THE NOETHER NUMBERS FOR CYCLIC GROUPS OF PRIME ORDER

P. FLEISCHMANN, M. SEZER, R.J. SHANK, AND C.F. WOODCOCK

Abstract. The Noether number of a representation is the largest degree of an element in a minimal homogeneous generating set for the corresponding ring of invariants. We compute the Noether number for an arbitrary representation of a cyclic group of prime order, and as a consequence prove the “2p − 3 conjecture”.

1. Introduction

Let V denote a finite dimensional representation of a finite group G over a field F, i.e., V is a finite module over the group ring FG. The action of G on V induces an action of G on the dual V^{∗} which extends to an action by degree preserving algebra automorphisms on the symmetric algebra of V^{∗}, F[V] := S(V^{∗}). The ring of invariants,

F[V]^{G} := {f ∈ F[V] | g(f) = f, ∀g ∈ G},

is a finitely generated subring of F[V], and can be interpreted as the ring of regular functions on the categorical quotient V//G. Since the action of G on F[V] preserves degree, the generators can be chosen to be homogeneous. The Noether number of the representation, β(V), is defined to be the least integer d such that F[V]^{G} is generated by homogeneous elements of degree less than or equal to d. Emmy Noether [13] proved that if F has characteristic zero, then β(V) ≤ |G|. This result has been extended to all non-modular representations by Fleischmann [8] and Fogarty [9]. It is clear that a non-trivial non-modular representation of a cyclic group of prime order has Noether number |G|. In fact, cyclic groups are the only groups with non-modular representations having |G| as the Noether number (see Schmid [17] for characteristic zero and Sezer [18] for the generalisation to non-modular representations). It follows from the work of Richman (see [15] and [16]), that |G| is not an upper bound for β(V) for modular representations. It has been conjectured that β(V) ≤ max{|G|, dim(V)(|G|−1)} (see [5, Remark 3.9.10]). Derksen & Kemper have shown that

β(V) ≤ n(|G| − 1) + |G|^{n·2^{n−1}+1} · n^{2^{n−1}+1}

with n := dim(V) [5, Theorem 3.9.11], and Karagueuzian & Symonds [11] have shown that if F = F_{q} then β(V) ≤ (q^{n} − 1)(nq − n − 1)/(q − 1) (for n > 2). The Noether number is an important characteristic of the ring of invariants. For example, a prior knowledge of the Noether number reduces the construction of a generating set to a problem in linear algebra.

Date: August 3, 2005.

1991 Mathematics Subject Classification. 13A50.

Research supported by EPSRC.

For the remainder of the paper, F will denote a field of characteristic p for a prime number p, and Z/p will denote the cyclic group of order p. Despite the simplicity of the representation theory of Z/p, the corresponding rings of invariants can be surprisingly complicated. From Ellingsrud & Skjelbred [7], we know that these rings of invariants almost always fail to be Cohen-Macaulay. While formulae for the Hilbert series are known for all representations of Z/p, see Almkvist & Fossum [1] and Hughes & Kemper [10], explicit generating invariants are known for only a few special cases.

