Invariant derivations and differential forms for reflection groups
aa r X i v : . [ m a t h . C O ] F e b INVARIANT DERIVATIONS AND DIFFERENTIAL FORMSFOR REFLECTION GROUPS
VICTOR REINER AND ANNE V. SHEPLER
To Peter Orlik and Hiroaki Terao on their 80th and 65th birthdays, respectively.
Abstract.
Classical invariant theory of a complex reflection group W highlights threebeautiful structures: • the W -invariant polynomials constitute a polynomial algebra, over which • the W -invariant differential forms with polynomial coefficients constitute an exterioralgebra, and • the relative invariants of any W -representation constitute a free module.When W is a duality (or well-generated ) group, we give an explicit description of the iso-typic component within the differential forms of the irreducible reflection representation.This resolves a conjecture of Armstrong, Rhoades and the first author, and relates to Lie-theoretic conjectures and results of Bazlov, Broer, Joseph, Reeder, and Stembridge, and alsoDeconcini, Papi, and Procesi. We establish this result by examining the space of W -invariantdifferential derivations; these are derivations whose coefficients are not just polynomials, butdifferential forms with polynomial coefficients.For every complex reflection group W , we show that the space of invariant differentialderivations is finitely generated as a module over the invariant differential forms by thebasic derivations together with their exterior derivatives. When W is a duality group, weshow that the space of invariant differential derivations is free as a module over the exteriorsubalgebra of W -invariant forms generated by all but the top-degree exterior generator.(The basic invariant of highest degree is omitted.)Our arguments for duality groups are case-free, i.e., they do not rely on any reflectiongroup classification. Introduction
A celebrated result of Solomon [29] exhibits the set of differential forms invariant underthe action of a complex reflection group as an exterior algebra . A similar result holds whenwe consider derivations instead of differential forms, i.e., elements of S ( V ∗ ) ⊗ V instead of S ( V ∗ ) ⊗ ∧ V ∗ , for a reflection representation V with symmetric algebra S ( V ∗ ). The polyno-mial degrees of generators of these sets of invariants are positioned into various combinatorialidentities expressing the geometry, topology, and representation theory of reflection groups.Recently, a theory of Catalan combinatorics for reflection groups (e.g., see [1]) has promptedquestions about the structure of invariant forms for other representations of a reflectiongroup. Of particular interest are the differential derivations , elements of S ( V ∗ ) ⊗ ∧ V ∗ ⊗ V .The first author together with Armstrong and Rhoades conjectured a formula [1, Conj. 11.5 ′ ]for the Poincar´e polynomial of the invariant differential derivations when the reflection group W is real, i.e., a finite Coxeter group. We verify this conjecture and show that the set of Key words and phrases.
Reflection groups. Invariant theory. Weyl groups. Coxeter groups.Research supported in part by NSF grants DMS-1601961,DMS-1101177, and Simons Foundation Grant invariant differential derivations, ( S ( V ∗ ) ⊗ ∧ V ∗ ⊗ V ) W , is a free module over an exterior algebra constructed from exterior derivatives df i of all butone of the basic invariants f i ; the last basic invariant of highest polynomial degree is omitted.In fact, we give the explicit structure of the invariant differential derivations for all complexreflection groups W that are duality groups. We also give a basis for nonduality groups. Weexplain these two main results next. Invariant theory of reflection groups.
Recall that a reflection on a finite dimensionalvector space V = C ℓ is a nonidentity general linear transformation that fixes a hyperplane in V pointwise. A complex reflection group W is a subgroup of GL( V ) generated by reflections ;we assume all reflection groups are finite. We fix a C -basis x , . . . , x ℓ of V ∗ with dual C -basis y , . . . , y ℓ of V and identify the symmetric algebra S := Sym( V ∗ ) with the polynomial ring C [ x , . . . , x ℓ ], which carries a W -action by linear substitutions. Shephard and Todd [26] andChevalley [8] showed that the W -invariant subalgebra S W is again polynomial : S W = C [ f , . . . , f ℓ ]for certain algebraically independent polynomials f , . . . , f ℓ called basic invariants . One canchoose f , . . . , f ℓ homogeneous; we assume deg( f ) ≤ · · · ≤ deg( f ℓ ) after re-indexing. Itfollows that for any W -representation U , the space of relative invariants ( S ⊗ U ) W forms afree S W -module of rank dim C ( U ) (see, e.g., Hochster and Eagon [15, Prop. 16]). Note thatwe take all tensor products and exterior algebras over C unless otherwise indicated. We alsoassume all representations are complex and finite dimensional. Differential forms and derivations.
Two particular cases have received much attention.First, when U = ∧ V ∗ , one may identify S ⊗ U = S ⊗ ∧ V ∗ with the S -module of differentialforms with polynomial coefficients generated by dx , . . . , dx ℓ . Solomon’s theorem [26, 29]asserts that ( S ⊗ ∧ V ∗ ) W is not just a free S W -module, but, in fact, an exterior algebra over S W on exterior generators df , . . . , df ℓ , where df j := P ℓi =1 ∂f j ∂x i ⊗ x i has degree e i := deg( f i ) − S ⊗ ∧ V ∗ ) W = ^ S W { df , . . . , df ℓ } . Second, when U = V , one may identify S ⊗ U = S ⊗ V with the set of derivations S → S on V generated by the partial derivatives ∂/∂x , . . . , ∂/∂x n . Here, a derivation θ = P ℓi =1 θ ( i ) ⊗ y i maps f in S to θ ( f ) = P ℓi =1 θ ( i ) ∂∂x i ( f ). One may choose a homogeneous basis θ , . . . , θ ℓ forthe free S W -module ( S ⊗ V ) W , i.e.,(1.1) ( S ⊗ V ) W = S W θ ⊕ . . . ⊕ S W θ ℓ for some basic derivations θ j = P ℓi =1 θ ( j ) i ⊗ y i with each θ ( j ) i in S homogeneous of fixed degree,say e ∗ j . Duality groups.
Our main results combine these contexts, with special results for dualitygroups. An irreducible complex reflection group W is a duality group if its coexponents e ∗ ≥ · · · ≥ e ∗ ℓ and exponents e ≤ · · · ≤ e ℓ determine each other via the relation e i + e ∗ i = h, where h := deg( f ℓ ) = e ℓ + 1 is the largest degree of a basic invariant, called the Coxeternumber for the duality group W . NVARIANT DERIVATIONS AND DIFFERENTIAL FORMS 3
Duality groups include all irreducible real reflection groups (i.e., finite Coxeter groups)as well as symmetry groups of regular complex polytopes [9]. It was observed by Orlikand Solomon [21] in a case-by-case fashion (using the classification [26]) that an irreduciblecomplex reflection group W is a duality group if and only if it is well-generated , that is,generated by ℓ = dim( V ) reflections, but we will not need this fact in the sequel. Main theorems.
We consider the set M of mixed forms, called differential derivations , M := S ⊗ ∧ V ∗ ⊗ V. We view M as an ( S ⊗ ∧ V ∗ )-module via multiplication in the first two tensor positions. Its W -invariant subspace M W is then a ( S ⊗ ∧ V ∗ ) W -module. In general, M W will not be a free module. Nevertheless, our first main result asserts that for duality groups, M W is a free R -module, where R is the subalgebra of the invariant forms generated by all df i but the last, R := ^ S W { df , . . . , df ℓ − } , with only df ℓ omitted. To give an R -basis, we extend the usual exterior derivative operator d on S to a function on S ⊗ ∧ V ∗ ⊗ V defining d ( f ⊗ ω ⊗ y ) = P ≤ i ≤ ℓ ∂f∂x i ⊗ ( x i ∧ ω ) ⊗ y. We identify S ⊗ V with the subspace S ⊗ ⊗ V of differential derivations, S ⊗ V ֒ → S ⊗ ∧ V ∗ ⊗ Vf ⊗ y f ⊗ ⊗ y, and apply d to derivations using this inclusion. Our first main result is shown with case-freearguments (it does not depend on any classification). Theorem 1.1.
For W a duality (well-generated) complex reflection group, ( S ⊗ ∧ V ∗ ⊗ V ) W forms a free R -module on the ℓ basis elements { θ , . . . , θ ℓ , dθ , . . . , dθ ℓ } . For arbitrary complex reflection groups, we have no uniform statement like Theorem 1.1.However, we give an explicit free S W -basis for ( S ⊗∧ V ∗ ⊗ V ) W in the remaining (non-duality)cases, showing this: Theorem 1.2.
For W any complex reflection group, ( S ⊗ ∧ V ∗ ⊗ V ) W is generated as amodule over the exterior algebra ( S ⊗ ∧ V ∗ ) W = V S W { df , . . . , df ℓ } by the ℓ generators { θ , . . . , θ ℓ , dθ , . . . , dθ ℓ } . To be clear: ( S ⊗ ∧ V ∗ ⊗ V ) W is not freely generated as a module over V S W { df , . . . , df ℓ } by { θ i , dθ i } ℓi =1 . Example 1.3.
A rank ℓ = 1 reflection group W ⊂ GL( V ) = GL ( C ) = C × is a cyclic group W = h ζ i ∼ = Z /h Z for some ζ = e πih in C . Let V = C x and V ∗ = C y with y dual to x . Underthe generator of W , x ζ − x and y ζ y . Then S = C [ x ] , S W = C [ f ] where f = f ℓ = x h has degree h. The S W -module of invariant forms, ( S ⊗ ∧ V ∗ ) W , is an exterior algebra over S W generatedby df = hx h − ⊗ x of degree e = h −
1, that is,( S ⊗ ∧ V ∗ ) W = ( S ⊗ ∧ V ∗ ) W ⊕ ( S ⊗ ∧ V ∗ ) W = S W (1 ⊗ | {z } := R ⊕ S W ( x h − ⊗ x ) = S W ⊕ S W df . VICTOR REINER AND ANNE V. SHEPLER
On the other hand, the S W -module of invariant derivations is( S ⊗ V ) W = S W ( x ⊗ y ) = S W θ , for θ = x ⊗ y of degree e ∗ = 1 . In particular, W is a duality group, since e ∗ + e = 1 + ( h −
1) = deg( f ℓ ). Now consider theinvariant differential derivations: An easy check confirms that M W = ( S ⊗ ∧ V ∗ ⊗ V ) W isa free module over R = S W with basis { θ , dθ } . Here, we have identified θ = x ⊗ y with x ⊗ ⊗ y and dθ = 1 ⊗ x ⊗ y . Thus( S ⊗ ∧ V ∗ ⊗ V ) W = S W ( x ⊗ ⊗ y ) ⊕ S W (1 ⊗ x ⊗ y ) . Outline.
Section 2 provides further context and implications of Theorem 1.1 while Section 3gives its relation to some theorems and conjectures in Lie theory. We collect some tools forestablishing helpful reflection group numerology (like Molien’s Theorem and the Gutkin-Opdam Lemma) in Section 4 and reap that numerology in Section 5. Section 6 is a slightdigression deriving basis conditions reminiscent of Saito’s Criterion for free arrangements. Alinear independence condition is given in Section 7. We complete the proof of Theorem 1.1in Section 9 after showing that duality groups exhibit auspicious numerology in Section 8.The remainder is about nonduality groups and Theorem 1.2. Section 10 addresses ranktwo reflection groups, while Section 11 considers the Shephard and Todd group G — theonly irreducible non-duality group that is neither of rank two, nor within the Shephardand Todd infinite family of monomial groups G ( r, p, n ). Section 12 assembles the proof ofTheorem 1.2, relegating the case of G ( r, p, n ) to Appendix 14. When p = 1 or p = r , theseare duality groups and covered by Theorem 1.1; when 1 < p < r , they are nonduality groupsand we give an alternate basis for the invariant differential derivations in this appendix.We consider some further questions in Section 13.2. Implications of Theorem 1.1
To provide context for Theorem 1.1, we first note a few consequences and special cases.
