GENERALIZED EXPONENTS AND FORMS

ANNE V. SHEPLER

Abstract. Real and complex finite reflection groups exhibit fascinating numerology. The exponents and coexponents of the group G arise as the degrees of the reflection representation V and its dual V^{∗} in the coinvariant algebra and also as the degrees of a generating set of invariant differential forms and invariant derivations. We consider generalized exponents (fake degrees) for any irreducible representation M of G, i.e., the degrees in which M occurs in the coinvariant algebra. Any basis of the S(V^{∗})^{G}-module of invariants in S(V^{∗}) ⊗ M^{∗} gives a linear basis for the isotypic component of the coinvariant algebra of type M in a natural way. Thus generalized exponents appear as the degrees of a generating set of G-invariants in S(V^{∗}) ⊗ M^{∗}. We investigate the case when M is a twisted reflection representation χV, where χ is a linear character of G, using the theory of semi-invariant differential forms. We give a formula for the case when G is generated by dim V reflections using Springer’s theory of regular numbers. Although our arguments are case-free, we also include the explicit exponents for each twisted reflection representation of each irreducible exceptional group. We also give a method (using differential operators) for computing semi-invariants and basic derivations. The data give an explicit basis for the corresponding isotypic component of the coinvariant algebra.

1. Introduction

The exponents and coexponents of a finite reflection group arise in various ways. Let V = ℂ^{ℓ} and let G be a finite reflection group acting on V, i.e., a finite subgroup of GL(V) generated by reflections. These groups are often called pseudo-reflection groups and they include the Weyl and Coxeter groups. We refer the reader to Orlik and Terao [12], Kane [5], and Smith [15] for basic notions. All G-modules will be ℂG-modules. For any G-module U and irreducible G-module M, let U^{M} be the isotypic component of U of type M, i.e., the direct sum of those G-submodules of U isomorphic to M. Let U^{G} = {u ∈ U : gu = u for all g ∈ G} denote the set of G-invariants (the isotypic component of the trivial representation). For any linear character χ : G → ℂ^{∗}, let ℂ_{χ} be a one-dimensional G-module affording χ and let M^{χ} = M^{ℂ_{χ}} = {m ∈ M : gm = χ(g)m for all g ∈ G} be the set of χinvariants. The group G acts contragradiently on V^{∗} and thus on the symmetric algebra S = S(V^{∗}), which we identify with the algebra of polynomial functions on V. The algebra S is naturally graded by polynomial degree. Let I ⊂ S be the ideal generated by the invariant polynomials of positive degree. Chevalley [4] (and Shephard and Todd [13]) showed that S^{G} = ℂ[f_{1}, …, f_{ℓ}] for some homogeneous polynomials f_{1}, …, f_{ℓ} called the basic invariants. Chevalley also showed

Date: April 14, 2004. Key words and phrases. reflection group, u.g.g.r, Coxeter group, Weyl group, Shephard group,

fake degree, exponents, semi-invariant forms, invariant derivations.

that S/I is isomorphic to the regular representation and that S ≃ S^{G} ⊗_{ℂ} S/I as G-modules. We call S/I the coinvariant algebra. The coinvariant algebra S/I inherits the grading on S and for any irreducible G-module M, the isotypic component (S/I)^{M} = M_{1} ⊕M_{2} ⊕⋯⊕M_{dim M} for some homogeneous subspaces M_{i} ≃ M of degree e_{i}(M). We call e_{1}(M), e_{2}(M), …, e_{dim M}(M) the M-exponents. For any linear character χ of G, we write e(χ) for the ℂ_{χ}-exponent and call it the χexponent. The V -exponents and the V^{∗}-exponents carry a special interpretation in terms of invariant vector fields and derivations which we discuss in section 3. The V^{∗}-exponents also express the cohomology of the complement of the hyperplane arrangement (see Orlik and Solomon [11, Cor. 6.62]).

Springer [18] studies these generalized exponents, and Stembridge [21] gives a combinatorial interpretation for the infinite family G(r, p, ℓ) and other wreath products. The associated generating function is called the “fake degree” (see Broué, Malle, Michel [3], for example). Ariki, Terasoma, and Yamada [1] give a basis for the coinvariant algebra for the monomial groups G(r, 1, ℓ) consisting of higher Specht polynomials associated to Young diagrams. Morita and Yamada [9] develop a theory of higher Specht polynomials for the groups G(r, p, ℓ). The exceptional reflection groups do not lend themselves to the same kind of combinatorial interpretation. We hope our approach here will be helpful in understanding the coinvariant algebra for the exceptional reflection groups.

We relate the isotypic component (S/I)^{M} with the space of “forms” S ⊗ M^{∗}. The group G acts naturally on S ⊗M^{∗} and the rank of (S ⊗M^{∗})^{G} as an S^{G}-module is dim_{ℂ} M (see [17, Lemma 2]). The module S ⊗M^{∗} also inherits a grading from S. Let q_{1}, …, q_{r} be a fixed basis of M^{∗} and suppose ω = ∑_{i} w_{i} ⊗ q_{i} ∈ S ⊗ M^{∗}. If the polynomial coefficients w_{i} are all homogeneous of degree p in S, then we say that ω is homogeneous of polynomial degree p. In section 2, we remark that the polynomial coefficients of any S^{G}-basis of (S ⊗ M^{∗})^{G} form a basis of the isotypic component (S/I)^{M}. Thus, the M-exponents are just the degrees of a homogeneous basis of (S⊗M^{∗})^{G} over S^{G}. We begin our investigation of twisted reflection representations of the form χV := V ⊗ ℂ_{χ}, where χ is a linear character of G, in Section 3 with some background and results on semi-invariant differential forms. In Section 4, we use information about semi-invariant polynomials, forms, and derivations to describe χV -exponents. The main result of this section is Corollary 13 relating χV - exponents to V -exponents, V^{∗}-exponents, χ-exponents, and \overline{χ} -exponents. We use Springer’s Theory of regular numbers in Section 4 to give additional information in the case that G is a duality group (i.e., generated by dim V reflections). In Section 5, we discuss a method for computing derivations and semi-invariants and show how this method fits with the numerology of previous sections. Computational results are given in tables at the end, although previous results are not obtained case-by-case. We include the explicit χV -exponents for all of the linear characters χ and exceptional irreducible reflection groups. Previous research has centered on Coxeter groups and the infinite family G(r, p, ℓ). We believe these explicit exponents for the exceptional groups may be helpful to others working on reflection groups and invariant theory.

2. Basis for the Isotypic Component of the Coinvariant Algebra

Solomon [17, Lemma 2] shows that for an irreducible G-module M, the Mexponents are the degrees of a homogeneous basis of (S/I ⊗ M^{∗})^{G} over S^{G}. We point out a slightly stronger result:

Proposition 1. Let M be an irreducible G-module. Then a natural G-isomorphism

M ⊗ (S ⊗ M^{∗})^{G} ≃ S^{G} ⊗ (S/I)^{M}

provides an injective map

S^{G}-bases of (S ⊗ M^{∗})^{G} ⟶ ℂ-bases of (S/I)^{M} :

the polynomial coefficients of an S^{G}-basis of (S ⊗ M^{∗})^{G} form a ℂ-basis of (S/I)^{M} when taken modulo I. Hence, the M-exponents are precisely the degrees of a homogeneous generating set of (S ⊗ M^{∗})^{G} over S^{G}.