Choose a generator σ for Z/p. In FZ/p, define Δ := σ − 1 and Tr := ∑_{i=1}^{p} σ^{i}. The isomorphism type of a representation of Z/p is determined by the Jordan canonical form of σ. If n ≤ p, then the n × n matrix over F consisting of a single Jordan block with eigenvalue 1, has order p and determines an indecomposable FZ/p-module, which we denote by V_{n}. Note that for n > p, this matrix has order at least p^{2} and, therefore, does not give a representation of Z/p. Recall that 1 is the only p^{th} root of unity in F. Thus V_{1}, V_{2}, …, V_{p} is a complete list, up to isomorphism, of the indecomposable FZ/p-modules. For a positive integer k, we will denote the direct sum of k copies of V_{n} by kV_{n}. Explicit generating sets for F[V_{2}]^{Z/p} and F[V_{3}]^{Z/p} were given by Dickson [6]. Generators for F[V_{4}]^{Z/p} and F[V_{5}]^{Z/p} can be found in [20]. The problem of finding an explicit generating set for F[V_{n}]^{Z/p} for n > 5, remains open. Even when the invariants of the indecomposable summands are well understood, it can be difficult to construct generating sets for decomposable representations. Campbell & Hughes, in [4], describe a generating set for F[kV_{2}]^{Z/p}, proving a conjecture of Richman [15]. Generating sets have recently been constructed for F[V_{2} +V_{3}]^{Z/p} [21] and F[2V_{3}]^{Z/p} [3]. Using the known generating sets, it is easy to see that β(V_{2}) = β(V_{3}) = β(2V_{2}) = p. It follows from [4] and [15], that β(kV_{2}) = k(p−1) for k > 2. By [19, Remark 3.3], β(V_{4}) = 2p − 3. This paper is devoted to computing the Noether number for all remaining modular representations of Z/p. An FZ/p-module is called reduced if it is has no trivial summands. Note that, for any FZ/p-module W, we have β(kV_{1} + W) = β(W), i.e., adding trivial summands does not change the Noether number. Therefore, it is sufficient to compute β for reduced FZ/p-modules. Also note that V_{n}^{Z/p}, the vector space of fixed points, has dimension one. Therefore, for any FZ/p-module W, dim(W^{Z/p}) is the number of indecomposable summands in the decomposition of W.

Proposition 1.1. Suppose that W is a reduced finite dimensional FZ/p-module.

(a) If W contains a summand isomorphic to V_{n} with n > 3, then

β(W) = (p − 1) dim(W^{Z/p}) + p − 2.

(b) If W is isomorphic to mV_{2} + ℓV_{3} with ℓ > 0, then

β(W) = (p − 1) dim(W^{Z/p}) + 1.

Proof. The first result follows from the lower bound given in Theorem 2.1 and the upper bound given in Corollary 3.4. The second result follows from the lower bound given in Theorem 2.1 and the upper bound given in Corollary 2.8 of [19]. □

We note that the “2p−3 conjecture”[21, Conjecture 6.1] is a special case of Proposition 1.1 (a), and that Conjecture 1.1 of [19] follows from Theorem 3.3.

For general facts concerning the invariant theory of finite groups, see [2], [5] or [12].

2. Lower bounds

It is well known that β(V_{2}) = β(V_{3}) = β(2V_{2}) = p. For all other reduced representations, W, the number (p − 1) dim(W^{Z/p}) is a lower bound for β(W) (see [21, Proposition 3.1]). If W = kV_{2} with k > 2, then this lower bound is sharp (see [4] and [15]). This section is devoted to proving the following theorem.

Theorem 2.1. Suppose that W is a non-trivial reduced FZ/p-module. Define k := dim(W^{Z/p}). If W contains a summand isomorphic to V_{n} with n > 3, then β(W) ≥ k(p − 1) + p − 2. If W contains a summand isomorphic to V_{n} with n > 2, then β(W) ≥ k(p − 1) + 1.

Recall that for a submodule U ≤ W, β(U) ≤ β(W) [21, Theorem 4.2]. Thus to construct a lower bound, it is sufficient to find a lower bound for a submodule. To prove the first result, we give a lower bound for β((k − 1)V_{2} + V_{4}) and to prove the second result, we give a lower bound for β ((k − 1)V_{2} + V_{3}).

Choose a basis {x_{1}, y_{1}, x_{2}, y_{2}, …, x_{k−1}, y_{k−1}, w, x, y, z} for ((k − 1)V_{2} + V_{4})^{∗} with Δ(y_{i}) = x_{i}, Δ(x_{i}) = 0, Δ(z) = y, Δ(y) = x, Δ(x) = w and Δ(w) = 0.

Lemma 2.2. Tr ((y_{1} ⋯ y_{k−1}z)^{p−1}y^{p−2}) is indecomposable in F[(k − 1)V_{2} + V_{4}]^{Z/p}.