The case of exterior degree zero.
Theorem 1.1 implies something that we already knewabout ( S ⊗ ∧ V ∗ ⊗ V ) W = ( S ⊗ V ) W , namely, that it is a free S W -module on the basis { θ j } j ∈{ ,...,ℓ } ; this is true even when W isnot a duality group. We will end up using this fact in the proof of the theorem. The case of top exterior degree.
At the opposite extreme, Theorem 1.1 asserts that( S ⊗ ∧ ℓ V ∗ ⊗ V ) W ∼ = ( S ⊗ det ⊗ V ) W is free as an S W -module on the basis { df · · · df ℓ − dθ k } k ∈{ ,...,ℓ } . This agrees with the polyno-mial degrees of a basis found in [27] for any reflection group W , as we may view ( S ⊗ det ⊗ V ) W as the space of invariant derivations for the “twisted reflection representation” det ⊗ V , wheredet : W C ∗ is the determinant character of W acting on V . Indeed, if J = det (cid:16) ∂f i ∂x j (cid:17) isthe Jacobian determinant of the basic invariants f , . . . , f ℓ (see Section 6), then the forms df · · · df ℓ − dθ k have degrees e + . . . + e ℓ − + e ∗ k − e + . . . + e ℓ ) − ( e ℓ + 1) + e ∗ k = deg J − ( h − e ∗ k ) = deg J − e k when W is a duality group (see [27, Cor. 13(e)]). NVARIANT DERIVATIONS AND DIFFERENTIAL FORMS 5
The Hilbert series consequence.
Theorem 1.1 has implications for
Hilbert series analo-gous to those given by the Shephard-Todd-Chevalley and Solomon theorems with S = ⊕ i ≥ S i graded by polynomial degree. Just as these classical results immediately imply thatHilb( S W ; q ) := X i ≥ q i dim S Wi = ℓ Y i =1 (1 − q e i +1 ) − , Hilb (cid:0) ( S ⊗ V ) W ; q (cid:1) := X i ≥ q i dim ( S i ⊗ V ) W = Hilb( S W ; q ) · ℓ X i =1 q e ∗ i , andHilb (cid:0) ( S ⊗ ∧ V ∗ ) W ; q, t (cid:1) := X i,j ≥ q i t j dim( S i ⊗ ∧ j V ∗ ) W = Hilb( S W ; q ) · ℓ Y i =1 (1 + q e i t ) , Theorem 1.1 analogously immediately implies that(2.1)Hilb (cid:16) ( S ⊗ ∧ V ∗ ⊗ V ) W ; q, t (cid:17) := X i,j ≥ q i t j dim (cid:0) S i ⊗ ∧ j V ∗ ⊗ V (cid:1) W = Hilb( S W ; q, t ) · ℓ X i =1 ( q e ∗ i + q e ∗ i − t )= Hilb( S W ; q ) · ( q + t ) · ℓ − Y i =1 q e i t ! ℓ X i =1 q e ∗ i − ! . Our original motivation, in fact, was the special case of (2.1) for real reflection groups W ,which appeared as [1, Conj. 11.5 ′ ], based on Coxeter-Catalan combinatorics and computerexperimentation. 3. The Lie theory connection
We now explain how the case of Theorem 1.1 when W is a Weyl group , i.e, a finitecrystallographic real reflection group, relates to Lie-theoretic results and work of Bazlov,Broer, Joseph, Reeder, and Stembridge, and also DeConcini, Papi, and Procesi. Let G be asimply-connected, compact simple Lie group with a choice of maximal torus T . Denote by g and V the complexification of their corresponding Lie algebras, and let W := N G ( T ) /T bethe associated Weyl group acting on a real vector space V . Then G acts on ∧ g ∗ , while W acts on S := S ( V ∗ ) and on its coinvariant algebra S/S W + ∼ = H ∗ ( G/T )where the last isomorphism to cohomology, due to Borel, is grade-doubling, and where S W + is the ideal generated by invariant polynomials of positive degree. Classical results (see [23])give isomorphisms(3.1) ( ∧ g ∗ ) G ∼ = H ∗ ( G ) ∼ = ( H ∗ ( G/T ) ⊗ H ∗ ( T )) W ∼ = (cid:0) S/S W + ⊗ ∧ V ∗ (cid:1) W exhibiting both of these rings as (isomorphic) exterior algebras, with exterior generators P , P , . . . , P ℓ where P i lies in ( ∧ e i +1 g ∗ ) G . The isomorphism is again homogeneous afterdoubling the grading in S/S W + . VICTOR REINER AND ANNE V. SHEPLER
Reeder [24] conjectured a similar relation between G -invariants and W -invariants, relatingtwo Hilbert series associated with a finite-dimensional G -representation M : P G ( M ; t ) := X j ≥ t j dim( ∧ j g ∗ ⊗ M ) G ,P W ( M T ; q, t ) := X i,j ≥ q i t j dim (cid:0) ( S/S W + ) i ⊗ ∧ j V ∗ ⊗ M T (cid:1) W . Conjecture 3.1. [24, Conj. 7.1] If M is small , meaning its weight space M α = 0 for allroots α , one has P G ( M ; t ) = P W ( M T ; q, t ) | q = t . Various special cases of Conjecture 3.1 were known at the time that it was formulated. Forexample, when M is the trivial G -representation it follows from (3.1) above. Reeder [24,Cor. 4.2] proved the t = 1 specialization of Conjecture 3.1 and credited it also to Kostant:For M small,(3.2) dim( ∧ g ∗ ⊗ M ) G = P G ( M ; t ) | t =1 = P W ( M T ; q, t ) | q = t =1 = dim (cid:0) S/S W + ⊗ ∧ V ∗ ⊗ M T (cid:1) W . The type A special case was also known to follow from the “first-layer” formulas of Stem-bridge [33]. The type B special case was recently verified in work of DeConcini and Papi,and Stembridge independently verified Conjecture 3.1 case-by-case for all types.A further bit of motivation comes from a generalization of Chevalley’s restriction theoremdue to Broer [5]. Chevalley’s result asserts that restriction g ∗ → V ∗ induces an isomorphismof polynomial rings(3.3) S ( g ∗ ) G → S W , while Broer [5] showed more generally that, for any small G -module M , restriction alsoinduces an isomorphism (of modules over the polynomial rings in (3.3))( S ( g ∗ ) ⊗ M ) G → ( S ⊗ M T ) W . Broer’s result suggested to the authors the following enhanced version of Conjecture 3.1.
Conjecture 3.2. (Enhanced Reeder Conjecture) For a small G -representation M , there isan isomorphism ( ∧ g ∗ ⊗ M ) G ∼ = (cid:0) S/S W + ⊗ ∧ V ∗ ⊗ M T (cid:1) W of modules over the exterior algebra in (3.1) which is degree-preserving after doubling thegrading in S/S W + . While this paper was under review, DeConcini and Papi [10, p. 259] showed that notall small G -representations M satisfy Conjecture 3.2, and one asks, “For which small G -representations does the Enhanced Reeder Conjecture 3.2 hold?” DeConcini and Papi [10,Thm. 2.2 and Cor. 6.6] proved the conjecture holds for two particularly important cases ofsmall G -representations, namely, the adjoint representation M = g and the little adjointrepresentation , which are the g -irreducibles whose highest weights are the highest root and DeConcini and Papi, personal communication, 2016. Stembridge, personal communication, 2016. It is exactly in the case of the adjoint representation that Bazlov [2] proved Conjecture 3.1, and he creditsthis special case of the conjecture to Joseph [17].
NVARIANT DERIVATIONS AND DIFFERENTIAL FORMS 7 highest short root, respectively. The adjoint case M = g connects our work to the followingresult of DeConcini, Papi, and Procesi. Theorem 3.3. [11, Thm 1.1]
Regard ( ∧ g ∗ ⊗ g ) G as a module over the exterior subalgebra R of ( ∧ g ∗ ) G generated by P , P , . . . , P ℓ − , via multiplication in the first tensor factor. Then ( ∧ g ∗ ⊗ g ) G is free as an R -module, with basis elements { f i , u i } ℓi =1 of degrees deg( f i ) =2 e i , deg( u i ) = 2 e i − . An alternate proof of Theorem 3.3 follows from our Theorem 1.1, using the adjoint specialcase of Conjecture 3.2, that is, [10, Thm. 2.2], after modding out by S W + and bearing inmind that { e ∗ i } ℓi =1 = { e i } ℓi =1 for Weyl groups W .4. Degree sums and the Gutkin-Opdam Lemma
Before determining bases for the invariant differential derivations, we recall some tools forinvestigating the relevant numerology, most notably a useful lemma for finding the sum ofdegrees in a basis.
Degree sum.
Let k be a field, and let A be a graded k -algebra and integral domain. Con-sider a free graded A -module M ∼ = A p of finite rank, say with homogeneous basis m , . . . , m p .The (unordered) list of degrees deg( m ) , . . . , deg( m p ) are uniquely determined by the quo-tient of Hilbert series p X i =1 q deg( m i ) = Hilb( M, q ) / Hilb(
A, q ) . Thus we may assign to any such M the degree sum ∆ A ( M ) := p X i =1 deg( m i ) = (cid:20) ∂∂q Hilb(
M, q )Hilb(
A, q ) (cid:21) q =1 . If one knows this degree sum a priori , then one may determine an explicit A -basis for M byjust checking independence over the fraction field: Lemma 4.1.
Let A be a graded k -algebra and integral domain and M ∼ = A p a free graded A -module. A set of homogeneous elements { n , . . . , n p } in M with P pi =1 deg( n i ) = ∆ A ( M ) isan A -basis for M if and only if it is linearly independent over the fraction field K = Frac( A ) .Proof. The forward implication is clear. For the reverse implication, note that linear in-dependence is equivalent to nonsingularity of the matrix B in A p × p with n = B m , for n = [ n , . . . , n p ] T and m = [ m , . . . , m p ] T . Since det( B ) = 0, its expansion contains anonzero term indexed by a permutation π with b i,π ( i ) = 0 for each i = 1 , , ..., p ; after re-indexing, one may assume π is the identity permutation. Hence deg( n i ) = deg( b i,i ) + deg( m i )for i = 1 , . . . , p . Since P i deg( n i ) = ∆ A ( M ) = P i deg( m i ), one has that deg( n i ) = deg( m i )for each i . Thus after re-ordering the rows and columns of B in increasing order of degree, B will be block upper triangular, with each diagonal block an invertible matrix with entriesin k . Therefore B gives an A -module automorphism of M sending the A -basis m to n . (cid:3) Modules over the Invariant Ring.
As mentioned in the Introduction, a result of Hochsterand Eagon implies that for any representation U of a complex reflection group W , the setof all relative invariants M = ( S ⊗ U ) W is a free module of finite rank p = dim C U over thegraded k -algebra A = S W . We introduce an abbreviation for the above degree sum: VICTOR REINER AND ANNE V. SHEPLER
Definition 4.2.
Let W be a complex reflection group. For any W -representation U , set∆( U ) := ∆ S W (cid:0) ( S ⊗ U ) W (cid:1) = X ≤ i ≤ p deg ψ i for any S W -basis { ψ i } pi =1 of ( S ⊗ U ) W . Local Data.