Proof: Note that (S ⊗ M^{∗})^{G} ≃ (S^{G} ⊗ S/I ⊗ M^{∗})^{G} ≃ S^{G} ⊗ (S/I ⊗ M^{∗})^{G} by Chevalley’s Theorem and (S/I ⊗ M^{∗})^{G} ≃ Hom_{G}(M, S/I) ≃ Hom_{G}(M, (S/I)^{M}). But M ⊗ Hom_{G}(M, (S/I)^{M}) ≃ (S/I)^{M} as M is irreducible. Suppose ω_{1}, …, ω_{r} is an S^{G}-basis for (S ⊗ M^{∗})^{G} and write each ω_{k} as ω_{k} = ∑_{j=1,...r} s_{jk} ⊗ m_{j} for some fixed basis m_{1}, …, m_{r} of M^{∗}. Then under a canonical composition

M ⊗ (S ⊗ M^{∗})^{G} ≃ S^{G} ⊗ (S/I)^{M} → (S/I)^{M}

(via a ⊗ (b + I) ↦ ab + I),

{m_{j} ⊗ ω_{k} : 1 ≤ k, j ≤ r} ↦ {s_{jk} + I : 1 ≤ k, j ≤ r}.

One may verify that this last set spans (S/I)^{M} over ℂ and hence forms a linear basis. □

3. Twisted Reflection Representations and Semi-Invariant Differential Forms

We consider the special case when M is a twisted reflection representation and relate the isotypic component (S/I)^{M} to semi-invariant differential forms. Let χ : G → ℂ^{∗} be a linear character. We call χV := V ⊗ ℂ_{χ} a twisted reflection representation. If G is irreducible, Proposition 1 implies that a S^{G}-basis of (S ⊗V^{∗} ⊗ℂ \overline{χ})^{G} ≃ (S ⊗V^{∗})^{χ} yields a ℂ-basis for the isotypic component (S/I)_{χV}.

Let Ω^{p} := S ⊗ ⋀^{p} V^{∗} and Ω := ⨁_{i=0,...,ℓ} Ω^{p}. We identify ⋀^{p} V^{∗} with the space of p-linear alternating functions on V so that Ω^{p} is the set of differential p-forms on V. Choose a coordinate system x_{1}, …, x_{ℓ} on V^{∗} and let d : Ω^{p} → Ω^{p+1} be the usual exterior derivative. Note that dx_{i} = d(x_{i}⊗1) = 1⊗x_{i} under the identification Ω^{0} = S. Set vol := dx_{1} ∧ ⋯ ∧ dx_{ℓ}, the volume form on V.

The exterior derivative d commutes with the group action on Ω, and hence df_{1}, …, df_{ℓ} are invariant 1-forms. Solomon [16] showed that df_{1}, …, df_{ℓ} generate the S^{G}-module Ω^{G} as an exterior algebra: for each p, the set

{df_{i_{1}} ∧ ⋯ ∧ df_{i_{p}} : 1 ≤ i_{1} < … < i_{p} ≤ ℓ} ⊂ Ω^{p}

generates (Ω^{p})^{G} over S^{G}. We recall a related result for Ω^{χ}, the S^{G}-module of χ-invariant forms. The space of χ-invariant polynomials, S^{χ}, has rank 1 as a S^{G}- module. Let Q_{χ} ∈ S be a (homogeneous) generator:

S^{χ} = Q_{χ} S^{G}.

(Note that Q_{χ} is only defined up to a nonzero scalar.) The degree of Q_{χ} is the χexponent, e_{χ} (see Lemma 5 below). One may show that the polynomial Q_{χ} divides the exterior product of any two χ-invariant forms (see Shepler [14]) and we define a multiplication on Ω^{χ} called χ-wedging:

(1)  ω ⋏ η := \frac{ω ∧ η}{Q_{χ}} .

Let det : G → C^{∗} be the determinant character of G on V. We recall a criterion from Shepler [14] for a set of χ-invariant forms to generate the algebra Ω^{χ} over S^{G}:

Theorem 2. Let χ be a linear character of G and let ω_{1}, …, ω_{ℓ} be homogeneous χ-invariant 1-forms. Then for p = 1, …, ℓ,

(Ω^{p})^{χ} = ⨁_{1≤i_{1}<...<i_{p}≤ℓ} S^{G} ω_{i_{1}} ⋏ ⋯ ⋏ ω_{i_{p}}

if and only if ω_{1} ⋏ ⋯ ⋏ ω_{ℓ} = Q_{χ det} vol up to a nonzero scalar. The forms ω_{k} exist, and Ω^{χ} has the structure of an exterior algebra:

(Ω^{p})^{χ} = Q_{χ}^{1−p} ⋀_{S^{G}}^{p} (Ω^{1})^{χ}.

We say that ω_{1}, …, ω_{ℓ} generate Ω^{χ} if they generate Ω^{χ} over S^{G} via χ-wedging in the sense of Theorem 2. We will assume that such generators are always homogeneous. Although the ω_{i} are not unique, their degrees are unique. Proposition 1 then implies

Corollary 3. Let χ be a linear character of G . Suppose that G is irreducible and that ω_{1}, …, ω_{ℓ} generate Ω^{χ}. Write each ω_{k} as ∑_{i=1,...,ℓ} w_{ik} dx_{i}. Then {w_{ik} + I : i, k = 1, …, ℓ} is a ℂ-basis for the isotypic component (S/I)_{χV}. The degrees of a generating set of Ω^{χ} are precisely the χV -exponents.

Since df_{1}, …, df_{ℓ} generate Ω^{V} = Ω^{G}, the integers m_{1} := deg f_{1} − 1, m_{2} := deg f_{2} − 1, …, m_{ℓ} := deg f_{ℓ} − 1 are the degrees in which the representation V appears in the coinvariant algebra. These integers are called the exponents of the group. The dual representation, V^{∗}, also has special significance. We may regard S⊗V as the S-module of derivations (or vector fields) on V. Generators of (S⊗V)^{G} over S^{G} are called basic derivations (see [12, p. 234]), and we assume they are homogeneous. Although the basic derivations are not unique, their polynomial degrees m_{1}^{∗}, …, m_{ℓ}^{∗} are unique, and we call them the coexponents of the group. Thus the coexponents of the group are just the V^{∗}-exponents and the exponents of the group are just the V -exponents.

Corollary 4. Let χ be a linear character of G and let e_{1}, …, e_{ℓ} be the degrees of generators of Ω^{χ}. Then a homogeneous S^{G}-basis of (S ⊗ V)^{χ det} has degrees deg Q_{χ det} + deg Q_{χ} − e_{i} for i = 1, …, ℓ.

