SEMIINVARIANTS OF FINITE REFLECTION GROUPS

ANNE V. SHEPLER

Abstract. Let G be a finite group of complex n × n unitary matrices generated by reflections acting on ℂ^{n}. Let R be the ring of invariant polynomials, and χ be a multiplicative character of G. Let Ω^{χ} be the R-module of χ-invariant differential forms. We define a multiplication in Ω^{χ} and show that under this multiplication Ω^{χ} has an exterior algebra structure. We also show how to extend the results to vector fields, and exhibit a relationship between χ-invariant forms and logarithmic forms.

1. Introduction

In 1989, P. Doyle and C. McMullen [2] solved the fifth degree polynomial with a highly symmetrical dynamical system which preserved the Galois group A_{5}. In 1997, S. Crass and P. Doyle [1] solved the sixth degree polynomial by again finding a dynamical system with special symmetry—this time A_{6} symmetry. Each dynamical system was formed by iterating a map that was equivariant under the projective action of the group. Such maps correspond naturally to semiinvariant differential forms. Because almost nothing was known about these forms, constructing the necessary dynamical systems was a difficult step in both cases.

We introduce here a general theory of semiinvariants. Specifically, we show that for any finite unitary reflection group G and multiplicative character χ of G, the module of χ-invariant differential forms has a natural multiplication which turns the module into an exterior algebra. This exterior algebra structure allows us to understand completely the forms that give rise to highly symmetrical dynamical systems, and gives us tools to compute these forms explicitly. We also show how to extend these results to vector fields (or derivations), and observe the relationship between semiinvariants and logarithmic forms.

The theory presented here builds on work by R. Stanley, who characterized the module of χ-invariant polynomials in 1977 [8]. It also builds on more recent work by Orlik, Saito, Solomon, Terao and others on invariant derivations and the theory of hyperplane arrangements (see [3], Chapter 6). Note that det-invariant forms have received attention under the name of anti-invariant forms in the context of Coxeter groups (see e.g. [7]).

2. notation

Let G be a finite group of complex n × n unitary matrices generated by reflections acting on V := ℂ^{n}. Recall that a unitary matrix is a reflection if it has finite order and fixes a hyperplane pointwise in V. Let S := ℂ[x_{1}, …, x_{n}] be the ring of polynomials of V. Let f_{1}, …, f_{n} ∈ S be basic invariants, and R = ℂ[f_{1}, …, f_{n}] be the ring of invariant polynomials. Let χ be a multiplicative character of G. Denote the module of differential p-forms on V by

Ω^{p} := ⨁_{1≤i_{1}<...<i_{p}≤n} Sdx_{i_{1}} ∧ … ∧ dx_{i_{p}}
≃ S ⊗ ⋀^{p}V^{∗}.

The group G acts contragradiently on V^{∗} and S, and Ω^{p} is a ℂ[G]-module. Define the R-module of χ-invariant differential p-forms as

(Ω^{p})^{χ} := {ω ∈ Ω^{p} : gω = χ(g)ω for all g ∈ G} .

Let

Ω^{χ} := ⨁_{0≤p}(Ω^{p})^{χ}.

It is convenient to define I^{p} as the set of multiindices of {1, ..., n} of length p:

I^{p} := {I = {I_{1}, ..., I_{p}} : 1 ≤ I_{1} < … < I_{p} ≤ n} .

For a multiindex I, let I^{c} denote the complementary index. Denote the volume form on V by vol := dx_{1} ∧ … ∧ dx_{n}. If f and g are differential forms, we write f ≐ g if f = cg for some c in ℂ^{∗}.

We recall some facts and notation from \underline{Arrangements of Hyperplanes} ([3], p. 228). Let A be the hyperplane arrangement defined by G. For each H ∈ A, define α_{H} ∈ S by ker(α_{H}) = H. Fix some H ∈ A, and let G_{H} be the cyclic subgroup of elements in G that fix H pointwise. Let s_{H} be a generator of G_{H} and let o(s_{H}) be the order of s_{H}. Define a_{H}(χ) as the least integer satisfying 0 ≤ a_{H}(χ) < o(s_{H}) and χ(s_{H}) = det(s_{H})^{−a_{H}(χ)}. Let

Q_{χ} = ∏_{H∈A} α_{H}^{a_{H}(χ)}.