Proof. Define H := Tr ((y_{1} ⋯ y_{k−1}z)^{p−1}y^{p−2}). The proof is by induction on k. For k = 1, the result follows from Remark 3.3 of [19]. Assume k > 1. Suppose, by way of contradiction, that H = ∑ f_{i}h_{i} for some f_{i}, h_{i} ∈ F[(k − 1)V_{2} + V_{4}]_{+}^{Z/p}. We may assume that f_{i}, h_{i} and f_{i}h_{i} are homogeneous with respect to multidegree, and that each f_{i}h_{i} has multidegree (p − 1, p − 1, …, p − 1, 2p − 3). Use the inclusion of V_{1} + (k − 2)V_{2} + V_{4} into (k − 1)V_{2} + V_{4} to define a projection π : F[(k − 1)V_{2} + V_{4}] → F[V_{1} + (k − 2)V_{2} + V_{4}]. Note that π is the algebra homomorphism taking x_{1} to 0, y_{1} to x_{1} and fixing the other variables. Since π is equivariant, π(H) = Tr (π ((y_{1} ⋯ y_{k−1}z)^{p−1}y^{p−2})) = x_{1}^{p−1} Tr ((y_{2} ⋯ y_{k−1}z)^{p−1}y^{p−2}). Collecting all factors of x_{1}, write π(f_{i}) = x_{1}^{n_{i}} f̃_{i} and π(h_{i}) = x_{1}^{m_{i}} h̃_{i}. Using the homogeneous multidegree assumption, n_{i}+m_{i} = p−1 and f̃_{i},h̃_{i} ∈ F[(k−2)V_{2}+V_{4}]^{Z/p}. Thus π(H) = x_{1}^{p−1} ∑ f̃_{i}h̃_{i}. Therefore H̃ := Tr ((y_{2} ⋯ y_{k−1}z)^{p−1}y^{p−2}) = ∑ f̃_{i}h̃_{i}. By the induction hypothesis, H̃ is indecomposable in F[(k −2)V_{2} +V_{4}]^{Z/p}. It is clear that f̃_{i}, h̃_{i} ∈ F[(k −2)V_{2} +V_{4}]^{Z/p}. Suppose that one of the factors, say f̃_{1}, is of degree zero. Then the multidegree of f_{1} is (n_{1}, 0, 0, …, 0) with n_{1} < p. Hence f_{1} is a homogeneous element of degree less than p in F[x_{1}, y_{1}^{p} − x_{1}^{p−1}y_{1}] ≅ F[V_{2}]^{Z/p}. Thus f_{1} = cx_{1}^{n_{1}} for some c ∈ F. Therefore π(f_{1}) = 0, giving f̃_{1} = 0. Hence ∑ f̃_{i}h̃_{i} determines a decomposition of H̃, providing the required contradiction. □

Choose a basis {x_{1}, y_{1}, …, x_{k−1}, y_{k−1}, x, y, z} for ((k − 1)V_{2} + V_{3})^{∗} with Δ(y_{i}) = x_{i}, Δ(x_{i}) = 0, Δ(z) = y, Δ(y) = x and Δ(x) = 0. We use the graded reverse lexicographic order with x_{1} < y_{1} < x_{2} < ⋯ < y_{k−1} < x < y < z.

Lemma 2.3. Tr ((y_{1} ⋯ y_{k−1}z)^{p−1}y) is indecomposable in F[(k − 1)V_{2} + V_{3}]^{Z/p}.

Proof. The proof is by induction on k. For k = 1, F[V_{3}]^{Z/p} is a hypersurface with generators in degrees 1, 2, p and p (see [6] or [14, Proposition 5.8]); the element Tr(yz^{p−1}) has lead monomial y^{p} and may be chosen to be one of the generators. The induction step is essentially the same as the induction step in the proof of Lemma 2.2. □

3. An upper bound

Suppose that W is a reduced finite dimensional FZ/p-module. The purpose of this section is to construct an upper bound on β(W). Decompose W into a direct sum of indecomposable modules: W = ⊕_{i=1}^{k}W_{i}. For each i, choose an FZ/p-module generator z_{i} for W_{i}^{∗}. Define d_{i} := dim(W_{i}). Then

⋃_{i=1}^{k}{Δ^{t}(z_{i}) | t = 0, 1, …, d_{i} − 1}

is a basis for W^{∗}. We use this basis to determine generating variables for F[W]. Let A be the subalgebra of F[W] generated by