Proof: Let Υ^{p} := S ⊗ ⋀^{p} V. The G-equivariant perfect pairing ⋀^{p} V ⊗ ⋀^{ℓ−p} V → ℂ\overline{det} gives a degree-preserving duality between semi-invariant differential forms and vector-field forms:

(Υ^{p})^{χ·det} ≃ (Ω^{n−p})^{χ}

as S^{G}-modules. Hence, by Theorem 2, (S ⊗ V)^{χ det} = (Υ^{1})^{χ det} ≃ (Ω^{n−1})^{χ} = Q_{χ}^{2−n} ⋀_{S^{G}}^{n−1}(Ω^{1})^{χ}. Theorem 2 also implies that e_{1} + … + e_{n} = (n − 1) deg Q_{χ} + deg Q_{det χ}. Hence generators of (S ⊗ V)^{χ det} have degrees (2 − n) deg Q_{χ} + (e_{1} + … + ê_{i} + … + e_{n}) = (2 − n) deg Q_{χ} + (e_{1} + … + e_{n}) − e_{i} = deg Q_{χ} + deg Q_{det χ} − e_{i} for i = 1, …, n. □

4. Exponents of twisted reflection representations

We now collect some observations about χ-invariant forms, where χ is any linear character of the reflection group G. These observations in turn provide various combinatorial relations among χ-exponents and χV -exponents. The main result of this section is Corollary 13. Our first observation relates the χV -exponents to the \overline{χ} V -exponents.

Let A denote the collection of reflecting hyperplanes in V. For each hyperplane H in A, let s_{H} be a reflection of maximal order fixing H pointwise. Let l_{H} in V^{∗} be a linear form with H = {l_{H} = 0}. Stanley [19] gives a formula for Q_{χ}:

(2)  Q_{χ} = ∏_{H∈A} l_{H}^{a_{H}(χ)},

where a_{H}(χ) is the unique integer satisfying 0 ≤ a_{H}(χ) < order(s_{H}) and χ(s_{H}) = det(g)^{−a_{H}(χ)}. Define

Q := Q\overline{det} = ∏_{H∈A} l_{H},

the polynomial which defines the hyperplane arrangement A. Steinberg gave a proof that the determinant of the Jacobian derivative of a set of basic invariants, J(f_{1}, …, f_{ℓ}), is Q_{det} up to a nonzero scalar (see [12, Thm. 6.42]). But J(f_{1}, …, f_{ℓ}) does not lie in I (for example, see [10, Lemma 6] or [15, Corollary 6.5.2]). Since each Q_{χ} divides Q_{det}, we have the following well-known fact:

Lemma 5. For any linear character χ of G, Q_{χ} does not lie in I, and the χexponent is e(χ) = deg Q_{χ}.

The following lemma follows directly from Stanley’s formula. The lemma after gives generators of Ω^{\overline{χ}} in terms of generators of Ω^{χ}. Corollary 8 is a result of Terao [22] (see [12, 6.61]).

Lemma 6. Let χ be a linear character of G. Up to a complex scalar,

Q_{χ} Q_{\overline{χ} ·det} = Q_{det}.

Lemma 7. Let χ be a linear character of G. Suppose ω_{1}, …, ω_{ℓ} generate Ω^{χ} and let η_{i} := (Q \overline{χ} /Q_{χ}) ω_{i}. Then η_{1}, …, η_{ℓ} generate Ω^{\overline{χ}}.

Proof: We first observe that Q_{χ} divides each Q \overline{χ} ω_{k}. Fix some ω := ω_{k} = ∑_{i=1}^{ℓ} w_{i}dx_{i}. Choose H in A with a := a_{H}(χ) ≠ 0 and choose coordinates so that l_{H} = x_{1} and the matrix of the reflection s_{H} is diagonal. Since ω is invariant, x_{1}^{a} divides w^{i} whenever i ≠ 1 and x_{1}^{a−1} divides w_{1}. From Stanley’s formula for Q_{χ} (Equation 2), x_{1} divides Q \overline{χ}, and hence l_{H}^{a} divides Q \overline{χ} ω. As H was arbitrary, Q_{χ} divides Q \overline{χ} ω, and each η_{i} is a \overline{χ} -invariant 1-form. By Lemma 6 and Theorem 2,

η_{1} ∧ ⋯ ∧ η_{ℓ} = Qℓ ; \overline{χ} Q_{χ}^{−ℓ} ω_{1} ∧ ⋯ ∧ ω_{ℓ} = Qℓ ; \overline{χ} Q_{χ}^{−ℓ} Q_{χ}^{ℓ−1} Q_{χ·det} vol
= Qℓ−1 ; \overline{χ} Q_{χ}^{−1} Q_{det} vol = Qℓ−1 ; \overline{χ} Q_{\overline{χ} ·det} vol

up to a nonzero scalar. Hence, by Theorem 2, η_{1}, …, η_{ℓ} generate Ω^{\overline{χ}}. □

Corollary 8. Generators of Ω^{det} have degrees deg Q_{det} − m_{i}^{∗}, for 1 ≤ i ≤ ℓ. Generators of Ω\overline{det} have degrees deg Q\overline{det} − m_{i}^{∗} for 1 ≤ i ≤ ℓ.

Proof: We apply Corollary 4 to the case χ = det^{−1} = \overline{det} and use the fact that the invariant derivations have degrees m_{1}^{∗}, …, m_{ℓ}^{∗}. The first claim then follows from Lemma 7. □

We now relate the χV -exponents to the exponents m_{i} and the coexponents m_{i}^{∗} of G. Label these integers so that m_{1} ≤ … ≤ m_{ℓ} and m_{1}^{∗} ≥ … ≥ m_{ℓ}^{∗}.

Proposition 9. Let χ be a linear character of G and let ω_{1}, …, ω_{ℓ} generate Ω^{χ} with deg ω_{i} ≤ deg ω_{i+1}. Then deg Q_{χ} − m_{i}^{∗} ≤ deg ω_{i} ≤ deg Q_{χ} + m_{i} for each i.

Proof: The forms Q_{χ}df_{1}, …, Q_{χ}df_{ℓ} are independent over S^{G}, and hence

deg ω_{i} ≤ deg Q_{χ} df_{i} = deg Q_{χ} + m_{i}

for each i. Let μ_{1}, …, μ_{ℓ} be a set of homogeneous generators of Ω^{det} with deg μ_{k} = deg Q_{det}−m_{k}^{∗} (by Corollary 8). Note that Q_{\overline{χ} ·det} ω_{1}, …, Q_{\overline{χ} ·det} ω_{ℓ} are det-invariant forms independent over S^{G}, and hence (by Lemma 6)

deg Q_{det} − m_{i}^{∗} = deg μ_{i} ≤ deg Q_{\overline{χ} ·det} ω_{i} = deg Q_{det} − deg Q_{χ} + deg ω_{i}.
□

Proposition 10. Let χ be a nontrivial linear character of G. Suppose ω_{1}, …, ω_{ℓ} generate Ω^{χ}. Then deg ω_{i} = deg Q_{χ} − 1 for some i.