The polynomial Q_{χ} is uniquely determined, upto a nonzero scaler multiple, by the group G.

R. Stanley [8] proved that (Ω^{0})^{χ} = R Q_{χ}, and since vol is (det^{−1})- invariant, it follows that

(Ω^{n})^{χ} = R Q_{χ·det} vol. (∗)

R. Steinberg [9] proved that Q_{det} = ∏_{H∈A} α_{H}^{o(s_{H})−1} is the determinant of

the Jacobian matrix \{\frac{∂}{∂x_{i}} f_{j}\}, upto a nonzero scalar multiple. Note also that Q_{det}_{−1} = ∏_{H∈A} α_{H} ([3], p. 229).

3. χ-wedging

The next lemma will be used to show that Q_{χ} divides the exterior product of any two χ-invariant forms.

Lemma 1. Suppose that μ is a χ-invariant p-form. Fix a hyperplane H ∈ A, and let a = a_{H}(χ). Choose coordinates in which x_{1} = α_{H} and s_{H} is diagonal. If

μ = ∑_{I∈I^{p}} μ_{I} dx_{I_{1}} ∧ … ∧ dx_{I_{p}}

in these coordinates, then x_{1}^{a−1} divides μ_{I} whenever I_{1} = 1 and x_{1}^{a} divides μ_{I} whenever I_{1} ≠ 1, for each I = {I_{1}, …, I_{p}} ∈ I^{p}.

Proof. Let s = s_{H} and ρ be the determinant of s. Then

s = (\begin{matrix}ρ \\ & 1 \\ & & . & . & . \\ & & & & & 1 \end{matrix}) ,

and s^{−1}dx_{1} = ρ dx_{1}, s^{−1}dx_{2} = dx_{2}, …, s^{−1}dx_{n} = dx_{n}.

Let I = {I_{1}, I_{2}, ..., I_{p}} ∈ I^{p}. If I_{1} = 1, then

\begin{aligned}s^{−1}(μ_{I} dx_{I_{1}} ∧ ⋯ ∧ dx_{I_{p}}) & = & s^{−1}μ_{I} s^{−1}dx_{1} ∧ ⋯ ∧ s^{−1}dx_{I_{p}} \\ & = & μ_{I} ∘ s ρ dx_{1} ∧ ⋯ ∧ dx_{I_{p}}. \end{aligned}

If I_{1} ≠ 1, then

\begin{aligned}s^{−1}(μ_{I} dx_{I_{1}} ∧ ⋯ ∧ dx_{I_{p}}) & = & s^{−1}μ_{I} s^{−1}dx_{I_{1}} ∧ ⋯ ∧ s^{−1}dx_{I_{p}} \\ & = & μ_{I} ∘ s dx_{I_{1}} ∧ ⋯ ∧ dx_{I_{p}}. \end{aligned}

But μ is χ-invariant, so ρ^{a}μ = det(s)^{a}μ = χ^{−1}(s)μ = s^{−1}μ. Hence if I_{1} = 1, then ρ^{a}μ_{I} = ρ μ_{I} ∘ s, i.e. ρ^{a−1}μ_{I} = μ_{I} ∘ s. Thus x_{1}^{a−1} divides μ_{I}. Similarly, if I_{1} ≠ 1, then ρ^{a}μ_{I} = μ_{I} ∘ s and x_{1}^{a} divides μ_{I}.

Lemma 2. Q_{χ} divides the exterior product of any two χ-invariant differential forms.

