CHARACTERIZING SEPARATING INVARIANTS

DEDICATED TO LUCHO AVRAMOV ON THE OCCASION OF HIS 60TH BIRTHDAY

MARA D. NEUSEL AND MÜFİT SEZER

Abstract. We study separating algebras for rings of invariants of finite groups. We give an algebraic characterization for these. Furthermore, we describe a particularly nice separating subalgebra for rings of invariants of p-groups in characteristic p. This leads to a characterization of subalgebras such that their p-root and integral closure is equal to the ring of invariants (see Property (⋆) below). Finally, we present separating sets for invariants rings of nonmodular representations of abelian groups whose size depends only on the degree of the representation.

Let 𝔽 be an algebraically closed field and let G be a finite group. Consider a faithful representation

ρ : G ↪ GL(n, 𝔽)

of degree n. It induces an action of the group G on the symmetric algebra on the dual space V^{∗}, which we denote by 𝔽[V]. The subring of G-invariants is denoted by 𝔽[V]^{G}. We note that the vector space V decomposes into disjoint G-orbits. We denote the orbit space by

V/G = {[v] = {gv|g ∈ G}|v ∈ V }.

Any invariant f ∈ 𝔽[V]^{G} is constant on the G-orbits [v]. Indeed, 𝔽[V]^{G} ⊆ 𝔽[V] is the largest subalgebra with this property. A finitely generated graded subalgebra A ⊆ 𝔽[V]^{G} (or more generally a subset in 𝔽[V]^{G}) is called separating if for any two distinct G-orbits [v] ≠ [w] there exists a function f ∈ A separating the two, i.e.,

f(v) ≠ f(w).

This notion was introducted in Definition 2.3.8 in [2]. Denote by \overline{A} the integral closure of the algebra A (in its field of fractions) and by \sqrt{A} its p-root closure in 𝔽[V], where p is the characteristic of 𝔽. In Theorem 2.3.12 ibid. it is shown that

(⋆)  \sqrt{\overline{A}} = 𝔽[V]^{G},

provided that A ⊆ 𝔽[V]^{G} is a finitely generated separating graded subalgebra. The converse is not valid, see Example 2.3.14 ibid.

Remark 1. We note that for fields that are not algebraically closed, the notion “separating” does not make sense. For example, consider the finite field 𝔽_{2} with two elements. The general linear group GL(2, 𝔽_{2}) is a finite group of order 6. Its ring of invariants 𝔽_{2}[x, y]^{GL(2,𝔽_{2})} is a polynomial ring generated by d_{2,0} = x^{2}y + xy^{2} and d_{2,1} = x^{2} + xy + y^{2}, see, e.g., Theorem 6.1.4 in [11]. The vector space V = span_{𝔽_{2}}{e_{1}, e_{2}} decomposes into the two orbits V \ 0 and {0}. Note that the subalgebra

Date: April 10, 2008, 14 h 29 min.

𝔽_{2}[d_{2,1}] ⊆ 𝔽_{2}[x, y]^{GL(2,𝔽_{2})}

is separating, but the extension is neither finite nor integral. Even worse, the

subgroup ℤ/3 generated by [\begin{matrix}0 & 1 \\ 1 & 1 \end{matrix}] has the same orbits on V. However, we could

consider the invariants over the algebraic closure of 𝔽_{2}. We obtain

\overline{𝔽}[x, y]^{GL(2,𝔽_{2})} = \overline{𝔽} ⊗_{𝔽_{2}} 𝔽_{2}[x, y]^{GL(2,𝔽_{2})}.

Taking into account the orbits of the group action on \overline{V} = span\overline{𝔽}\{e_{1}, e_{2}\} we see that the subalgebra generated by the degree two invariant is, as expected, no longer separating: d_{2,1} vanishes on the orbit [(1, ω)] for a primitive 3rd root of unity ω.

Separating invariants have been studied by several people, see, e.g., [7], [3], [4],

[2] and the references there. All of these studies show that separating invariants are often better behaved than the ring of invariants itself, e.g., there are always separating algebras that satisfy Noether’s bound, see Corollary 3.9.14 in [2], or, separating invariants of vector invariants can be obtained by polarizations, see [4]. In this paper we continue the study of separating invariants.

