THE MODULE STRUCTURE OF A GROUP ACTION ON A POLYNOMIAL RING: EXAMPLES, GENERALIZATIONS, AND APPLICATIONS

DIKRAN B. KARAGUEUZIAN AND PETER SYMONDS

Abstract. The proof of a module-structure result for group actions on polynomial rings, which the authors published in a previous paper, is discussed in some detail. Also examined are possible generalizations of the result and an application to invariant theory.

1. Introduction

This article is a commentary on the authors’ paper [12]. We address three main themes. First, the difficulties in the proof of the main theorem (9.1) of [12]. Second, possible generalizations of this module-structure result. Third, applications of the result to invariant theory.

The main theorem of [12] can be stated as follows.

Let k be a finite field of characteristic p and order q and let U_{n} = U_{n}(k) denote the group of n × n upper triangular matrices over k with 1s on the diagonal. This acts on the polynomial ring in n variables S = k[x_{1}, …, x_{n}]. The ring of invariants under this group, for which we write S^{U_{n}}, is known to be polynomial in generators {d_{i} : i = 1, …, n}, where the degree of d_{i} is q^{i−1}. Note that any p-group P acting faithfully on S may be considered to be a subgroup of U_{n} after a change of variables. Our structure theorem describes the kP -module structure of S for any such P:

Theorem 1.1. There is an isomorphism of graded kP -modules:

S ≅ ⨁_{J⊆I} k[d_{i} : i ∈ I ∪ {n} − J] ⊗_{k} X̄_{J}(I),

where X̄_{J}(I) is a finite dimensional graded kP -module, P acts trivially on k[d_{i}], and I is the set {1, 2, …, n − 1}.

This should be read as saying that S contains one copy of X̄_{J}(I) for each monomial in the d_{i} with i ∈ I ∪ {n} − J.

One consequence of this theorem is a finiteness result.

Corollary 1.2. For any group G (not necessarily a p-group) acting on S, only a finite number of isomorphism classes of indecomposable kG-modules appear as a summand of S.

Section 2 presents a worked example, illustrating the proof of 1.1 and Section 3 comments on the trace lemma ([12], 15.1), which is the most complicated part of the proof.

Next, in Section 4, we comment on several possible generalizations of the structure theorem or its finiteness corollary; in particular Section 5 calculates some examples of a structure theorem for GL_{n}(k).

Some partial results in this direction are proved in Sections 6 and 7. Finally, in Section 8, we sketch how studying the ring S as a module for kG has been used in calculations in invariant theory.

2. Extended Example

In this section we present an extended example to help the reader sort through the details of the proof of the main theorem (9.1) of [12]. We take n = 3, q = 3, and I = ⟨1, 2⟩; i.e. we consider the proof of the theorem for the group U_{3} over the field k with three elements. For simplicity of notation we write U_{1,2} instead of U_{{1,2}} = U_{3}, and similarly for U_{1} = U_{{1}} and U_{2} = U_{{2}} (even though this conflicts with the upper triangular matrices definition of these groups).

In this case the polynomial ring S = k[x_{1}, x_{2}, x_{3}] contains the invariant ring S^{U_{1,2}} = k[d_{1}, d_{2}, d_{3}], where the degrees of d_{1}, d_{2}, and d_{3} are 1, 3, and 9 respectively. We study the kU_{1,2}-module structure of S by first reducing to the study of the kU_{1,2}-module structure of T = S/(d_{3}), using Lemma 5.4 of [12] which tells us that, as kU_{1,2}-modules, S = k[d_{3}] ⊗ T.

Now we examine the statement of the main theorem (9.1) of [12] in this case. Specializing the claims, the result tells us that there is a decomposition

T = X_{⟨1,2⟩}(⟨1, 2⟩) ⊕ X_{⟨1⟩}(⟨1, 2⟩) ⊕ X_{⟨2⟩}(⟨1, 2⟩) ⊕ X_{∅}(⟨1, 2⟩)

with certain properties. (In what follows we will eliminate the angle-brackets in the notation.)

Perhaps the simplest property to understand is the statement about Poincaré series. To unwind this statement ([12], 9.1,(4)) we note that the sets S_{J}(I) here live in P (I) = ℕ_{0}^{I} = ℕ_{0}^{2}. These four sets S_{1,2}(1, 2), S_{1}(1, 2), S_{2}(1, 2), and S_{∅}(1, 2) are defined by the inequalities in [12], 4.6. In the present case, with the natural coordinates a_{1}, a_{2} on ℕ_{0}^{2}, we have

S_{1,2}(1, 2) = {(a_{1}, a_{2}) | a_{2} < 2 and a_{1} + 2a_{2} < 6},
S_{1}(1, 2) = {(a_{1}, a_{2}) | a_{2} ≥ 2 and a_{1} < 4},
S_{2}(1, 2) = {(a_{1}, a_{2}) | a_{2} < 2 and a_{1} + 2a_{2} ≥ 6}, and
S_{∅}(1, 2) = {(a_{1}, a_{2}) | a_{2} ≥ 2 and a_{1} ≥ 4}.

Note that these sets form a partition of ℕ_{0}^{2}, as shown in Proposition 4.10 of [12].

It follows from the definition of the κs ([12], 7.5) that the sequence κ_{i}^{(3−1)} is 1, 3, 6, 8, 9, 9, 9, … and that the sequence κ_{j}^{(3−2)} is 1, 2, 3, 3, 3, . . .. We are interested in the matrix of products

κ_{a_{1}}^{(3−1)}κ_{a_{2}}^{(3−2)}, which looks like this:

1 3 6 8 9 9 9 9 9
2 6 12 16 18 18 18 18 18
3 9 18 24 27 27 27 27 27
3 9 18 24 27 27 27 27 27

By noting the regions corresponding to the S_{J}(I), we can obtain explicitly the Poincaré Series for the X_{J}(I) claimed in the result ([12], 9.1). For example, the Poincaré Series of X_{1,2}(1, 2) is 1 + 3t + 6t^{2} + (8 + 2)t^{3} + (9 + 6)t^{4} + (9 + 12)t^{5} + 16t^{6}. This is the only Poincaré series which is a polynomial; the others are

PS(X_{2}(1, 2)) = \frac{9t^{6} + 18t^{7}}{1 − t} ,
PS(X_{1}(1, 2)) = \frac{3t^{6} + 9t^{7} + 18t^{8} + 24t^{9}}{1 − t^{3}} , and
PS(X_{∅}(1, 2)) = \frac{27t^{10}}{(1 − t)(1 − t^{3})}.