Proof. Let μ be a χ-invariant p-form and ω be a χ-invariant q-form. Fix H ∈ A. Let s = s_{H} and a = a_{H}(χ). Assume that a ≠ 0. We show that α_{H}^{a} divides μ ∧ ω by choosing coordinates from Lemma 1 in which α_{H} = x_{1}. Let

\begin{aligned}μ & = & ∑_{I∈I^{p}} & μ_{I} dx_{I_{1}} ∧ … ∧ dx_{I_{p}}, \\ ω & = & ∑_{J∈I^{q}} & ω_{J} dx_{J_{1}} ∧ … ∧ dx_{J_{q}}, and \\ μ ∧ ω & = & ∑_{K∈I^{p+q}} & γ_{K} dx_{K_{1}} ∧ … ∧ dx_{K_{p+q}} \end{aligned}

in these coordinates. Then x_{1}^{a} divides μ_{I} whenever I_{q} ≠ 1 and x_{1}^{a} divides ω_{J} whenever J_{1} ≠ 1.

Hence, for I ∈ I^{p} and J ∈ I^{q}, the polynomial μ_{I}ω_{J} is divisible by x_{1}^{a} given that not both I_{1} and J_{1} are 1. Since each γ_{K} is either zero or a sum of terms of the form ±μ_{I}ω_{J} where the multiindices I and J are disjoint, x_{1}^{a} divides each γ_{K} and hence μ ∧ ω. Thus, μ ∧ ω is divisible by α_{H}^{a} = α_{H}^{a_{H}(χ)}. Since H was arbitrary, Q_{χ} divides μ ∧ ω.

Lemma 2 prompts us to define the following multiplication in Ω^{χ}: For differential forms μ and ω, define the χ-wedge of μ and ω as

μ ⋏ ω := \frac{μ ∧ ω}{Q_{χ}} .

If μ and ω are χ-invariant forms, μ⋏ω is again χ-invariant. Thus, Lemma 2 implies

Corollary 1. The R-module Ω^{χ} is closed under χ-wedging.

The following proposition gives a condition (similar to Saito’s Criterion) for n 1-forms to generate Ω^{χ}. The proof is similar to Solomon’s original argument [6] that df_{1}, …, df_{n} generate the module of invariant differential forms.

Proposition 1. Let ω_{1}, …, ω_{n} be χ-invariant 1-forms. The forms ω_{I_{1}} ⋏ … ⋏ ω_{I_{p}}, for I ∈ I^{p} and p ≥ 0, generate Ω^{χ} over R if and only if

ω_{1} ⋏ … ⋏ ω_{n} ≐ Q_{χ·det} vol.

Proof. Assume that ω_{1}⋏…⋏ω_{n} ≐ Q_{χ·det} vol. The p-forms ω_{I_{1}}⋏…⋏ω_{I_{p}}, I ∈ I^{p}, are χ-invariant by Corollary 1.

Since ω_{1} ⋏ … ⋏ ω_{n} ≠ 0, ω_{1} ∧ … ∧ ω_{n} ≠ 0, and the forms ω_{I_{1}} ∧ … ∧ ω_{I_{p}}, I ∈ I^{p}, are linearly independent over F := ℂ(x_{1}, …, x_{n}). If not, there exist rational functions r_{I} with

0 = ∑_{I∈I^{p}} r_{I} ω_{I_{1}} ∧ ⋯ ∧ ω_{I_{p}}.

Fix J ∈ I^{p} and J^{c} ∈ I^{n−p}. Then

0 = ( ∑_{I∈I^{p}} r_{I} ω_{I_{1}} ∧ ⋯ ∧ ω_{I_{p}}) ∧ ω_{J_{1}^{c}} ∧ ⋯ ∧ ω_{J_{n−p}^{c}}
= ± r_{J} ω_{1} ∧ ⋯ ∧ ω_{n},

and r_{J} must be zero. Hence the forms

ω_{I_{1}} ⋏ … ⋏ ω_{I_{p}} = (Q_{χ})^{1−p} ω_{I_{1}} ∧ … ∧ ω_{I_{p}}, I ∈ I^{p},

are also linearly independent over F, and thus span

Ω^{p}(V) := ⨁_{I∈I^{p}} F dx_{I_{1}} ∧ … ∧ dx_{I_{p}}

since Ω^{p}(V) has dimension \binom{n}{p}.

Choose an arbitrary χ-invariant p-form μ. Then there exist rational functions t_{I} ∈ F with

μ = ∑_{I∈I^{p}} t_{I} ω_{I_{1}} ⋏ ⋯ ⋏ ω_{I_{p}}.