In Section 1 we will describe a separating subalgebra for the ring of invariants of a finite p-group P over a field of characteristic p. We note that generating invariants of p-groups are usually difficult to describe. Indeed, apart from individual cases, the only large families of modular representations of finite p-groups for which complete (but maybe not minimal) generating sets for the invariants are known are the (all of them) representations of cyclic groups of order p, see [6], and the indecomposable representations of cyclic groups of order p^{2}, see [10]. In both cases, the rings of invariants are generated by norms, transfers, and invariants up to a certain degree. The reason for including all invariants up to some degree is that norms and transfers can be employed to decompose invariants usually only after some degree and not all invariants at small degrees are norms or (relative) transfers. We want to show that in contrast norms and transfers suffice to separate orbits for all representations of any p-group. This study, moreover, will enable us to characterize subalgebras A ⊆ 𝔽[V]^{P} satisfying Property (⋆).

In Section 2 we turn to the other extreme: We will consider nonmodular representations of abelian groups. With Noether’s bound, generating sets of the ring of invariants can be easily described leading to an upper estimate on the number of generators depending on the group order and the degree of the representation. However, again separating invariants have yet a simpler structure: We will describe separating sets whose size depends only on the degree of the representation.

We close the introduction with a characterization of separating subalgebras in algebraic terms.

Theorem 2. Let ρ : G ↪ GL(n, 𝔽) be a faithful representation of a finite group G over some algebraically closed field 𝔽. Then a finitely generated subalgebra A ⊆ 𝔽[V]^{G} is separating if and only if the following two conditions are satisfied:

(1) The extension ϕ : A ↪ 𝔽[V]^{G} is integral.

(2) The induced map

ϕ^{∗} : Max(𝔽[V]^{G}) ⟶ Max(A)

between the spectra of maximal ideals is bijective.

Proof. Choose coordinates for V^{∗} and write 𝔽[V] = 𝔽[x_{1}, …, x_{n}]. For any point v = (v_{1}, …, v_{n}) ∈ V we obtain a maximal ideal

𝔪_{v} = (x_{1} − v_{1}, …, x_{n} − v_{n}) ⊊ 𝔽[V].

Two maximal ideals 𝔪_{v} and 𝔪_{w} coincide upon contraction to 𝔽[V]^{G},

𝔪_{v} ∩ 𝔽[V]^{G} = 𝔪_{w} ∩ 𝔽[V]^{G}

if and only if v and w lie in the same G-orbit. Thus the spectrum of maximal ideal in 𝔽[V]^{G} is given by

Max(𝔽[V]^{G}) = {𝔪_{v} ∩ 𝔽[V]^{G}|[v] ∈ V/G}.

If A is separating, then the extension A ↪ 𝔽[V]^{G} is integral by Theorem 2.3.12 in [2]. Thus, the induced map on the spectra of maximal ideals is surjective. Furthermore, for any two points v, w ∈ V lying in different G-orbits, there is a function f ∈ A such that

f(v) ≠ f(w).

The function f − f(w) separates the two points as well and moreover vanishes at w. Thus we may assume that f ∈ J (w) = 𝔪_{w} ⊆ 𝔽[V] is contained in the vanishing ideal of the point w, but f ∉ J (v) = 𝔪_{v}. Thus

f ∈ 𝔪_{w} ∩ A but f ∉ 𝔪_{v} ∩ A.

Therefore, the induced map on the spectra of maximal ideals is also injective.

Conversely, if A satisfies the two conditions mentioned above, then for any two distinct G-orbits [v] and [w] we find

𝔪_{v} ∩ A ≠ 𝔪_{w} ∩ A.

Therefore, there is a function f ∈ A vanishing on v but not on w (and vice versa). Thus A is separating. □

Since p-root extensions form a relatively nice subset of integral extensions we have the following characterization.

Proposition 3. Let ρ : G ↪ GL(n, 𝔽) be a faithful representation of a finite group G over some algebraically closed field 𝔽. Then a finitely generated integrally closed graded subalgebra A ⊆ 𝔽[V]^{G} is separating if and only if \sqrt{A} = 𝔽[V]^{G}. Furthermore, in that case the induced map on the prime ideal spectra

Spec(𝔽[V]^{G}) ⟶ Spec(\overline{A})

is bijective.