It follows from Lemma 7.8 of [12] that these four Poincaré series add up to the Poincaré series of T; here this can be checked directly by adding the four rational functions.

Properties (1), (2), (3), and (4) of the decomposition of [12], 9.1 will be treated as we examine the proof of this result in the case at hand. (Observe that thus far we have only written down the desired Poincaré series, rather than constructing modules which actually have the corresponding series, so further discussion of (4) is required.) We note that the proof of this decomposition depends through the induction on other decompositions (for smaller groups), and so we describe these now.

We consider S as a module for the group U_{1} (of order 9); to understand its structure we study T (1) = k[x_{1}, x_{2}, x_{3}]/(d_{2}, e_{3}), where S^{U_{1}} = k[x_{1}, d_{2}, e_{3}]. (In fact e_{3} may be written explicitly as x_{3}^{3} − x_{1}^{2}x_{3}.) As a kU_{1}-module, T (1) splits into a projective part and a non-projective part B_{1}(1) with Poincaré series 1 + 3t + 6t^{2} + 8t^{3}. The module B_{1}(1) is easy to describe: it is just the part of T (1) with degree less than 4.

The kU_{1,2} module X̄_{1}(1, 2) is obtained by induction from B_{1}(1): as noted in the proof of Proposition 8.4 of [12], there is a map Ind_{U_{1}}^{U_{1,2}} B_{1}(1) → T defined by g ⊗ b ↦ g · (G(1, 2 | 1) · b). The proof that this map is injective requires the Trace Lemma ([12], 15.1); in this special case that lemma states that

Tr_{U_{2}} \frac{d_{3}}{d_{2}} · x_{1}^{i} = −d_{2}^{2} · d_{1}^{i},

for i = 0, 1, 2, 3. We omit any discussion of filtration and terms lying in the “error ideal” since in this case, and in the other cases of the trace lemma being treated here, there are no such terms, and we may arrange the modules coming from lower stages of the induction to have monomial socles also. The proof of the injectivity gives an identification (generally in the sense of leading terms, but here exactly) of the socle of X̄_{1}(1, 2); since the module X_{1}(1, 2) is obtained from X̄_{1}(1, 2) by propagating by d_{2}, it follows that we have identified the socle of X_{1}(1, 2).

This means that X_{1}(1, 2) has the properties (1), (2), (3), and (4) required by the statement of [12], 9.1: (1) is automatic from the propagation construction, (2) follows from the construction of X̄_{1}(1, 2) by mapping in an induced module, while (3) comes from the proof of injectivity in Proposition 8.4 of [12], which gives an identification of the socle of X̄_{1}(1, 2). Finally, property (4) follows from our knowledge of the Poincaré series of B_{1}(1).

The constructions of X_{2}(1, 2) and X_{∅}(1, 2) are similar; we abbreviate the treatments because we have already discussed X_{1}(1, 2) in detail.

To construct X̄_{2}(1, 2) we start with the preliminary module B_{2}(2), which is the part of T (2) with degree less than 2; this module has Poincaré series 1 + 2t. Note that T (2) = k[x_{1}, x_{2}, x_{3}]/(x_{2}, f_{3}), where f_{3} = x_{3}^{3} − x_{2}^{2}x_{3}. B_{2}(2) is a U_{2}-module, which we induce to U_{1,2}, and map to T by g ⊗ b ↦ g · (G(1, 2 | 2) · b). We prove that this map is injective and indentify (in the sense of leading terms) the socle using the following special case of the Trace Lemma:

Tr_{U_{1}} \frac{d_{3}}{x_{3}^{3} − x_{2}^{2}x_{3}} · x_{2}^{i} = −d_{1}^{6−2i} · d_{2}^{i},

where i = 0, 1. We then propagate by d_{1}; the resulting module X_{2}(1, 2) has the desired properties.

To construct X̄_{∅}(1, 2) we need only note that the following special case of the Trace Lemma implies the existence of a projective module in the degree 10 part of T:

Tr_{U_{1,2}} G(1, 2 | ∅) = Tr_{U_{1,2}} \frac{d_{3}}{x_{3}} \frac{d_{2}}{x_{2}} = d_{1}^{4} · d_{2}^{2}.

We then propagate by d_{1} and d_{2}; the resulting module X_{∅}(1, 2) has the desired properties.

We turn to the construction of X_{1,2}(1, 2), the complement to the direct sum of the other three. The existence and properties of this module follow from Proposition 8.1 of [12]. We review the proof of this proposition to call attention to a subtle point. To show the existence of the complement X_{1,2}(1, 2), we note that the other modules we have constructed do not intersect (because it is clear from our bookkeeping that their socles do not), and in large degrees, their Poincaré series add up to the Poincaré series of T. It follows that the modules X_{1}(1, 2), X_{2}(1, 2), and X_{∅}(1, 2) are summands of T in large degrees, and from this it easily follows that they are individually summands in small degrees ([12], 6.18). The subtle point is that our control of the socles is sufficient to show that these modules are simultaneously summands in small degrees, i.e. that the complement X_{1,2}(1, 2) exists. This is handled in Lemma 6.19 of [12].

From the decomposition of T we have studied, a similar decomposition for S follows, using [12], 5.4. This completes our study of the extended example.

3. Remarks on the Trace Lemma

The proof of the main result of [12] is a very detailed exercise in bookkeeping, and it is natural to wonder if the effort required could somehow be reduced by a slightly different approach. Many of the complications of the trace lemma ([12]) are connected with the “higher terms” which must be handled in the accounting.

The reader of Section 2 may wonder if such higher terms are ever nonzero. Below we give an explicit example in which these higher terms are not zero.

Example 3.1. Let n = 4, q = 3. We write U for the upper-triangular group over k, which we take to act on k[w, x, y, z] in such a way as to preserve the flag of subspaces ⟨w⟩ ⊂ ⟨w, x⟩ ⊂ ⟨w, x, y⟩ ⊂ ⟨w, x, y, z⟩. We write d_{w}, d_{x}, d_{y}, and d_{z} for the orbit polynomials of w, x, y, and z. In the notation of [12], we will study Tr_{U_{1}} : k[w, x, y, z]^{U_{23}} → k[w, x, y, z]^{U}. We recall for the reader’s convenience that U_{1} is the “add w” group, isomorphic to (k^{+})^{3}, and U_{23} is the “add x,y” group, isomorphic to the group of upper-triangular 3 × 3 matrices over k. This group acts on k[x, y, z]; its invariants are generated by the orbit polynomials (under U_{23}!) of x, y, and z, which we write e_{x}, e_{y}, and e_{z}. In this notation we have G(1, 2, 3 | 2, 3) = d_{z}/e_{z}. We now give a short table of traces:

\begin{aligned}& i = 0 & 1 & 2 & 3 & 4 & 5 \\ j = 0 & −d_{w}^{18} & d_{w}^{16}d_{x} + d_{w}^{10}d_{x}^{3} & −d_{w}^{14}d_{x}^{2} + d_{w}^{8}d_{x}^{4} & d_{w}^{12}d_{x}^{3} & −d_{w}^{10}d_{x}^{4} & d_{w}^{8}d_{x}^{5} \\ j = 1 & d_{w}^{12}d_{y} − d_{w}^{6}d_{x}^{2}d_{y} & −d_{w}^{10}d_{x}d_{y} & d_{w}^{8}d_{x}^{2}d_{y} & −d_{w}^{6}d_{x}^{3}d_{y} & 0 & 0 \end{aligned} Table 3.2. Tr_{U_{1}} G(1, 2, 3 | 2, 3) · e_{x}^{i}e_{y}^{j}