Fix J ∈ I^{p} and its complementary index J^{c}. We will show that t_{J} ∈ R.

By Corollary 1, the n-form (ω_{J_{1}^{c}} ⋏ ⋯ ⋏ ω_{J_{n−p}^{c}}) ⋏ μ is χ-invariant. Thus by Equation (∗) above, there exists a polynomial f ∈ R with

(ω_{J_{1}^{c}} ⋏ ⋯ ⋏ ω_{J_{n−p}^{c}}) ⋏ μ = f Q_{χ·det} vol.

On the other hand,

(ω_{J_{1}^{c}}⋏ ⋯ ⋏ ω_{J_{n−p}^{c}}) ⋏ μ

= (ω_{J_{1}^{c}} ⋏ ⋯ ⋏ ω_{J_{n−p}^{c}}) ⋏ ∑_{I∈I^{p}} t_{I} ω_{I_{1}} ⋏ ⋯ ⋏ ω_{I_{p}}

= (Q_{χ}^{1−n}) (ω_{J_{1}^{c}} ∧ ⋯ ∧ ω_{J_{n−p}^{c}}) ∧ ∑_{I∈I^{p}} t_{I} ω_{I_{1}} ∧ ⋯ ∧ ω_{I_{p}}

= (Q_{χ}^{1−n}) ± t_{J} ω_{1} ∧ ⋯ ∧ ω_{n}

= ± t_{J} ω_{1} ⋏ ⋯ ⋏ ω_{n}

≐ ± t_{J} Q_{χ·det} vol.

Thus f Q_{χ·det} ≐ ±t_{J} Q_{χ·det}. Hence, t_{J} ∈ R. Since J was arbitrary, μ is in the R-span of {ω_{I_{1}} ⋏ … ⋏ ω_{I_{p}}, I_{p} ∈ I^{p}}.

The converse follows from Equation (∗) above.

4. Condition satisfied

Since Ω^{p} has rank \binom{n}{p}, the R-module (Ω^{p})^{χ} is also free of rank \binom{n}{p} (this follows from Lemma 6.45 of [3], p. 232). We will show that the generators of (Ω^{1})^{χ} satisfy the condition given in Proposition 1, but first we must gather some preliminary facts.

We recall some results about invariant vector fields. There exist n invariant vector fields, called basic derivations, that generate the module of invariant vector fields over R (see [3], Section 6.3). Using Saito’s Criterion, H. Terao showed that the coefficient matrix of the basic derivations has determinate Q_{det}_{−1} upto a nonzero scaler multiple (see [3], p. 238). Using the minors of this coefficient matrix, we construct (det^{−1})-invariant 1-forms, μ_{1}, …, μ_{n}, that satisfy

μ_{1} ∧ ⋯ ∧ μ_{n} = Q_{det^{−1}}^{n−1} vol.

The forms μ_{1}, …, μ_{n} thus generate Ω^{det^{−1}} over R by Proposition 1. We will use these forms to give an argument for arbitrary χ.

We also note the relationship between Q_{χ·det} and Q_{χ}: Fix H ∈ A with a_{H}(χ) ≠ 0. The exponent a_{H}(χ · det) is the least nonnegative integer satisfying

det(s_{H})^{−a_{H}(χ·det)} = (χ · det)(s_{H})
= χ(s_{H}) det(s_{H})
= det(s_{H})^{−a_{H}(χ)} det(s_{H})
= det(s_{H})^{−(a_{H}(χ)−1)}.

Hence, a_{H}(χ · det) = a_{H}(χ) − 1. Now fix H ∈ A with a_{H}(χ) = 0. Then

det(s_{H})^{−a_{H}(χ·det)} = (χ · det)(s_{H})
= χ(s_{H}) det(s_{H})
= det(s_{H})
= det(s_{H})^{−(o(s_{H})−1)},

and a_{H}(χ · det) = o(s_{H}) − 1. Thus,

Q_{χ·det} = ∏_{H∈A} α_{H}^{a_{H}(χ·det)}
= _{∏_{H∈A} ; χ(s_{H})≠1}α_{H}^{a_{H}(χ)−1} _{∏_{H∈A} ; χ(s_{H})=1}α_{H}^{o(s_{H})−1}.

