THE MODULE STRUCTURE OF A GROUP ACTION ON A POLYNOMIAL RING: A FINITENESS THEOREM

DIKRAN B. KARAGUEUZIAN AND PETER SYMONDS

Abstract. Draft, 24 March 2006. A more recent version might be obtainable at http://www.ma.umist.ac.uk/pas/preprints. Consider a group acting on a polynomial ring over a finite field. We study the polynomial ring as a module for the group and prove a structure theorem with several striking corollaries. For example, any indecomposable module that appears as a summand must also appear in low degree and so the number of isomorphism types of such summands is finite. There are also applications to invariant theory, giving a priori bounds on the degrees of the generators.

1. Introduction

We consider a polynomial ring S in n variables over a finite field k of characteristic p and an action of a finite group G on S by homogeneous linear substitutions. This is equivalent to taking the symmetric powers of an n-dimensional kG-module.

We want to understand S as a kG-module in a manner as explicit as possible. The ideal solution would be to give a decomposition into indecomposable summands. We are primarily interested in the modular case, when p divides the order of G, so the problem is much harder than that of determining the composition factors.

The case of two variables was studied by Glover [13] and Alperin and Kovacs [2] and the case of three variables by the authors in [11]. This paper generalizes the results of [11] to any number of variables and we prove a strong finiteness property as a consequence.

Theorem (17.1). The kG-module S has only finitely many isomorphism types of indecomposable summands, provided that k is finite.

Particularly notable are the applications to invariant theory.

Theorem (17.4). The invariants S^{G} are generated as a ring by elements in degrees less than or equal to \frac{q^{n}−1}{q−1}(nq − n − 1), (2q^{2} − q − 2 if n = 2), where q is the order of k.

It is well known that the invariants are finitely generated and, since k is finite, there are only finitely many possible actions for a given n, so some bound on the degrees of the generators must exist for given n and k. The point of the corollary is that it gives an explicit bound (|G| is such a bound in characteristic 0). For a long time no such bound was known (except in particular cases), but recently one was given by Derksen and Kemper, although it is very large (see Section 17 for more details).

There are more precise results if we replace the general finite group G by a p-group P.

Theorem (17.2). Any indecomposable kP -module which is a summand of S is isomorphic to one occuring in degree less than or equal to \frac{q^{n}−1}{q−1} − n.

Theorem (17.3). If we can compute the decomposition into indecomposables of S as a kP -module in degrees less than or equal to q^{n−1} − n then we can compute it in all degrees.

Thus what is a priori an infinite problem becomes a finite one. Unfortunately the computation in the range required is beyond the capabilities of most computers in interesting cases with more than three variables.

All of the above results are obtained as corollaries of a Structure Theorem for the action of the group of upper triangular matrices with 1’s on the diagonal, U_{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 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 (§10). 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 {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.

In fact we have a lot more information about the modules X̄_{J}(I). For example, they are induced from certain subgroups and we have explicit bounds on the degrees of the elements.

The idea behind the proof of the Main Theorem is that first we formulate a version for certain subgroups U_{I} of U_{n} that correspond to a subset I ⊆ {1, …, n − 1} and in which only the rows indexed by elements of I can have non-zero off-diagonal entries. Next we construct the pieces X̄_{J}(I) by induction on |I|, controlling them via the leading monomials (in an appropriate sense) of the elements of their socles, and by their Poincaré series. To show that the pieces fit together exactly is a complicated exercise in accounting.

At one point in the construction we need to know that a certain monomial is the leading monomial of a trace (or orbit sum) over a subgroup. That this is the case is guaranteed by the Trace Lemma 16.1. The proof of this lemma fills a seemingly disproportionate part of the paper, comprising sections 11 to 16. Although the Trace Lemma appears as a difficult technical point in this context, it might be of independent interest in invariant theory.

It is a pleasure to thank all our colleagues who have listened patiently to our ideas on the subject, especially Stephen Siegel, who introduced us to the problem and suggested Theorem 17.1 in 1995. The first author would also like to thank Matt Brin for listening to a very detailed series of lectures on trace lemmas. The Trace Lemma and its proof are based on extensive calculations with Magma [4], in particular code for working with rings of invariants which was developed by Gregor Kemper (and has since been incorporated into Magma with help from Allan Steel).

Finally we thank the referee for his meticulous reading of the manuscript.

2. Background

The fundamental result about group actions on rings from our point of view is the Normal Basis Theorem in Galois Theory. Let S be a field and G a finite group of automorphisms of S and let R denote the invariant subfield. The Normal Basis Theorem states that S is a free RG-module of rank one.

One generalization of this in algebraic number theory is a theorem of Noether. Let S be the ring of integers in a number field and G a group of automorphisms with R as ring of invariants. If the extension S/R is tamely ramified then S is locally free of rank one as an RG-module. The study of this sort of problem is known as Galois Module Theory and contains a large number of deep results from number theory, particularly class field theory, and algebraic geometry.

Another generalization is described in [10, 5, 18]. If S is a polynomial ring over a field k and G is a finite group of automorphisms of S with R as ring of invariants then if we consider S as a kG-module we find that it is mostly projective in an asymptotic sense; it is even mostly free when viewed in the right way.

We emphasize that we are really only concerned with the modular case, when the characteristic p of k divides the order of G, otherwise all modules would be projective.

Of course, the study of group actions on polynomial rings leads to the vast classical subject of invariant theory, which we will not discuss here. We just mention that for the groups U_{I} that we consider the rings of invariants are easy to calculate and are polynomial. In general, neither property holds.

It is essential to distinguish between the two different problems of describing the composition factors of S and describing its indecomposable summands. The former is dealt with in principle by a generalization of Molien’s formula (see e.g. [7] 3.2.5), although it can still be very difficult. It is the latter that concerns us here. For most groups it is not even possible to classify the indecomposable representations, so the problem could become very complicated.

The case of two variables was considered in detail by Glover [13], when p = q, and for general q by Alperin and Kovacs [2]. It turns out that S is periodic modulo projectives.

Rather than restricting the number of variables, one can restrict the group G. The case when G is of prime order p was dealt with by Almkvist and Fossum [1]. If the homogeneous component of S of degree 1 is indecomposable then again S is periodic of period p modulo projectives, although the periodic part is complicated to describe. There are also partial results for other cyclic groups [1, 14, 15, 19]. In this case, if S is indecomposable in degree 1 then the action is periodic modulo summands induced up from proper subgroups. This is a conjecture of Kemper that was proved in [19] as a corollary of the Main Theorem of the present paper. This periodicity does not extend to more complicated groups, however.