Proof. If 𝔽 has characteristic zero, then an integrally closed separating subalgebra A is equal to the ring of invariants, since there are no nontrivial purely inseparable extensions.

Thus let 𝔽 have finite characteristic. If A is separating, then

\sqrt{\overline{A}} = \sqrt{A} = 𝔽[V]^{G}

by Theorem 2.3.12 in [2]. Conversely, if \sqrt{A} = 𝔽[V]^{G}, then there exists an integer s ∈ ℕ_{0} such that

\sqrt{A}^{p^{s}} ⊆ A ⊆ \sqrt{A} = 𝔽[V]^{G}.

Since with \sqrt{A} also \sqrt{A}^{p^{s}} is separating, so is A.

The last statement follows by Exercise 15 in Chapter 5 of [1]. □

Corollary 4. Let ρ : G ↪ GL(n, 𝔽) be a faithful representation of a finite group G over some algebraically closed field 𝔽. Let A ⊆ 𝔽[V]^{G} be a finitely generated graded subalgebra. If \sqrt{A} = 𝔽[V]^{G} then A is separating.

Proof. Immediate from the preceding result. □

Example. Let ρ : G ↪ GL(n, 𝔽) be a representation of a finite group G. Denote by 𝔽G the group algebra and let

V (G) = 𝔽G ⊗ V

be the induced module. The group G acts on V (G) by left multiplication on the first component. We obtain a surjective G-equivariant map between the rings of polynomial functions

η_{G} : 𝔽[V (G)] ⟶ 𝔽[V].

By restriction to the induced ring of invariants, we obtain the classical Noether map, see Section 4.2 in [11],

η_{G}^{G} : 𝔽[V (G)]^{G} ⟶ 𝔽[V]^{G}.

We note that V (G) is the n-fold regular representation of G. Thus 𝔽[V (G)]^{G} are the n-fold vector invariants of the regular representation of G. In the classical nonmodular case the map η_{G}^{G} is surjective, see Proposition 4.2.2 in [11]. This does not remain true in the modular case. However, as shown in Proposition 2.2 of [9] the p-root closure of the image of the Noether map is equal to 𝔽[V]^{G}. Thus, by the preceding result, the image of the Noether map is separating.

1. Separating subalgebras for modular p-Groups

In this section we want to present a new construction for separating subalgebras of rings of invariants of finite p-groups over an algebraically closed field 𝔽 of characteristic p. This result is particularly interesting because it gives rise to a characterization of subalgebras satisfying Property (⋆).

We start with a recollection of two methods to construct invariants. For f ∈ 𝔽[V], we define the norm of f, denoted N(f), by

∏_{g∈G} g(f) ∈ 𝔽[V]^{G}.

Furthermore, the transfer is defined by

Tr^{G} : 𝔽[V] ⟶ 𝔽[V]^{G}, f ↦ ∑_{g∈G} g(f).

We obtain a relative version in the following way: Let H be a subgroup of G. Then the relative transfer (from H to G) is given by

Tr_{H}^{G} : 𝔽[V]^{H} → 𝔽[V]^{G}, Tr_{H}^{G}(f) = ∑_{ḡ∈G/H} ḡ(f),

where the sum runs over a set of coset representatives of H in G. We set

I = ∑_{H<G, max} Im(Tr_{H}^{G}) ⊆ 𝔽[V]^{G},

i.e., I is the ideal in 𝔽[V]^{G} generated by the image of the relative transfers for all maximal subgroups H < G.

As mentioned in the introduction, norms and transfers usually^{1} do not suffice to generate the entire ring of invariants 𝔽[V]^{G}, but play a crucial role for invariants of p-groups as they appear in every known list of generating invariants. We proceed by showing that, in contrast, norms and transfers suffice to separate orbits for any representation of a p-group.

For a subset X ∈ 𝔽[V]^{G} we define its zero set in V/G by

V(X) = {[v] ∈ V/G|f(v) = 0 ∀f ∈ X}.