We next review an a priori calculation for the degree sum ∆( U ), Lemma 4.3below, due originally to Gutkin [14], and later rediscovered by Opdam [20, Lemma 2.1]; seealso Brou´e [6, Prop. 4.3.3 and eqn. (4.6)].The formula for ∆( U ) is expressed in what is sometimes called the local data for U at eachreflecting hyperplane H of W . The pointwise stabilizer subgroup W H in W of H is cyclic,say of order e H ; note that e H is the maximal order of a reflection in W fixing H pointwise.The W H -irreducible representations are the powers { det j } e H − j =0 of the 1-dimensional (linear)character det := det ↓ WW H restricted from W to W H acting on V . It is convenient to introducethe representation ring R ( W H ) := Z [ v ] / ( v e H − , where v j represents the class of the 1-dimensional representation det j , and define a Z -linearfunctional D H : R ( W H ) → Z , v j j . Then for any W -representation U , the functional D H on the restricted representation U ↓ WW H can be expressed in terms of the inner products µ H,j := h U ↓ WW H , det j i W H as D H (cid:0) U ↓ WW H (cid:1) = e H − X j =0 j · µ H,j . Lemma 4.3. (Gutkin-Opdam Lemma) Let U be a representation of a complex reflectiongroup W . Then ∆( U ) = X H D H (cid:0) U ↓ WW H (cid:1) , where the sum runs over all reflecting hyperplanes H for W . Lemma 4.3 can be deduced (see Brou´e [6, § Lemma 4.4. [6, Lem. 3.28]
For any W -representation U , (4.1) Hilb (cid:0) ( S ⊗ U ) W , q (cid:1) = | W | X w ∈ W Tr U ( w − )det(1 − qw ) . For a complex reflection group W , taking U to be the trivial representation in Lemma 4.1immediately implies the well-known fact that ℓ Y i =1 − q e i +1 = Hilb( S W , q ) = | W | X w ∈ W − qw )as S W = C [ f , . . . , f ℓ ] with deg( f i ) = e i + 1. As noted by Shephard and Todd [26, § q = 1 on the left and NVARIANT DERIVATIONS AND DIFFERENTIAL FORMS 9 right immediately gives these facts: | W | = ℓ Y i =1 ( e i + 1) , (4.2) N := { reflections in W } = ℓ X i =1 e i . (4.3) 5. Numerology from the Gutkin-Opdam lemma
Again, W is a complex reflection group. This section harvests numerology from Lemma 4.3. Number of Reflecting Hyperplanes.
We first see that Lemma 4.3 implies that the co-exponents e ∗ i sum to the number N ∗ of reflecting hyperplanes for W . Example 5.1.
Let U = V , the reflection representation. Each reflecting hyperplane H has µ H,j ( V ) = 0 for j = 0 ,
1, with µ H, = ℓ − µ H, = 1, and hence D H ( V ↓ WW H ) = 1. Thusin this case, Lemma 4.3 implies that the coexponents e ∗ i := deg( θ i ) for the S W -basis { θ i } ℓi =1 of ( S ⊗ V ) W satisfy the well-known formula (e.g., see [7, p. 130])(5.1) N ∗ := { reflecting hyperplanes in W } = X H V ) = ℓ X i =1 e ∗ i . Graded representations.
In order to apply Lemma 4.3 to graded W -representations,we consider the graded representation ring R ( W H )[[ t ]], that is, the ring of graded virtual W H -characters. We also extend D H to a map R ( W H )[[ t ]] → Z coefficientwise, defining D H (cid:0)P k ≥ t k χ (cid:1) := P k ≥ t k D H ( χ ). The sum in the following corollary is over all reflectinghyperplanes H of W . Corollary 5.2.
For any W -representation U , ℓ X m =0 ∆( ∧ m V ∗ ⊗ U ) t m = (1 + t ) ℓ − X H D H (1 + v e H − t ) e H − X j =0 µ H,j v j ! . Proof.
Recall that each U ↓ WW H = P e H − j =0 µ H,j v j in R ( W H ). The restriction V ∗ ↓ WW H is a sumof ℓ − e H − . Hence the restriction ∧ V ∗ ↓ WW H will be represented by (1+ t ) ℓ − (1+ v e H − t ), and ( ∧ V ∗ ⊗ U ) ↓ WW H will be representedby (1+ t ) ℓ − (1+ v e H − t ) P e H − j =0 µ H,j v j in R ( W H )[[ t ]]. The result then follows from Lemma 4.3. (cid:3) Solomon’s theorem.
We illustrate in the next example how Corollary 5.2 gives Solomon’sresult [29] that the space of W -invariant differential forms is generated by df , . . . , df ℓ as anexterior algebra . Example 5.3.
We show that ( S ⊗ ∧ m V ∗ ) W has S W -basis { df I } I ∈ ( [ ℓ ] m ) where (cid:0) [ ℓ ] m (cid:1) denotesthe collection of all m -element subsets I = { i , . . . , i ℓ } of the set [ ℓ ] := { , , . . . , ℓ } , with1 ≤ i < . . . < i m ≤ ℓ , and where df I := df i ∧ · · · ∧ df i m . For an alternate geometric proof sketch of this result, see Berest, Etingof, and Ginzburg [3, Remark 1.17].
Apply Corollary 5.2 to the trivial representation U , obtaining ℓ X m =0 ∆( ∧ m V ∗ ) t m = (1 + t ) ℓ − X H D H (1 + v e H − t ) = (1 + t ) ℓ − X H ( e H − t = N t (1 + t ) ℓ − where the last equality uses Equation (4.3). Therefore ∆( ∧ m V ∗ ) = (cid:0) ℓ − m − (cid:1) N. Note that thesum of degrees of the elements in the alleged basis { df I } I ∈ ( [ ℓ ] m ) matches this: X I ∈ ( [ ℓ ] m ) X i ∈ I e i = ℓ X i =1 e i n I ∈ (cid:0) [ ℓ ] m (cid:1) : i ∈ I o = N (cid:0) ℓ − m − (cid:1) = ∆( ∧ m V ∗ ) . Hence by Lemma 4.1, it suffices only to check the linear independence of df I over K =Frac( S ). As observed by Solomon, the m -fold wedge products of the elements df , . . . , df ℓ form a K -basis for K ⊗ ∧ m V ∗ if and only their top wedge is nonvanishing: df ∧ · · · ∧ df ℓ = det (cid:18) ∂f i ∂x j (cid:19) ⊗ x ∧ · · · ∧ x ℓ = 0 . But this follows immediately from the Jacobi Criterion [16]: The algebraic independence of f , . . . , f ℓ implies that the matrix of coefficients ( ∂f i ∂x j ) of df , . . . , df ℓ is nonsingular. Numerology of differential derivations.
Consider the W -representation U := ∧ m V ∗ ⊗ V of dimension p := ℓ (cid:0) ℓm (cid:1) = dim C ( ∧ m V ∗ ⊗ V ) . We show that ∆( ∧ m V ∗ ⊗ V ) depends only on N , N ∗ , and ℓ := dim C V . Proposition 5.4.
For any complex reflection group W and ≤ m ≤ ℓ , ∆( ∧ m V ∗ ⊗ V ) = ( ℓ − (cid:0) ℓ − m − (cid:1) N + (cid:0) ℓ − m (cid:1) N ∗ . Proof.
We take U = V in Corollary 5.2: ℓ X m =0 ∆( ∧ m V ∗ ⊗ V ) t m = (1 + t ) ℓ − X H D H (cid:0) (1 + v e H − t )( ℓ − v ) (cid:1) = (1 + t ) ℓ − X H D H (cid:0) ℓ − ℓ − tv e H − + v + t (cid:1) = (1 + t ) ℓ − X H (cid:0) ( ℓ − t ( e H −
1) + 1 (cid:1) = (1 + t ) ℓ − (cid:0) ( ℓ − N t + N ∗ (cid:1) , using Equations (4.3) and (5.1). The proposition now follows from the binomial theorem. (cid:3) Proposition 5.4 and Lemma 4.1 imply a corollary used repeatedly in the proofs of Theo-rems 1.1 and 1.2.
Corollary 5.5.
Any set of homogeneous elements { ψ i } i =1 , ,...,ℓ ( ℓm ) in ( S ⊗ ∧ m V ∗ ⊗ V ) W with X i deg( ψ i ) = ∆( ∧ m V ∗ ⊗ V ) = ( ℓ − (cid:0) ℓ − m − (cid:1) N + (cid:0) ℓ − m (cid:1) N ∗ forms an S W -basis for ( S ⊗ ∧ m V ∗ ⊗ V ) W if and only if they are linearly independent over K = Frac( S ) . NVARIANT DERIVATIONS AND DIFFERENTIAL FORMS 11 Digression: A Saito Criterion
This section, although not needed for the sequel, extends Corollary 5.5 to a condition sim-ilar to
Saito’s criterion for free hyperplane arrangements [22, § W . We first recall some facts about the coefficient matrices for the differential forms df i and basic derivations θ i . Jacobian matrix and product of hyperplanes.
Recall the defining polynomial Q andthe Jacobian polynomial J for a complex reflection group W : Q := Y H l H , and J := Y H l e H − H . Here, the product is taken over all reflecting hyperplanes H = ker l H in V for some choiceof linear forms l H ∈ V ∗ with e H = | Stab W ( H ) | . Note that Q and J are only well-defined upto nonzero scalars. Let Jac( f ) and M ( θ ), respectively, be the matrices in S ℓ × ℓ that express { df i } i ∈ [ ℓ ] and { θ i } i ∈ [ ℓ ] in the S -bases { ⊗ x i } i ∈ [ ℓ ] and { ⊗ y i } i ∈ [ ℓ ] for S ⊗ V ∗ and S ⊗ V ,respectively. Steinberg [32] and Orlik and Solomon [21, § J = det(Jac( f )) and Q = det( M ( θ )) , so that(6.1) N = ℓ X i =1 e i = deg( J ) = { reflections in W } and N ∗ = ℓ X i =1 e ∗ i = deg( Q ) = { reflecting hyperplanes for W } . Note that Terao [34] also showed that invariant derivations θ i also give an S -basis for the module of derivations of the reflection hyperplane arrangement of W ; see also [22, § § W is a duality group, observe that (by definition)(6.2) h := deg( f ℓ ) = 1 ℓ ℓ X i =1 ( e i + e ∗ i ) = N + N ∗ ℓ . Matrix of Coefficients.
We capture the coefficients of any set of invariant differentialderivations in a matrix of coefficients. Consider the obvious free S -basis for S ⊗ ∧ m V ∗ ⊗ V given by(6.3) { dx I ⊗ y j : 1 ≤ i, j ≤ ℓ } with dx I := 1 ⊗ x i ∧ · · · ∧ x i m for m -subsets I = { i < · · · < i m } of [ ℓ ] and [ ℓ ] := { , . . . , ℓ } . Given a collection B ⊂ ( S ⊗ ∧ m V ∗ ⊗ V ) W , let Coef( B ) denote its coefficient matrix in S p × p with respect to the S -basis in (6.3). Lemma 6.1.
For any B = { ψ i } pi =1 ⊂ ( S ⊗∧ m V ∗ ⊗ V ) W , the product J ( ℓ − ( ℓ − m − ) Q ( ℓ − m ) divides det Coef( B ) .Proof. Fix a reflecting hyperplane H in V for W and a reflection s in W of maximal order e H fixing H . Choose coordinates x , . . . , x ℓ of V ∗ so that l H = x and s acts diagonally with nonidentity eigenvalue ξ : s ( y i ) = ( ξ − y if i = 1 ,y i if i = 1 , and s ( x i ) = ( ξx if i = 1 ,x i if i = 1 . Each row of Coef( B ) lists the coefficients f I,j of some invariant differential derivation ψ i = P f I,j dx I ⊗ y j in ( S ⊗ ∧ m V ∗ ⊗ V ), while each column of Coef( B ) is indexed by a pair ( I, j )for I an m -subset of [ ℓ ] and j ∈ [ ℓ ]. Note two observations: • For each pair (
I, j ) with j = 1 but 1 ∈ I , the polynomial x e H − divides f I,j since s ( dx I ⊗ y j ) = ξ ( dx I ⊗ y j ) implies that s ( f I,j ) = ξ − f I,j ;thus ( ℓ − (cid:0) ℓ − m − (cid:1) different columns of the matrix Coef( B ) are divisible by x e H − . • For each pair (
I, j ) with j = 1 but 1 ∈ I , the polynomial x divides f I,j , since s ( dx I ⊗ y j ) = ξ − ( dx I ⊗ y j ) implies that s ( f I,j ) = ξf I,j ;thus (cid:0) ℓ − m (cid:1) different columns of the matrix Coef( B ) are divisible by x .Hence ℓ H = x when raised to the power ( e H − ℓ − (cid:0) ℓ − m − (cid:1) + (cid:0) ℓ − m (cid:1) divides det Coef( B ).This holds for each reflecting hyperplane H , and therefore unique factorization implies that J ( ℓ − ( ℓ − m − ) Q ( ℓ − m ) divides det Coef( B ). (cid:3) Saito-like Criterion.