⋃_{i=1}^{k}{Δ^{t}(z_{i}) | t = 1, …, d_{i} − 1},

i.e., the subalgebra formed by omitting the “terminal variables”. Suppose that m = u_{1}u_{2} ⋯ u_{p−1} is a monomial of degree p − 1 in A. For each u_{i}, there exists a variable, say w_{i}, with Δ(w_{i}) = u_{i}. Define m′ = w_{1}w_{2} ⋯ w_{p−1}. Note that m′ is a monomial of degree p − 1 in F[W]. Define

F := ∑_{ℓ∈F_{p}} ∏_{j=1}^{p−1} (w_{j} − σ^{ℓ}(w_{j})) ,

where F_{p} denotes the field with p elements. For S ⊆ {1, 2, …, p − 1}, define S′ = {1, 2, …, p − 1} \ S. Further define X_{S} = ∏_{j∈S} w_{j} and X_{S′} = ∏_{j∈S′} w_{j}.

Lemma 3.1.

F = ∑_{S⊆{1,2,...,p−1}}(−1)^{|S|}X_{S′} Tr(X_{S}).

Proof. Expanding, and using the fact that σ^{ℓ} is an algebra automorphism, gives

∏_{j=1}^{p−1} (w_{j} − σ^{ℓ}(w_{j})) = ∑_{S⊆{1,2,...,p−1}}(−1)^{|S|}X_{S′}σ^{ℓ}(X_{S}).

Summing over ℓ ∈ F_{p} gives the required formula. □

We use a graded reverse lexicographic order with Δ^{t}(z_{i}) > Δ^{t+1}(z_{i}) for t < d_{i} − 1, and denote the lead term of a polynomial f by LT(f).

Lemma 3.2. LT(F) = −m.

Proof. Note that σ^{ℓ}(w_{j}) = w_{j} + ℓu_{j} + ⋯, where the missing terms are lower in the order. Thus the lead term of w_{j} − σ^{ℓ}(w_{j}) is −ℓu_{j}. Since the lead term of a product is the product of the lead terms, we have LT (∏_{j=1}^{p−1} (w_{j} − σ^{ℓ}(w_{j}))) = (−ℓ)^{p−1} ∏_{j=1}^{p−1} u_{j} = ℓ^{p−1}m. Thus F is a sum of polynomials all with lead monomial m. The result follows from the fact that ∑_{ℓ∈F_{p}} ℓ^{p−1} = −1. □

Recall that the Hilbert ideal of W is the ideal in F[W] generated by homogeneous invariants of positive degree. The ring of coinvariants, denoted by F[W]_{Z/p}, is the finite dimensional graded algebra consisting of the quotient of F[W] by the Hilbert ideal.

Theorem 3.3. The top degree of F[W]_{Z/p} is bounded above by k(p − 1) + p − 2.

Proof. The Hilbert series of a graded ideal in F[W], and the series for the corresponding ideal of lead terms, coincide. Thus it is sufficient to bound the top degree of the quotient by the lead term ideal of the Hilbert ideal. It follows from Lemma 3.2 that every monomial of degree p − 1 in A is in the lead term ideal. Furthermore, z_{i}^{p} is the lead term of the orbit product of z_{i}. Thus the monomials lying outside the lead term ideal are of the form γ ∏_{i=1}^{k} z_{i}^{n_{i}} were γ is a monomial in A with degree at most p − 2, and n_{i} ≤ p − 1. □

Corollary 3.4. The Noether number of W is bounded above by k(p − 1) + p − 2.

Proof. The top degree of the coinvariants gives an upper bound for the degrees of a generating set of the image of transfer. It follows from [10, Lemma 2.12] that F[W]^{Z/p} is generated by transfers, orbit products and elements of at most degree k(p−1)−(dim(W)−k). Thus the upper bound for the top degree of the coinvariants given in Theorem 3.3 gives an upper bound for the Noether number of W. □

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Department of Mathematics, Boğaziçi University, TR-34342 Bebek, Istanbul, Turkey

  E-mail address: mufit.sezer@boun.edu.tr

Institute of Mathematics, Statistics & Actuarial Science, University of Kent at Canterbury, CT2 7NF, UK

  E-mail address: P.Fleischmann@kent.ac.uk
  E-mail address: R.J.Shank@kent.ac.uk
  E-mail address: C.F.Woodcock@kent.ac.uk