For v ∈ V, let G_{v} denote the stabilizer of v in G. The following is a part of Theorem 12.4 in [5] generalizing Feshbach’s Transfer Theorem.

Lemma 5. The zero set of I in V/G is equal to the fixed point space of G

V(I) = {[v] ∈ V/G | G_{v} = G} = V^{G}.

Proof. Let v ∈ V such that G_{v} = G. Let H < G be a maximal subgroup and f ∈ 𝔽[V]^{H}. Then

Tr_{H}^{G}(f)(v) = ( ∑_{ḡ∈G/H} g(f))(v) = ∑_{ḡ∈G/H} f(g^{−1}v) = |G : H|f(v) = 0.

Conversely pick v ∈ V such that G_{v} ≠ G. Since G is finite, there exists f ∈ 𝔽[V] such that f(v) ≠ 0 and f(gv) = 0 for all g ∉ G_{v}. Let

N = ∏_{h∈G_{v}} h(f) ∈ 𝔽[V]^{G_{v}}.

Note that N(v) ≠ 0 and N(gv) = 0 for all g ∉ G_{v}. Moreover

Tr_{G_{v}}^{G}(N)(v) = ∑_{\overline{g}∈G/G_{v}} \overline{g}N(v) = ∑_{\overline{g}∈G/G_{v}} N(\overline{g}^{−1}v) = N(v) ≠ 0.

Let H be a maximal subgroup of G containing G_{v}. Since G_{v} ⊆ H, we have Im(Tr_{G_{v}}^{G}) ⊆ Im(Tr_{H}^{G}). It follows that v ∉ V(Im(Tr_{H}^{G})) and accordingly, v ∉ V(I) as desired. □

Let e_{1}, e_{2}, …, e_{k} be a basis for V^{G}, and let x_{1}, x_{2}, …, x_{k} denote the corresponding basis elements in the dual space.

Theorem 6. Let ρ : P ↪ GL(n, 𝔽) be a faithful representation of a finite p-group P over a field 𝔽 of characteristic p. Then the subalgebra in 𝔽[V]^{P} generated by I and N(x_{i}), i = 1, …, k is separating.

Proof. Assume that v, w ∈ V are in different P -orbits. Then there exists an invariant f ∈ 𝔽[V]^{P} such that f(v) ≠ f(w).

If one of them, say v, lies outside of V(I), then there exists a maximal subgroup Q in P and an invariant h ∈ 𝔽[V]^{Q} such that Tr_{Q}^{P}(h)(v) ≠ 0.

If Tr_{Q}^{P}(h)(v) ≠ Tr_{Q}^{P}(h)(w) we are done. Otherwise, we find that

f · Tr_{Q}^{P}(h) = Tr_{Q}^{P}(f · h) ∈ Im(Tr_{Q}^{P})

separates v and w.

1An exception would be vector invariants of the regular representation of the cyclic group of order p. Indeed, a very special case. See [8]

Thus, we may assume that both, v as well as w, lie in V(I). From the previous lemma we have v, w ∈ V^{P}. Since the fixed point space V^{P} is spanned by {e_{1}, …, e_{k}} we can write v = ∑_{i=1}^{k} α_{i}e_{i} and w = ∑_{i=1}^{k} β_{i}e_{i} for suitable α_{i}, β_{i} ∈ 𝔽, 1 ≤ i ≤ k. Since v ≠ w there is a i_{0} ∈ {1, …, k} such that α_{i_{0}} ≠ β_{i_{0}}. Thus

N(x_{i_{0}})(v) = α_{i_{0}}^{p^{r}} and N(x_{i_{0}})(w) = β_{i_{0}}^{p^{r}}.

Since 𝔽 has characteristic p, N(x_{i_{0}})(v) ≠ N(x_{i_{0}})(w). Thus N(x_{i_{0}}) separates v and w as desired. □

The preceding result enables us to describe the radical of the ideal I, see Corollary 12.3 [5] for the special case of cyclic p-groups. We denote by J (V^{P}) ⊆ 𝔽[V] the vanishing ideal of the fixed point set of P.