But cyclic groups have only finitely many isomorphism classes of indecomposable modules. Siegel and Totaro calculated many examples for the Klein four group, where there are infinitely many isomorphism types of indecomposables, but they can be classified. They observed that whenever they decomposed S as a sum of indecomposables only a finite number of isomorphism types of indecomposable summands actually occurred.

Motivated by this, we produced a theorem for polynomial rings in three variables [11], showing, in particular, that only a finite number of isomorphism types of indecomposable summands could arise. Instead of periodicity modulo projectives there is a more complicated system of multiple periodicities, which we called a structure theorem.

The main result of the present paper is a generalization of this structure theorem to an arbitrary number of variables, Theorem 10.1.

A commentary on the proof, together with some examples, generalizations and applications is given in [12].

When the field k is infinite and the group G is allowed to be infinite then the situation is quite different. Doty [8] calculates the submodule lattice for the natural action of SL(n, k) when k is algebraically closed, and it follows that S is indecomposable in each degree. In particular, there are infinitely many non-isomorphic summands.

The theory has been generalized in two directions. One is in [20], where finitely generated graded modules over a ring RG are considered. These do not have to satisfy any finiteness property on the summands, but it is shown that the property of having only a finite number of isomorphism types of indecomposable summands is equivalent to the seemingly stronger property that there is a structure theorem of a similar form to that of Theorem 1.1.

The other direction is due to Bleher and Chinberg [3]. The polynomial ring S can be considered as the ring of regular functions on projective space ℙ^{n−1}(k), and the group G acts on ℙ^{n−1}(k). This generalizes to the ring of regular functions on any projective variety with an action of a finite group G. The authors of [3] prove that there are only finitely many isomorphism types of indecomposable kG-summands for curves, and for surfaces where the Sylow p-subgroup has a fixed point.

3. Organization of the Paper

Section 4 is devoted to various subgroups of U_{n} and their invariants. Section 5 introduces the parameter space P (I) which indexes the monomials in the invariants, and we show how to partition it into pieces S_{J}(I). Section 6 explains a convenient reduction from S to a truncated version T.

In Section 7 we order the monomials in the invariants and and show how to deal with the socle of a submodule in terms of the least monomials of its elements. This is one of the tools that we use to control the pieces; the other is their Poincaré series, and the lemmas that we will need to manipulate these are proved in Section 8.

Sections 9 and 10 use all this machinery to define the pieces X_{J}(I) of T that correspond to the S_{J}(I) and to prove the Main Theorem, assuming the validity of the Trace Lemma. Here X_{J}(I) for J ≠ I is produced inductively from X_{J}(J) using the Trace Lemma. We then show that ⊕_{J⫋I}X_{J}(I) is a summand of T using our control over the socles to show that the pieces are linearly independent and our computation of the Poincaré series to show that they span in high degrees. Then X_{I}(I) is defined to be the complement.

Sections 11 through 16 contain the proof of the Trace Lemma, and Section 17 contains the proofs of the corollaries.

The reader might first wish to examine the proof of the Main Theorem in the case when n = 3 and q = 3, as is presented in [12]; this case should help shed light on many of the intricacies of the paper, even though the higher terms in the trace lemma are zero in this case. (But note that in all references in [12] to specific results in this paper the section number should be increased by 1.)

There is an index of notation at the end.

4. Groups and Polynomials

In this section we define the groups whose representations we will be studying and describe their invariants.

Let k be the finite field of q elements, where q = p^{s} for some prime p. Let S = ⊕S_{r} = k[x_{1}, …, x_{n}] denote the polynomial ring over k in n variables, graded by degree. This notation will remain fixed throughout the paper.

All the groups that we study are subgroups of the upper-triangular n × n matrices over k with 1’s on the diagonal, denoted by U_{n}. This group is a Sylow p-subgroup of GL_{n}(k). The group U_{n} is taken to act on the degree-1 part of the polynomial ring S_{1} (not on its dual, as is often the case in invariant theory) in the natural way and this extends to the whole of S.

To be explicit, let a_{ij}(λ) be the matrix with λ ∈ k in row i, column j, 1’s on the diagonal and 0’s elsewhere: then a_{ij}(λ) · x_{j} = x_{j} + λx_{i}, while a_{ij}(λ) · x_{l} = x_{l} if l ≠ j.

Notation 4.1. [I, J, K] I, J, K will always be subsets of {1, …, n−1} such that I = J ⨿ K. We enumerate their elements in ascending order: I = ⟨i_{1}, …, i_{|I|}⟩, J = ⟨j_{1}, …, j_{|J|}⟩, K = ⟨k_{1}, …, k_{|K|}⟩. The notation ⟨…⟩ is used to imply that the elements are listed in order and without repetition.

Thus, by convention, K = I − J throughout this paper.

Notation 4.2. [U_{I}] U_{I} is the subgroup of U_{n} with non-zero off-diagonal entries only in the rows corresponding to I.

Thus U_{I} contains the a_{ij}(λ) with i ∈ I and j > i.

Notation 4.3. [J^{<i}] For any i ∈ ℕ, let J^{<i} = {j ∈ J | j < i}, and similarly for >, ≤, ≥.

Notation 4.4. [θ(z; I)] If I = ⟨i_{1}, …, i_{|I|}⟩ ⊆ {1, 2, … n − 1}, we define θ(z; I) ∈ k[z, x_{i_{1}}, …, x_{i_{|I|}}] by

θ(z; I) = ∏_{(λ_{i_{1}},...,λ_{i_{|I|}})∈k^{I}}(z + λ_{i_{1}}x_{i_{1}} + ⋯ + λ_{i_{|I|}}x_{i_{|I|}}).

When I = ∅ we interpret this to mean that θ(z; ∅) = z.

Definition 4.5. [d_{i}] For i = 1, …, n let d_{i} = θ(x_{i}; I^{<i}).

Proposition 4.6. The d_{i} are algebraically independent and generate the invariant ring of the group U_{I}, i.e. S^{U_{I}} = k[d_{1}, …, d_{n}].

Proof. This result is well known in invariant theory: see, for example, [16] or [7] 3.7.5. For the convenience of the reader we sketch a proof.

In order to simplify the notation, let U = U_{I} and let R be the purported ring of invariants and R̄ = S^{U} the true ring of invariants. Clearly R ⊆ R̄. We will use the letter Q to denote the field of fractions of a ring.

From Lemma 6.4 below we know that S is a free R-module with a basis consisting of |U| elements. It follows that Q(S) is a Q(R)-vector space of dimension |U| so, by Galois Theory, Q(R) = Q(S)^{U} = Q(R̄).