In this table of values there are three examples where the trace is not a monomial in the chosen generators of the invariant ring; i.e. where the “higher terms” are nonzero. The energetic reader may wish to verify that the leading terms are as specified in the trace lemma ([12], 15.1).

The proof of the trace lemma would be vastly simplified if one could produce a basis with the convenient property that the required traces took basis elements to basis elements (as opposed to linear combinations of these). Such a basis would eliminate all of the complications of “higher terms”. Needless to say, the authors have not managed to produce such basis.

The authors noted in [11] that a very appealing proof of the main theorem of [12], based on a “factorization” of the module structure, was possible for n = 3, q = 2, but that this proof did not generalize to other values of n and q. (The proof was given in detail in [2].) One possible avenue to creating a generalization of the proof in [2] would be to take the components of the factorization, which are n submodules, each with Krull dimension 1, then invent a construction which takes these components as input and yields the full polynomial ring. However, it is not clear what sort of construction this might be. For example, when n = 3 and q = 3, one would expect to construct k[x, y, z] out of three components, one having Poincaré series (1 + 3t + 6t^{2} + 8t^{3} + 9t^{4} + 9t^{5} + ⋯), and corresponding to the successive radicals of the module k[U_{1,2}/U_{2}] in the notation of Section 2. Thus, one would expect the Jacobson radical J_{8} of k[U_{1,2}/U_{2}] to figure in the construction of S^{3}. However, it can be verified with Magma that J_{8} is neither a submodule nor a quotient of S^{3}.

4. Examples of Possible Generalizations

The reader of [12] will note that the proof of the main theorem in that paper seems to depend on a remarkable collection of coincidences; it is not easy to see how to generalize the modulestructure finiteness result.

In this section we give some examples which suggest that the phenomenon of module-structure finiteness may hold in a more general setting than that of our theorem. There are limits to this generality, as seen in Examples 4.4 and 4.5.

Example 4.1 (Hopf Algebras). Let B = Λ(a, b), an exterior algebra on two generators over the field of two elements, which we write k in this example. We regard B as an algebra of cohomology operations by taking a to act as Sq^{1} and b to act as Sq^{3} + Sq^{2} Sq^{1}. We let this algebra act on the cohomology of ℤ_{2} × ℤ_{2}, which we write k[x, y]. We have the following module-structure decomposition:

k[x, y] ≅ k ⊕ (k[x^{2}, y^{2}] ⊗ F ) ⊕ (L_{1} ⊕ L_{2} ⊕ L_{3}) ,

where k = ⟨1⟩ is the trivial B-module, F = ⟨xy, x^{2}y + xy^{2}, x^{4}y + xy^{4}, x^{4}y^{2} + x^{2}y^{4}⟩ is a free B-module of rank 1, L_{1} = ⟨x, x^{2}, x^{3}, …⟩, L_{2} = ⟨y, y^{2}, y^{3}, …⟩, and L_{3} = ⟨x^{2}y, x^{2}y^{2}, x^{2}y^{3}, …⟩. (We have given vector-space bases for each module appearing.)

We note that, as B-modules, L_{1} ≅ L_{2} ≅ L_{3}, thus only the three isomorphism classes k, F, and L_{1} appear.

We sketch a proof of the above assertions. Note that the Poincaré series of all these modules add up to the Poincaré series of the polynomial ring. So, to get the decomposition result, we need only prove the surjectivity of the map from the decomposition to the polynomial ring.

Using the given basis of the module L_{3}, plus an induction, we can show that the free module x^{2i}y^{2j}F lets us obtain the monomials x^{2i+1}y^{2j+1}, x^{2i+1}y^{2j+2}, x^{2i+4}y^{2j+1}, and x^{2i+4}y^{2j+2}. This gives us every monomial of x^{α}y^{β}, where α > 0 and β > 0. The rest of the monomials are covered by L_{1}, L_{2}, and k.

To see that all of the given modules are indecomposable is standard; The only tricky part is to show that the L_{i} are indecomposable. Perhaps the simplest method of doing this is to use the isomorphism Λ(a, b) ≅ 𝔽_{2}[ℤ_{2} × ℤ_{2}] and quote the results on infinitely generated modules for this group ring in [5].

Remark 4.2. The reader should be aware that there is some question of whether a modulestructure decomposition is well-defined in a context involving modules which are not finitedimensional. Since we have not chosen hypotheses or made specific conjectures, we will not address this issue.

The finiteness corollary still holds over the algebraic closure of a finite field, because any representation over such a field can be written inside a finite field. It is not clear that this remains the case for other infinite fields.

The next example, which was considered indepedently by Derksen and Kemper, shows that the finiteness result can continue to hold even when the field contains transcendentals.

Example 4.3 (Infinite Fields). Let K = k(s, t) ⊃ k be the extension of k generated by two algebraically independent elements s, t. Then G = {y ↦ y + (λs + μt)x, x ↦ x | λ, μ ∈ k} is a group of automorphisms of S = K[x, y]. We note that S^{G} = K[x, d_{y}], where d_{y} = ∏_{(λ,μ)∈k^{2}}(y + (λs + μt)x).

We will show, by the method of [12], that

S ≅ {K[x, d_{y}] ⊗ S^{q^{2}−1}} ⊕ {K[d_{y}] ⊗ (⊕_{0}^{q^{2}−2}S^{i})} ,

i.e. we will prove that the main theorem of [12] holds in this context.

To see this, note that by [7], Proposition 9.5, we have that Tr_{G} y^{q^{2}−1} = αx^{q^{2}−1}, where α is a nonzero element of K. Thus, the submodule of S^{q^{2}−1} generated by y^{q^{2}−1} is free. By dimensioncounting, this module is S^{q^{2}−1} itself. In this setting, projective and injective are the same, so we may propagate S^{q^{2}−1} by any monomial in x and d_{y} to obtain new summands, which do not intersect as their socles are different.