Proposition 7. Let ρ : P ↪ GL(n, 𝔽) be a faithful representation of a finite p-group P over a field 𝔽 of characteristic p. Then we obtain the prime ideal

\sqrt{I} = J (V^{P}) ∩ 𝔽[V]^{P} .

Proof. By Lemma 5, we have V(I) = V^{P}. On the other hand, it is clear that V(J (V^{P})) = V^{P}. Since J (V^{P}) is generated by the linear forms x_{i} such that i ∉ {1, …, k}, it is a prime ideal. Therefore, the Nullstellensatz yields

\sqrt{I𝔽[V]} = J (V^{P}),

where I𝔽[V] ⊆ 𝔽[V] denotes the extension of I in 𝔽[V]. Thus we obtain

\sqrt{I} ⊆ \sqrt{I𝔽[V]} ∩ 𝔽[V]^{P} = J (V^{P}) ∩ 𝔽[V]^{P} .

Since finite p-groups are reductive it follows that

J (V^{P}) ∩ 𝔽[V]^{P} ⊆ \sqrt{I}

by Lemma 3.4.2 in [12]. □

Theorem 6 also enables us to characterize the set of algebras A with Property (⋆) in the case of p-groups. We need a preliminary lemma.

Lemma 8. Let A ⊆ 𝔽[V] be a finitely generated 𝔽-algebra. Then \sqrt{\overline{A}} is integrally closed (in its field of fractions).

Proof. Denote by FF(−) the field of fraction functor.

If FF(A) = FF(\sqrt{\overline{A}}), then \overline{A} = \sqrt{\overline{A}}, and thus the latter is integrally closed.

In general though

FF(A) = FF(\overline{A}) ↪ FF(\sqrt{\overline{A}})

is a nontrivial purely inseparable extension. We note that by definition the extension

A ↪ \overline{A} ↪ \sqrt{\overline{A}}

is integral. Now, let a ∈ FF(\sqrt{\overline{A}}) be integral over \sqrt{\overline{A}}. Let

a^{m} + α_{m−1}a^{m−1} + ⋯ + α_{1}a + α_{0} = 0

be a relation of integral dependence, where α_{i} ∈ \sqrt{\overline{A}} for all i. Thus, α_{i}^{p^{s}} ∈ \overline{A} and a^{p^{s}} ∈ FF(A) for some large s ∈ ℕ. In orther words, a^{p^{s}} ∈ FF(A) is integral over \overline{A} and hence in \overline{A}. Thus a ∈ \sqrt{\overline{A}} by definition. □

Theorem 9. Let ρ : P ↪ GL(n, 𝔽) be a faithful representation of a finite p-group P over a field 𝔽 of characteristic p. Let A ⊆ 𝔽[V]^{P} be a subalgebra. Then \sqrt{\overline{A}} = 𝔽[V]^{P} if and only if the following two conditions are satisfied:

(1) The extension A ↪ 𝔽[V]^{P} is integral.

(2) There exists a t ∈ ℕ_{0} such that I^{p^{t}} ⊆ \overline{A}.

Proof. If A ⊆ 𝔽[V]^{P} has Property (⋆) then the extension

A ↪ 𝔽[V]^{G}

is integral. Furthermore, \sqrt{\overline{A}} = 𝔽[V]^{P} contains the image of the relative transfers Tr_{Q}^{P} for every maximal subgroup Q ≤ P. Since rings of invariants are Noetherian, the image of any (relative) transfer is finitely generated. Thus there exist s = s(Q) ∈ ℕ_{0} such that

(Im(Tr_{Q}^{P}))^{p^{s}} ⊆ \overline{A}

for all maximal subgroups Q ≤ P. Furthermore, a finite group has only finitely many maximal subgroups. Thus setting

t = max{s(Q)|Q < P maximal}

yields a universal power such that

I^{p^{t}} ⊆ \overline{A}.

To prove the converse we assume that the two properties listed above are satisfied. Thus the image of the relative transfers Tr_{Q}^{P} is contained in \sqrt{\overline{A}} ↪ 𝔽[V]^{P}. We obtain an algebraic extension of the associated fields of fractions as follows

FF(𝔽[Im(∑_{Q} Tr_{Q}^{P})]) ⊆ FF(\sqrt{\overline{A}}) ⊆ 𝔽(V)^{P} .