Again, let K = C ( x , . . . , x ℓ ) be the fraction field of S = C [ x , . . . , x ℓ ]. Corollary 6.2.
For a homogeneous subset B = { ψ i } pi =1 of ( S ⊗ ∧ m V ∗ ⊗ V ) W , the followingare equivalent: (a) B forms an S W -basis for ( S ⊗ ∧ m V ∗ ⊗ V ) W . (b) det Coef( B ) is nonzero of degree ( ℓ − (cid:0) ℓ − m − (cid:1) N + (cid:0) ℓ − m (cid:1) N ∗ . (c) det Coef( B ) = c · J ( ℓ − ( ℓ − m − ) Q ( ℓ − m ) for some nonzero scalar c in C . (d) P pi =1 deg( ψ i ) = ( ℓ − (cid:0) ℓ − m − (cid:1) N + (cid:0) ℓ − m (cid:1) N ∗ and B is K -linearly independent in thespace K ⊗ ∧ m V ∗ ⊗ V .Proof. Corollary 5.5 gives the equivalence of (a) and (d), linear algebra gives the equivalenceof (d) and (b), and Lemma 6.1 gives the equivalence of (b) and (c). (cid:3)
Remark 6.3.
An argument with Cramer’s rule shows that Corollary 6.2 (c) implies (a)directly without using Corollary 5.5, or appealing to any Hilbert series argument. Indeed, (c)implies that B spans ( S ⊗ ∧ m V ∗ ⊗ V ) W over S W , as we explain next. Label the S -basiselements dx I ⊗ y j of S ⊗ ∧ m V ∗ ⊗ V as z , . . . , z p for convenience. Then the matrix Coef( B )in S p × p expresses the elements B = { ψ i } pi =1 in the S -basis { z i } pi =1 :(6.4) ψ j = p X i =1 Coef( B ) ij · z i . To show that a typical element P pi =1 s i z i in ( S ⊗ ∧ m V ∗ ⊗ V ) W lies in the S W -span of B , find k i in the fraction field K of S (using det Coef( B ) = 0) with(6.5) p X i =1 s i z i = p X j =1 k j ψ j . NVARIANT DERIVATIONS AND DIFFERENTIAL FORMS 13
We may assume each k j lies in K W , else apply the symmetrizer | W | P g ∈ W g ( − ) to (6.5) anduse the W -invariance of P i s i z i and of each ψ j . To show the k j actually lie in S W , substi-tute (6.4) into (6.5), giving a matrix equation relating the column vectors s = [ s , . . . , s p ] t and k = [ k , . . . , k p ] t in K p : s = Coef( B ) · k . Cramer’s rule then implies that(6.6) k i = det Coef( B ( i ) )det Coef( B )where the numerator matrix Coef( B ( i ) ) is obtained from Coef( B ) by replacing its i th columnwith s . Then since Coef( B ( i ) ) expresses the elements ψ , . . . , ψ i − , P i s i z i , ψ i +1 , . . . , ψ p of( S ⊗ ∧ m V ∗ ⊗ V ) W in terms of the basis { z i } , Lemma 6.1 implies that its determinant isdivisible by the nonzero polynomial det Coef( B ). Thus the right side of (6.6) lies in S , sothat its left side k i lies in K W ∩ S = S W , as desired.7. Independence over the fraction field
In this section, we use Springer’s theory of regular elements to investigate differentialderivations with coefficients in the fraction field K = C ( x , . . . , x ℓ ) of S = C [ x , . . . , x ℓ ]. Wewill later show that Theorem 1.1 (for duality groups) follows from a more general statementestablished in this section for arbitrary reflection groups, Theorem 7.3, describing a K -vectorspace basis for ( K ⊗ ∧ m V ∗ ⊗ V ) W .We first give a definition and a lemma. Recall the notation V reg for the complementwithin V of the union of all reflecting hyperplanes for W , that is, the subset of vectors in V having regular W -orbit. Recall the notation df I := df i ∧ · · · ∧ df i m ∈ S ⊗ ∧ m V ∗ for subsets I = { i < . . . < i m } ⊂ [ ℓ ] := { , , . . . , ℓ } , and the notation (cid:0) [ ℓ ] m (cid:1) for the collection of all m -subsets I ⊂ [ ℓ ]. Definition 7.1.
Define a K -linear map K ⊗ V ∗ E −→ K by E ( dx i ) = E (1 ⊗ x i ) = x i . One also has a K -linear map K ⊗ V ∗ ⊗ V E ⊗ V −−−−→ K ⊗ V .
Note that by
Euler’s identity , for any homogeneous f in S ,(7.1) E ( df ) = deg( f ) · f. Likewise, for θ = P ℓj =1 θ ( j ) ⊗ y j in S ⊗ V with each θ ( j ) homogeneous of fixed degree deg( θ ),(7.2) ( E ⊗ V )( dθ ) = deg( θ ) · θ. We first give a K -basis for differential derivations with coefficients in K . Recall from (1.1)that θ , . . . , θ ℓ are any choice of basic derivations, i.e., any choice of homogeneous S W -basisof ( S ⊗ V ) W = ( S ⊗ ∧ V ∗ ⊗ V ) W (identifying S ⊗ V with S ⊗ ⊗ V ). Lemma 7.2.
For each ≤ m ≤ ℓ , the following set gives a K -basis for K ⊗ ∧ m V ∗ ⊗ V : (7.3) e B ( m ) := (cid:8) df I θ k (cid:9) I ∈ (cid:0) [ ℓ ] m (cid:1) , k ∈ [ ℓ ] . Furthermore, elements of S ⊗ V ∗ ⊗ V can be expressed in the K -basis e B (1) with coefficientsin ( J Q ) − ℓ S . Proof.
The matrix that expresses e B ( m ) in the usual S -basis { dx I ⊗ y j : I ∈ (cid:0) [ ℓ ] m (cid:1) , j ∈ [ ℓ ] } of S ⊗ ∧ m V ∗ ⊗ V is the tensor product of the matrices ∧ m (Jac( f )) ⊗ M ( θ ), where ∧ m (Jac( f ))is the m th exterior power of Jac( f ). The invertibility of Jac( f ) and functoriality of ∧ m ( − )imply the invertibility of ∧ m (Jac( f )). Then since M ( θ ) is also invertible, so is the tensorproduct ∧ m (Jac( f )) ⊗ M ( θ ), and hence e B ( m ) is another K -basis. The last assertion of theproposition then follows, since in the m = 1 case,det(Jac( f ) ⊗ M ( θ )) = det(Jac( f )) ℓ · det( M ( θ )) ℓ = J ℓ Q ℓ . (cid:3) We will show that Theorem 1.1 follows from the next theorem.
Theorem 7.3.
Let W be a complex reflection group with homogeneous basic invariants f , . . . , f ℓ , and an index i in , , . . . , ℓ that satisfies (7.4) V reg ∩ \ i = i f − i { } 6 = ∅ . Then for each m = 0 , , . . . , ℓ , the following set gives a K -vector space basis for K ⊗∧ m V ∗ ⊗ V : (7.5) B ( m ) := n df I dθ k o I ∈ (cid:0) [ ℓ ] \{ i } m − (cid:1) , k ∈ [ ℓ ] ⊔ n df I θ k o I ∈ (cid:0) [ ℓ ] \{ i } m (cid:1) , k ∈ [ ℓ ] . Proof of Theorem 7.3.