Suppose that R̄ ≠ R, so there is an r ∈ R̄ that is not in R. By expressing r in terms of the basis we see that it is not in Q(R), hence not in Q(R̄), a contradiction. □

5. The Parameter Space

Here we develop the machinery needed to describe the pieces X̄_{J}(I) in the Main Theorem. We maintain control of these recursively constructed pieces in part by knowing the leading monomials of bases of their socles. The parameter space P (I) (5.2) will index the monomials, and X̄_{J}(I) will correspond to a subset S̄_{J}(I) ⊆ P (I).

Definition 5.1. [μ] Let J ⊆ {1, …, n − 1} and l ∈ ℤ. Write J^{>l} = ⟨j_{1}, …, j_{m}⟩ (4.3). Then μ(J, l) ∈ ℤ is defined by

μ(J, l) = [(n + 1 − j_{m} − 1)q^{m} + (j_{m} − j_{m} − 1)q^{m−1} + ⋯ + (j_{2} − j_{1} − 1)q + (j_{1} − l − 1)](q − 1).

(If m = 0, then μ(J, l) = (n − l)(q − 1).)

Equivalently, μ(J, l) = nq^{m}(q − 1) − q^{m} + 1 − (j_{m}q^{m−1} + ⋯ + j_{1})(q − 1)^{2} − l(q − 1).

Definition 5.2. [P (I)] For any I ⊆ {1, …, n−1}, the parameter space on I is P (I) = ℕ_{0}^{I}, i.e. the set of all ℕ_{0}-valued functions on I.

We shall often write a⃗ = (a_{i_{1}}, …, a_{i_{|I|}}) for an indeterminate element of P (I). The restriction of a⃗ to P (J) will be denoted by a⃗_{J}, or just by a⃗ when no no confusion is likely to arise (and similarly for any other subset).

Definition 5.3. [ψ] If J = ⟨j_{1}, …, j_{|J|}⟩ and a⃗ ∈ P (I) we define ψ(J, a⃗) ∈ ℤ[a_{j_{1}}, …, a_{j_{|J|}}] to be (a_{j_{1}} + qa_{j_{2}} + ⋯ + q^{|J|−1}a_{j_{|J|}})(q − 1). (So ψ(∅, a⃗)) = 0.)

Equivalently, ψ(J, a⃗) = (q − 1) ∑_{j∈J} q^{|J^{<j}|}a_{j}.

Definition 5.4. [λ] λ(J, a⃗, l) = μ(J, l) − ψ(J^{>l}, a⃗). We can also regard this as a function in the last coordinate to obtain λ⃗(J, a⃗) ∈ P (I).

Definition 5.5. [S_{J}(I), S̄_{J}(I), IE_{J,i}, E_{J,i}] We associate to J ⊆ I a set S_{J}(I) ⊆ P (I) defined by |I| linear inequalities IE_{J,i} in the coordinates of a⃗ ∈ P (I), one for each i ∈ I.

IE_{J,i} is \begin{cases}a_{i} + ψ(J^{>i}, a⃗) < μ(J, i) if i ∈ J, \\ a_{i} + ψ(J^{>i}, a⃗) ≥ μ(J, i) if i ∉ J. \end{cases}

Similarly S̄_{J}(I) is defined by |I| linear (in)equalities E_{J,i}, where

E_{J,i} is \begin{cases}a_{i} + ψ(J^{>i}, a⃗) < μ(J, i) if i ∈ J, \\ a_{i} + ψ(J^{>i}, a⃗) = μ(J, i) if i ∉ J. \end{cases}

In a context where I is fixed, we write S_{J} for S_{J}(I).

It bears emphasizing that μ(J, i) is a constant. It depends on the elements of J, but not on the coordinates a_{j}.

Lemma 5.6. If we consider P (K) as the subset of P (I) consisting of the vectors with coordinates 0 on elements of J, then there is a bijective map S̄_{J}(I)×P (K) → S_{J}(I) given by (s⃗, n⃗) ↦ s⃗ + n⃗.

Proof. Notice that a_{k} for k ∈ K appears only in the defining inequality IE_{J,k} of S_{J}(I), and this is a_{k} + ψ(J^{>i}, a⃗) ≥ μ(J, k), whereas a_{k} appears only in the defining (in)equality E_{J,k} of S̄_{J}(I) and this is a_{k} + ψ(J^{>i}, a⃗) = μ(J, k). □

Lemma 5.7. S_{J}(J) = S̄_{J}(J) and S̄_{J}(I) = {a⃗ ∈ P (I) | a⃗_{J} ∈ S_{J}(J), a⃗_{K} = λ⃗(J, a⃗_{J})}.

Proof. The first claim follows from the definitions (or from 5.6).

S̄_{J}(I) is defined by |I| linear (in)equalities. The inequalities are the E_{J,j} for j ∈ J and these are exactly the |J| (in)equalities that define S_{J}(J). The equalities are the E_{J,k} for k ∈ K and these are of the form a_{k} + ψ(J^{>k}, a⃗) = μ(J, i), which can be written a_{k} = λ(J, a⃗, k). □

Lemma 5.8. (1) IE_{J,i} depends only on the set J^{≥i} and i; i.e. it is independent of J^{<i}.

(2) If i ∉ J then IE_{J,i} and IE_{J∪{i},i} differ only in that the inequality sign ≥ is replaced by <.

Proof. Examine the definitions. □

Definition 5.9. [X(L, v)] If I = ⟨i_{1}, …, i_{t}⟩ and L ⊂ ⟨i_{v}, …, i_{t}⟩ where 1 ≤ v ≤ t + 1, we write X(L, v) for the subset of P (I) defined by IE_{L,i_{v}}, …, IE_{L,i_{t}}.

Proposition 5.10. [Partitions] Let I = ⟨i_{1}, …, i_{t}⟩ be a sequence of positive integers, and {S_{J} | J ⊆ I} the collection of 2^{t} subsets of P (I) defined above. Then this collection of subsets is a partition of P (I), i.e. if J and J′ are distinct subsets of I, then S_{J} ∩ S_{J′} = ∅, and ⋃_{J⊆I} S_{J} = P (I).

We are not simply cutting up P (I) along t hyperplanes, so this is not obvious.

Proof. This is a special case of the following lemma, since P (I) = X(∅, t + 1). □

Notation 5.11. [Q(L, v)] If v ≤ t + 1 and L ⊂ {i_{v}, …, i_{t}} we write Q(L, v) for the following collection of subsets of P (I): {S_{J}(I) | L ⊆ J ⊆ L ∪ {i_{1}, …, i_{v−1}}}.

