REGULARITY OF MULTIPLICATIVE INVARIANTS

Martin Lorenz

Department of Mathematics Temple University Philadelphia, PA 19122-2585

                          e-mail: lorenz@math.temple.edu

Abstract. The group GL_{d}(ℤ) acts on the Laurent polynomial ring k[X_{1}^{±1}, …, X_{d}^{±1}] over the field k via its natural action on the multiplicative group generated by the variables X_{1}, …, X_{d} (≅ ℤ^{d}). For G a finite subgroup of GL_{d}(ℤ) with order |G| not divisible by char k, we show that the algebra of Ginvariants in k[X_{1}^{±1}, …, X_{d}^{±1}] is regular precisely if G is a reflection group and H^{1}(G/D, (ℤ^{d})^{D}) = 0. Here, D is the subgroup of G that is generated by the reflections which are diagonalizable over ℤ, and (ℤ^{d})^{D} are the D-invariants in ℤ^{d}. The results of some machine computations are also discussed.

Introduction

This note continues our investigaton of multiplicative invariants in [L] and ultimately depends on Farkas’s work in [F]. In fact, our main result does little more than add some finishing ring theoretic touches to [F], Corollary 13 (which is largely credited to Steinberg [St] by Farkas). In particular, we use the structure of the class group of multiplicative invariants ([L]) to give the result a form that lends itself conveniently to computational applications.

Specifically, let S = kA denote the group algebra of a finitely generated free abelian group A over the field k and let G be a finite subgroup of GL(A). Then G acts on S by means of the unique extension of the natural G-action on A. Our aim is to characterize regularity of the algebra of invariants R = S^{G} of this action:

Theorem. Assume that the order of G is not divisible by char k. Then the following assertions are equivalent:

i. The algebra of invariants R = S^{G} is regular; ii. S is free as R-module; iii. R is a mixed polynomial algebra, i.e., R ≅ k[X_{1}^{±1}, …, X_{r}^{±1}, X_{r+1}, …, X_{d}] for some r, d; iv. G is a reflection group and H^{1}(G, A^{D}) = 0, where D is the subgroup of G generated by the reflections that are diagonalizable over ℤ and A^{D} denotes the D-invariants in A.

We remark that, in (iii), one necessarily has d = rank A and r = rank A^{G}. This follows from dimension considerations and the fact that all units of S have the form ca (c ∈ k^{∗}, a ∈ A), and hence the units of R have the same form, with a ∈ A^{G}. Thus regularity completely determines the structure of R. Also, in (ii), the rank of S as R-module equals the order of G, by Galois theory.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 13A50, 13F20, 13H05, 13-04, 16W20, 16S34, 20H15.

Key words and phrases. multiplicative group action, algebra of invariants, regular ring, unique factorization domain, reflection group, root system.

The author was supported in part by NSF Grant DMS–9400643.

As is often the case with multiplicative actions, the above Theorem is similar to, but more complicated than the corresponding result for “linear” actions of a finite group G′ ⊆ GL(V) on the symmetric (polynomial) algebra S(V) of a finite dimensional k-vector space V. Indeed, under the same assumption on the order of G′, the classical theorem of Shephard-Todd, Chevalley, and Serre ([Bou], p. 115 or [BH], p. 275) establishes the equivalence of (i), (ii), (iii)’: the algebra of invariants is a polynomial algebra, and (iv)’: G′ is a pseudoreflection group.

Condition (iv) of the Theorem is quite amenable to machine computations. To illustrate this point, we list in §2 all reflection groups G in ranks 2 and 3 which yield nonregular invariant algebras.

Notations and conventions. The above notations A, G ⊆ GL(A), S = kA, and R = S^{G} will retain their meaning throughout this article. In addition, we put

p = char k (≥ 0)  and  d = rank A .

So A ≅ ℤ^{d} and G can be viewed as a subgroup of the group GL_{d}(ℤ) of integer d × d-matrices of determinant ±1. We will work under the standing hypothesis that p does not divide the order of G.

1. Proof of the Theorem

1.1 If R is regular then G is a reflection group. Let Ω = (a − 1| a ∈ A) denote the augmentaton ideal of S. Since Ω is G-invariant, G acts on the localized algebra S′ = S_{Ω} and the algebra of invariants R′ = (S′)^{G} is canonically isomorphic with R localized at Ω ∩ R ([Bou2], Prop. 23, p. 34). Thus S′ and R′ are both local rings with residue field k, and our assumption on p implies that S′ is a Noetherian R′-module (e.g., [M], Theorem 7.6). Furthermore, S′ and R′ are both regular, the latter by assumption on R. Therefore, by [S], Théorème 2’ (or see [Bou], Exerc. 7(b), p. 138), G acts as a pseudoreflection group on the Zariski tangent space Ω/Ω^{2} of S′. Using the kG-isomorphism