The other summand in the decomposition, K[d_{y}]⊗(⊕_{0}^{q^{2}−2}S^{i}), can be mapped to the polynomial ring by multiplication, and does not intersect the first summand. (Again, consider the socles.) Thus, the map we have constructed from the decomposition to the polynomial ring is injective. By counting graded dimensions using Poincaré series, we see that the map is surjective and therefore an isomorphism.

The point of this example is not so much that the finiteness result continues to hold, but that the method of proof of [12] still works. This is because (as noted in [11]) cases where there are three or fewer variables in the polynomial ring have special characteristics which may make alternate proofs of the finiteness result possible. In fact, in Section 6 we will prove that the finiteness result holds for the action of a finite group on a polynomial ring in three variables over any field.

The next example shows that the finiteness result fails for free modules over a polynomial ring.

Example 4.4 (Free Modules over a Polynomial Ring). Let k be the field of two elements and let V = ⟨α, β⟩ ≅ ℤ_{2} × ℤ_{2} act on k[x, y]⟨1, z⟩ by α: z ↦ z + x and β: z ↦ z + y. We can regard k[x, y]⟨1, z⟩ as a subset of k[x, y, z] or as a free k[x, y]-module of rank two.

If n ∈ ℕ_{0}, we write Ω^{n}k for the n-th Heller translate of the trivial kV -module k. These modules are pairwise nonisomorphic, since their ranks over k are all different. Ω^{n}k can be pictured using the diagrams popularized by Benson and Carlson [4]. Writing a = 1 + α ∈ kV, b = 1 + β we have:

Ω^{1}k = •[dl]_{a}[dr]^{b} • • Ω^{2}k = •[dl]_{a}[dr]^{b} • [dl]_{a}[dr]^{b} • • •
Ω^{3}k = •[dl]_{a}[dr]^{b} • [dl]_{a}[dr]^{b} • [dl]_{a}[dr]^{b} • • • • etc.

The dots represent basis vectors and the arrows the action of the group ring; see [4].

Now we note that if we attach a grading to our module by assigning x, y, and z grading 1, then M = k[x, y]⟨1, z⟩ is the direct sum ⊕_{i=0}^{∞}M_{i}. Further, the basis of M_{i} consisting of the monomials of total degree i has exactly the same diagram as Ω^{i}k:

M_{1} = z[dl]_{a}[dr]^{b}xy  M_{2} = zx[dl]_{a}[dr]^{b}zy[dl]_{a}[dr]^{b}x^{2}xyy^{2}  etc.

This gives an explicit isomorphism M_{i} ≅ Ω^{i}k, and therefore

M ≅ ⊕_{i=0}^{∞}M_{i} ≅ ⊕_{i=0}^{∞}Ω^{i}k.

Thus, the module-structure finiteness property does not hold for M.

Example 4.5 (Tensor Algebras). In this example once again k is the field of two elements and V the elementary abelian group of order 4. We write T for the tensor algebra of Ω^{1}k, so that T_{n} = (Ω^{1}k)^{⊗n}. Then T_{n} ≅ Ω^{n}k ⊕ F, where F is a free kV -module. As noted above, the modules Ω^{n}k are pairwise nonisomorphic, so that the finiteness result does not hold in this context either.

Various results about the modules for which a finiteness result does hold for the tensor algebra are given by Feit in [10], II.5. The obvious generalization of the example above to any prime and the results of Feit show that the only groups for which there is a finiteness result for all modules are those of finite representation type.

5. The General Linear Group

In this section we indicate the form a module-structure theorem for the general linear group should take by giving two examples. We will study the cases of two and three variables over the field of two elements. Thus, throughout this section, we take k = 𝔽_{2}. The reader should note that in our examples, we will be proving “best possible” module-structure theorems in the sense that all the summands exhibited are indecomposable.

Our first goal is to handle the case of two variables. We want to describe the module structure that results when G = GL_{2}(k) acts on k[y, z] by homogeneous linear substitutions. We begin by recalling the following standard result:

Lemma 5.1. There are exactly three indecomposable kG-modules: the Steinberg module St, the trivial module k and the free module for the quotient C_{2}, M.

Now we identify instances of these modules in k[y, z]. Recall that k[y, z]^{G} = k[c_{2}, c_{3}], where c_{2} = y^{2} + yz + z^{2}, c_{3} = y^{2}z + yz^{2}.

Lemma 5.2. ⟨y, z⟩ ≅ ⟨y^{2}, z^{2}⟩ ≅ St, ⟨y^{2} + yz + z^{2}⟩ ≅ k, and ⟨y^{3} + y^{2}z + z^{3}⟩ ≅ M.

Writing St^{(1)}, St^{(2)} and M^{(3)} for the copies of these modules we have identified in k[y, z], we have

Proposition 5.3. k[y, z] ≅ k[c_{2}] ⊕ k[c_{2}, c_{3}] ⊗ (St^{(1)} ⊕ St^{(2)}) ⊕ k[c_{2}, c_{3}] ⊗ M^{(3)}.

Proof. We construct a map from the decomposition on the right-hand-side to the polynomial ring using the inclusion of the submodules exhibited in Lemma 5.2 and multiplication by the invariants. We note that the Poincaré Series of the right-hand-side is

\frac{2t + 2t^{2}}{(1 − t^{2})(1 − t^{3})} + \frac{2t^{3}}{(1 − t^{2})(1 − t^{3})} + \frac{1}{1 − t^{2}} = \frac{1}{(1 − t)^{2}} ,

i.e. the same as the Poincaré Series of the left-hand side. So, to prove that the map is an isomorphism, it suffices to show that it is injective.

To show this, we show that there is no linear relation between the socles of the three parts of our decomposition. Such a relation would have the form

f(c_{2}) + c_{3} · g(c_{2}, c_{3}) + (ax + by + cx^{2} + dy^{2}) · h(c_{2}, c_{3}) = 0,

where f, g, and h are polynomials and a, b, c, and d are elements of k. If we rearrange the equation so that all the invariants are on one side, we see that (ax + by + cx^{2} + dy^{2}) must be invariant, and it follows that a, b, c, and d are all zero. Then we have reduced to f(c_{2}) + c_{3} · g(c_{2}, c_{3}) = 0, and since the Dickson invariants are algebraically independent, it follows that f and g are zero.