Lemma 5.12. If L ⊂ ⟨i_{v}, …, i_{t}⟩ then Q(L, v) is a partition of X(L, v).

Proof. We work by induction on v (but for all eligible L). The initial case (v = 1) is immediate since both X(L, 1) and Q(L, 1) are just S_{L}(I).

To prove the general case we note that, for L ⊂ ⟨i_{v}, …, i_{t}⟩, X(L, v) = X(L∪{i_{v−1}}, v − 1) ⨿ X(L, v − 1), because IE_{L,i} and IE_{L∪{i_{v−1}},i} are the same for i ≥ i_{v}, while IE_{L,i_{v−1}} and IE_{L∪{i_{v−1}},i_{v−1}} are opposites, by 5.8.

Also it is easy to verify that Q(L, v) = Q(L ∪ {i_{v−1}}, v − 1) ⨿ Q(L, v − 1). By the induction hypothesis, Q(L ∪ {i_{v−1}}, v − 1) is a partition of X(L ∪ {i_{v−1}}, v − 1) and Q(L, v − 1) is a partition of X(L, v − 1). □

Lemma 5.13. S̄_{J}(I) is a finite set.

Proof. All the defining inequalities are of the form < and all the a_{i} and their coefficients are non-negative, so certainly 0 ≤ a_{i} ≤ μ(J, i) for each i ∈ I. □

Lemma 5.14. S̄_{∅}(I) consists of just one point p⃗ = p⃗_{I}, with coordinates (p⃗_{I})_{i} = (n − i)(q − 1). (We will mostly omit the subscript I.)

Proof. The defining equalities reduce to this. □

It will be convenient to make a change of coordinates on our parameter space P (I). We think of (a_{i_{1}}, …, a_{i_{|I|}}) as the standard coordinates, and for each J ⊆ I, we define new coordinates (b_{i_{1}}^{J}, …, b_{i_{|I|}}^{J}) in terms of these.

Definition 5.15. [b_{i}^{J}] For any J ⊆ I and i ∈ I we set b_{i}^{J} = a_{i} − λ(J, a⃗, i) = a_{i} + ψ(J^{>i}, a⃗) − μ(J, i).

We can evaluate the b’s at the point p⃗ from 5.14.

Lemma 5.16. b_{i}^{J}(p⃗) = q^{|J^{>i}|} − 1.

Proof. Let J^{>i} = ⟨j_{1}, …, j_{s}⟩ and substitute the values given in 5.14 into the definitions of the constituent parts of b_{i}^{J} (5.15, 5.3, 5.1):

p_{i} + ψ(J^{>i}, p⃗) = (n − i)(q − 1) + [(n − j_{1}) + ⋯ + (n − j_{s})q^{s−1}](q − 1)^{2}
= −i(q − 1) − [j_{1} + ⋯ j_{s}q^{s−1}](q − 1)^{2} + nq^{s}(q − 1),
μ(J, i) = nq^{s}(q − 1) − [j_{s}q^{s−1} + ⋯ + j_{1}](q − 1) − i(q − 1) − q^{s} + 1.

The difference is b_{i}^{J}(p⃗) = q^{s} − 1, as required. □

Lemma 5.17. For any J ⊆ I and i ∈ I we have a_{i}−p_{i} = (b_{i}^{J} −b_{i}^{J}(p⃗))−(q−1) ∑_{j∈J^{>i}}(b_{j}^{J} − b_{j}^{J}(p⃗)).

Proof. Both sides are linear combinations of a_{i}, the a_{j} for j ∈ J^{>i} and a constant.

Clearly the coefficients of a_{i} match, and the constant will look after itself by construction, so we calculate the coefficient of a_{l} on the right hand side for l ∈ J^{>i}. If we set J̄ = (J^{>i})^{<l}, for convenience, then the coefficient of a_{l} in b_{j}^{J} for j ∈ J̄ or j = i is (q−1)q^{|J̄^{>j}|} and the coefficient of a_{l} in b_{l}^{J} is 1.

Thus the total coefficient of a_{l} on the right hand side is (q − 1)q^{|J̄|} − (q − 1)((∑_{j∈J̄}(q − 1)q^{|J̄^{>j}|}) + 1). But this is equal to (q − 1)(q^{|J̄|} − ((q − 1)(∑_{u=1}^{|J̄|} q^{u−1}) + 1)), which collapses to 0. □

Lemma 5.18. If a⃗ ∈ S_{J}(I) then a_{k} ≥ p_{k} for k ∈ K.

Proof. By definition of S_{J}(I), we know that b_{k}^{J} ≥ 0 for k ∈ K and b_{j}^{J} ≤ −1 for j ∈ J. Now substitute this information into the statement of 5.17 with k in place of i to obtain a_{k} − p_{k} ≥ −b_{k}^{J}(p⃗) − (q − 1) ∑_{j∈J^{>k}}(−1 − b_{j}^{J}(p⃗)).

By 5.16, the right hand side is 1−q^{|J^{>k}|} +(q −1) ∑_{j∈J^{>k}} q^{|J^{>j}|}, which collapses to 0. □

Definition 5.19. [deg_{I}(a⃗)] For a⃗ ∈ P (I) we define deg_{I}(a⃗) = \frac{1}{q−1}ψ(I, a⃗).

deg_{I}(a⃗) will be the degree of the monomial associated to a⃗ when this is defined in 7.1. The next few lemmas are elementary, but tricky, and will only be used to obtain explicit

bounds in the corollaries to the Main Theorem. The reader who is not interested in this, or who is willing to take Lemmas 5.21 and 5.22 on trust, can skip to 5.23.

Lemma 5.20. For any J ⊆ I we have deg_{I}(a⃗) − deg_{I}(p⃗) = ∑_{i∈I} ω_{i}^{J}(b_{i}^{J} − b_{i}^{J}(p⃗)), where ω_{i}^{J} = 1 if i ∈ J and ω_{i}^{J} = q^{|I^{<i}|} if i ∉ J.

Proof. The right hand side is a linear combination of the b_{i}^{J} and the constant term is correct by construction. Using the definitions and 5.17 we obtain

deg_{I}(a⃗) − deg_{I}(p⃗) = ∑_{i∈I}(a_{i} − p_{i})q^{|I^{<i}|}

≡ ∑_{i∈I} [b_{i}^{J} − (q − 1) ∑_{j∈J^{>i}} b_{j}^{J}]q^{|I^{<i}|} modulo a constant.