V :=A ⊗_{ℤ} k ⟶^{≅}  Ω/Ω^{2}
a ⊗ 1 ⟼ a − 1 + Ω^{2}  (a ∈ A) ,

we conclude that G acts as a pseudoreflection group on V. In other words, letting π : GL(A) → GL(V) denote the canonical map (“reduction mod p”), π(G) is generated by pseudoreflections. Our hypothesis on p implies that π is mono on G (cf. [Bou], Exerc. 7(c), p. 138). Thus it suffices to show that, if g ∈ G is such that π(g) is a pseudoreflection, that is, rank(π(g) − Id_{V}) = 1, then g is a reflection. Indeed, since det(g) = ±1, we have μ = det(π(g)) = ±1 in k, and since π(g) is conjugate in GL(V) to the diagonal matrix diag(μ, 1, …, 1), we actually have μ = −1 and g^{2} = 1 (and p ≠ 2). Thus, putting A_{±} = ann_{A}(g ∓ 1) and V_{±} = ann_{V} (π(g) ∓ 1), we have 2A ⊆ A_{+} ⊕ A_{−} ⊆ A and A_{±} ⊗_{ℤ} k = V_{±}. Since dim_{k} V_{+} = d − 1 and dim_{k} V_{−} = 1, we conclude that rank(A_{+}) = d − 1 and rank(A_{−}) = 1, which proves that g is a reflection.

1.2 If R is regular then R is a mixed polynomial algebra. By §1.1, we know that G is a reflection group which enables us to use the results of [F]. In particular, it has been shown in [F], proof of Theorem 10, that the invariant algebra R of any finite reflection group is a semigroup algebra kC, where C is a multiplicative semigroup with 1 that embeds as a subsemigroup into a free abelian group of finite rank. (We remark that, even though Farkas works over k = ℂ, his methods pertain more generally to the situation where p does not divide the order of G.) Furthermore, since R is an affine k-algebra, by Noether’s theorem ([Bou2], Théorème 2, p. 33), the semigroup C is finitely generated. Thus C is an affine semigroup in the sense of [BH], §6.1. It is known (cf. [BH], Exercise 6.1.11) that kC is regular if and only if C is of the form ℤ^{u} ⊕ ℕ^{v}, in additive notation. In other words, kC is regular precisely if kC ≅ k[X_{1}^{±1}, …, X_{u}^{±1}, X_{u+1}, …, X_{u+v}], whence our assertion.

1.3 R is regular iff S is free as R-module. Inasmuch as projective modules over mixed polynomial algebras are known to be free (e.g., [Lam], p. 144), §1.2 reduces the assertion to the following one:

R is regular iff S is projective as R-module.

The validity of this is a consequence of a general fact from commutative algebra which we include for the reader’s convenience.

Lemma. Let U ⊆ V be an extension of commutative Noetherian domains such that V is finitely generated as U-module.

(a) If V is Cohen-Macaulay and U is regular then V is projective as U-module.

(b) Suppose that V is regular and there is a Reynolds operator ρ : V → U (i.e., ρ is a U-module map that is the identity on U). Then U is regular if and only if V is projective as U-module.

Proof. (a) We have to show that, for each maximal ideal 𝔪 of U, the localized ring V_{𝔪} is free over U_{𝔪}. Note that U_{𝔪} ⊆ V_{𝔪} is a finite extension of Noetherian domains, with V_{𝔪} Cohen-Macaulay and U_{𝔪} regular local. Freeness of V_{𝔪} over U_{𝔪} now follows from [Mat], (18.H) Theorem 46.

(b) The “only if” direction is a special case of (a). Conversely, assume that V is projective as U-module and fix a maximal ideal 𝔪 of U. Then V_{𝔪} is free over U_{𝔪} and the extension U_{𝔪} ⊆ V_{𝔪} has Reynolds operator ρ_{𝔪} = ρ ⊗_{U} Id_{U_{𝔪}}. Therefore, [Kap], Theorem 5, p. 173 implies that gldim U_{𝔪} ≤ gldim V_{𝔪}. Finally, V_{𝔪} is a regular semi-local ring and so gldim V_{𝔪} is finite. Hence gldim U_{𝔪} is finite as well, which proves regularity of U. □

The Lemma applies to the extension R ⊆ S, since S = kA is a regular affine k-algebra and, by Noether’s theorem, R is affine and S is finitely generated as R-module. (All this is independent of our hypothesis on p.) Furthermore, a Reynolds operator ρ : S → R is given by ρ(s) = |G|^{−1} ∑_{g∈G} s^{g}.

1.4 Conclusion. §1.2 proves that (i) implies (iii) and the converse is obvious. Thus (i) and (iii) are equivalent and so are (i) and (ii), by §1.3. In view of [L], Theorem 2.5, assertion (iv) is equivalent with

G is a reflection group and R is a UFD.