Since the transfer is surjective at the level of fields of fractions, see Proposition 1.1 in [9], we have

𝔽(V)^{P} = FF(𝔽[Im(∑_{Q} Tr_{Q}^{P})]).

Thus FF(\sqrt{\overline{A}}) = 𝔽(V)^{P}. Since \sqrt{\overline{A}} is integrally closed by Lemma 8 it follows that \sqrt{\overline{A}} = 𝔽[V]^{P}. □

2. Separating subsets for non-modular abelian groups

In this section we consider finite abelian groups G and nonmodular representations thereof, i.e., |G| ∈ 𝔽^{×}. The field 𝔽 continues to be algebraically closed, thus 𝔽 contains all |G|-th roots of unity. As it turns out, in this case we can not only describe a separating subset (and thus separating subalgebra), but also give a sharp upper bound on its order.

Let κ(G) denote the character group of G over 𝔽. Since every representation of G over 𝔽 is diagonalizable, there exists a basis {x_{1}, x_{2}, …, x_{n}} of V^{∗} such that g(x_{i}) = χ_{i}(g)x_{i}, with χ_{i} ∈ κ(G) for 1 ≤ i ≤ n.

The corresponding ring of invariants 𝔽[V]^{G} is generated by monomials, see, e.g., Lemma 7.3.5 in [11]. Furthermore, a monomial m = x_{1}^{e_{1}}x_{2}^{e_{2}} ⋯ x_{n}^{e_{n}} is invariant if and only if e_{1}χ_{1} + e_{2}χ_{2} + ⋯ + e_{n}χ_{n} = 0 in κ(G).

To each subset S ⊆ {1, 2, …, n} we associate an invariant monomial in the following way. Set

M(S) = {x_{1}^{e_{1}}x_{2}^{e_{2}} ⋯ x_{n}^{e_{n}} ∈ 𝔽[V]^{G}| e_{j} = 0 for j ∉ S} ⊆ 𝔽[V]^{G}.

Denote by i = i(S) the smallest integer in S. Define A = A(S) ⊆ ℕ to be the set of positive integers a such that there exists a monomial x_{i}^{e_{i}} ⋯ x_{n}^{e_{n}} in M(S) such that e_{i} = a.

We note that A is not empty since it contains o_{i}, the order of χ_{i} in κ(G).

For a, b ∈ A with a > b, we have that a − b ∈ A as can be seen as follows. By construction there are two invariants

x_{i}^{e_{i}} ⋯ x_{n}^{e_{n}} and x_{i}^{f_{i}} ⋯ x_{n}^{f_{n}} ∈ M(S)

and thus we obtain two equations

e_{i}χ_{i} + ⋯ + e_{n}χ_{n} = 0 and f_{i}χ_{i} + ⋯ + f_{n}χ_{n} = 0

such that e_{i} = a, f_{i} = b, and e_{j} = f_{j} = 0 for j ∉ S. Taking the difference of these equations yields

(e_{i} − f_{i})χ_{i} + ⋯ + (e_{n} − f_{n})χ_{n} = 0,

with e_{i} − f_{i} = a − b. The coefficients of this equation are not necessarily nonnegative. However, since G is finite, we can choose for each 1 ≤ j ≤ n, a positive integer (namely, the order of χ_{j}) o_{j} such that o_{j}χ_{j} = 0. Therefore by adding enough positive multiples of o_{j}χ_{j} for j ∈ S \ {i}, we get an equation

h_{i}χ_{i} + ⋯ + h_{n}χ_{n} = 0

with h_{i} = a − b, h_{j} ≥ 0 for j ∈ S and h_{j} = 0 for j ∉ S. It follows that A ∪ {0} is a lattice in ℕ_{0} and hence generated by its smallest positive member, say a_{min}. Let

m_{S} = x_{i}^{e_{i}} ⋯ x_{n}^{e_{n}} ∈ M(S)

be the smallest monomial in M(S) w.r.t. lexicographic order with x_{1} > x_{2} > ⋯ > x_{n} such that e_{i} = a_{min}. We show that the collection of these monomials m_{S}, for every S ⊆ {1, …, n} is separating.