There is nothing to prove in the case m = 0. We consider first theextreme case m = 1, then the opposite extreme case m = ℓ , and finally the intermediatecases with 2 ≤ m ≤ ℓ − The case m = 1 . Note that the set B (1) that we want to show is a K -basis for K ⊗ V ∗ ⊗ V , B (1) = { df i θ k : i ∈ [ ℓ ] \ { i } , k ∈ [ ℓ ] } ⊔ { dθ k : k ∈ [ ℓ ] } , has substantial overlap with the known K -basis e B (1) for K ⊗ V ∗ ⊗ V given in Lemma 7.2, e B (1) = { df i θ k : i, k ∈ [ ℓ ] } = { df i θ k : i ∈ [ ℓ ] \ { i } , k ∈ [ ℓ ] } ⊔ { df i θ k : k ∈ [ ℓ ] } . Thus we need only show that when working in the quotient of K ⊗ V ∗ ⊗ V by the K -subspacespanned by B (1) ∩ e B (1) = { df i θ k : i ∈ [ ℓ ] \ { i } , k ∈ [ ℓ ] } , a nonsingular matrix in K ℓ × ℓ expresses the images of the elements { dθ k : k ∈ [ ℓ ] } = B (1) \ B (1) ∩ e B (1) uniquely in terms of the images of the elements { df i θ k : k ∈ [ ℓ ] } = e B (1) \ B (1) ∩ e B (1) . Here is how one produces this ℓ × ℓ matrix. First use Lemma 7.2 to uniquely write(7.6) dθ k = X i,j ∈ [ ℓ ] r i,j,k df i θ j for each k ∈ [ ℓ ] , with r i,j,k in ( J Q ) − ℓ S . Then the matrix in K ℓ × ℓ that we wish to show is nonsingular is( r i ,j,k ) j,k ∈ [ ℓ ] . NVARIANT DERIVATIONS AND DIFFERENTIAL FORMS 15
To this end, apply to (7.6) the map E ⊗ V from Definition 7.1, giving a system of equationsin K ⊗ ⊗ V : e ∗ k θ k = X i,j ∈ [ ℓ ] r i,j,k deg( f i ) f i θ j for each k ∈ [ ℓ ] . Since { θ j } j ∈ [ ℓ ] forms a K -basis for K ⊗ V , this gives a linear system in K :(7.7) e ∗ k δ j,k = X i ∈ [ ℓ ] r i,j,k deg( f i ) f i for each j, k ∈ [ ℓ ] , where δ j,k denotes the Kronecker delta function.To show that ( r i ,j,k ) j,k ∈ [ ℓ ] in K ℓ × ℓ is nonsingular, we will evaluate each of its entries at acarefully chosen vector v . By the hypothesis (7.4), one can choose a vector v in V reg withthe property that f i ( v ) = 0 for i = i . Since the coefficients r i,j,k lie in ( J Q ) − ℓ S , and since J, Q vanish nowhere on V reg , one may evaluate the linear system (7.7) at v to obtain a linearsystem over C :(7.8) e ∗ k δ j,k = r i ,j,k ( v ) deg( f i ) f i ( v ) for each j, k ∈ [ ℓ ] . We claim f i ( v ) = 0: otherwise f i ( v ) = 0 for every i in [ ℓ ], meaning v is in the commonzero locus within V of the homogeneous system of parameters f , . . . , f ℓ in S , forcing thecontradiction v = 0. As the coexponents e ∗ k are also nonzero, (7.8) shows that the specializedmatrix ( r i ,j,k ( v )) j,k ∈ [ ℓ ] in C ℓ × ℓ is diagonal with nonzero determinant. Hence it is nonsingular,and so is the unspecialized matrix ( r i ,j,k ) j,k ∈ [ ℓ ] in K ℓ × ℓ , as desired. The case m = ℓ . To show that B ( ℓ ) = { df [ ℓ ] \{ i } dθ k } k ∈ [ ℓ ] is K -linearly independent, considera dependence 0 = X k ∈ [ ℓ ] c k df [ ℓ ] \{ i } dθ k . Substitute the expressions for dθ k from Equation (7.6) to obtain0 = X i,j,k ∈ [ ℓ ] c k r i,j,k df [ ℓ ] \{ i } df i θ j = X j,k ∈ [ ℓ ] c k ( − i r i ,j,k df [ ℓ ] θ j . But { df [ ℓ ] θ j } j ∈ [ ℓ ] is a K -basis for K ⊗ ∧ ℓ ( V ∗ ) ⊗ V by Lemma 7.2, hence,0 = X k ∈ [ ℓ ] c k ( − i r i ,j,k for each j ∈ [ ℓ ] . The matrix ( r i ,j,k ) j,k ∈ [ ℓ ] was already shown nonsingular in the m = 1 case, and hence c k = 0for each k . The intermediate cases ≤ m ≤ ℓ − . To show that B ( m ) is K -linearly independent, considera dependence(7.9) 0 = X I ∈ (cid:0) [ ℓ ] \{ i } m − (cid:1) k ∈ [ ℓ ] c I,k df I dθ k + X I ∈ (cid:0) [ ℓ ] \{ i } m (cid:1) k ∈ [ ℓ ] c I,k df I θ k . It suffices to show all coefficients c I,k in the first sum vanish: If so, then the second sum givesa dependence among a subset of the K -basis e B ( m ) from Lemma 7.2, and hence its coefficients c I,k must also vanish. To this end, fix a subset I ∈ (cid:0) [ ℓ ] \{ i } m − (cid:1) and consider its complementarysubset within [ ℓ ] \ { i } , namely, I c := [ ℓ ] \ { i } \ I . Since | I c | = ( ℓ − − ( m −
1) = ℓ − m, we note that • I c ∩ I = ∅ for each I ⊂ [ ℓ ] \ { i } with | I | = m , and • I c ∩ I = ∅ for each I ⊂ [ ℓ ] \ { i } with | I | = m − I = I .Consequently, multiplying both sides of Equation (7.9) by df I c causes all terms in the secondsum to vanish, as well as most of the terms in the first sum, leaving only0 = X k ∈ [ ℓ ] ± c I ,k df [ ℓ ] \{ i } dθ k , with sign corresponding to that in df I c ∧ df I = ± df [ ℓ ] \{ i } . But then by the case m = ℓ already proven, the coefficients c I ,k = 0 all vanish, and thus the coefficients in the first sumof Equation (7.9) vanish, as desired. This completes the proof of Theorem 7.3. (cid:3) Numerology of duality groups
We now fix our focus on duality groups and the candidate basis for ( S ⊗ ∧ m V ∗ ⊗ V ) W given in Theorem 1.1. We check in this section that these sets, comprising the putative basis,have appropriate degree sum. Let(8.1) B ( m ) = n df I θ k o I ∈ (cid:0) [ ℓ − m (cid:1) , k ∈ [ ℓ ] ⊔ n df I dθ k o I ∈ (cid:0) [ ℓ − m − (cid:1) , k ∈ [ ℓ ] for 0 ≤ m ≤ ℓ . Here one interprets the second set as empty when m = 0 and the first set as empty when m = ℓ . Lemma 8.1.
For a duality group W , the sum of the degrees of elements in B ( m ) above is ∆( ∧ m V ∗ ⊗ V ) = ( ℓ − (cid:0) ℓ − m − (cid:1) N + (cid:0) ℓ − m (cid:1) N ∗ . Proof.
Using the shorthand notation e I := P i ∈ I e i for subsets I ⊂ [ ℓ ], note thatdeg( df I θ k ) = e I + e ∗ k and deg( df I dθ k ) = e I + e ∗ k − , and therefore the sum of degrees for B ( m ) is X I ∈ (cid:0) [ ℓ − m (cid:1) ,k ∈ [ ℓ ] ( e I + e ∗ k ) + X I ∈ (cid:0) [ ℓ − m − (cid:1) ,k ∈ [ ℓ ] ( e I + e ∗ k − X k ∈ [ ℓ ] X I ∈ (cid:0) [ ℓ − m (cid:1) e ∗ k + X I ∈ (cid:0) [ ℓ − m − (cid:1) ( e ∗ k − + X k ∈ [ ℓ ] X I ∈ (cid:0) [ ℓ − m (cid:1) e I + X I ∈ (cid:0) [ ℓ − m − (cid:1) e I . Now the first sum over k can be rewritten as (cid:0) ℓ − m (cid:1) N ∗ + (cid:0) ℓ − m − (cid:1) ( N ∗ − ℓ ) = (cid:0) ℓm (cid:1) N ∗ − ℓ (cid:0) ℓ − m − (cid:1) , NVARIANT DERIVATIONS AND DIFFERENTIAL FORMS 17 while the second sum over k can be rewritten as ℓ X i ∈ [ ℓ − (cid:0) ℓ − m − (cid:1) e i + X i ∈ [ ℓ − (cid:0) ℓ − m − (cid:1) e i = ℓ (cid:0) ℓ − m − (cid:1) ( N − ( h − P i ∈ [ ℓ − e i = ( P i ∈ [ ℓ ] e i ) − e ℓ = N − ( h −
1) by Equation (4.3), as e ℓ + 1 = deg( f ℓ ) = h by definition. Hence the degree sum is (cid:0) ℓm (cid:1) N ∗ + ℓN (cid:0) ℓ − m − (cid:1) − ℓh (cid:0) ℓ − m − (cid:1) = (cid:0) ℓ − m (cid:1) N ∗ + ( ℓ − (cid:0) ℓ − m − (cid:1) N = ∆( ∧ m V ∗ ⊗ V ) , where the first equality used the duality group equation N + N ∗ = hℓ (see (6.2)). (cid:3) Duality groups and proof of Theorem 1.1
We now investigate differential derivations invariant under duality groups and prove Theo-rem 1.1. We combine the linear independence results from Section 7 with the numerology ofthe last section. We first check that the hypothesis (7.4) in Theorem 7.3 holds for all dualitygroups when one chooses the index i = ℓ . We emphasize that although both Lemma 9.1 andits consequence Corollary 9.3 below could easily be checked case-by-case, we give case-freeproofs so that Theorem 1.1 relies on no classification of reflection groups. Lemma 9.1.
Irreducible complex reflection groups have exactly one coexponent equal to .Proof. Since V is a nontrivial irreducible W -representation, and since the polynomial ring S = Sym( V ∗ ) carries the trivial representation in its degree zero component S and therepresentation V ∗ in its degree one component S , Schur’s Lemma impliesdim( S ⊗ V ) W = dim( ⊗ V ) W = dim V W = 0 , dim( S ⊗ V ) W = dim( V ∗ ⊗ V ) W = dim Hom C ( V, V ) W = dim Hom C W ( V, V ) = 1 . Hence among the S W -basis elements θ , . . . , θ ℓ for ( S ⊗ V ) W , there must be none of degreezero, and exactly one of degree one; in fact, the latter must be a multiple of the Eulerderivation θ E := x ⊗ y + · · · + x ℓ ⊗ y ℓ . (cid:3) Remark 9.2.
The above proof shows more generally that even for non-reflection (finite)groups W acting nontrivially and irreducibly on V = C ℓ , there will be, up to scaling, onlythe Euler derivation θ E as a W -invariant derivation in ( S ⊗ V ) W of degree one. Corollary 9.3.
For any duality group W , there is a unique highest exponent e ℓ and accom-panying unique highest degree h = deg( f ℓ ) = e ℓ + 1 . Lemma 9.4.
Duality groups W satisfy hypothesis (7.4) in Theorem 7.3 with i = ℓ , that is, V reg ∩ ℓ − \ i =1 f − i { } 6 = ∅ . Proof. (cf. [4, p. 4]) Springer [30, Prop. 3.2(i)] showed that if W is a complex reflectiongroup with basic invariants f , . . . , f ℓ and ζ is any primitive d th root of unity in C , then(9.1) \ i =1 ,...,ℓ : d ∤ deg( f i ) f − i { } = [ g ∈ W ker( ζ V − g ) . On the other hand, Lehrer and Michel [18, Thm. 1.2] showed existence of g in W withker( ζ V − g ) ∩ V reg = ∅ if and only if d divides as many degrees deg( f i ) = e i + 1 as codegrees e ∗ i −
1. For a duality group W , the equations e i + e ∗ i = h = deg( f ℓ ) imply h divides as manydegrees as codegrees. Also, Corollary 9.3 implies that f ℓ is the only basic invariant of degree h , so that (9.1) gives the result. (cid:3) We can now deduce the two equivalent statements of our main result. Recall that R is theexterior subalgebra of the W -invariant forms ( S ⊗ ∧ V ∗ ) W = V S W { df , . . . , df ℓ } generated byall df i except for the last one df ℓ , that is, R := V S W { df , . . . , df ℓ − } . Theorem 1.1.
For W a duality (well-generated) complex reflection group, ( S ⊗ ∧ V ∗ ⊗ V ) W forms a free R -module on R -basis { θ , . . . , θ ℓ , dθ , . . . , dθ ℓ } . Equivalently, ( S ⊗ ∧ m V ∗ ⊗ V ) W for ≤ m ≤ ℓ has S W -basis B ( m ) = n df I θ k o I ∈ (cid:0) [ ℓ − m (cid:1) , k ∈ [ ℓ ] ⊔ n df I dθ k o I ∈ (cid:0) [ ℓ − m − (cid:1) , k ∈ [ ℓ ] . Proof of Theorem 1.1.
Lemma 9.4 and Theorem 7.3 imply that B ( m ) has nonsingular coeffi-cient matrix. Indeed, when i = ℓ , the set B ( m ) in (7.5) agrees with that in (8.1). Note thatthis set has cardinality (cid:0) ℓ − m − (cid:1) ℓ + (cid:0) ℓ − m (cid:1) ℓ = (cid:0) ℓm (cid:1) ℓ = dim K ( K ⊗ ∧ m V ∗ ⊗ V ) = rank S W ( S ⊗ ∧ m V ∗ ⊗ V ) W . Lemma 8.1 shows that their degree sum is appropriate, and the theorem then follows fromCorollary 5.5. (cid:3)
Remark 9.5.
Theorem 1.1 has an amusingly compact rephrasing: defining by convention f := 1, the last assertion of the theorem is equivalent to the assertion that ( S ⊗ ∧ m V ∗ ⊗ V ) W is a free S W -module on basis(9.2) { df i m · · · df i · d ( f i θ k ) : 0 ≤ i < i < · · · < i m ≤ ℓ − k ∈ [ ℓ ] } . The reason is that the elements in (9.2) with i = 0 coincide with the basis elements { df I dθ k } in the second part of B ( m ) , while the elements in (9.2) with i ≥ { df I θ k } in the first part of B ( m ) , but differ from them by f i times an elementin the second part of B ( m ) .10. Two dimensional reflection groups
We now consider reflection groups acting on 2-dimensional complex space, i.e., the casewhen ℓ = 2. We found in Theorem 1.1 an S W -basis for ( S ⊗ ∧ V ∗ ⊗ V ) W when W is a dualitygroup. Here, we find a different choice of basis that works for any rank 2 complex reflectiongroup W , duality or not. Suppose we have basic derivations in ( S ⊗ V ) W = ( S ⊗ ⊗ V ) W (10.1) θ := x ⊗ ⊗ y + x ⊗ ⊗ y (= θ E ) , and θ := a ⊗ ⊗ y + b ⊗ ⊗ y , for some a, b in C [ x , x ]. With this indexing, e ∗ = 1 and e ∗ = deg( a ) = deg( b ). Theorem 10.1.