Proposition 2. If ω_{1}, …, ω_{n} generate (Ω^{1})^{χ} over R, then

ω_{1} ⋏ … ⋏ ω_{n} ≐ Q_{χ·det} vol.

Proof. Let M be the coefficient matrix of ω_{1}, …, ω_{n}, i.e. ω_{1} ∧ … ∧ ω_{n} = det M vol. Suppose that det M = 0. Then one row of M is a linear combination of the other rows over F = ℂ(x_{1}, …, x_{n}). Multiplying by a least common multiple yields a relation over S: ∑_{i=1}^{n} s_{i} ω_{i} = 0. To get a relation over R, apply a group element g, multiply by χ^{−1}(g), and then sum over G:

0 = ∑_{g∈G} ∑_{i=1}^{n} χ^{−1}(g) gs_{i} gω_{i}

= ∑_{i=1}^{n} ∑_{g∈G} χ^{−1}(g) gs_{i} χ(g) ω_{i}

= ∑_{i=1}^{n} (∑_{g∈G} gs_{i}) ω_{i}.

This contradicts the fact that (Ω^{1})^{χ} is free over R with basis ω_{1}, …, ω_{n}. Hence, det M ≠ 0.

By Corollary 1, ω_{1}⋏⋯⋏ω_{n} is a χ-invariant n-form. Thus (from Equation (∗)) there exists a nonzero f ∈ R with

(Q_{χ})^{1−n} det M vol = (Q_{χ})^{1−n} ω_{1} ∧ … ∧ ω_{n} = ω_{1} ⋏ ⋯ ⋏ ω_{n} = f Q_{χ·det} vol.

Hence, det M = f Q_{χ·det} (Q_{χ})^{n−1}.

We show that f is constant by finding two polynomials that share no factors, yet are each divisible by f. Since each df_{i} is invariant, each Q_{χ} df_{i} is χ-invariant and hence a combination of ω_{1}, …, ω_{n} over R. There exists a matrix of coefficients, N, with entries in S, such that

Q_{χ}df_{1} ∧ ⋯ ∧ Q_{χ}df_{n} = det M det N vol
= f Q_{χ·det} (Q_{χ})^{n−1} det N vol.

But, df_{1} ∧ ⋯ ∧ df_{n} ≐ Q_{det}, so

Q_{χ}df_{1} ∧ ⋯ ∧ Q_{χ}df_{n} ≐ (Q_{χ})^{n} Q_{det} vol.

Hence,

f Q_{χ·det} det(N) ≐ Q_{χ} Q_{det}

and since det N ∈ S, f divides Q_{χ} Q_{det} (Q_{χ·det})^{−1}.

Since each μ_{i} (introduced above) is (det^{−1})-invariant, each Q_{χ·det} μ_{i} is χ-invariant, and thus a R-combination of ω_{1}, …, ω_{n}. There exists a matrix of coefficients, N′, with coefficients in S, such that

Q_{χ·det}μ_{1} ∧ ⋯ ∧ Q_{χ·det}μ_{n} = det M det N′ vol
= f Q_{χ·det} (Q_{χ})^{n−1} det N′ vol.

But we choose the μ_{i} so that

Q_{χ·det}μ_{1} ∧ ⋯ ∧ Q_{χ·det}μ_{n} = (Q_{χ·det})^{n} (Q_{det}_{−1})^{n−1} vol.

Hence,

f Q_{χ·det} (Q_{χ})^{n−1} det N′ = (Q_{χ·det})^{n} (Q_{det}_{−1})^{n−1}

and since det N′ ∈ S, (Q_{χ·det}Q_{det}_{−1})^{n−1} (Q_{χ})^{1−n} is divisible by f.