Proposition 10. The set T = {m_{S}|S ⊆ {1, 2, …, n}} is separating. In particular, the minimal size of a separating set is at most 2^{n} − 1.

Proof. We assume to the contrary that the monomials in T do not separate the distinct orbits [v], [w] ∈ V/G. We will show that this implies that m(v) = m(w) for any invariant monomial m, and hence for any invariant, which is the desired contradiction. Let

m = x_{1}^{e_{1}}x_{2}^{e_{2}} ⋯ x_{n}^{e_{n}} ∈ 𝔽[V]^{G}.

Denote by S the complement in {1, 2, …, n} of {j | e_{j} = 0}. Thus m ∈ M(S). We proceed by induction on the order of S.

If |S| = 1, say S = {j}, then m = x_{j}^{t·o_{j}} for some positive integer t, since m is invariant. Furthermore, m_{S} = x_{j}^{o_{j}} ∈ T . Since we are assuming the monomials in T do not separate v = (v_{1}, …, v_{n}) and w = (w_{1}, …, w_{n}) we find that

m(v) = v_{j}^{t·o_{j}} = w_{j}^{t·o_{j}} = m(w).

Since this is true for any choice of j we are done.

Next, we assume that |S| > 1, and the result has been proven for sets of smaller size.

Let i denote the smallest integer in S. By construction there exists a positive integer r such that the monomial

m_{S}^{r} = x_{i}^{f_{i}} ⋯ x_{n}^{f_{n}}

satisfies e_{i} = f_{i}. Hence,

\frac{m}{m_{S}^{r}} = \frac{x_{i+1}^{e_{i+1}} ⋯ x_{n}^{e_{n}}}{x_{i+1}^{f_{i+1}} ⋯ x_{n}^{f_{n}}} ∈ 𝔽(V)^{G}.

is a rational invariant.

Let J denote the set of indices j such that x_{j} appears in the denominator of \frac{m}{m_{S}^{r}}. Since x_{j}^{o_{j}} is an invariant for all j ∈ J, it follows that for some suitably large t ∈ ℕ

m′ := \frac{m}{m_{S}^{r}} ∏_{j∈J} x_{j}^{to_{j}} ∈ 𝔽[V]^{G}

is an invariant monomial. Moreover, since x_{i} does not appear in m′ and all the indices of the variables that appear in m′ come from S, we have m′ ∈ M(S′) for some S′ ⊊ S. Consider

m = \frac{m′ · m_{S}^{r}}{∏_{j∈J} x_{j}^{o_{j}}} .

Since m_{S} ∈ T , the monomial m_{S}^{r} does not separate v and w. Moreover, by our induction hypothesis m′ ∈ M(S′) and ∏_{j∈T} x_{j}^{o_{j}} ∈ M(J) do not separate v and w either, because S′, J ⊊ S. But the value of m at a point is uniquely determined by m′, m_{S} and ∏_{j∈J} x_{j}^{o_{j}} if ∏_{j∈J} x_{j}^{o_{j}} is non-zero at that point. Therefore m does separate v and w if ∏_{j∈J} x_{j}^{o_{j}} is non-zero at one (hence both) of v and w.

On the other hand if ∏_{j∈J} x_{j}^{o_{j}} vanishes at a point, then m also vanishes at that point because J ⊆ S, namely if a variable appears in ∏_{j∈J} x_{j}^{o_{j}}, it also appears in m.

Finally, the monomial m_{∅} corresponding to empty set is just 1, hence it is not needed in a separating set. This completes the proof. □

We demonstrate in the following example that the bound of the previous proposition is sharp.