If l ∉ J then b_{l}^{J} appears only once, with coefficient q^{|I^{<l}|}. If l ∈ J then the total coefficient of b_{l}^{J} is q^{|I^{<l}|} − (q − 1) ∑_{i∈I^{<l}} q^{|I^{<i}|}, which collapses to 1. □

Lemma 5.21. For a⃗ ∈ S̄_{J}(I) we have deg_{I}(a⃗) ≤ deg_{I}(p⃗) ≤ \frac{q^{n}−1}{q−1} − n.

Proof. By definition of S̄_{J}(I) we know that each b_{i}^{J} ≤ 0. Using 5.16, we see that b_{i}^{J}(p⃗) ≥ 0, hence the right hand side of the expression in 5.20 is less than or equal to 0, and the first inequality follows.

From 5.14 we obtain the formula deg_{I}(p⃗_{I}) = ∑_{i∈I}(n − i)(q − 1)q^{|I^{<i}|} and we see that if we add an element to I then each term corresponding to i ∈ I can not decrease and the new term is positive. Repeating this we see that deg_{I}(p⃗_{I}) ≤ deg_{⟨1,...,n−1⟩}(p⃗_{⟨1,...,n−1⟩}). Finally, deg_{⟨1,...,n−1⟩}(p⃗_{⟨1,...,n−1⟩}) = ∑_{i=1}^{n−1}(n − i)(q − 1)q^{i−1} = (∑_{i=0}^{n−1} q^{i}) − n = \frac{q^{n}−1}{q−1} − n. □

Lemma 5.22. For a⃗ ∈ S̄_{J}(I) we have deg_{I}(a⃗) ≤ deg_{I}(p⃗) − ∑_{j∈J} q^{|I^{<j}|} and for a⃗ ∈ S̄_{I}(I) we have deg_{I}(a⃗) ≤ (n − |I|)q^{|I|} − n ≤ q^{n−1} − n.

Proof. From 5.16 and 5.20, we know that

deg_{I}(a⃗) − deg_{I}(p⃗) = ∑_{i∈J}(b_{i}^{J} − q^{|J^{>i}|} + 1) + ∑_{i∈K} q^{|I^{<k}|}(b_{i}^{J} − q^{|J^{>i}|} + 1).

Since a⃗ ∈ S̄_{J}(I), we know from 5.5 and 5.15 that if i ∈ J then b_{i}^{J} < 0, hence b_{i}^{J} ≤ −1, and if i ∈ K then b_{i}^{J} = 0.

Thus deg_{I}(a⃗) − deg_{I}(p⃗) ≤ − ∑_{i∈J} q^{|J^{>i}|} − ∑_{i∈K} q^{|I^{<i}|}(q^{|J^{>i}|} − 1), so

deg_{I}(a⃗) − deg_{I}(p⃗) + ∑_{j∈J} q^{|I^{<j}|} ≤ − ∑_{i∈J}(q^{|J^{>i}|} − q^{|I^{<i}|}) − ∑_{i∈K} q^{|I^{<i}|}(q^{|J^{>i}|} − 1).

Let A denote the negative of the right hand side. We need to show that A ≥ 0, but in fact we claim that A = 0.

Since I^{<i} = J^{<i} ⨿ K^{<i} and for i ∈ K we have J = J^{<i} ⨿ J^{>i} we find that

A = ∑_{i∈J}(q^{|J^{>i}|} − q^{|I^{<i}|}) + ∑_{i∈K}(q^{|K^{<i}|}q^{|J|} − q^{I^{<i}|})

= ∑_{i∈J} q^{|J^{>i}|} + q^{|J|} ∑_{i∈K} q^{|K^{<i}|} − ∑_{i∈I} q^{|I^{<i}|}

= \frac{1}{q−1}[(q^{|J|} − 1) + q^{|J|}(q^{|K|} − 1) − (q^{|I|} − 1)] = 0.

For the second claim of the lemma we have J = I, so the first claim yields deg_{I}(a⃗) − deg_{I}(p⃗) ≤ − ∑_{i∈I} q^{|I^{<i}|}.

From 5.14 and 5.19, we obtain deg_{I}(p⃗) = ∑_{i∈I} p_{i}q^{|I^{<i}|} = ∑_{i∈I}(n − i)(q − 1)q^{|I^{<I}|}.

So deg_{I}(a⃗) ≤ ∑_{i∈I}((n−i)(q −1)−1)q^{|I^{<i}|}. Observe that, as I varies but t = |I| remains constant, the sum on the right hand side is largest when I = ⟨1, …, t⟩ and then its value is B = ∑_{i=1}^{t}((n − i)(q − 1) − 1)q^{i−1}. Notice, for later use, that this increases with t.

B = [n(q − 1) − 1] ∑_{i=1}^{t} q^{i−1} − ∑_{i=1}^{t} i(q − 1)q^{i−1}

= [n(q − 1) − 1] ∑_{i=1}^{t} q^{i−1} − [tq^{t} − ∑_{i=1}^{t} q^{i−1}]

= (n − t)q^{t} − n.

We have remarked that this increases with t: the largest value of t allowed is n − 1, which gives q^{n−1} − n. □

The remaining results in this section are technical observations, which will only be used in the proof of the Trace Lemma.

We define a slight variation on λ(·, ·, ·) (5.4) to emphasize the dependency on the last variable. This function will frequently occur as an exponent in the proof of the Trace Lemma.

Definition 5.23. [f(l)] For fixed J and a⃗ ∈ P (J) we define f(l) = λ(J, a⃗, l − 1). (See 5.4.)

Thus f(l) depends implicitly on J^{≥l} and a⃗_{J^{≥l}} as well as on l.

Lemma 5.24. f(n + 1) = 0 and f satisfies:

f(l) = \begin{cases}qf(l + 1) − a_{l}(q − 1) & l ∈ J \\ f(l + 1) + q − 1 & l ∉ J. \end{cases}

Proof. The first part follows from the definitions and the second from Lemma 5.25, which is straightforward to prove. □

Lemma 5.25. μ satisfies

μ(J, l − 1) = \begin{cases}qμ(J, l) & l ∈ J \\ μ(J, l) + q − 1 & l ∉ J \end{cases}

and ψ satisfies

ψ(J^{>l−1}, a⃗) = \begin{cases}qψ(J^{>l}, a⃗) + (q − 1)a_{l} & l ∈ J, \\ ψ(J^{>l}, a⃗) & l ∉ J. \end{cases}

We will also need another variation on this theme. Recall our convention that I = J⨿K.