We show that the two polynomials

Q_{χ} Q_{det} (Q_{χ·det})^{−1} and (Q_{χ·det}Q_{det}_{−1})^{n−1} (Q_{χ})^{1−n}

have no common factors by writing them both in terms of the α_{H}. We expand the factors:

Q_{χ} = _{∏_{H∈A} ; χ(s_{H})≠1}α_{H}^{a_{H}(χ)} ,

Q_{det} = _{∏_{H∈A} ; χ(s_{H})≠1}α_{H}^{o(s_{H})−1} _{∏_{H∈A} ; χ(s_{H})=1}α_{H}^{o(s_{H})−1} ,

Q_{χ·det} = _{∏_{H∈A} ; χ(s_{H})≠1}α_{H}^{a_{H}(χ)−1} _{∏_{H∈A} ; χ(s_{H})=1}α_{H}^{o(s_{H})−1} ,

Q_{det}_{−1} = _{∏_{H∈A} ; χ(s_{H})≠1}α_{H}  _{∏_{H∈A} ; χ(s_{H})=1}α_{H} .

The first polynomial, Q_{χ} Q_{det} (Q_{χ·det})^{−1}, simplifies to

∏_{H∈A} ; χ(s_{H})≠1α_{H}^{o(s_{H})} ,

and the second polynomial, (Q_{χ·det} Q_{det}_{−1})^{n−1} (Q_{χ})^{1−n}, simplifies to

(_{∏_{H∈A} ; χ(s_{H})=1}α_{H}^{o(s_{H})})^{n−1} .

Since f divides both polynomials, f must be constant. Thus, ω_{1}, …, ω_{n} satisfy the criterion of Proposition 1.

Corollary 2. There exist n 1-forms ω_{1}, …, ω_{n} such that Ω^{χ} is generated over R by the forms ω_{I_{1}} ⋏…⋏ω_{I_{p}}, I ∈ I^{p}, p ≥ 0. Thus Ω^{χ} has the structure of an exterior algebra.

5. Example: G_{26}

For an example, let us take a three dimensional complex reflection group, G_{26}. This group is the symmetry group of a regular complex polyhedron, and is number 26 in Shephard and Todd’s enumeration of finite irreducible unitary groups generated by reflections [4]. The group G_{26} consists of 1,296 complex 3 × 3 matrices and is generated by reflections of order two and three. The associated collineation group (which results from moding out by the scaler matrices) is the Hessian group of order 216.

The group is generated by the matrices

( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & α^{2} \end{matrix} ) , and \frac{i}{\sqrt{3}} ( \begin{matrix}α & α^{2} & α^{2} \\ α^{2} & α & α^{2} \\ α^{2} & α^{2} & α \end{matrix} ) ,

where α is a primitive cube root of unity.

The character table for this group reveals six multiplicative characters, each a power of the determinate character. Choose χ = det^{3}. Note that

Q_{det}_{3} = (x^{3} − y^{3})(x^{3} − z^{3})(y^{3} − z^{3}),

and

Q_{det}_{4} = x^{2}y^{2}z^{2}(x^{9} + 3x^{6}(y^{3} + z^{3}) + (y^{3} + z^{3})^{3} + 3x^{3}(y^{6} − 7y^{3}z^{3} + xz^{6}))^{2}.

The following 1-forms are det^{3}-invariant:

ω_{1} = x^{2}(y − z)(y^{2} + yz + z^{2})( 2x^{3} − y^{3} − z^{3}) dx
−y^{2}(x − z)(x^{2} + xz + z^{2})(−x^{3} + 2y^{3} − z^{3}) dy
−z^{2}(x − y)(x^{2} + xy + y^{2})( x^{3} + y^{3} − 2z^{3}) dz,

ω_{2} = x^{2}(x^{3} − y^{3})(x^{3} − z^{3})(y^{3} − z^{3})( x^{3} − 5y^{3} − 5z^{3}) dx
y^{2}(x^{3} − y^{3})(x^{3} − z^{3})(y^{3} − z^{3})(−5x^{3} + y^{3} − 5z^{3}) dy
z^{2}(x^{3} − y^{3})(x^{3} − z^{3})(y^{3} − z^{3})(−5x^{3} − 5y^{3} + z^{3}) dz,