Proof: Since the 1-form dQ_{χ} is χ-invariant, dQ_{χ} = ∑_{i} h_{i}ω_{i} for some homogeneous h_{i} in S^{G}. Suppose none of the h_{i} lie in ℂ^{∗}. Then each ∂/∂x_{j}(Q_{χ}) lies in I. Hence by Euler’s formula, (deg Q_{χ}) Q_{χ} = (deg Q_{χ}) ∑_{i} x_{i} \frac{∂}{∂x_{i}} (Q_{χ}) also lies in I, contradicting Lemma 5. So some h_{i} ∈ C^{∗}, and {ω_{1}, …, ω_{i−1}, dQ_{χ}, ω_{i+1}, …, ω_{ℓ}} also generates Ω^{χ}. □

We say that the character χ is wholly non-trivial (borrowing terminology from Victor Reiner) when χ(s_{H}) ≠ 1 for each H in A. Thus χ is wholly non-trivial exactly when Q divides Q_{χ}. Stanley’s formula (Equation 2) for Q_{χ} directly implies

Lemma 11. Let χ be a linear character of G. Then χ is wholly nontrivial if and only if Q_{χ det}Q_{\overline{χ} det} = Q_{det^{2}} up to a nonzero scalar.

Proposition 12. Let χ be a linear character of G. Then χ is wholly nontrivial if and only if generators of Ω^{χ} have degrees deg Q_{χ} − m_{k}^{∗} for k = 1, …, ℓ. Furthermore, χ is trivial if and only if generators of Ω^{χ} have degrees deg Q_{χ} + m_{k} for k = 1, …, ℓ.

Proof: Recall that generators of Ω^{det} have degrees deg Q_{det} − m_{i}^{∗} for i = 1, …, ℓ (Corollary 8). Let ω_{1}, …, ω_{ℓ} generate Ω^{χ} with deg ω_{i} ≤ deg ω_{i+1}. Then

ω_{1} ∧ ⋯ ∧ ω_{ℓ} = Q_{χ}^{ℓ−1} Q_{χ det} vol

by Theorem 2, and hence

Q_{\overline{χ} det} ω_{1} ∧ ⋯ ∧ Q_{\overline{χ} det} ω_{ℓ} = Q_{\overline{χ} det} Q_{det}^{ℓ−1} Q_{χ det}vol

by Lemma 6. On the other hand, the S^{G}-module of det-invariant ℓ-forms is generated by Q_{det}^{2} vol, and thus

Q_{\overline{χ} det} ω_{1} ∧ ⋯ ∧ Q_{\overline{χ} det} ω_{ℓ} = f Q_{det}^{ℓ−1} Q_{det}^{2} vol

for some f in S^{G} (see Equation 1). Hence, Q_{\overline{χ} det} Q_{χ det} = f Q_{det}^{2}. But χ is wholly nontrivial exactly when f is a nonzero constant (by Lemma 11), exactly when the Q_{\overline{χ} det} ω_{i} generate Ω^{det} (by Theorem 2), exactly when the degree of each Q_{\overline{χ} det} ω_{i} is deg Q_{det} − m_{i}^{∗}, and thus exactly when the degree of each ω_{i} is deg Q_{χ} − m_{i}^{∗} (by Lemma 6). Also note that if each deg ω_{i} = deg Q_{χ} + m_{i}, then

deg Q_{χ}^{ℓ−1} + deg Q_{χ det} = deg ω_{1} ∧ ⋯ ∧ ω_{ℓ}
= deg Q_{χ}^{ℓ} + m_{1} + … + m_{ℓ} = deg Q_{χ}^{ℓ} + deg Q_{det},

and deg Q_{χ det} = deg Q_{χ} + deg Q_{det}. But Stanley’s formula for Q_{χ} (Equation 2) implies that deg Q_{χ det} < deg Q_{χ} + deg Q_{det} unless χ is trivial. Conversely, if χ is trivial, then Q_{χ} = 1 and we may take ω_{i} := df_{i}. □

Recall that the M-exponents of an irreducible G-module are e_{1}(M), …, e_{r}(M) and e(χ) = e_{1}(ℂ_{χ}) for any linear character χ of G. We apply Corollary 3 and Lemma 5 to Theorem 2, Propositions 9, 10, 12, Lemma 7, and Corollary 4 (using Lemma 6):

Corollary 13. Assume G is irreducible and let χ be a linear character of G. Then:

(a) e_{1}(χV) + … + e_{ℓ}(χV) = ℓe(χ) + e(χ det).

(b) e(χ) − e_{i}(V^{∗}) ≤ e_{i}(χV) ≤ e(χ) + e_{i}(V) for i = 1, …, ℓ.

(c) If χ is nontrivial, then some e_{i}(χV) = e(χ) − 1.

(d) χ is wholly nontrivial if and only if e_{i}(χV) = e(χ) − e_{i}(V^{∗}) for i = 1, …, ℓ.

(e) χ is trivial if and only if e_{i}(χV) = e(χ) + e_{i}(V) for i = 1, …, ℓ.

(f) e_{i}( \overline{χ} V) = e( \overline{χ}) − e(χ) + e_{i}(χV) for i = 1, …, ℓ.

(g) e_{i}(χ det V^{∗}) = e(det) − (e( \overline{χ}) − e(χ) + e_{i}(χV)) for i = 1, …, ℓ.

5. Springer’s Theory of Regular Elements

We are particularly interested in the case when G is generated by ℓ = dim V reflections. We recall Springer’s theory of regular elements. A vector v in V is regular if its isotropy group in G is trivial. Steinberg [20, Theorem 1.5] showed that v is regular if and only if v does not lie on a reflecting hyperplane H in A. When g in G has a regular eigenvector, then g is a regular element and the order of g is a regular number for G. Springer (see [18, Theorem 4.2]) showed that d is regular if and only if d divides some d_{i} = deg f_{i}. Springer [18, Prop. 4.5] also showed

Theorem 14. Let g be a regular element of G with order d. Let ξ = e^{\frac{2πi}{d}}. Let M be any irreducible representation of G. Then the eigenvalues of the action of g on M are ξ^{−e_{1}}, …, ξ^{−e^{deg M}} where e_{1}, …, e_{deg M} are the M-exponents.

Corollary 15. Let G be an irreducible reflection group and let χ be a linear character of G. Let d be a regular number for G. The exponents of the twisted reflection representation χV are deg Q_{χ} + m_{1}, deg Q_{χ} + m_{2}, …, deg Q_{χ} + m_{ℓ} modulo d.