Definition 5.26. [S_{J}(J)-k-compatible] Let a⃗ ∈ P (J) and let k ∈ I. We say that a⃗ is S_{J}(J)-k-compatible if IE_{J,j} is satisfied for each j ∈ J^{>k}.

Lemma 5.27. If a⃗ is S_{J}(J)-(l − 1)-compatible then f(l + 1) < f(l) and if also l ∈ J then a_{l} < f(l + 1).

Proof. By hypothesis and the definition of IE we know that if l ∈ J then a_{l} + ψ(J^{>l}, a⃗) < μ(J, l). Now move the ψ term to the other side of the inequality and apply the definition of f (5.23) to see that a_{l} < f(l + 1).

For the first inequality use 5.24. If l ∉ J the result is clear and if l ∈ J then f(l) = qf(l + 1) − a_{l}(q − 1) > qf(l + 1) − f(l + 1)(q − 1) = f(l + 1). □

Lemma 5.28. If a⃗ is S_{J}(J)-k-compatible, then it is S_{J}(J)-l-compatible for any l ≥ k and f(l) ≥ 0 for any l > k.

Proof. The first claim follows from the definitions. The second is because f(n + 1) = 0 (5.24) and f(l) > f(l + 1) > ⋯ > f(n + 1), by 5.27. □

Lemma 5.29. The S_{J}(J)-k-compatibility of a⃗ ∈ P (J) depends only on the values of a_{j} for j ∈ J^{>k}.

Proof. Examine the definitions. □

For the next two lemmas we write K = ⟨k_{1}, …, k_{|K|}⟩ and suppose that 1 < m ≤ |K|. We are given a⃗ ∈ P (J) and define α⃗ ∈ P (J ∪ K^{≤k_{m−1}}) to have i-th coordinate a_{i} if i ∈ J and λ(J, a⃗, i) if i ∈ K^{≤k_{m−1}}.

Lemma 5.30. λ(J ∪ K^{≤k_{m−1}}, α⃗, k_{m}) = λ(J, a⃗, k_{m}).

Proof. Note that (J ∪ K^{≤k_{m−1}})^{>k_{m}} = J^{>k_{m}} and apply the definition of λ. □

Lemma 5.31. If a⃗ is S_{J}(J)-k_{1}-compatible, then α is S_{J∪K^{≤k_{m−1}}}(J∪K^{≤k_{m−1}})-k_{m}-compatible.

Proof. Since none of k_{1}, …, k_{m} are strictly greater than k_{m}, the k_{i}-coordinates do not matter (5.29), and to see that α⃗ is S_{J∪K^{≤k_{m−1}}}(J ∪ K^{≤k_{m−1}})-k_{m}-compatible it is enough to show that a⃗ is S_{J}(J)-k_{m}-compatible. But a⃗ is S_{J}(J)-k_{1}-compatible by hypothesis, and this is a stronger condition (5.28). □

6. The Splitting of the Polynomial Ring

There is a certain easily defined kU_{I}-submodule of S, which we denote by T (I), from which we can recover the whole of S as a kU_{I}-module. Recall that the invariants under U_{I} are denoted by d’s.

Definition 6.1. [T (I)] Let T (I) = ⊕T_{r} be the k-subspace of k[x_{1}, …, x_{n}] spanned by {x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}} | a_{u} < deg d_{u} for u ∉ I}.

Lemma 6.2. T (I) is a kU_{I}-submodule of k[x_{1}, …, x_{n}]. It is closed under multiplication by x_{i} for each i ∈ I.

Proof. Note that if g ∈ U_{I} and v ∉ I, then g · x_{v} = x_{v} + ∑_{i∈I^{<v}} c_{i}x_{i} for some c_{i} ∈ k. □

Lemma 6.3. T (I) is closed under multiplication by d_{i} for i ∈ I, and also under multiplication by h_{j} for j ∈ J.

Proof. Apply Lemma 6.2 and the formula for the invariants (4.5). □

If we order the x’s by x_{1} < x_{2} < ⋯ < x_{n}, and monomials lexicographically, then d_{u} = x_{u}^{deg d_{u}} + (lower terms), and the lower terms are monomials in x_{u} and the x_{i} with i ∈ I.

Lemma 6.4. k[x_{1}, …, x_{n}] ≅ k[d_{u} | u ∉ I] ⊗ T (I) as a graded k[d_{i_{1}}, …, d_{i_{|I|}}]U_{I}-module.

Proof. The multiplication map from the right hand side to the left is k[d_{i_{1}}, …, d_{i_{|I|}}]U_{I}- equivariant, so we need only show that it is a bijection.

Let H ⊆ {1, …, n} be disjoint from I. Let T (I, H) be the subspace of S spanned by {x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}} | a_{u} < deg d_{u} for u ∈ H}.

We claim that S ≅ k[d_{i} | i ∈ H] ⊗ T (I, H) by the multiplication map. The proof is by induction on |H|; the case H = ∅ is trivial and the case I ⨿ H = {1, …, n} is the statement of the lemma.

Let u ∈ H and let H′ = H − {u}. For the induction step it suffices to show that T (I, H) ≅ k[d_{u}] ⊗ T (I, H′). But we mentioned above that d_{u} = x_{u}^{deg d_{u}} + ( lower terms ), and the lower terms are in T (I) ⊆ T (I, H′). Also, it is clear from the definitions that T (I, H) = k[x_{u}^{deg d_{u}}] ⊗ T (I, H′). It is easy to check that d_{u}T (I, H) ∩ T (I, H′) = 0, hence T (I, H) = d_{u}T (I, H) ⊕ T (I, H′), by considering the dimension in each degree. The induction step follows by repeated substitution for T (I, H). □

Lemma 6.5. T (I)^{U_{I}} = k[d_{i_{1}}, …, d_{i_{|I|}}].

Proof. T (I)^{U_{I}} = k[x_{1}, …, x_{n}]^{U_{I}} ∩ T (I) = k[d_{1}, …, d_{n}] ∩ T (I) = k[d_{i_{1}}, …, d_{i_{|I|}}]. □

Lemma 6.6. If J ⊆ I, then T (J) is a U_{J}-submodule of T (I).

Proof. By Lemma 6.2, T (J) is a U_{J}-module and so is T (I), by restriction. From the definition of T (J) (6.1), it is clear that T (J) ⊆ T (I). □

7. Monomials

In this section we make the association of a collection of monomials to a submodule of one of our polynomial rings precise. We also prove that we can perform certain constructions (most notably that of direct complements, 7.19) in a way that preserves this leading-monomial information.