In this form, the implication (iv) ⇒ (iii) is part of [F], Corollary 13. Since mixed polynomial algebras are UFD, (iii) ⇒ (iv) follows from §1.1 (or from [F2], Theorem 8+). This completes the proof of the Theorem.

2. Computations

2.1. In this section, we list the results of computations that were carried out using the computer group theory package GAP version 3.4, in particular its crystallographic groups library which contains all finite subgroups of GL_{d}(ℤ) for d = 2, 3, 4, up to conjugation. The program multinv.g is available via anonymous ftp to ftp.math.temple.edu in the directory /pub/lorenz/programs. Besides our standing hypothesis on p = char k, the field k is assumed throughout to be a splitting field for (G/N)^{ab}, where N denotes the subgroup of G that is generated by the reflections in G. Furthermore, the matrices given below are understood to act from the right on rows.

2.2 Rank 2 (cf. [L], §2.7). Among the 12 conjugacy classes of nontrivial finite subgroups of GL_{2}(ℤ), exactly 8 consist of reflection groups, and 2 of these give rise to a nonregular invariant algebra R. They are represented by the groups

G = ⟨y, −y⟩ ≅ C_{2} × C_{2} :  Cl R ≅ ℤ/2ℤ ,
G = ⟨x, y⟩ ≅ D_{6} :  Cl R ≅ ℤ/3ℤ ,

Here, Cl R denotes the class group of R, and x = ( \begin{matrix}0 & −1 \\ 1 & −1 \end{matrix} ), y = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ). In both cases, a suitable GL_{2}(ℚ)-conjugate of G in GL_{2}(ℤ) yields a regular invariant algebra. All instances where R is a UFD come from reflection groups G, and so R is actually a mixed polynomial algebra.

2.3 Rank 3. There are 72 conjugacy classes of nontrivial finite subgroups of GL_{3}(ℤ) of which 28 consist of reflection groups. Among the latter, 11 give rise to a nonregular invariant algebra. The above two rank 2 groups G yield 4 of these: ( \begin{matrix}G \\ & 1 \end{matrix} ) produces the invariant algebra R[X^{±1}]

and ( \begin{matrix}G \\ & ±1 \end{matrix} ) produces R[X], both with the same class group as R. In addition, there are the following 7 classes of ℤ-indecomposable groups, given by a representative group:

G_{1} = ⟨x, −sxs^{−1}⟩ ≅ C_{2} × C_{2} :  Cl R ≅ ℤ/2ℤ ,
G_{2} = ⟨− Id, x, sxs^{−1}⟩ ≅ C_{2} × C_{2} × C_{2} : Cl R ≅ ℤ/2ℤ ⊕ ℤ/2ℤ ,
G_{3} = G_{2}^{tr} ≅ C_{2} × C_{2} × C_{2} :  Cl R ≅ ℤ/2ℤ ,
G_{4} = ⟨− Id, x^{tr}, y⟩ ≅ D_{8} × C_{2} :  Cl R ≅ ℤ/2ℤ ,
G_{5} = ⟨s, e, d⟩ ≅ S_{4} :  Cl R ≅ ℤ/2ℤ ,
G_{6} = ⟨r, s, −x^{tr}⟩ ≅ S_{4} :  Cl R ≅ ℤ/4ℤ ,
G_{7} = ⟨− Id, r, s, x^{tr}⟩ ≅ S_{4} × C_{2} :  Cl R ≅ ℤ/2ℤ .

Here, tr denotes transpose matrices, and x = ( \begin{matrix}0 & 0 & −1 \\ 1 & 1 & 1 \\ −1 & 0 & 0 \end{matrix} ), y = ( \begin{matrix}0 & 1 & 0 \\ 0 & 1 & −1 \\ −1 & 1 & 0 \end{matrix} ), e = ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ), d =

( \begin{matrix}−1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 1 \end{matrix} ), s = ( \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ), r = ( \begin{matrix}1 & 0 & −1 \\ 0 & 1 & −1 \\ 0 & 0 & −1 \end{matrix} ). In GL_{3}(ℚ), the group G_{1} is conjugate to ( \begin{matrix}G \\ & 1 \end{matrix} ), and

G_{2} and G_{3} are conjugate to ( \begin{matrix}G \\ & ±1 \end{matrix} ), where G is the first group (≅ C_{2} ×C_{2}) in §2.2. Furthermore, G_{5} and G_{6} are also conjugate in GL_{3}(ℚ). Again, whenever R is a UFD then R is actually a mixed polynomial algebra.