Proof: Suppose ξ = e^{\frac{2πi}{d}} where d is the order of a regular element g. By Lemma 5, e(χ) = deg Q_{χ}. We apply Theorem 14 to M = ℂ_{χ}, M = V, and M = χV respectively: χ(g) = ξ^{− deg Q_{χ}} and the eigenvalues of g on V are ξ^{m_{i}}, hence the eigenvalues of g on χV are χ(g)ξ^{m_{i}} = ξ^{−m_{i}−deg Q_{χ}} for i = 1, …, ℓ. □

Corollary 15 above implies that if d is regular, then there is a permutation π of 1, …, ℓ such that the exponents and coexponents of G satisfy m_{i}+m_{π(i)}^{∗} ≡ 0 modulo d (also see Lehrer and Springer [8, Cor. 4.6]). Again, assume that m_{1} ≤ … ≤ m_{ℓ} and m_{1}^{∗} ≥ … ≥ m_{ℓ}^{∗}. We say that G is a duality group when m_{i} + m_{i}^{∗} = deg f_{ℓ} for each i. Examples include Coxeter groups and Shephard groups. Theorem 16 below implies that if G is a duality group, then d_{ℓ} is a regular number. The converse is not true, e.g., d_{ℓ} is regular for the group G_{31}, but G_{31} is not a duality group.

Orlik and Solomon [11, Theorem 5.5] observe (among other equivalences) that G is a duality group if and only if G can be generated by ℓ = dim V reflections. They examine the irreducible groups case-by-case. Bessis [2] gives a proof for this result which avoids case-by-case analysis using an observation by Lehrer and Springer [8]. Lehrer and Michel [7] give a case-free proof of this observation, which is our next theorem. The degrees of G are the integers d_{i} = m_{i} + 1. The codegrees of G are the integers d_{i}^{∗} = m_{i}^{∗} − 1.

Theorem 16. An integer d is a regular number for G if and only if d divides as many degrees as codegrees.

The following result is false for many non-duality groups.

Corollary 17. Let G be a duality group and let χ be a linear character of G. Let ω_{1}, …, ω_{ℓ} be homogeneous generators of Ω^{χ} with deg ω_{i} ≤ deg ω_{i+1}. Then deg ω_{i} = deg Q_{χ} +m_{i} or deg Q_{χ} −m_{i}^{∗} for each i. Hence, the χV -exponents are each of the form deg Q_{χ} + m_{i} or deg Q_{χ} − m_{i}^{∗} for i = 1, …, ℓ.

Proof: Since G is a duality group, d_{ℓ} is a regular number by Theorem 16 and deg ω_{i} ≡ deg Q_{χ} + m_{i} ≡ deg Q_{χ} − m_{i}^{∗} modulo d_{ℓ} by Corollary 15 and Corollary 7. The result then follows from Proposition 9. □

6. Constructing semi-invariant forms

We now give a method to construct generators for Ω^{χ} using differential operators. This method produces an explicit ℂ-basis for the isotypic component (S/I)_{χV} of the coinvariant algebra (by Proposition 1). We list the χV -exponents explicitly for the irreducible reflection groups (except the infinite family) and indicate the explicit generating χ-invariant forms in tables at the end.

We may assume that G preserves a Hermitian inner product, V × V → ℂ. The inner product induces a natural map from S(V) to S = S(V^{∗}), say p ↦ ∂p. We identify S(V) with the algebra of differential operators to obtain a natural map

\begin{matrix}S × S & ⟶ S \\ (p, f) & ↦ (∂p)f \end{matrix}

(where (∂p)f is the result of applying the differential operator ∂p to f). This map preserves the group action: (g∂p)(gf) = g(∂p(f)) for every g in G and polynomials p, f in S. This implies that the induced “star and bar” map from the product space of derivations and polynomials to the space of differential forms preserves semi-invariance:

Proposition 18. Let χ and τ be linear characters of G. The natural map

(S ⊗ V) × S ⟶ (S ⊗ V^{∗})
given by  (p ⊗ v, f) ↦ (∂p)f ⊗ ∂v
induces a map (S ⊗ V)^{τ} × S^{χ} ⟶ (S ⊗ V^{∗})^{χ \overline{τ}} .

Let θ̃f (a differential form) denote the image of a derivation θ and a polynomial f ∈ S under this map. Let ω̃f (a derivation) denote the image of a differential form ω and a polynomial f ∈ S under the analogous map (S ⊗V^{∗})^{τ} ×S^{χ} ⟶ (S ⊗V)^{χ \overline{τ}} .

Corollary 19. If f_{i} and f_{j} are basic invariants, then d̃f_{i}f_{j} is an invariant derivation. If θ is a basic derivation, then θ̃Q_{χ} is a χ-invariant 1-form.

We use the above corollary to construct basic derivations and generators for Ω^{χ}. These techniques are suggested by the numerology of Corollaries 18 and 22. Shephard and Todd [13] classify the irreducible reflection groups: there is an infinite family G(r, p, ℓ) and thirty-three exceptional groups labeled G_{4} through G_{37}. Let θ_{1}, …, θ_{ℓ} be a set of basic derivations. Assume that m_{i} = deg f_{i} ≤ deg f_{i+1} = m_{i+1} and m_{i}^{∗} = deg θ_{i} ≥ deg θ_{i+1} = m_{i+1}^{∗} . When G is a duality group, deg(d̃f_{i}f_{ℓ}) = deg f_{ℓ} − deg df_{i} = deg θ_{i}. Hence, d̃f_{1}f_{ℓ}, …, d̃f_{ℓ}f_{ℓ} form a set of invariant derivations with the same (polynomial) degrees as θ_{1}, …, θ_{ℓ}. Do they form a set of basic derivations? We may also form two sets of χ-invariant 1-forms, {θ̃_{1}Q_{χ}, …, θ̃_{ℓ}Q_{χ}} and {Q_{χ}df_{1}, …, Q_{χ}df_{ℓ}}, and then ask whether they generate Ω^{χ}. Corollary 20 suggests that a generating set of Ω^{χ} may be chosen from {θ̃_{1}Q_{χ}, …, θ̃_{ℓ}Q_{χ} , Q_{χ}df_{1}, …, Q_{χ}df_{ℓ}} when G is a duality group. We verify this suggestion in the observation below using basic invariants from Shephard and Todd [13]. The observation after suggests a pattern for nonduality groups as well. Both observations seem likely for the family G(r, p, ℓ) although we have not checked details.

Observation 20. Let G be an irreducible duality group, G ≠ G(r, p, ℓ). The basic invariants, f_{1}, …, f_{ℓ}, may be chosen so that {d̃f_{1}f_{ℓ}, …, d̃f_{ℓ}f_{ℓ}} is a set of basic derivations. Let θ_{i} := d̃f_{i}f_{ℓ} and let χ be a linear character of G. A generating set of Ω^{χ} may be chosen from {θ̃_{1}Q_{χ}, …, θ̃_{ℓ}Q_{χ}, Q_{χ}df_{1}, …, Q_{χ}df_{ℓ}}.

Observation 21. Let G be an irreducible reflection group, G ≠ G(r, p, ℓ), and let χ be a linear character of G. The basic invariants, f_{1}, …, f_{ℓ}, may be chosen so that {d̃f_{1}F_{1}, …, d̃f_{ℓ}F_{ℓ}} is a set of basic derivations, where each F_{i} is an invariant polynomial.