We will apply the results of this section to several different groups and their rings of invariants; for convenience here we will take the group to be U_{I} and so the polynomial generators of the invariant ring are d’s.

Notation 7.1. [d_{I}, da⃗ ; I] We write d_{I} = {d_{i} | i ∈ I}. For a⃗ ∈ P (I), we write da⃗ ; I for the monomial ∏_{i∈I} d_{i}^{a_{i}}.

The map a⃗ ↦ da⃗ ; I gives a one-to-one correspondence between the elements of P (I) and monomials in the d_{i} for i ∈ I. Notice that this is compatible with 5.19 in the sense that

deg(da⃗ ; I) = deg_{I}(a⃗).

Definition 7.2. [Ordering] We legislate that the d_{i} are ordered as d_{1} < d_{2} < ⋯ < d_{n} and then the monomials in k[d_{1}, …, d_{n}] are lexicographically ordered. We regard 0 as the least element of the set of monomials. This ordering is transferred to P (I).

Lemma 7.3. Suppose that v⃗ = (v_{l+1}, …, v_{n}) and a⃗ = (a_{l+1}, …, a_{n}) are in P ({l + 1, …, n}) and that m, m′ are any monomials involving only d_{1}, …, d_{l}. If dv⃗ > da⃗, then we have m · dv⃗ > m′ · da⃗.

Definition 7.4. [H(a⃗)] Given a⃗ ∈ P (I), we write H(a⃗) for the vector space spanned by {dv⃗ ; I | v⃗ ∈ P (I) and v⃗ > a⃗}. Sometimes for clarity we write H(a⃗, k[d_{I}]), or H(da⃗, k[d_{I}]).

Lemma 7.5. H(a⃗) is an ideal in k[d_{I}].

We refer to H(a⃗) as the ideal of “higher terms”, or the “error ideal” corresponding to a⃗.

Notation 7.6. [LM] For f ∈ k[d_{I}], we write LM(f) for the least monomial of f in the ordering described above, i.e. the monomial lowest in the ordering which appears with non-zero coefficient in the expression for f as a sum of monomials. If f = 0 then LM(f) = 0.

Notice that LM ignores any coefficient that this monomial may have. We may choose to regard the value of LM as an element of P (I) when it is convenient to do so.

Lemma 7.7. Let {f_{1}, …, f_{r}} ⊂ k[d_{I}] and suppose that the LM(f_{i}) are distinct and nonzero. Let λ_{i} ∈ k for i = 1, …, r. Then

(1) LM(∑_{i=1}^{r} λ_{i}f_{i}) = min{LM(f_{i}) | λ_{i} ≠ 0}.

(2) The f_{i} are linearly independent.

Proof. The first assertion is clear. The second follows from the first since ∑ λ_{i}f_{i} = 0 implies that 0 = LM(∑ λ_{i}f_{i}) = min{LM(f_{i}) | λ_{i} ≠ 0}, so all the λ_{i} must be 0. □

Lemma 7.8. If f, m ∈ k[d_{I}] and m is a monomial, then LM(mf) = m LM(f).

Definition 7.9. [soc] Recall that the socle of a kG-module M, denoted by soc M, is the sum of the irreducible submodules.

The socle has the following well-known properties.

Lemma 7.10. Let G be a finite group and let M and N be kG-modules. Then (1) If soc M = 0 then M = 0.

(2) If f : M → N is injective on soc M then it is injective.

(3) If G is a p-group then soc M is just the submodule of invariants, M^{G}.

The next lemma is also well known.

Lemma 7.11. Let G be a finite group, H ≤ G and let M be a kH-module. Then (Ind_{H}^{G} M)^{G} = {∑_{g∈G/H} g ⊗ m | m ∈ M^{H}} ≅ M^{H}, by the morphism m ↦ ∑_{g∈G/H} g ⊗ m, m ∈ M^{H}.

Proof. We concentrate on the equality, since the isomorphism is clear. Certainly the right hand side is contained in the left hand side. Fix a set of representatives {g_{i}} for G/H with g_{1} = 1, so any x ∈ Ind_{H}^{G} M can be written uniquely as ∑_{i=1}^{|G/H|} g_{i} ⊗ m_{i} for some m_{i} ∈ M.

If x is invariant under H then m_{1} must be invariant under H and if x is invariant under g_{i} then g_{i} ⊗ m_{i} = g_{i}(1 ⊗ m_{1}) = g_{i} ⊗ m_{1}, so m_{i} = m_{1}. Thus if x is invariant under G it must have the form claimed. □

Notation 7.12. [supp X] Let X be a graded kU_{I}-submodule of T (I), so soc X ⊆ k[d_{I}]. We write supp X for {LM(x) | x ∈ soc X, x ≠ 0}.

Lemma 7.13. Suppose that for each m ∈ supp X we are given an element s_{m} ∈ soc X with LM(s_{m}) = m. Then {s_{m} | m ∈ supp X} is a basis for soc X.

Proof. {s_{m} | m ∈ supp X} is a linearly independent set by 7.7. To see that this set spans soc X, fix a degree r (in order to make soc X_{r} finite-dimensional) and let s ∈ soc X_{r} be outside the span of {s_{m}} with LM(s) maximal. Then there is a c ∈ k such that s−cs_{LM(s)} has a higher least monomial and is also outside the span of {s_{m}}, a contradiction. □

Lemma 7.14. If X and Y are graded submodules of T (I) then (1) supp(X ∩ Y) ⊆ supp X ∩ supp Y.

(2) If supp X ∩ supp Y = ∅ then X + Y is a direct sum and supp(X + Y) = supp X ⨿ supp Y.

Proof. Since soc(X ∩ Y) = soc X ∩ soc Y, (1) follows from the definition of supp.

To prove (2), note that X and Y are linearly independent by part (1) and Lemma 7.10. Now the result follows from 7.7(1). □

Lemma 7.15. Let {X_{i} | i = 1, …, r} be graded submodules of T (I) with supp X_{i} pairwise disjoint. Then the sum of the X_{i} is direct.

Proof. We prove by induction on r that ∑_{i=1}^{r} X_{i} is direct and that supp(∑_{i=1}^{r} X_{i}) = ⨿_{i=1}^{r} supp X_{i}. The case r = 1 is vacuous, and the case r = 2 is Lemma 7.14.

The general case follows by applying the case r = 2 to the modules ∑_{i=1}^{r−1} X_{i} and X_{r}. □

Suppose that X ⊆ T (I) is a U_{I}-submodule and D ⊆ d_{I}. Then k[D] · X is also a kU_{I}-submodule of T (I).