ω_{3} = x^{2}(x^{3} − y^{3})(x^{3} − z^{3})(y^{3} − z^{3})(x^{9} + 3y^{9} + 61y^{6}z^{3} + 61y^{3}z^{6} + 3z^{9}
+9x^{6}(y^{3} + z^{3}) + x^{3}(−13y^{6} + 122y^{3}z^{3} − 13z^{6})) dx +

y^{2}(x^{3} − y^{3})(x^{3} − z^{3})(y^{3} − z^{3})(3x^{9} + y^{9} + 9y^{6}z^{3} − 13y^{3}z^{6} + 3z^{9}

+x^{6}(−13y^{3} + 61z^{3}) + x^{3}(9y^{6} + 122y^{3}z^{3} + 61z^{6})) dy +

z^{2}(x^{3} − y^{3})(x^{3} − z^{3})(y^{3} − z^{3})(3x^{9} + 3y^{9} − 13y^{6}z^{3} + 9y^{3}z^{6} + z^{9}

+x^{6}(61y^{3} − 13z^{3}) + x^{3}(61y^{6} + 122y^{3}z^{3} + 9z^{6})) dz.

The polynomial Q_{det}_{3} divides ω_{1} ∧ω_{2}, ω_{2} ∧ω_{3}, and ω_{1} ∧ω_{3}. The determinate of the coefficient matrix of ω_{1}, ω_{2}, and ω_{3} is (−16)Q_{det}_{4} Q_{det^{3}}^{2}, hence ω_{1}, ω_{2}, and ω_{3} χ-wedge to a multiple of Q_{χ·det} = Q_{det}_{4}. Proposition 1 then implies that ω_{1}, ω_{2}, and ω_{3} generate the entire module of det^{3}-invariants over the ring of invariants via det^{3}-wedging.

6. Logarithmic forms

We have so far only discussed regular differential forms; we now consider rational differential forms. The S-module of logarithmic p-forms with poles along A (see also [3], p. 124) is defined as

Ω^{p}(A) := \{\frac{ω}{Q_{det}_{−1}} : ω ∈ Ω^{p} and ω ∧ dα_{H} ∈ α_{H} Ω^{p+1} for all H ∈ A\}.

Ziegler [10] extends this definition to multiarrangements of hyperplanes, hyperplane arrangements in which each hyperplane has a positive integer multiplicity. We apply his definitions to our context of reflection groups and semiinvariants: Let A_{χ} be the multiarrangement consisting of hyperplanes H ∈ A each with multiplicity α_{H}(χ), i.e. the multiarrangement defined by Q_{χ}. We define (as in [10]) the module of logarithmic p-forms of A_{χ}:

Ω^{p}(A_{χ}) := \{\frac{ω}{Q_{χ}} : ω ∈ Ω^{p} and ω ∧ dα_{H} ∈ α_{H}^{a_{H}(χ)} Ω^{p+1} for all H ∈ A\}.

Let

Ω(A_{χ}) := ⨁_{p≥0} Ω^{p}(A_{χ}).

Corollary 3.

Ω^{χ} ⊂ Q_{χ} Ω(A_{χ}).

Proof. Choose ω in (Ω^{p})^{χ} and fix H ∈ A. Using Lemma 1, choose coordinates in which x_{1} = α_{H}, ω = ∑_{I∈I^{p}} ω_{I} dx_{I_{1}} ∧ … ∧ dx_{I_{p}}, and x_{1}^{a_{H}(χ)} divides ω_{I} if 1 ∉ I. Then dα_{H} = dx_{1}, and ω ∧ dα_{H} = ω ∧ dx_{1} = ∑_{I,1∉I} ω_{I} ∧ dx_{1},

which is divisible by x_{1}^{a_{H}(χ)}. Hence, ω ∧ dα_{H} ∈ α_{H}^{a_{H}(χ)} Ω^{p+1}. As H was arbitrary, \frac{ω}{Q_{χ}} ∈ Ω(A_{χ}).

This relationship is stronger when χ = det^{−1}. In this case, the forms that generate Ω^{χ} via χ-wedging over R also generate Ω(A) over S (see [5] for an independent proof).