We give the explicit χ and χV -exponents and some illustrative examples in tables below. Klein’s invariants [6] appear in Table 1. Table 2 gives basic derivations in terms of differential operators for the exceptional groups. (We omit the Coxeter groups since the coefficients of each θ_{i} are just the coefficients of df_{n−i}.) Tables 3 and 4 list the exceptional groups and give the polynomial Q_{χ}, its degree (the χ-exponent e(χ)), and generators of Ω^{χ} and their degrees (the χV -exponents) for each linear character χ of G. We omit those duality groups whose only linear characters are \overline{det} and the trivial character, since these two cases are well understood. The symbol ▶ indicates a nonduality group throughout. The χV -exponents were first computed from character tables using a version of Molien’s theorem and the software GAP and Mathematica. It may be interesting to note that for a fixed two-dimensional exceptional group, one may compute all the semi-invariant forms and derivations from just one polynomial.

Acknowledgments. The author thanks Gus Lehrer for pointing out a shorter version of the proof of Proposition 1 and Hiroaki Terao for helpful discussions.

10 ANNE V. SHEPLER

Table 1. Klein’s Invariants for 2-dim. Groups

\begin{matrix}Φ = & x_{1}^{4} + 2i\sqrt{3} x_{1}^{2}x_{2}^{2} + x_{2}^{4} \\ Ψ = & x_{1}^{4} − 2i\sqrt{3} x_{1}^{2}x_{2}^{2} + x_{2}^{4} \\ t = & x_{1}x_{2}(x_{1}^{4} − x_{2}^{4}) \\ W = & x_{1}^{8} + 14x_{1}^{4}x_{2}^{4} + x_{2}^{8} \\ X = & x_{1}^{12} − 33x_{1}^{8}x_{2}^{4} − 33x_{1}^{4}x_{2}^{8} + x_{2}^{12} \\ f = & x_{1}x_{2}(x_{1}^{10} + 11x_{1}^{5}x_{2}^{5} − x_{2}^{10}) \\ H = & x_{1}^{20} − 228x_{1}^{15}x_{2}^{5} + 494x_{2}^{10}x_{2}^{10} + 228x_{1}^{5}x_{2}^{15} + x_{2}^{20} \\ T = & x_{1}^{30} + 522x_{1}^{25}x_{2}^{5} − 10005x_{1}^{20}x_{2}^{10} − 10005x_{1}^{10}x_{2}^{20} − 522x_{1}^{5}x_{2}^{25} + x_{2}^{30} \end{matrix}

Table 2. Basic Invariants and Basic Derivations

Group Basic Inv.  Basic Der.
f_{1}  f_{2}  θ_{1}  θ_{2}
4  Φ  t  d̃f_{1}f_{2}  d̃f_{2}f_{2}
5  t  Φ^{3}  d̃f_{1}f_{2}  d̃f_{2}f_{2}
6  t^{2}  Φ  d̃f_{1}f_{2}  d̃f_{2}f_{2}
▶ ▶ ▶ 7  Φ^{3}  t^{2}  d̃f_{1}f_{2}^{2}  d̃f_{2}f_{2}
8  W  X  d̃f_{1}f_{2}  d̃f_{2}f_{2}
9  W  X^{2}  d̃f_{1}f_{2}  d̃f_{2}f_{2}
10  X  W^{3}  d̃f_{1}f_{2}  d̃f_{2}f_{2}
▶ ▶ ▶ 11  W^{3} X^{2}  d̃f_{1}f_{2}^{2}  d̃f_{2}f_{2}
▶ ▶ ▶ 12  t  W  d̃f_{1}f_{2}^{2}  d̃f_{2}f_{2}
▶ ▶ ▶ 13  W  t^{2}  d̃f_{1}f_{2}^{2}  d̃f_{2}f_{2}
14  t  X^{2}  d̃f_{1}f_{2}  d̃f_{2}f_{2}
▶ ▶ ▶ 15  t^{2}  X^{2}  d̃f_{1}(f_{1}f_{2}) d̃f_{2}f_{2}

Group Basic Inv.  Basic Der.
f_{1}  f_{2}  θ_{1}  θ_{2}
16  H  T  d̃f_{1}f_{2}  d̃f_{2}f_{2}
17  H  T^{2}  d̃f_{1}f_{2}  d̃f_{2}f_{2}
18  T  H^{3}  d̃f_{1}f_{2}  d̃f_{2}f_{2}
▶ ▶ ▶ 19  H^{3} T^{2}  d̃f_{1}f_{2}^{2}  d̃f_{2}f_{2}
20  f  T  d̃f_{1}f_{2}  d̃f_{2}f_{2}
21  f  T^{2}  d̃f_{1}f_{2}  d̃f_{2}f_{2}
▶ ▶ ▶ 22  f  H  d̃f_{1}f_{2}^{2}  d̃f_{2}f_{2}
Group  Basic Der.θ_{i}
24–27, 29, 32–24  d̃f_{1}f_{n}  d̃f_{2}f_{n}  …  d̃f_{n}f_{n}
▶^{▶} ▶31  d̃f_{3}f_{4}^{2}  d̃f_{1}f_{4}  d̃f_{2}f_{4}  d̃f_{4}f_{4}

GENERALIZED EXPONENTS AND FORMS 11

Table 3. Semi-invariants and χV -Exponents.

Group  Q_{χ}  Deg Generators of Ω^{χ} χV Exp
G^{4}  Ψ^{2}  8  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  5, 7
Ψ  4  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  1, 3
G^{5}  ΦΨ^{2}  12  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  5, 11
Φ^{2}Ψ  12  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  5, 11
Φ  4  θ̃_{2}Q_{χ} df_{1}Q_{χ}  3, 9
Φ^{2}Ψ^{2}  16  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  9, 15
Ψ  4  θ̃_{2}Q_{χ} df_{1}Q_{χ}  3, 9
Ψ^{2}  8  θ̃_{2}Q_{χ} df_{1}Q_{χ}  7, 13
Φ^{2}  8  θ̃_{2}Q_{χ} df_{1}Q_{χ}  7, 13
ΦΨ  8  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  1, 7
G^{6}  t  6  θ̃_{2}Q_{χ} df_{1}Q_{χ}  5, 9
Ψt  10  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  1, 9
Ψ^{2}t  14  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  5, 13
Ψ  4  θ̃_{2}Q_{χ} df_{1}Q_{χ}  3, 7
Ψ^{2}  8  θ̃_{2}Q_{χ} df_{1}Q_{χ}  7, 11
▶ G_{7}  t  6  θ̃_{2}Q_{χ} df_{1}Q_{χ}  5, 17
Φ^{2}Ψt  18  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  5, 17
ΦΨ^{2}t  18  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  5, 17
Φ^{2}Ψ  12  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 11, 11
ΦΨ^{2}  12  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 11, 11
Φ^{2}Ψ^{2}t  22  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  9, 21
Φt  10  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ})  9, 9
Ψt  10  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ})  9, 9
Φ^{2}Ψ^{2}  16  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 15, 15
Φ  4  θ̃_{2}Q_{χ} df_{2}Q_{χ}  3, 15
Ψ  4  θ̃_{2}Q_{χ} df_{2}Q_{χ}  3, 15
ΦΨt  14  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  1, 13
ΨΨt  14  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 13, 13
Φ^{2}t  14  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 13, 13
ΦΨ  8  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ})  7, 7
Φ^{2}  8  θ̃_{2}Q_{χ} df_{2}Q_{χ}  7, 19
Ψ^{2}  8  θ̃_{2}Q_{χ} df_{2}Q_{χ}  7, 19