Example. Let G = ℤ_{3} be the cyclic group of order 3 acting diagonally on the polynomial ring ℂ[x_{1}, x_{2}] with complex coefficients by σ(x_{i}) = λx_{i} for 1 ≤ i ≤ 1, where λ is a primitive 3rd root of unity and σ is a generator of G. The invariant ring is minimally generated by {x_{1}^{3}, x_{1}^{2}x_{2}, x_{1}x_{2}^{2}, x_{2}^{3}}. As the previous proposition predicts, the set {x_{1}^{3}, x_{1}x_{2}^{2}, x_{2}^{3}} is separating. If there were a separating set consisting of two elements, say f_{1}, f_{2}, then ℂ[f_{1}, f_{2}] ⊆ ℂ[x_{1}, x_{2}]^{G} is a finite extension and moreover ℂ[x_{1}, x_{2}]^{G} is the normalization of ℂ[f_{1}, f_{2}], because there are no nontrivial purely inseparable extensions in characteristic zero. But this is impossible because ℂ[f_{1}, f_{2}] is a regular ring and hence is integrally closed.

Meanwhile the separating set in Proposition 10 can be refined substantially for cyclic groups of prime order as we show next.

Proposition 11. Let G be a cyclic group of prime order. Furthermore assume that n ≥ 2 and χ_{j} ≠ 0 for 1 ≤ j ≤ n. Then

T = {m_{S}|S ⊆ {1, 2, …, n} and |S| = 1, 2}

is separating. Note that the order of T does not exceed \frac{n^{2}+n}{2}.

Proof. Let |S| = 1, say S = {j}, then m_{S} = x_{j}^{o_{j}}. Assume next that |S| = 2 with S = {i, j} and i < j. Since κ(G) is cyclic of prime order and χ_{i}, χ_{j} ≠ 0 there exists a unique positive integer a_{i,j} < o_{j} such that

χ_{i} + a_{i,j}χ_{j} = 0 ∈ κ(G).

Hence x_{i}x_{j}^{a_{i,j}} is an invariant monomial. Since a_{i,j} is the smallest among the positive integers k such that x_{i}x_{j}^{k} is invariants it follows that m_{S} = x_{i}x_{j}^{a_{i,j}}. Thus we have obtained

T = {x_{j}^{o_{j}}}_{1≤j≤n} ∪ {x_{i}x_{j}^{a_{i,j}}}_{1≤i<j≤n}.

From this point on, the proof of the previous proposition carries over: We assume that the monomials in T do not separate the vectors v, w ∈ V with distinct Gorbits. Let m = x_{1}^{e_{1}}x_{2}^{e_{2}} ⋯ x_{n}^{e_{n}} be an arbitrary invariant monomial and let S denote the complement in {1, 2, …, n} of {j | e_{j} = 0}. Then m ∈ M(S). Let i be the smallest integer in S. We proceed by induction on |S|.

If |S| = 1, then m = x_{i}^{t·o_{i}} for some positive integer o_{i}. But, being in T , x_{i}^{o_{i}} does not separate v and w. Therefore m = x_{i}^{t·o_{i}} does not either.

Assume next |S| ≥ 2. Pick j ∈ S with j > i. Then x_{i} does not appear in

\frac{m}{(x_{i}x_{j}^{a_{i,j}})^{e_{i}}} = \frac{x_{i+1}^{e_{i+1}} ⋯ x_{n}^{e_{n}}}{x_{j}^{a_{i,j}e_{i}}} .

It follows that for sufficiently large t ∈ ℕ

m = \frac{m′(x_{i}x_{j}^{a_{i,j}})^{e_{i}}}{x_{j}^{to_{j}}} ,

for some m′ that lies in M(S′) for some proper subset S′ in S. The value of m at a point is uniquely determined by m′, (x_{i}x_{j}^{a_{i,j}})^{e_{i}} and x_{j}^{o_{j}}, if j-th coordinate of that point is non-zero. In this case m does not separate v and w by induction since m′ ∈ M(S′) and x_{j}^{o_{j}}, x_{i}x_{j}^{a_{i,j}} ∈ T . On the other hand if x_{j}^{o_{j}}(v) = 0 (hence x_{j}^{o_{j}}(w) = 0), then m(v) = m(w) = 0 as well since j ∈ S, i.e., x_{j} appears in m. □

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Department of Math. and Stats., Texas Tech University, MS 1042 Lubbock, TX 79409, USA

   E-mail address: Mara.D.Neusel@ttu.edu

Department of Mathematics, Bilkent University, Ankara 06800, Turkey

   E-mail address: sezer@fen.bilkent.edu.tr