For a complex reflection group W acting on C , and ≤ m ≤ , thefollowing sets B ( m ) give free S W -bases for ( S ⊗ ∧ m V ∗ ⊗ V ) W : (10.2) B (0) := { θ , θ }B (1) := { df θ , df θ , dθ , dθ }B (2) := { df dθ , df dθ } . NVARIANT DERIVATIONS AND DIFFERENTIAL FORMS 19
Proof.
The m = 0 case is immediate. For m = 1 ,
2, the basic derivations as in (10.1) giveCoef( B (2) ) = df dθ df dθ ⊗ x ∧ x ⊗ y − ∂f ∂x − ∂f ∂x ⊗ x ∧ x ⊗ y ∂f ∂x ∂f ∂x , Coef( B (1) ) = df θ df θ dθ dθ ⊗ x ⊗ y x ∂f ∂x x ∂f ∂x ∂a∂x ⊗ x ⊗ y x ∂f ∂x x ∂f ∂x ∂b∂x ⊗ x ⊗ y x ∂f ∂x x ∂f ∂x ∂b∂x ⊗ x ⊗ y x ∂f ∂x x ∂f ∂x ∂a∂x . One now computes that B (1) , B (2) have the right degree sums and satisfy the hypotheses ofCorollary 5.5:det Coef( B (2) ) = ∂f ∂x ∂f ∂x − ∂f ∂x ∂f ∂x = det Jac( f , f ) = J = J ( ℓ − ( ℓ − m − ) Q ( ℓ − m ) for ℓ = 2 = m, det Coef( B (1) ) = (cid:16) ∂f ∂x ∂f ∂x − ∂f ∂x ∂f ∂x (cid:17) (cid:16) x (cid:16) x ∂b∂x + x ∂b∂x (cid:17) − x (cid:16) x ∂a∂x + x ∂a∂x (cid:17)(cid:17) = J ( x e ∗ b − x e ∗ a )= e ∗ J det( M ( θ , θ )) = e ∗ J Q = e ∗ J ( ℓ − ( ℓ − m − ) Q ( ℓ − m ) for ℓ = 2 , m = 1 . (cid:3) This gives an immediate Hilbert series corollary when ℓ = 2. Corollary 10.2.
For a complex reflection group W acting on C , Hilb(( S ⊗ ∧ V ∗ ⊗ V ) W ; q, t )Hilb( S W ; q ) = t ( q + q e ∗ ) + t (1 + q e ∗ − + q e +1 + q e +1 ) + t ( q e + q e ) . In the case of a duality group W with ℓ = 2, one can check that this agrees with description(2.1), bearing in mind that e ∗ = 1 and e ∗ + e = h = e + 1 with the above conventions.11. The reflection group G The group W = G is an irreducible complex reflection group of rank 4 containing 60reflections, each of order 2 (so N ∗ = N = 60), although it is not the complexification of aCoxeter group. It is not a duality group; the exponents are (7 , , ,
23) and the coexponentsare (1 , , , W = G (taking U = V = C ), one obtains(11.1) Hilb(( S ⊗ ∧ V ∗ ⊗ V ) W ; q, t )Hilb( S W ; q ) = (1 + q t )(1 + q t )( q + t )(1 + q )(1 + q t + q + q t ) . It is not hard to see that this is inconsistent with a description of ( S ⊗ ∧ m V ∗ ⊗ V ) W exactlyas in Theorem 1.1. However, rewriting the right side of (11.1) as(1 + q t )(1 + q t ) · ( q + t ) (cid:2) (1 + q + q + q ) + ( q t + q t + q + q t ) (cid:3) suggests a modified statement. Let R ′ := ^ S W { df , df } as a subalgebra of ( S ⊗ ∧ V ∗ ) W = V S W { df , df , df , df } . Theorem 11.1.
For W = G , the R ′ -module ( S ⊗ ∧ V ∗ ⊗ V ) W is free with R ′ -basis (cid:8) θ i , dθ i (cid:9) i =1 , , , ⊔ ( df θ , df θ , df θ , df θ ,df dθ , df dθ , df dθ , df dθ ) . Proof.
One can check that the elements listed above that lie in S ⊗ ∧ m V ∗ ⊗ V have degreesadding to ∆( ∧ m V ∗ ⊗ V ) = 60 (cid:18) (cid:18) m − (cid:19) + (cid:18) m (cid:19)(cid:19) , for 0 ≤ m ≤ . Thus the theorem follows from Corollary 5.5 after one checks that each matrix of coefficientsfor 0 ≤ m ≤ Mathematica , using explicit choices of basicinvariant polynomials f , f , f , f of degrees 8 , , ,
24 and basic derivations θ , θ , θ , θ of degrees 1 , , ,
29, constructed as prescribed by Orlik and Terao (using Maschke’s [19]invariants F , F , F , with F = det Hessian( F ); see also Dimca and Sticlaru [12] andalso [22, p. 285]). (cid:3) Proof of Theorem 1.2
We first recall the statement of the theorem.
Theorem 1.2.
For W any complex reflection group, ( S ⊗ ∧ V ∗ ⊗ V ) W is generated asa module over the exterior algebra ( S ⊗ ∧ V ∗ ) W = V S W { df , . . . , df ℓ } by the ℓ generators { θ , . . . , θ ℓ , dθ , . . . , dθ ℓ } . Proof.
The general statement follows from the case where W is irreducible. For irreducible W , we proceed case-by-case, taking advantage of the fact that the irreducible non-duality(that is, not well-generated) reflection groups fall into three camps: • The 2-dimensional groups ( ℓ = 2). • The exceptional group G (with ℓ = 4). • The infinite family of monomial groups G ( r, p, ℓ ) for 1 < p < r .Reflection groups of dimension 2 were considered in Section 10; Theorem 10.1 gives a basis.The group G was considered in Section 11; Theorem 11.1 gives a basis. The groups G ( r, p, ℓ )are considered in the appendix, as some direct computation is required to prove the patternin this general case; Theorem 14.2 gives a basis. In each case, we provided an explicit S W -module basis for ( S ⊗ ∧ V m ⊗ V ) W whose elements all have either the form df I θ k or df I dθ k for various subsets I ⊂ [ ℓ ] and k in [ ℓ ]. (cid:3) Remarks and questions
What about U = ∧ k V ? One might wonder whether for complex reflection groups W , oreven just duality groups, one can factor the Hilbert series more generally for (cid:0) S ⊗ ∧ V ∗ ⊗ ∧ k V (cid:1) W NVARIANT DERIVATIONS AND DIFFERENTIAL FORMS 21 when k takes values besides k = 0 ,
1. One can manipulate Molien-style computations usingthis consequence of Lemma 4.1:Hilb (cid:16) ( S ⊗ ∧ V ∗ ⊗ ∧ V ) W ; q, t, u (cid:17) := X i,j,k (cid:0) dim S i ⊗ ∧ j V ∗ ⊗ ∧ k V (cid:1) W q i t j u k = | W | X w ∈ W det(1 + uw − ) det(1 + tw )det(1 − qw ) . Things seem not to factor so nicely unless k ∈ { , , ℓ − , ℓ } , but at least we have a reciprocity: Proposition 13.1.
Let W be a complex reflection group and set τ ( q, t, u ) := Hilb (cid:0) ( S ⊗ ∧ V ∗ ⊗ ∧ V ) W ; q, t, u (cid:1) . Then τ satisfies the reciprocity τ ( q, t, u ) = t ℓ u ℓ τ ( q, t − , u − ) . Proof.
Let C det be a 1-dimensional W -module carrying the determinant character of W acting on V , and likewise for C det − . The W -equivariant perfect pairings ∧ j V ∗ ⊗ ∧ ℓ − j V ∗ −→ ∧ ℓ V ∗ ∼ = C det − and ∧ k V ⊗ ∧ ℓ − k V −→ ∧ ℓ V ∼ = C det imply that S ⊗ ∧ j V ∗ ⊗ C det − ∼ = S ⊗ ∧ ℓ − j V and ∧ k V ⊗ C det ∼ = ∧ ℓ − k V ∗ as W -modules (see [27], proof of Corollary 4), since V ∼ = V ∗∗ as W -modules. The result thenfollows from the isomorphisms of W -modules S ⊗ ∧ j V ∗ ⊗ ∧ k V ∼ = ( S ⊗ ∧ j V ∗ ⊗ C det − ) ⊗ ( ∧ k V ⊗ C det ) ∼ = ( S ⊗ ∧ ℓ − j V ) ⊗ ( ∧ ℓ − k V ∗ ) . (cid:3) A similar argument confirms the following.
Proposition 13.2.
Let W be a complex reflection group and set τ ( χ, q, t, u ) := Hilb (cid:0) ( S ⊗∧ V ∗ ⊗∧ V ⊗ C χ ) W ; q, t, u (cid:1) = | W | X w ∈ W χ − ( w ) det(1 + uw − ) det(1 + tw )det(1 − qw ) for any character χ : W → C ∗ afforded by a -dimensional W -module C χ . Then τ satisfiesthe reciprocity τ ( χ ; q, t, u ) = t ℓ u ℓ τ ( χ ; q, t − , u − ) . Remark 13.3.
The last two results generalize an observation for real reflection groupsfrom [13, Eqn. (1.24)].
Example 13.4.
For the Weyl group W = W ( F ), with exponents (1 , , , V ∗ ∼ = V ,a computation in Mathematica gives
Hilb (cid:16) ( S ⊗ ∧ V ⊗ ∧ V ) W ; q, t, u (cid:17) / Hilb( S W , q )= u (1 + qt )(1 + q t )(1 + q t )(1 + q t )+ u ( q + t )(1 + q + q + q )(1 + qt )(1 + q t )(1 + q t )+ u ( q + t )(1 + qt )(1 + q ) (cid:16) ( q + q − q + q + q )(1 + t ) + (1 + q + q + q + q + q ) t (cid:17) + u (1 + qt )(1 + q + q + q )( q + t )( q + t )( q + t )+ u ( q + t )( q + t )( q + t )( q + t ) . The coefficient of u does not seem to factor further, but Proposition 13.1 explains theduality between the coefficients of u k and of u ℓ − k .14. Appendix: The case of G ( r, p, ℓ )The Shephard and Todd infinite family of reflection groups G ( r, p, ℓ ) includes the Weylgroups of types B ℓ , D ℓ , the dihedral groups, and symmetric groups. To define these groups,fix an integer r ≥
1. Then G ( r, , ℓ ) is the set of ℓ × ℓ monomial matrices (i.e., matriceswith a single nonzero entry in each row and column) whose nonzero entries are complex r -throots of unity. The group G ( r, , ℓ ) is the wreath product of the symmetric group of order ℓ ! and a cyclic group: G ( r, , ℓ ) ∼ = Sym ℓ ≀ Z /r Z ∼ = Sym ℓ ⋉ ( Z /r Z ) ℓ . Each group G ( r, , ℓ ) acts on V = C ℓ as a reflection group generated by complex reflectionsof order 2 and order r . In fact, the group G ( r, , ℓ ) is the symmetry group of the complexcross-polytope in C ℓ , a regular complex polytope as studied by Shephard [25] and Coxeter [9].For integers p ≥ r , the group G ( r, p, ℓ ) consists of those matrices in G ( r, , ℓ )whose product of nonzero entries is an ( r/p )-th root-of-unity. Both G ( r, , ℓ ) and G ( r, r, ℓ )are duality groups, and hence covered by Theorem 1.1. When 1 < p < r , the group G ( r, p, ℓ )is a nonduality-group.We record here a convenient choice of basic invariant polynomials, derivations for G ( r, p, ℓ ). Proposition 14.1.