Group  Q_{χ}  Deg Generators of Ω^{χ} χV Exp
G^{8}  t^{2}  12  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  7, 11
t  6  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  1, 5
t^{3}  18  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  13, 17
G^{9}  t^{2}  12  θ̃_{2}Q_{χ} df_{1}Q_{χ}  11, 19
X  12  θ̃_{2}Q_{χ} df_{1}Q_{χ}  11, 19
t^{2}X  24  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  7, 23
t^{3}X  30  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  13, 29
t  6  θ̃_{2}Q_{χ} df_{1}Q_{χ}  5, 13
t^{3}  18  θ̃_{2}Q_{χ} df_{1}Q_{χ}  17, 25
tX  18  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  1, 17
G^{10}  t^{2}  12  θ̃_{2}Q_{χ} df_{1}Q_{χ}  11, 23
t^{3}  18  θ̃_{2}Q_{χ} df_{1}Q_{χ}  17, 29
t  6  θ̃_{2}Q_{χ} df_{1}Q_{χ}  5, 17
t^{3}W  26  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  13, 25
tW  14  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  1, 13
t^{2}W  20  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  7, 19
W  8  θ̃_{2}Q_{χ} df_{1}Q_{χ}  7, 19
t^{3}W^{2}  34  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  21, 33
tW^{2}  22  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  9, 21
t^{2}W^{2}  28  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  15, 27
W^{2}  16  θ̃_{2}Q_{χ} df_{1}Q_{χ}  15, 27
▶ G_{15}  t  6  θ̃_{2}Q_{χ} df_{2}Q_{χ}  5, 29
tX  18  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 17, 17
X  12  θ̃_{2}Q_{χ} df_{1}Q_{χ}  11, 23
tW^{2}  22  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 21, 21
tW^{2}X  34  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  9, 33
W^{2}X  28  θ̃_{2}Q_{χ} df_{1}Q_{χ}  15, 27
W^{2}  16  θ̃_{2}Q_{χ} df_{1}Q_{χ}  15, 27
tW  14  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 13, 13
tW X  26  θ̃_{2}Q_{χ} df_{1}Q_{χ}  1, 25
W X  20  θ̃_{2}Q_{χ} df_{1}Q_{χ}  7, 19
W  8  θ̃_{2}Q_{χ} df_{1}Q_{χ}  7, 19

Group  Q_{χ}  Deg Generators of Ω^{χ} χV Exp
▶ G_{11}  X  12  θ̃_{2}Q_{χ} df_{1}Q_{χ}  11, 35
t^{2}  12  θ̃_{2}Q_{χ} df_{2}Q_{χ}  11, 35
t^{2}X  24  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 23, 23
t^{3}X  30  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 29, 29
t  6  θ̃_{2}Q_{χ} df_{2}Q_{χ}  5, 29
t^{3}  18  θ̃_{2}Q_{χ} df_{2}Q_{χ}  17, 41
tX  18  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 17, 17
t^{3}W X  38  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  13, 37
tW  14  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 13, 13
t^{3}W  26  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 25, 25
tW X  26  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  1, 25
t^{2}W  20  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 19, 19
W X  20  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 19, 19
t^{2}W X  32  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  7, 31
W  8  θ̃_{2}Q_{χ} df_{2}Q_{χ}  7, 31
t^{3}W^{2}X  46  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  21, 45
tW^{2}  22  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 21, 21
t^{3}W^{2}  34  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 33, 33
tW^{2}X  34  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  9, 33
W^{2}X  28  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 27, 27
t^{2}W^{2}  28  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 27, 27
t^{2}W^{2}X  40  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  15, 39
W^{2}  16  θ̃_{2}Q_{χ} df_{2}Q_{χ}  15, 39
▶ G_{12}  X  12  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  1, 11
▶ G_{13}  tX  18  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  1, 17
t  6  θ̃_{2}Q_{χ} df_{1}Q_{χ}  5, 13
X  12  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 7, 11
G^{14}  X  12  θ̃_{2}Q_{χ} df_{1}Q_{χ}  11, 17
W X  20  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  1, 19
W  8  θ̃_{2}Q_{χ} df_{1}Q_{χ}  7, 13
W^{2}X  28  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  9, 27
W^{2}  16  θ̃_{2}Q_{χ} df_{1}Q_{χ}  15, 21

12 ANNE V. SHEPLER

Table 4. More Semi-invariants and χV -Exponents

Group Q_{χ}  Deg Generators of Ω^{χ} χV Exp
G^{16}  f^{3}  36  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  25, 35
f  12  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  1, 11
f^{4}  48  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  37, 47
f^{2}  24  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  13, 23
G^{17}  T  30  θ̃_{2}Q_{χ} df_{1}Q_{χ}  29, 49
f^{4}T  78  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  37, 77
f^{3}T  66  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  25, 65
f^{2}T  54  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  13, 53
fT  42  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  1, 41
f^{4}  48  θ̃_{2}Q_{χ} df_{1}Q_{χ}  47, 67
f^{3}  36  θ̃_{2}Q_{χ} df_{1}Q_{χ}  35, 55
f^{2}  24  θ̃_{2}Q_{χ} df_{1}Q_{χ}  23, 43
f  12  θ̃_{2}Q_{χ} df_{1}Q_{χ}  11, 31
G^{18}  f^{2}  24  θ̃_{2}Q_{χ} df_{1}Q_{χ}  23, 53
f^{4}  48  θ̃_{2}Q_{χ} df_{1}Q_{χ}  47, 77
f  12  θ̃_{2}Q_{χ} df_{1}Q_{χ}  11, 41
f^{3}  36  θ̃_{2}Q_{χ} df_{1}Q_{χ}  35, 65
H  20  θ̃_{2}Q_{χ} df_{1}Q_{χ}  19, 49
f^{2}H  44  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  13, 43
f^{4}H  68  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  37, 67
fH  32  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  1, 31
f^{3}H  56  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  25, 55
H^{2}  40  θ̃_{2}Q_{χ} df_{1}Q_{χ}  39, 69
f^{2}H^{2}  64  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  33, 63
f^{4}H^{2}  88  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  57, 87
fH^{2}  52  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  21, 51
f^{3}H^{2}  76  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  45, 75