On a similar note, we have

Proposition 3. Ω(A_{χ}) is closed under the exterior product.

Proof. Let ω/Q_{χ} and μ/Q_{χ} be in Ω(A_{χ}). Fix H in A and let a_{H}(χ) = a. Choose coordinates such that x_{1} = α_{H}, and write ω = ∑_{I∈I^{p}} ω_{I} dx_{I_{1}} ∧ … ∧ dx_{I_{p}} and μ = ∑_{J∈I^{q}} μ_{J} dx_{J_{1}} ∧ … ∧ dx_{J_{q}} in these coordinates. Since ω ∧ dx_{1} = ω ∧ dα_{H} ∈ α_{H}^{a}Ω = x_{1}^{a}Ω, ω_{I} is divisible by x_{1}^{a} as long as 1 ∉ I. Similarly, μ_{J} is divisible by x_{1}^{a} whenever 1 ∉ J. As in the proof of Lemma 2, it follows that Q_{χ} divides ω ∧ μ. Whenever 1 ∉ I and 1 ∉ J, x_{1}^{2a} divides ω_{I}μ_{J}, and thus

\frac{ω ∧ μ}{Q_{χ}} ∧ dx_{1}

is also divisible by x_{1}^{a}. Hence α_{H}^{a} divides (1/Q_{χ}) ω ∧ μ ∧ dα_{H}, and as H was arbitrary, (ω/Q_{χ}) ∧ (μ/Q_{χ}) is in Ω(A_{χ}).

7. Remarks

Analogous results hold for vector fields, or derivations. Let Υ^{χ} be the module of χ-invariants in the exterior algebra of derivations. Because the group action differs here, Lemma 1 is slightly different, with a + 1 taking the place of a − 1 when I_{1} = 1. The case where I_{1} ≠ 1 is the same as in the original lemma, and hence Q_{χ} also divides the exterior product of two elements in Υ^{χ} (the proof is analogous to the case of Ω^{χ}). The criterion for n derivations to generate Υ^{χ} via χ-wedging is also slightly different: they must χ-wedge to Q_{χ·det^{−1}} \frac{∂}{∂x_{1}} ∧ … ∧ \frac{∂}{∂x_{n}} instead of Q_{χ·det} dx_{1} ∧ … ∧ dx_{n}. This follows from the fact that dx_{1}∧…∧dx_{n} is (det^{−1})-invariant while \frac{∂}{∂x_{1}} ∧ … ∧ \frac{∂}{∂x_{n}} is det-invariant. Finally, we note that the correspondence between differential p-forms (in Ω^{p}) and (n − p)-forms in Υ (the exterior algebra of derivations) induces a module isomorphism between Ω^{χ} and Υ^{χ·det}.

8. Acknowledgments

The author is grateful to Peter Doyle, Hiroaki Terao, and Nolan Wallach for their helpful comments.

References

[1] Crass, S. and Doyle, P., Solving the sextic by iteration: a complex dynamical approach. Internat. Math. Res. Notices, 1997, no. 2, 83–99.

[2] Doyle, P. and McMullen, C., Solving the quintic by iteration. Acta Math., 163, no.3–4 (1989), 151–180.

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

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

[5] Shepler, A. and Terao, H., Logarithmic forms and anti-invariant forms of reflection groups, to appear. Singularities and Arrangements, Sapporo-Tokyo 1998, Advanced Studies in Pure Mathematics, North-Holland.

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

[7] Solomon, L., and Terao, H., The double Coxeter arrangements. Commentarii Math. Helvetica, 73 (1998), 237–258.

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

[9] Steinberg, R., Invariants of finite reflection groups. Canad. J. Math., 12 (1960), 616– 618.

[10] Ziegler, G., Multiarrangements of hyperplanes and their freeness. Singularities, Contemporary Math. Amer. Math. Soc., 90 (1989), 345–359.

12 ANNE V. SHEPLER

Department of Mathematics, University of California at San Diego, La Jolla, California, 92093-0112

E-mail address: ashepler@euclid.ucsd.edu