This proves that the map is injective and therefore an isomorphism. □

Now we turn to the result for three variables, that is, we study the module structure resulting from the action of G = GL_{3}(k) on k[x, y, z] by homogeneous linear transformations. Recall that the Dickson invariants k[x, y, z]^{G} are a polynomial algebra k[c_{4}, c_{6}, c_{7}] on homogeneous generators of degree 4, 6, and 7.

The result is quite complex, and to state it, we introduce a notational convention: M_{r}^{(d)} denotes a module of rank r (over k) which is homogeneous of degree d. We write St for the Steinberg module, which has dimension 8 over k. The proof, which we omit, is handled using Magma.

Our decomposition is:

Proposition 5.4. There is an isomorphism of GL_{3}(k)-modules:

k[x, y, z] ≅ k[c_{4}] ⊗ (k ⊕ M_{3}) ⊕ k[c_{4}, c_{6}] ⊗ (N_{6}^{(2)} ⊕ N_{6}^{(4)} ⊕ X_{10}^{(3)} ⊕ X_{10}^{(5)} ⊕ Z_{14}^{(6)} ⊕ Y_{10}^{(7)})

⊕ k[c_{4}, c_{6}, c_{7}] ⊗ (St^{(4)} ⊕ St^{(5)} ⊕ St^{(6)} ⊕ St^{(7)⊕2} ⊕ St^{(8)} ⊕ St^{(9)} ⊕ St^{(10)}

⊕ P_{16}^{(8)} ⊕ P_{16}^{(9)} ⊕ P_{16}^{(11)} ⊕ P_{16}^{∗(10)} ⊕ P_{16}^{∗(12)} ⊕ P_{16}^{∗(13)} ⊕ W_{8}^{(14)}).

The modules P_{16}, P_{16}^{∗}, St, and W_{8} are projective.

These two examples suggest that the result for GL_{n}(k), where k = 𝔽_{q}, should have n parts, propagated by the subsets of the Dickson invariants

{c_{q^{n}−q^{n−1}}}, {c_{q^{n}−q^{n−1}}, c_{q^{n}−q^{n−2}}}, …, {c_{q^{n}−q^{n−1}}, c_{q^{n}−q^{n−2}}, …, c_{q^{n}−1}}.

One can make more specific conjectures, but since we do not have a completely precise conjecture on the “best possible” decomposition of k[x_{1}, …, x_{n}] as a GL_{n}(k)-module, we will stop here.

6. A General Proof for Three Variables

Here we present a version of the structure theorem for three variables that applies to any finite group and any field. In particular, this result (6.1) gives a decomposition for the action of GL_{3}(k) on k[x_{1}, …, x_{n}], where k is a finite field (see Proposition 5.4 and the remarks following). Further, Theorem 6.1 applies to a field that contains transcendentals (see also Example 4.3).

Theorem 6.1. If a finite group G acts on S = k[x_{1}, x_{2}, x_{3}] by homogeneous transformations then there are homogeneous invariant elements u, v ∈ S^{G} and graded kG-submodules P, U, B of S such that P is projective (but infinite dimensional) and U, B are finite dimensional and there is an isomorphism of kG-modules

S ≅ P ⊕ (U ⊗ k[u, v]) ⊕ (B ⊗ k[u]).

The proof depends on several lemmas, which hold for any number of variables. There is no content when k has characteristic 0, so we assume that the characteristic is p.

Lemma 6.2. Let R be a commutative graded ring, finitely-generated as an algebra over R_{0} = k by homogeneous elements of positive degree. Let M be a finitely generated graded RG-module for which there is a number N such that dim_{k} M_{r} ≤ N for all r ∈ ℤ. Then there are finitedimensional graded kG-submodules B, U of M and a homogeneous element v ∈ R such that M = B ⊕ (U ⊗ k[v]).

Proof. By Noether normalization there is a homogeneous element v ∈ R of degree s, say, such that M is finitely-generated over k[v]. It follows that there is some N such that if r ≥ N then M_{r+s} = vM_{r}. Let B = ⨁_{r<n} M_{r} and U = ⨁_{r=N}^{N+s−1} M_{r}. □

Lemma 6.3. If G acts on S = k[x_{1}, …, x_{n}] then there is an invariant homogeneous element u ∈ S^{G} such that, as kG-modules, S ≅ (S/uS) ⊗ k[u].

Proof. Let Q be the Sylow p-subgroup of G. We may assume that the matrices have triangular form with respect to the basis x_{1}, …, x_{n} (such that x_{1} is fixed). Let R = S^{G}. Since S is finitely generated over R and S contains x_{n}^{r} for all r ∈ ℕ there must be a homogeneous element u ∈ R that contains the monomial x_{n}^{r} for some r ∈ ℕ.

Let T denote the k-subspace of S spanned by all monomials not divisible by x_{n}^{r}. Then S = T ⊕ uS as kQ-modules. Thus the inclusion of uS in S is split over Q, so it is split over G: let T′ be a complement as a kG-module. Then S = T′ ⊕ uS = T′ ⊗ k[u]. □

Remark 6.4. If G is a p-group in triangular form then we may take u to be the orbit product of x_{n}. If k is finite we may take u to be the bottom Dickson invariant for GL_{n}(k).

Lemma 6.5. Returning to G acting on S = k[x_{1}, x_{2}, x_{3}], with u as in lemma 6.3, the nonprojective part of S/uS, which we denote by (S/uS)/(proj), has dimension bounded by some N in each degree.

Proof. Since S/(proj) ≅ ((S/uS)/(proj))⊗k[u] and dim_{k} S/(proj)_{r} is bounded by a linear function in r by [16], the result follows. □

Proof. of 6.1. By lemma 6.3 we can write S = (S/uS) ⊗ k[u] and, by lemma 6.5, we can apply lemma 6.2 to obtain (S/uS)/(proj) = B ⊕ (U ⊗ k[v]). □

7. Conlon’s Induction Theorem

One approach to obtaining a structure theorem for general G is to use Conlon’s Induction Theorem to reduce it to the case of subgroups that are cyclic modulo p.

Let T_{n} = T_{n}(k) be the subgroup of GL_{n}(k) consisting of upper triangular matrices where any non-zero entries are allowed on the diagonal. The generators d_{i} of the invariants under U_{n} are eigenvectors for T_{n}.

Proposition 7.1. The main structure formula 1.1 remains valid over T_{n}, with T_{n} acting on the d_{i} as above.

Proof. Examine the proof in [12] to verify that the pieces X̄_{I}(J) can be defined over T_{n}. □

Rather than a structure theorem we will produce one of a result that would be one of its main consequences: a formula for the multiplicity as a summand of any indecomposable kG-module.