Let W = G ( r, p, ℓ ) with ≤ p < r and p dividing r . One may choosebasic W -invariant polynomials { f i } ℓi =1 in S and derivations { θ i } ℓi =1 in S ⊗ ⊗ V as follows: f k = x rk + · · · + x rkℓ for k = 1 , , . . . , ℓ − , and f ℓ = ( x · · · x ℓ ) rp ,θ k = x ( k − r +11 ⊗ ⊗ y + · · · + x ( k − r +1 ℓ ⊗ ⊗ y ℓ for k = 1 , , . . . , ℓ . In particular, W hasexponents ( e , . . . , e ℓ − , e ℓ ) = ( r − , r − , . . . , ( ℓ − r − , ℓrp − , coexponents ( e ∗ , . . . , e ∗ ℓ ) = (1 , r + 1 , r + 1 , . . . , ( ℓ − r + 1) , number of reflections N = (cid:0) ℓ (cid:1) r + ℓ (cid:16) rp − (cid:17) , andnumber of hyperplanes N ∗ = (cid:0) ℓ (cid:1) r + ℓ . Proof.
The W -invariant derivations { θ i } ℓi =1 above are the usual choice, for example, as inOrlik and Terao [22, Prop. 6.77]. (Or use the m = 0 case of Corollary 5.5 to verify the θ i are basic derivations, since their associated coefficient matrix is an easy variant of aVandermonde matrix.) The W -invariant polynomials { f i } ℓi =1 above are closely related to amore usual choice { f ′ i } ℓi =1 of basic W -invariants e ( x r , . . . , x rℓ ) , e ( x r , . . . , x rℓ ) , . . . , e ℓ − ( x r , . . . , x rℓ ) , ( x · · · x ℓ ) rp , in which e k ( x , . . . , x ℓ ) is the k th elementary symmetric polynomial in x , . . . , x ℓ , the sum ofall square-free monomials of degree k ; e.g., see Smith [28, § { f i } ℓ − i =1 and { f ′ i } ℓ − i =1 generate the same subalgebra of polynomials when working over a field of character-istic zero because the collection of power sums and elementary symmetric functions can beexpressed as polynomials in each other; e.g., see [31, Thm. 7.4.4, Cor. 7.7.2, Prop. 7.7.6]. (cid:3) NVARIANT DERIVATIONS AND DIFFERENTIAL FORMS 23
Theorem 14.2.
Let W = G ( r, p, ℓ ) with ≤ p < r and p dividing r . Then for ≤ m ≤ ℓ ,the set B ( ℓ,m ) = { df I θ k } I ∈ (cid:0) [ ℓ ] m (cid:1) , k ∈ [ ℓ − m ] ⊔ { df I dθ k } I ∈ (cid:0) [ ℓ ] m − (cid:1) , k ∈ [ ℓ − m +1] gives a free basis for the S W -module of invariants (cid:0) S ⊗ ∧ m V ∗ ⊗ V ) W .Proof. We will apply Corollary 5.5 to B ( ℓ,m ) . There are several steps. Step 1.
First note that B ( ℓ,m ) has the correct cardinality: |B ( ℓ,m ) | = ( ℓ − m ) (cid:0) ℓm (cid:1) + ( ℓ − m + 1) (cid:0) ℓm − (cid:1) = ( ℓ − m ) (cid:0) ℓm (cid:1) + m (cid:0) ℓm (cid:1) = ℓ (cid:0) ℓm (cid:1) = rank S W ( S ⊗ ∧ V ∗ ∧ V ) W . Step 2.
We check that the sum of the degrees of the elements in B ( ℓ,m ) is ∆( ∧ m V ∗ ⊗ V ),which is straightforward albeit tedious. Again using the shorthand notation e I := P i ∈ I e i for I ⊂ [ ℓ ], this sum is X I ∈ (cid:0) [ ℓ ] m (cid:1) k ∈ [ ℓ − m ] ( e I + e ∗ k ) + X I ∈ (cid:0) [ ℓ ] m − (cid:1) k ∈ [ ℓ − m +1] ( e I + e ∗ k − , which one can rewrite as (cid:0) ℓm (cid:1) X k ∈ [ ℓ − m ] e ∗ k + (cid:0) ℓm − (cid:1) X k ∈ [ ℓ − m +1] ( e ∗ k −
1) +( ℓ − m ) X I ∈ (cid:0) [ ℓ − m (cid:1) e I + X I ∈ (cid:0) [ ℓ − m − (cid:1) ( e I + e ℓ ) + ( ℓ − m + 1) X I ∈ (cid:0) [ ℓ − m − (cid:1) e I + X I ∈ (cid:0) [ ℓ − m − (cid:1) ( e I + e ℓ ) . Bearing in mind that e i = ir − ≤ i ≤ ℓ −
1, we employ a shorthand notation(14.1) g ( n, m ) := X I ∈ (cid:0) [ k ] m (cid:1) i ∈ I e i = X I ∈ (cid:0) [ k ] m (cid:1) i ∈ I ( ir −
1) = (cid:0) k − m − (cid:1) X i ∈ I ( ir −
1) = (cid:0) k − m − (cid:1) (cid:0) r (cid:0) k +12 (cid:1) − k (cid:1) to rewrite the degree sum as (cid:0) ℓm (cid:1) (cid:0) r (cid:0) ℓ − m (cid:1) + ℓ − m (cid:1) + (cid:0) ℓm − (cid:1) (cid:0) r (cid:0) ℓ − m +12 (cid:1)(cid:1) +( ℓ − m ) (cid:0) g ( ℓ − , m ) + g ( ℓ − , m −
1) + (cid:0) ℓ − m − (cid:1) e ℓ (cid:1) +( ℓ − m + 1) (cid:0) g ( ℓ − , m −
1) + g ( ℓ − , m −
2) + (cid:0) ℓ − m − (cid:1) e ℓ (cid:1) . (Here, we use the fact that e ∗ i = ( i − r + 1 for all i .) Finally, substituting in the right sideof (14.1) for all g ( n, m ), and ℓrp − e ℓ , we obtain( ℓ − (cid:0) ℓ − m − (cid:1) (cid:16)(cid:0) ℓ (cid:1) r + ℓ (cid:16) ℓrp − (cid:17)(cid:17) + (cid:0) ℓ − m (cid:1) (cid:0)(cid:0) ℓ (cid:1) r + ℓ (cid:1) = ( ℓ − (cid:0) ℓ − m − (cid:1) N + (cid:0) ℓ − m (cid:1) N ∗ = ∆( ∧ m V ⊗ V ∗ ) . The first equality here was checked by hand and corroborated in computer algebra packages.
Step 3.
At this stage, to apply Corollary 5.5, we need only show that the set B ( ℓ,m ) is K -linearly independent in K ⊗ ∧ m V ∗ ⊗ V . We will use this to reduce to the case where p = 1,that is, W = G ( r, , ℓ ).Note that the formulas for θ k , dθ k , df k in W = G ( r, p, ℓ ) depend on the parameter p in onlyone place, namely, in the definition of df ℓ :(14.2) df k = kr P ℓj =1 x kr − j ⊗ x j for 1 ≤ k ≤ ℓ − ,df ℓ = rp ( x · · · x ℓ ) rp P ℓj =1 x − i ⊗ x i ,θ k = P ℓj =1 x ( k − r +1 j ⊗ ⊗ y j for 1 ≤ k ≤ ℓ ,dθ k = (( k − r + 1) P ℓj =1 x ( k − rj ⊗ x j ⊗ y j for 1 ≤ k ≤ ℓ. In checking whether the elements of B ( ℓ,m ) are K -linearly independent, we are free to scalethem by elements of the (rational function) field K = C ( x , . . . , x ℓ ). Hence, in (14.2), we maydivide each element dθ k by ( k − r + 1 for 1 ≤ k ≤ ℓ , we may also divide each element df k by kr for 1 ≤ k ≤ ℓ −
1, and lastly we may divide df ℓ by rp ( x · · · x ℓ ) rp . Hence Theorem 14.2is equivalent to asserting K -linearly independence of the set B ( ℓ,m ) = { df I θ k } I ∈ (cid:0) [ ℓ ] m (cid:1) , k ∈ [ ℓ − m ] ⊔ { df I dθ k } I ∈ (cid:0) [ ℓ ] m − (cid:1) , k ∈ [ ℓ − m +1] for θ k , dθ k , df k for 1 ≤ k ≤ ℓ redefined (from 14.2) to give a simple and uniform family: df k := P ℓj =1 x ( k − r − j ⊗ x j ,θ k := P ℓj =1 x ( k − r +1 j ⊗ ⊗ y j ,dθ k := P ℓj =1 x ( k − rj ⊗ x j ⊗ y j . Note that we have also employed a cyclic shift of the indexing, that is, the old df ℓ has beenreplaced by the new df , the old df by the new df , etc.This new K -linear independence assertion does not involve the parameter p . ThereforeTheorem 14.2 for W = G ( r, p, ℓ ) with 1 ≤ p < r follows upon proving it for W = G ( r, , ℓ ),that is, with p = 1. Step 4.
We rescale the K -basis elements { ⊗ dx I ⊗ y k } I ∈ (cid:0) ℓm (cid:1) ,k ∈ [ ℓ ] in K ⊗ ∧ m V ∗ ⊗ V as follows: dx I ⊗ y k x − k x I ⊗ dx I ⊗ y k (recall that dx I ⊗ y k = 1 ⊗ x i ∧ · · · ∧ x i m ⊗ y k for I = { i < · · · < i m } ). Then Theorem 14.2is equivalent to the assertion that B ( ℓ,m ) is K -linearly independent after redefining(14.3) df k := P ℓj =1 x ( k − rj ⊗ x j = P ℓj =1 z k − j ⊗ x j ,θ k := P ℓj =1 x ( k − rj ⊗ ⊗ y j = P ℓj =1 z k − j ⊗ ⊗ y j ,dθ k := P ℓj =1 x ( k − rj ⊗ x j ⊗ y j = P ℓj =1 z k − j ⊗ x j ⊗ y j , for k = 1 , , . . . , ℓ , where we have set z j := x rj in K . Step 5.
Consider the matrix B ( ℓ,m ) with entries in C ( z , . . . , z ℓ ) whose columns express eachelement of B ( ℓ,m ) , defined via (14.3), in terms of { dx I ⊗ y k } for I ∈ (cid:0) [ ℓ ] m (cid:1) and k ∈ [ ℓ ]. Sinceeach dθ k is a K -linear combination of terms of the form 1 ⊗ x j ⊗ y j , the expansion of each NVARIANT DERIVATIONS AND DIFFERENTIAL FORMS 25 df I dθ k in B ( ℓ,m ) has nonzero coefficient of dx I ⊗ y k only when k ∈ I . This leads to a blockupper-triangular decomposition: B ( ℓ,m ) = { df I dθ k } { df I θ k }{ dx I ⊗ y k : k ∈ I } C ( ℓ,m ) ∗ { dx I ⊗ y k : k I } D ( ℓ,m ) . By convention here, C ( ℓ, and D ( ℓ,ℓ ) are 0 × C ( ℓ,m ) and D ( ℓ,m ) are invertible. We reduce this to showinginvertibility of only D ( ℓ,m ) , since we claim that for m = 1 , , . . . , ℓ , the matrices C ( ℓ,m ) and D ( ℓ,m − differ only by row-scalings. To justify this claim, consider a pair ( J, j ) with J = { j < · · · < j m − } for an ( m − ℓ ] and j ∈ [ ℓ − m + 1]. Then ( J, j ) indexesboth a column in C ( ℓ,m ) , that lists the expansion coefficients in(14.4) df J dθ j = X i z j − i ⊗ x i ! · · · X i z j m − − i ⊗ x i ! X i z j − i ⊗ x i ⊗ y i ! = X ( i ,...,i m − ,i ) z j − i · · · z j m − − i m − z j − i ⊗ x i ∧ · · · ∧ x i m − ∧ x i ⊗ y i , and a column in D ( ℓ,m − , that lists the expansion coefficients in(14.5) df J θ j = X i z j − i ⊗ x i ! · · · X i z j m − − i ⊗ x i ! X i z j − i ⊗ ⊗ y i ! = X ( i ,...,i m − ,i ) z j − i · · · z j m − − i m − z j − i ⊗ x i ∧ · · · ∧ x i m − ⊗ y i . On the other hand, a pair (
I, k ) where I is an m -subset of [ ℓ ] and k ∈ I will index botha row for dx I ⊗ y k in C ( ℓ,m ) and a row for dx I \{ k } ⊗ y k in D ( ℓ,m − . If one assumes that I = { i , i , . . . , i m − } in the two expansions (14.4), (14.5) above, then one can see that thesetwo rows will differ by a sign; this sign is the product of the signs of two permutations, namely,those permutations that sort the ordered sequences ( i , . . . , i m − , i ) and ( i , . . . , i m − ) intothe usual integer orders on I and I \ { k } , respectively. Step 6.