Group  Q_{χ}  Deg Generators of Ω^{χ} χV Exp
▶ G_{19}  T  30  θ̃_{2}Q_{χ} df_{1}Q_{χ}  29, 89
f^{4}T  78  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 77, 77
f^{3}T  66  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 65, 65
f^{2}T  54  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 53, 53
fT  42  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 41, 41
f^{4}  48  θ̃_{2}Q_{χ} df_{2}Q_{χ}  47, 107
f^{3}  36  θ̃_{2}Q_{χ} df_{2}Q_{χ}  35, 95
f^{2}  24  θ̃_{2}Q_{χ} df_{2}Q_{χ}  23, 83
f  12  θ̃_{2}Q_{χ} df_{2}Q_{χ}  11, 71
HT  50  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 49, 49
f^{4}HT  98  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  37, 97
f^{3}HT  86  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  25, 85
f^{2}HT  74  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  13, 73
fHT  62  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  1, 61
H  20  θ̃_{2}Q_{χ} df_{2}Q_{χ}  19, 79
f^{4}H  68  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 67, 67
f^{3}H  56  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 55, 55
f^{2}H  44  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 43, 43
fH  32  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 31, 31
H^{2}T  70  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 69, 69
f^{4}H^{2}T 118  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  57, 117
f^{3}H^{2}T 106  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  45, 105
f^{2}H^{2}T  94  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  33, 93
fH^{2}T  82  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  21, 81
H^{2}  40  θ̃_{2}Q_{χ} df_{2}Q_{χ}  39, 99
f^{4}H^{2}  88  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 87, 87
f^{3}H^{2}  76  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 75, 75
f^{2}H^{2}  64  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 63, 63
fH^{2}  52  θ̃_{2}Q_{χ} θ̃_{1}(f_{2}Q_{χ}) 51, 51

Group Q_{χ}  Deg Generators of Ω^{χ} χV Exp
G^{20}  H^{2}  40  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  21, 39
H  20  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  1, 19
G^{21}  T  30  θ̃_{2}Q_{χ} df_{1}Q_{χ}  29, 41
H^{2}T  70  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  21, 69
H^{2}  40  θ̃_{2}Q_{χ} df_{1}Q_{χ}  39, 51
HT  50  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  1, 49
H  20  θ̃_{2}Q_{χ} df_{1}Q_{χ}  19, 31
▶ G_{22}  T  30  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  1, 29
Group deg Q_{χ}  Generators of Ω^{χ}  χV Exp
G^{25}  12  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  θ̃_{3}Q_{χ}  5, 8, 11
24  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  θ̃_{3}Q_{χ} 17, 20, 23
G^{26}  21  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  θ̃_{3}Q_{χ}  8, 14, 20
33  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ}  θ̃_{3}Q_{χ} 20, 26, 32
24  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ} df_{1}Q_{χ} 17, 23, 29
12  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ} df_{1}Q_{χ} 5, 11, 17
9  θ̃_{1}Q_{χ} df_{1}Q_{χ} df_{2}Q_{χ} 8, 14, 20
Group deg Q_{χ}  Generators of Ω^{χ}  χV Exp
G^{28}  24  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ} θ̃_{3}Q_{χ}  θ̃_{4}Q_{χ} 13, 17, 19, 23
12  θ̃_{3}Q_{χ} θ̃_{4}Q_{χ} df_{1}Q_{χ} df_{2}Q_{χ} 7, 11, 13, 17
12  θ̃_{3}Q_{χ} θ̃_{4}Q_{χ} df_{1}Q_{χ} df_{2}Q_{χ} 7, 11, 13, 17
▶ G_{31}  60  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ} θ̃_{3}Q_{χ}  θ̃_{4}Q_{χ} 31, 43, 47, 59
G^{32}  80  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ} θ̃_{3}Q_{χ}  θ̃_{4}Q_{χ} 61, 67, 73, 79
40  θ̃_{1}Q_{χ} θ̃_{2}Q_{χ} θ̃_{3}Q_{χ}  θ̃_{4}Q_{χ} 21, 27, 33, 39

References

[1] Ariki, S., Terasoma, T., and Yamada, H.-F. Higher Specht polynomials. Hiroshima Math. J. 27, 1 (March, 1997), 177–188.

[2] Bessis, D. Zariski theorems and diagrams for braid groups. To appear in Invent. Math.

[3] Broué, M., Malle, G., and Michel, J. Towards Spetses I. Transform. Groups 4 (1999), 157–218.

[4] Chevalley, C. Invariants of finite groups generated by reflections. Amer. J. Math. 77 (1955), 778–782.

[5] Kane, R. Reflection Groups and Invariant Theory, vol. 5 of CMS Books in Mathematics. Springer-Verlag, 2001.

[6] Klein, F. Gesammelte Mathematische Abhandlungen. Vol. 2. Springer, 1922.

[7] Lehrer, G., and Michel, J. Invariant theory and eigenspaces for unitary reflection groups. Submitted.

[8] Lehrer, G., and Springer, T. Reflection subquotients of unitary reflection groups. Canad. J. Math. 51, 6 (1999), 1175–1193.

[9] Morita, H., and Yamada, H.-F. Higher Specht polynomials for the complex reflection groups g(r, p, n). Hokkaido Math. J. XXVII, no. 3 (1998), 505–515.

[10] Orlik, P., Reiner, V., and Shepler, A. The sign representation for Shephard groups. Mathematische Annalen 322 (2002), 477–492.

[11] Orlik, P., and Solomon, L. Unitary reflection groups and cohomology. Invent. Math. 59 (1980), 77–94.

[12] Orlik, P., and Terao, H. Arrangements of Hyperplanes. Springer-Verlag, Berlin, 1992.

[13] Shephard, G., and Todd, J. Finite unitary reflection groups. Canad. J. Math. 6 (1954), 274–304.

[14] Shepler, A. Semi-invariants of finite reflection groups. J. Algebra 220 (1999), 314–326.

[15] Smith, L. Polynomial Invariants of Finite Groups, vol. 6 of Research Notes in Mathematics. A K Peters, 1995.

[16] Solomon, L. Invariants of finite reflection groups. Nagoya Math. J. 22 (1963), 57–64.

[17] Solomon, L. Invariants of Euclidean reflection groups. Trans. Amer. Math. Soc. 113 (1964), 274–286.

[18] Springer, T. Regular elements of finite reflection groups. Invent. Math. 25 (1974), 159–198.

[19] Stanley, R. Relative invariants of finite groups. Journal of Algebra 49 (1977), 134–148.

[20] Steinberg, R. Differential equations invariant under finite reflection groups. Trans. Amer. Math. Soc. 112 (1964), 392–400.

[21] Stembridge, J. On the eigenvalues of representations of reflection groups and wreath products. Pacific Journal of Math. 140, 2 (1989), 353–396.

[22] Terao, H. The Jacobians and the discriminants of finite reflection groups. Tohoku Math. J. 41 (1989), 237–247.

Anne V. Shepler, University of North Texas, Denton, Texas, U.S.A.