S(V * ) χ = S(V * ) G ∆ χ with respect to the Hermitian form, and let g α ∆ χ = g α where {g α } is a basis for S(V * ) G . Linear independence is unaffected if we replace {f a 1 1 · · · f an n } with {g α }. If Now use (g β , g β ) > 0. Recall the G-equivariant F -algebra homomorphism from Lemma 2.7.
Proof. Since M G = 0, we have (F ⊗ M) G = 0, so deg S(V * )ω M i ≥ 1. By (38), deg L ij ≥ 2, so L ij ∈ I. Suppose ω ∈ H(S(V * ) ⊗ M * ), or equivalently δ * j ω = 0 and ∂ z j ω = 0. We must show d * We now restate and prove our main result from the introduction.  We have ∆ M * ∈ H(S(V * ) ⊗ ∧M * ) since δ * i ∆ M * = 0 trivially. By Corollary 5.6, the proposed elements belong to H(S(V * ) ⊗ ∧M * ). The result hence follows from Theorem 4.5. We may pass to the quotient by (35).  Proof (Sketch). The irreducible S n -representations are indexed by integer partitions λ ⊢ n. The number of exponents of the irreducible indexed by λ is the degree f λ . Since there are sub-exponentially many λ ⊢ n and λ⊢n (f λ ) 2 = n!, one heuristically expects f λ to grow superexponentially in n, making J M = ∆ M quite rare. One may for instance apply the arguments in [25, p. 15] to make this intuition rigorous, though we omit the details.
If G is a dihedral group, then every irreducible representation M takes reflections to reflections or the identity, so by Corollary 3.8(d), J M = ∆ M up to a non-zero scalar. We now give somewhat less trivial examples.
Example 6.2. Let B n ⊂ O(n) be the Weyl group of type B n realized as n × n signed permutation matrices whose non-zero entries are ±1. Let Z n ⊂ B n be the group of diagonal matrices with diagonal entries ±1. Note that S n = B n /Z n . Thus we may consider the standard representation M of S n as an irreducible B n -representation. The exponents of M are the degrees in which M appears in the coinvariant algebra R[B n ]/I, or equivalently in the harmonics H(B n ). Since M is Z n -invariant, these occur in H(B n ) Zn . By Chevalley's result applied to B n (see Remark 2.14), multiplication gives an isomorphism i → x i . In particular, J M = ∆ M up to a non-zero multiple, and both have degree 2 + 4 + · · · + 2(n − 1) = 2 n 2 = n(n − 1). Example 6.3. Let D n ⊂ B n ⊂ O(n) be the Weyl group of type D n consisting of signed permutation matrices with evenly many negative entries. Let Z ′ n = D n ∩ Z n . Note that B n /Z ′ n ∼ = S n . It is easily checked that R[x 1 , . . . , x n ] Z ′ n = R[x 2 1 , . . . , x 2 n ] ⊕ R[x 2 1 , . . . , x 2 n ]x 1 · · · x n and R[x 1 , . . . , x n ] Dn = R[x 2 1 , . . . , x 2 n ] Sn ⊕ R[x 2 1 , . . . , x 2 n ] Sn x 1 · · · x n . Using the same argument as in Example 6.2, the standard representation M of S n yields an irreducible representation of D n with J M = ∆ M up to a non-zero multiple, and both have degree n(n − 1).
The Weyl group F 4 of type F 4 is isomorphic with S 3 ⋉ D 4 . One may then view the standard representation M of S 3 as an