2.4 Rank 4. The number of rank 4 cases makes their explicit listing prohibitive: GL_{4}(ℤ) has 709 conjugacy classes of nontrivial finite subgroups including 101 conjugacy classes of reflection groups. Of these, 51 give rise to a nonregular invariant algebra. In rank 4, for the first time, one finds a nonregular invariant algebra that is a UFD. The group in question is isomorphic with A_{5}

and is generated by the matrices ( \begin{matrix}0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ −1 & −1 & −1 & −1 \\ 0 & 0 & 1 & 0 \end{matrix} ) and ( \begin{matrix}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) .

2.5 Reflection groups and root systems. Following the referee’s suggestions, we briefly sketch how the groups exhibited in §§2.2, 2.3 can also be arrived at non-computationally working in the context of root systems. So let G ⊆ GL(A) be a finite reflection group and assume, for simplicity, that G is effective, that is, A^{G} = 0. Then, by [F2], Theorem 16 (or [St]), the invariant algebra R is regular or, equivalently, a polynomial algebra, precisely if there exists a root system Φ with Weyl group G so that A is G-isomorphic to the weight lattice L̂(Φ) of Φ. Now there always is a root system Φ so that G is the Weyl group of Φ and L(Φ) ⊆ A ⊆ L̂(Φ), where L(Φ) denotes the root lattice of Φ (cf. [F2], Proposition 12). As we are looking for nonregular invariant algebras, we must certainly choose A to be a proper subgroup of L̂(Φ). The structure of the groups \overline{L}(Φ) = L̂(Φ)/L(Φ) is known; it is listed, for all irreducible root systems Φ, in [Bou], pp. 250–275. For example, if Φ has type B_{n} then \overline{L}(Φ) has order 2 so that the only choice for A in this case would be the root lattice. However, the root lattice for type B_{n} is isomorphic, as a module over its Weyl group, to the weight lattice for type C_{n} (cf. [F3], Note added in proof). So type B_{n} never leads to a nonregular invariant algebra. Further, the root and weight lattices for A_{1} are clearly isomorphic as well.

In case rank A = 2, the possible root systems are of types A_{2}, B_{2}, or A_{1} × A_{1}, and the two groups in §2.2 come from taking A to be the preimage of the “diagonal” subgroup of \overline{L}(A_{1}×A_{1}) ≅ ℤ/2ℤ × ℤ/2ℤ or of the trivial subgroup of \overline{L}(A_{2}) ≅ ℤ/3ℤ.

If rank A = 3, the available root systems Φ are of types A_{3}, B_{3}, C_{3}, A_{2} ×A_{1}, B_{2} ×A_{1}, G_{2} ×A_{1}, or A_{1} × A_{1} × A_{1}, with B_{3} being irrelevant for our purposes. Likewise, G_{2} × A_{1} can be discarded, because \overline{L}(G_{2}) = 0 and so the only choice for A is L̂(G_{2})×L(A_{1}) which, however, is isomorphic to the weight lattice L̂(G_{2}) × L̂(A_{1}). For A_{3}, we have \overline{L}(A_{3}) ≅ ℤ/4ℤ, and the two proper subgroups lead to G_{6} and G_{5} in §2.3. Since \overline{L}(C_{3}) ≅ ℤ/2ℤ, type C_{3} produces only one example, which turns out to be the group G_{7}. The three proper subgroups of \overline{L}(A_{2} × A_{1}) ≅ ℤ/3ℤ × ℤ/2ℤ yield the lattices L(A_{2}) × L(A_{1}), L(A_{2}) × L̂(A_{1}), and L̂(A_{2}) × L(A_{1}). The latter is isomorphic with the weight lattice L̂(A_{2}) × L̂(A_{1}), and the former two are isomorphic to each other and yield the group ( \begin{matrix}G \\ & ±1 \end{matrix} ), where G is the second group (≅ D_{6}) of §2.2. Similarly, the group G_{4} in §2.3 comes from type B_{2} × A_{1}, and type A_{1} × A_{1} × A_{1} is responsible for the groups G_{1}, G_{2}, G_{3}, and ( \begin{matrix}G \\ & ±1 \end{matrix} ), where G is the first group (≅ C_{2} × C_{2}) in §2.2. The remaining two groups in §2.3 are non-effective and thus, strictly speaking, they are not covered by the above discussion. However, non-effective groups can in fact also be treated in a very similar manner by using Farkas’s method of “rooting sections” (see [F]).

Acknowledgments

It is a pleasure to thank Kenny Brown (Glasgow) for an informative correspondence concerning regularity of invariant algebras (in a broader context than discussed here). In particular, I learned the equivalence 1.3 from him. I would also like to thank Professor Neubüser and Dr. Werner Nickel (both Aachen) for help with GAP. The program multinv.g incorporates preliminary versions of some GAP functions that were generously made available to the author by Dr. Werner Nickel. Finally, I gratefully acknowledge the referee’s comments about root systems which lead to §2.5.

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