It remains to show invertibility for 0 ≤ m ≤ ℓ of the square matrix D ( ℓ,m ) , that is, thesubmatrix whose columns give the expansion coefficients in C ( z , . . . , z ℓ ) for each element of { df I θ k : I ∈ (cid:0) [ ℓ ] m (cid:1) , k ∈ [ ℓ − m ] } in terms of the basis elements { dx I ⊗ y k : I ∈ (cid:0) [ ℓ ] m (cid:1) , k I } , ignoring coefficients on all other K -basis elements dx I ⊗ y k .In fact, we will show that det D ( ℓ,m ) has coefficient ± lexicographically-largest monomial, that is, the monomial z a ℓ ℓ z a ℓ − ℓ − · · · z a z a that achieves the maximum exponent a ℓ ,and among all such monomials with maximum a ℓ , also maximizes a ℓ − , and so on. We argue via induction on ℓ by considering the following block decomposition of D ( ℓ,m ) : { df I θ k : ℓ ∈ I } { df I θ ℓ − m : ℓ I } { df I θ k : ℓ I, k = ℓ − m }{ dx I ⊗ y k : ℓ ∈ I } α ∗ ∗ { dx I ⊗ y ℓ } δ β ∗ { dx I ⊗ y k : ℓ I ⊔ { k }} φ ǫ γ . We note two degenerate cases when m = ℓ or m = ℓ −
1: if m = ℓ then D ( ℓ,ℓ ) is 0 ×
0, aspointed out in Step 5, leaving nothing to prove; if m = ℓ −
1, the sets indexing the rightmostblock of columns and the bottommost block of rows are empty, so that D ( ℓ,ℓ − = ( α ∗ δ β ) . As all rows dx I ⊗ y k have k I , the highest powers of x ℓ in entries of det D ( ℓ,m ) are • at most z ℓ − ℓ in the top block of rows ( α, ∗ , ∗ ), and z ℓ − ℓ occurs only in the block α , • at most z ℓ − − mℓ in the middle rows ( δ, β, ∗ ), and z ℓ − − mℓ occurs only in blocks δ, β , • only z ℓ in the bottom block of rows ( φ, ǫ, γ ), that is, no z ℓ ’s occur at all there.We examine the terms in the permutation expansion of det D ( ℓ,m ) that achieve the highestpower of z ℓ . Since 1 ≤ m ≤ ℓ −
1, without loss of generality, z ℓ − ℓ is a strictly higher powerthan z ℓ − − mℓ , and thus these terms must use only entries from the block α in the topmostblock of rows ( α, ∗ , ∗ ), that is, they must be terms from the product det α · det (cid:0) β ∗ ǫ γ (cid:1) . Inthe degenerate case m = ℓ −
1, they must be terms from det α · det β . In the nondegeneratecases 1 ≤ m ≤ ℓ −
2, they must be terms from det α · det β · det γ since z ℓ − − mℓ is a strictlyhigher power than z ℓ ; furthermore, these terms must always pick up entries from α divisibleby z ℓ − ℓ and entries from β divisible by z ℓ − − mℓ .Upon examining α, β, γ more closely, one finds that(14.6) γ = D ( ℓ − ,m ) ,β = z ℓ − − mℓ · ∧ m VM ( ℓ − ,α = z ℓ − ℓ D ( ℓ − ,m − + O ( z ℓ − ℓ )where ∧ m A is the m th exterior power of the matrix A , and VM ( n ) := [ z j − i ] i,j =1 , ,...,n is an n × n Vandermonde matrix . For example, when ℓ = 4 , m = 2, the matrix β is df df θ df df θ df df θ ⊗ x ∧ x ⊗ y ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ⊗ x ∧ x ⊗ y ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ⊗ x ∧ x ⊗ y ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z = z − − · ∧ VM (3) , and the matrix α (with terms with the highest power z = z ℓ − ℓ underlined) is NVARIANT DERIVATIONS AND DIFFERENTIAL FORMS 27 df df θ df df θ df df θ df df θ df df θ df df θ ⊗ x ∧ x ⊗ y ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ⊗ x ∧ x ⊗ y ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ⊗ x ∧ x ⊗ y ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ⊗ x ∧ x ⊗ y ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ⊗ x ∧ x ⊗ y ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ⊗ x ∧ x ⊗ y ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z ( z z − z z ) z . Note that here α = z − · D (3 , + O ( z − ), as asserted in (14.6).As the lex-largest monomial in det D ( ℓ,m ) has the same coefficient as in det α · det β · det γ ,the descriptions in (14.6) imply that this monomial is a power of z ℓ times the product of thelex-largest monomials indet D ( ℓ − ,m ) , det D ( ℓ − ,m − , det ∧ m VM ( ℓ − . By induction on ℓ , the coefficient on the lex-largest monomials in det D ( ℓ − ,m ) and det D ( ℓ − ,m − are both ±
1. For det ∧ m VM ( ℓ − , the Sylvester-Franke Theorem says det ∧ m A = (det A )( n − m − ),and since one has coefficient ± z ℓ − ℓ − · · · z z in det VM ( ℓ − , thesame holds for det ∧ m VM ( ℓ − . Thus this also holds for det D ( ℓ,m ) , completing the proof. (cid:3) Acknowledgments
The authors thank Corrado DeConcini, Paolo Papi, Mark Reeder and John Stembridgefor helpful conversations and references, and making them aware of their unpublished work.They also thank an anonymous referee for helpful comments.
References [1] D. Armstrong, V. Reiner and B. Rhoades, Parking spaces.
Adv. Math. (2015), 647–706.[2] Y. Bazlov, Graded multiplicities in the exterior algebra.
Adv. Math. (2001), 129–153.[3] Y. Berest, P. Etingof, and V. Ginzburg, Finite-dimensional representations of rational Cherednik algebras.
Int. Math. Res. Not. (2003), 1053–1088.[4] D. Bessis and V. Reiner, Cyclic sieving of noncrossing partitions for complex reflection groups. Ann.Comb. (2011), 197–222.[5] A. Broer, The sum of generalized exponents and Chevalley’s restriction theorem for modules of covariants. Indag. Math. (N.S.) (1995), 385–396.[6] M. Brou´e, Introduction to complex reflection groups and their braid groups. Lecture Notes in Mathematics . Springer-Verlag, Berlin, 2010.[7] M. Brou´e, G. Malle, and R. Rouquier, Complex reflection groups, braid groups, and Hecke algebras.
J.Reiner. Angew. Math. (1998), 127-190.[8] C. Chevalley, Invariants of finite groups generated by reflections.
Amer. J. Math. (1955), 778–782.[9] H.S.M. Coxeter, Regular complex polytopes, 2nd edition. Cambridge University Press, Cambridge, 1991.[10] C. De Concini and P. Papi, On some modules of covariants for a reflection group.Trans. Moscow Math.Soc. Tom 78 (2017), vyp. 2 2017, Pages 257273[11] C. De Concini, P. Papi, and C. Procesi, The adjoint representation inside the exterior algebra of asimple Lie algebra Adv. Math. (2015) 21–46.[12] A. Dimca and G. Sticlaru, On the Milnor monodromy of the exceptional reflection arrangement of type G . Doc. Math. (2018), 1–14. [13] A. Gyoja, K. Nishiyama, and H. Shimura, Invariants for representations of Weyl groups and two sidedcells, J. Math. Soc. Japan (1999), 1–34.[14] E.A. Gutkin, Matrices connected with groups generated by mappings, Funct. Anal. Appl. (Funkt. Anal.i Prilozhen) (1973) , 153–154 (81–82).[15] M. Hochster and J.A. Eagon, Cohen-Macaulay rings, invariant theory, and the generic perfection ofdeterminantal loci. Amer. J. Math. (1971), 1020–1058.[16] C.G.J. Jacobi, “De determinantibus functionalibus”, J. Reine Angew. Math. (1841), no. 4, 319–359.[17] Joseph, Anthony, Sur l’annulateur d’un module de Verma. NATO Adv. Sci. Inst. Ser. C Math. Phys.Sci., 514, Representation theories and algebraic geometry (Montreal, PQ, 1997), 237–300, Kluwer Acad.Publ., Dordrecht, 1998.[18] G.I. Lehrer and J. Michel, Invariant theory and eigenspaces for unitary reflection groups. C. R. Math.Acad. Sci. Paris (2003), 795–800.[19] H. Maschke, Ueber die quatern¨are, endliche, lineare Substitutionsgruppe der Borchardt’schen Moduln,
Math. Ann. (1887), 496–515.[20] E.M. Opdam, Complex reflection groups and fake Degrees. Technical report of the Mathematics Dept.,Univ. of Leiden, W98-17 (1998); arXiv:math/9808026 [21] P. Orlik and L. Solomon, Unitary reflection groups and cohomology,
Invent. Math. (1980), 77–94.[22] P. Orlik and H. Terao, Arrangements of hyperplanes. Grundlehren der Mathematischen Wissenschaften , . Springer-Verlag, Berlin, 1992.[23] M. Reeder, On the cohomology of compact Lie groups. Enseign. Math. (2) 41 (1995), 181–200.[24] M. Reeder, Exterior powers of the adjoint representation.
Canad. J. Math. (1997), 133–159.[25] G.C. Shephard, Regular complex polytopes. Proc. London Math. Soc. (1952), 82–97.[26] G.C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. (1954), 274–304.[27] A.V. Shepler, Generalized exponents and forms, J. of Algebraic Combin. , (2005), no. 1, 115–132.[28] L. Smith, Polynomial invariants of finite groups, Research Notes in Mathematics . A K Peters, Ltd.,Wellesley, MA, 1995.[29] L. Solomon, Invariants of finite reflection groups, Nagoya Math J. (1963), 57–64.[30] T.A. Springer, Regular elements of finite reflection groups. Invent. Math. (1974), 159–198.[31] R.P. Stanley, Enumerative combinatorics, Vol. 2. Cambridge Studies in Advanced Mathematics .Cambridge University Press, Cambridge, 1999.[32] R. Steinberg, Invariants of finite reflection groups, Canad. J. Math. (1960), 616–618.[33] J. Stembridge, First layer formulas for characters of SL ( n, C ), Trans. Amer. Math. Soc. (1987),319–350.[34] H. Terao, Free arrangements of hyperplanes and unitary reflection groups.
Proc. Japan Acad. Ser. AMath. Sci. (1980), no. 8, 389–392. E-mail address ::