For any indecomposable kG-module M let PS_{M}(S, t) be the power series in which the coefficient of t^{r} is the multiplicity of M as a summand of S_{r}.

By Conlon’s Induction Theorem ([8]; [9], 80.61), there are cyclic modulo p subgroups C_{i} of G and rational numbers a_{i} such that

S = ∑_{i} a_{i} Ind_{C_{i}}^{G} Res_{C_{i}}^{G} S

in the Green ring.

Let ℓ be an extension of k such that all the p′ elements of G become diagonalizable in ℓ. We write ℓS = ℓ ⊗_{k} S. The image of each C_{i} in GL_{n}(ℓ) can be conjugated in to T_{n}(ℓ), so we can apply 7.1 to Res_{C_{i}}^{G} ℓS to obtain

ℓS = ∑_{j} PS_{N_{ij}}(ℓS, t)N_{ij},

where N_{ij} runs through the finite set of indecomposable summands of ℓS. Putting this together we obtain

ℓS = ∑_{ij} a_{i} PS_{N_{ij}}(ℓS, t) Ind_{C_{i}}^{G} N_{ij}.

Now we restrict scalars and, to simplify the notation, we let b_{ij}(M) denote the multiplicity of the indecomposable kG-module M as a summand of Ind_{C_{i}}^{G} Res_{C_{i}}^{G} Res_{k}^{ℓ} N_{ij}. We obtain:

Theorem 7.2.

PS_{M}(S, t) = \frac{1}{[ℓ : k]} ∑_{ij} a_{i}b_{ij}(M) PS_{N_{ij}}(ℓS, t),

where PS_{N_{ij}}(ℓS, t) encodes the multiplicity of N_{ij} as a summand of ℓS as an ℓC_{i}-module.

To use this formula involves calculating the C_{i} and the a_{i}, which is routine, and also the PS_{N_{ij}}(ℓS, t). The latter can all be calculated by restricting the formula for the decomposition of ℓS as an ℓT_{n}(ℓ)-module. According to [12] this involved calculating ℓS up to degree r^{n−1} − n, where r is the order of ℓ.

Such a procedure is probably uneconomical, but it does yield an explicit algorithm.

8. Applications to Vector Invariants

In this section we present an application of module-structure theory to invariant theory. We show how to obtain the Poincaré Series of the vector invariants k[V ⊕ V ⊕ ⋯ ⊕ V]^{G} using the G-module structure of k[V]. We also show how to take this technique further and in some cases obtain a Poincaré series for what we call tensor invariants, that is, k[V ⊗ W]^{G×H}, where W carries a H-module structure. Most of these techniques were shown to the authors by Milgram [14], who states that their use was standard in certain types of homotopy-theoretic calculations about thirty years ago, but offers no specific attribution.

The basic idea for vector invariants is simple: the G-module structure of k[V] determines the Poincaré Series of k[V]^{G}. Since the G-module structure of k[V] determines the G-module structure of k[V^{⊕n}] ≅ k[V]^{⊗n}, it also determines the Poincaré series of k[V^{⊕n}]^{G}. Similarly, if W is a monomial representation of H, we will be able to determine the G × H-module structure of k[V ⊗ W] in terms of the G-module structure of k[V] and the H-module structure of k[W]; the vector invariants are simply the special case where W is the trivial H-module of dimension n.

Because the idea is not deep, and because the appropriate implementation will vary with the particular application, we will confine ourselves to illustrating the method by the study of certain special cases. We begin with the case of vector invariants of the regular representation F of the cyclic group of order 2, C_{2}, over the field of two elements k.

Example 8.1 (Vector invariants: the regular representation of G = C_{2}, n = 2.). The C_{2}-module structure of the polynomial ring k[F] is (k[d_{2}] ⊗ k) ⊕ (k[d_{1}, d_{2}] ⊗ F). From this fact we have

k[F ⊕ F] ≅ k[F] ⊗ k[F] ≅ {(k[d_{2}] ⊗ k) ⊕ (k[d_{1}, d_{2}] ⊗ F)} ⊗ {(k[e_{2}] ⊗ k) ⊕ (k[e_{1}, e_{2}] ⊗ F)} .

Noting that F ⊗ F ≅ F ⊕ F, we may expand this decomposition to yield

(8.2) k[F ⊕F] ≅ (k[d_{2}, e_{2}]⊗k)⊕(k[d_{1}, d_{2}, e_{2}]⊗F)⊕(k[d_{2}, e_{1}, e_{2}]⊗F)⊕(k[d_{1}, d_{2}, e_{1}, e_{2}]⊗F ⊕F)

Now we note that dim_{k} F^{C_{2}} = dim_{k} k^{C_{2}} = 1, so that 8.2 gives us an expression for the Poincaré Series of k[F ⊕ F]^{C_{2}}:

(8.3) \frac{1}{(1 − t^{2})^{2}} + \frac{t}{(1 − t)(1 − t^{2})^{2}} + \frac{t}{(1 − t)(1 − t^{2})^{2}} + \frac{2t^{2}}{(1 − t)^{2}(1 − t^{2})^{2}} = \frac{1 + t^{2}}{(1 − t)^{2}(1 − t^{2})^{2}} .

The skeptical reader will be pleased to note that our series in Equation 8.3 can be, and in fact has been, checked using Kemper’s software [13], which has been incorporated into Magma [6].

Now that we have illustrated the basic idea, we proceed to some refinements. Our next example studies tensor invariants; in the notation of the opening paragraph of this section we take G ≅ H ≅ C_{2} and V and W to be the regular representations of G and H.

Example 8.4 (Tensor invariants: G = C_{2}, H = C′_{2}, V = F, and W = F′.). We regard F ⊗ F′ ≅ F ˜⊕F as a C_{2} × C′_{2} module, where the C′_{2} exchanges the F s. This is the same as the C_{2} × C′_{2}-module obtained by taking the tensor product of the regular representations F, F′ of C_{2}, C′_{2}. We study k[F ⊗ F′] as a C_{2} × C′_{2}-module by writing k[F ⊗ F′] ≅ k[F ˜⊕F] ≅ k[F] ˜⊗k[F], where the tilde indicates that the action of C′_{2} switches the factors in the direct sum or tensor product. Of course underneath all the notation we are just studying the regular representation of C_{2} × C′_{2}; the point is to illustrate the method.

If we write down the structure of k[F] ˜⊗k[F] as a C_{2}-module in the first few degrees, we obtain

Degree  0  1  2  2
Degree Module  k  F  F  k
0  k  k ˜⊗k k ⊗ F k ⊗ F \underline{k ⊗ k}
1  F  F ⊗ k F ˜⊗F F ⊗ F F ⊗ k
2  F  F ⊗ k F ⊗ F F ˜⊗F F ⊗ k
2  k  \underline{k ⊗ k} k ⊗ F k ⊗ F k ˜⊗k

Table 8.5. C_{2}-module structure of k[F] ˜⊗k[F]

We note that the action of C′_{2} swaps the entries in the table across the main diagonal. Thus, as representations of C_{2} × C′_{2}, the boxed modules combine to form Ind_{C_{2}}^{C_{2}×C′_{2}}(F ⊗ k), while the

underlined modules combine to form Ind_{C_{2}}^{C_{2}×C′_{2}}(k ⊗ k). Similar remarks apply to the other offdiagonal entries in the table. The diagonal entries are marked with tildes to indicate that the action of C′_{2} switches the factors in the tensor product.

These regularities allow us to determine the C_{2} × C′_{2}-module structure of k[F] ˜⊗k[F] in the following way. We first write down for each C_{2}-isomorphism type [M] of summands of k[F] a Poincaré series P_{M}(t) for its multiplicity. This collection of isomorphism types [M_{i}] gives rise to three classes of C_{2} × C′_{2}-isomorphism types of summands of k[F] ˜⊗k[F], whose forms are listed below:

(1) (boxed type): Ind_{C_{2}}^{C_{2}×C′_{2}}(M_{i} ⊗ M_{j})

(2) (underlined type): Ind_{C_{2}}^{C_{2}×C′_{2}}(M_{i} ⊗ M_{i})

(3) (diagonal type): M_{i} ˜⊗M_{i} The Poincaré series for the multiplicities of each of these types of modules as summands k[F] ˜⊗k[F] of can be obtained easily in terms of the P_{M_{i}}(t) as follows:

(1) (boxed type): P_{M_{i}}(t) · P_{M_{j}}(t)

(2) (underlined type): (1/2)[P_{M_{i}}(t)^{2} − P_{M_{i}}(t^{2})]

(3) (diagonal type): P_{M_{i}}(t^{2}) In the case at hand we see from our decomposition of k[F] that P_{k}[t] = (1 − t^{2})^{−1} and that P_{F} (t) = t · (1 − t)^{−1} · (1 − t^{2})^{−1}. We then obtain Poincaré series for the multiplicities of the summands of k[F] ˜⊗k[F]:

Module  Poincaré series
Ind_{C_{2}}^{C_{2}×C′_{2}}(k ⊗ F)  \frac{t}{(1−t)(1−t^{2})^{2}}
Ind_{C_{2}}^{C_{2}×C′_{2}}(F ⊗ F) \frac{t^{3}}{(1−t)^{2}(1−t^{2})(1−t^{4})}
F ˜⊗F  \frac{t^{2}}{(1−t^{2})(1−t^{4})}
Ind_{C_{2}}^{C_{2}×C′_{2}}(k ⊗ k)  \frac{t^{2}}{(1−t^{2})(1−t^{4})}
k ˜⊗k  \frac{1}{(1−t^{4})}

Table 8.6. Poincaré series for the multiplicities of summands in k[F] ˜⊗k[F]

Finally, as in Example 8.2 we need to know the dimension of the invariant subspaces.

Module  Ind_{C_{2}}^{C_{2}×C′_{2}}(k ⊗ F) Ind_{C_{2}}^{C_{2}×C′_{2}}(F ⊗ F) F ˜⊗F Ind_{C_{2}}^{C_{2}×C′_{2}}(k ⊗ k) k ˜⊗k ; dim_{k}(M^{C_{2}×C′_{2}})  1  2  2  1  1

Table 8.7. Dimensions of invariant subspaces

This allows us to add rational functions to obtain the Poincaré series of k[F ⊗ F′]^{C_{2}×C′_{2}}.

(8.8) \frac{t}{(1 − t)(1 − t^{2})^{2}} + \frac{2t^{3}}{(1 − t)^{2}(1 − t^{2})(1 − t^{4})} + \frac{2t^{2}}{(1 − t^{2})(1 − t^{4})}+

\frac{t^{2}}{(1 − t^{2})(1 − t^{4})} + \frac{1}{(1 − t^{4})} = \frac{t^{2} − t + 1}{(1 − t)^{2}(1 − t^{2})^{2}}

This example, like the previous one, has been checked against the result obtained by Kemper’s software.

Although the techniques of Example 8.4 may seem specialized to the case of tensor products with the regular representation of C_{2}, they can be adapted to arbitrary permutation representations by a more sophisticated version of the analysis following Table 8.5. Next we will see that how to handle monomial representations. In the next example we will consider the case of a tensor product with the natural representation St of GL_{2}(k), i.e. in the setting of the opening paragraph of this section we take W = St and H = GL_{2}(k). Note that GL_{2}(k) ≅ S_{3} = ⟨a, b | a^{3} = 1, b^{2} = 1, bab = a^{2}⟩. We write C_{2} for the subgroup ⟨b⟩ ⊂ S_{3}.

Let K = {0, 1, ω, ω^{2}} be the field of four elements. If V is a representation of G, we can describe the G × S_{3}-module structure of K[V ⊗ St] by regarding V ⊗ St as V ˜⊕V, where b switches the two copies of V and a acts on the first as multiplication by ω and on the second by multiplication by ω^{2}. Notice that St can be defined over k (it is the natural representation of GL_{2}(k)) and that if we are only interested in the Poincaré series of the invariant ring, it does not matter whether we work over k or K.

We can adapt the results of Example 8.4 to the C_{2}-module structure of S^{∗}(V ˜⊕V). This gives us three types of submodules of S^{∗}(V ˜⊕V). The first type is V ⊗ W ˜⊕W ⊗ V, the second is V ⊗ V ˜⊕V ⊗ V, and the third is V ˜⊗V. Here we have used the tilde to indicate the action of C_{2}, as in Example 8.4.

We observe that all of these module types are actually S_{3}-invariant; i.e. working over K, the decomposition of the C_{2}-module structure is also a decomposition of the S_{3}-module structure. In the application of this result we used a compact description of the multiplicities of these three types of modules in S^{∗}(V ˜⊕V) in terms of their multiplicities in S^{∗}(V). We can give a similar description of the multiplicities of the different S_{3}-module types. To do this, we first refine our description of the decomposition. Modules of the first type are either St ⊗V ⊗ W or M ⊗ V ⊗ W, regarded as a S_{3} × G-module. (Here M is the free module for the quotient C_{2} of S_{3}.) We have the second case if deg W + 2 deg V ≡ 0 mod 3, and the first case if deg W + 2 deg V ≢ 0 mod 3. The same holds for modules of the second type. Modules of the third type always have trivial C_{3}-action. Note that, for modules of the first and second type, if we are in the first case (i.e. a tensor product with St) there are no invariants, since St^{S_{3}} = 0.)

Observe that if f(t) = ∑_{i} a_{i}t^{i}, g(t) = ∑_{j} b_{j}t^{j}, then ∑_{i+2j≡0 mod 3} a_{i}b_{j}t^{i+j} can be expressed in terms of f and g as

(8.9)  \frac{1}{3} [f(t)g(t) + f(ζt)g(ζ^{2}t) + f(ζ^{2}t)g(ζt)] ,

where ζ ∈ ℂ is a primitive cube root of 1. This discussion can be generalized to other roots of 1, and by combining this with arbitrary permutation representations, we can get results for monomial representations. However, the calculations required for such examples quickly become tedious.

We now proceed to an application of this general discussion.

Example 8.10 (Tensor invariants: G = C_{2}, V = F, H = S_{3}, and W = St). Take G = C_{2} and V = F, the regular representation of G. In the decomposition of K[V ⊗ St] we obtain modules of all five types mentioned above.

We recall from our discussion of the decomposition for S^{∗}(F) that P_{k}(t) = 1/(1 − t^{2}) and P_{F} (t) = t/(1 − t)(1 − t^{2}). Thus in S^{∗}(F ˜⊕F), regarded as an C_{2} × S_{3}-module, we have that the modules of type M ⊗ F ⊗ k have the following multiplicity:

\frac{1}{3} [P_{F} (t)P_{k}(t) + P_{F} (ωt)P_{k}(ω^{2}t) + P_{F} (ω^{2}t)P_{k}(ωt)]

These modules have C_{2} × S_{3}-fixed set of dimension 1, so the function above gives exactly their contribution to the Poincare series of S^{∗}(F ˜⊕F)^{C_{2}×S_{3}}

Next we have to consider modules of type M ⊗ F ⊗ F and M ⊗ k ⊗ k. In the first case we get the multiplicity

\frac{1}{2} · \{\frac{1}{3} [P_{F} (t)^{2} + 2P_{F} (ωt)P_{F} (ω^{2}t)] − P_{F} (t^{2})\}.

These modules have C_{2} × S_{3}-fixed set of dimension 2, so we need to multiply their contribution to the Poincare series by 2. In the second case we have

\frac{1}{2} · \{\frac{1}{3} [P_{k}(t)^{2} + 2P_{k}(ωt)P_{k}(ω^{2}t)] − P_{k}(t^{2})\},

and the modules have C_{2} × S_{3}-fixed set of dimension 1.

Finally there are two cases of modules of the third type. The F ˜⊗F -modules have multiplicity function P_{F} (t^{2}), and C_{2}×S_{3}-fixed set of dimension 2. The k ˜⊗k-modules have multiplicity function P_{k}(t^{2}), and C_{2} × S_{3}-fixed set of dimension 1.

Now we add all of this up and get

\frac{(t^{8} − t^{7} + t^{6} + t^{4} + t^{2} − t + 1)(t + 1)(t^{2} + 1)}{(1 − t^{2})(1 − t^{3})(1 − t^{4})(1 − t^{6})},

which agrees with the result obtained from Kemper’s program.

Now we apply these ideas to an example which is of interest in its own right and not just as an application of these techniques.

Example 8.11 (“Double Dickson” invariants). Take G = GL_{3}(k) and let V be the natural Gmodule. We compute the Poincaré series of the tensor invariants k[V ⊗ St]^{GL_{3}(k)×GL_{2}(k)}. Using the techniques described above, the decomposition of k[V] as a GL_{3}(k)-module (5.4), and Magma we find that our series is

(t^{46} + t^{43} + t^{42} + 2t^{41} + t^{40} + 3t^{38} + 2t^{36} + 3t^{35} + 5t^{34} + t^{33} + 4t^{32} + 3t^{31} + 3t^{30}+

t^{29} + 5t^{28} + 4t^{27} + 4t^{26} + 4t^{25} + 5t^{24} + 3t^{23} + 2t^{22} + 3t^{21} + 3t^{20} + t^{19} + 5t^{18} + 2t^{17}+
3t^{16} + t^{15} + 2t^{14} + t^{12} + t^{11} + 2t^{10} − t^{9} + 2t^{8} + t^{6} − t + 1)·
(t^{58} − t^{57} − t^{54} + t^{53} − t^{52} + t^{51} + t^{48} − t^{47} − t^{46} + t^{45} − t^{44} + t^{43} + t^{42} − t^{41} + 2t^{40}−
2t^{39} + t^{38} − 2t^{37} + t^{35} − t^{34} + 2t^{33} − t^{30} − t^{28} + 2t^{25} − t^{24} + t^{23} − 2t^{21} + t^{20}−

2t^{19} + 2t^{18} − t^{17} + t^{16} + t^{15} − t^{14} + t^{13} − t^{12} − t^{11} + t^{10} + t^{7} − t^{6} + t^{5} − t^{4} − t + 1)^{−1}

Admittedly this is not very appealing. One recalls the remark of Neusel and Smith ([15], p. 25) that “[Vector invariants] seem to have fallen out of favor, except as a means of providing nasty examples.”

Part of the interest of this example is that it is very close to the limit of what can be done with current invariant-theory software. The authors were unable to verify the result using Kemper’s software, though it may be possible to do this with better computing resources or algorithms adapted to specific features of this example. However, the series has been expanded and checked for correctness through degree 12.

This example was originally studied in the context of cohomology of groups. One of the maximal elementary abelian subgroups of M_{24}, E, is of rank 6, and the action of its normalizer realizes this representation. Further, if S is the Sylow 2-subgroup of M_{24}, then E ⊂ S ⊂ M_{24} is a weakly closed system [3], and so the restriction map from H^{∗}(M_{24}, k) to H^{∗}(E, k)^{N_{M_{24}}(E)} is surjective, i.e. the cohomology of M_{24} maps onto our invariant ring.

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[13] Kemper, G. Calculating invariant rings of finite groups over arbitrary fields. J. Symbolic Comput. 21 (1996), no. 3, pp. 351–366.

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[16] Symonds, P., Group actions on polynomial and power series rings. Pacific J. Math. 195 (2000), pp. 225–230.

Mathematics Department, Binghamton University, P. O. Box 6000, Binghamton, New York, 13902-6000

E-mail address: dikran@math.binghamton.edu

Department of Mathematics, U.M.I.S.T., P.O. Box 88, Manchester M60 1QD, England

E-mail address: Peter.Symonds@umist.ac.uk