Notation 7.16. [Propagation] When the multiplication map k[d_{K}] ⊗_{k} X → T (I) is an injection we say that X is propagated by d_{K}, and we call k[d_{K}] · X the propagation of X by d_{K}.

Lemma 7.17. The support of the propagation of X by d_{K} is k[d_{K}] · supp X.

Proof. Apply Lemma 7.8. □

Lemma 7.18. Let X ⊆ T (I)_{m} be a kG-submodule and let d be a monomial of k[d_{I}] of degree M. Suppose that d · X has a complement Y ⊆ T (I)_{m+M}. Then X has a complement Y′ ⊆ T (I)_{m}. If supp(d · X) ∩ supp Y = ∅ then Y′ can be chosen to satisfy supp X ∩ supp Y′ = ∅.

Proof. By hypothesis we have (d·X)⊕Y = T (I)_{m+M}; there is a projection π: T (I)_{m+M} → d · X corresponding to this decomposition with ker π = Y. Then the map d^{−1} · π(d · ( )) on T (I)_{m} is idempotent with image X, so we can take Y′ = ker π(d · ( )).

To prove the last claim, note that if x ∈ supp X ∩ supp Y′ then d · x ∈ supp(d · X) ∩ supp Y = ∅, a contradiction. □

Lemma 7.19. Suppose that X_{i} and Y_{i} are graded kU_{I}-submodules of T (I) such that T (I) = X_{i} ⊕ Y_{i} for i = 1, …, m, such that

(1) supp X_{i} ∩ supp X_{j} = ∅ when i ≠ j and

(2) supp T (I) = supp X_{i} ⨿ supp Y_{i} for each i. Then

(1) T (I) = (⊕_{i=1}^{m}X_{i}) ⊕ (∩_{i=1}^{m}Y_{i}) as kU_{I}-modules and

(2) supp T (I) = (⨿_{i=1}^{m} supp X_{i}) ⨿ supp(∩_{i=1}^{m}Y_{i}).

Proof. It is sufficient to prove this degree by degree, so we may restrict our attention to T (I)_{r}. Using the given conditions we see that supp T (I)_{r} = (⨿_{i=1}^{m} supp(X_{i})_{r}) ⨿ (∩_{i=1}^{m} supp(Y_{i})_{r}). (This is just a statement about sets.) So the sum of the modules ∩_{i=1}^{m}(Y_{i})_{r}, (X_{1})_{r}, …, (X_{m})_{r} is direct, by 7.15.

Writing codim for the codimension of a subspace of T (I)_{r}, we have

codim ∩_{i=1}^{m}(Y_{i})_{r} ≤ ∑_{i=1}^{m} codim(Y_{i})_{r} = ∑_{i=1}^{m} dim(X_{i})_{r}.

So, by dimension-counting, T (I)_{r} = (∑_{i=1}^{m}(X_{i})_{r}) + (∩_{i=1}^{m}(Y_{i})_{r}) for each degree r, proving (1). Part (2) now follows from 7.14(2). □

8. Poincaré Series Lemmas

Notation 8.1. [PS] If Y = ⊕_{n∈ℤ}Y_{n} is a graded k-module, finite dimensional in each degree, then we define its Poincaré series to be the formal power series in t defined by PS(Y) = ∑_{n∈ℤ}(dim Y_{n})t^{n}. (In some contexts this is traditionally called the Hilbert series instead.)

Lemma 8.2. The propagation of the X by d_{K} (in 7.16) has Poincaré series (∏_{k∈K} \frac{1}{1−t^{deg d_{k}}} )· PS(X).

Proof. By the definition of propagation, it is isomorphic to k[d_{K}] ⊗ X. □

Lemma 8.3. Let W be a graded submodule of T (I), and let X_{1}, …, X_{r} be graded submodules of W such that

PS(X_{1}) + ⋯ + PS(X_{r}) = PS(W) and supp(X_{i}) ∩ supp(X_{j}) = ∅ when i ≠ j.

Then W is the direct sum of the submodules X_{1}, …, X_{r}.

Proof. The sum of the X_{i} is direct, by 7.15, and ∑ dim X_{i} = dim W in each degree by hypothesis. □

Definition 8.4. [γ(t)] Let γ(t) = 1 + t + ⋯ + t^{q−1}.

Definition 8.5. [κ_{a}^{(n−i)}] Let κ_{a}^{(n−i)} be the coefficient of t^{a} in the formal power series \frac{γ(t)^{n−i}}{1−t} = ∑_{a≥0} κ_{a}^{(n−i)}t^{a}.

Lemma 8.6. If a ≥ (n − i)(q − 1) then κ_{a}^{(n−i)} = q^{n−i}.

Proof. Multiplying out the denominator in 8.5 we have γ(t)^{n−i} = κ_{0}^{(n−i)} + ∑_{a≥1}(κ_{a}^{(n−i)} − κ_{a−1}^{(n−i)})t^{a}.

But γ(t)^{(n−i)} is a polynomial of degree (q − 1)(n − i), so we must have κ_{a}^{(n−i)} = κ_{a−1}^{(n−i)} for a > (q − 1)(n − i) and also γ(t)^{n−i} = κ_{0}^{(n−i)} + ∑_{a=1}^{(q−1)(n−1)}(κ_{a}^{(n−i)} − κ_{a−1}^{(n−i)})t^{a}.. Setting t = 1 gives q^{n−i} = κ_{0}^{(n−i)} + ∑_{a=1}^{(q−1)(n−1)}(κ_{a}^{(n−i)} − κ_{a−1}^{(n−i)}) = κ_{(q−1)(n−i)}^{(n−i)}. □

Lemma 8.7. If a⃗ ∈ S_{J}(I) then κ_{a_{k}}^{(n−k)} = q^{n−k} for each k ∈ K.

Proof. Just combine 8.6, 5.18 and 5.14. □

Lemma 8.8. We have PS(T (I)) = ∑_{a⃗∈P (I)} ∏_{i∈I} κ_{a_{i}}^{(n−i)} ∏_{i∈I} t^{q^{|I^{<i}|}a_{i}}.

Proof. It follows from Definition 6.1 that

PS(T (I)) = ∏_{i∈I}(\frac{1}{1 − t^{deg d_{i}}}) · ∏_{l=1}^{n}(1 + t + ⋯ + t^{deg d_{l}−1}).

We study the second part of the product and find that, since deg(d_{l}) = q^{|I^{<l}|},

1 + t + ⋯ + t^{deg d_{l}−1} = ∏_{a=0}^{|I^{<l}|−1} γ(t^{q^{a}}) = ∏_{i∈I^{<l}} γ(t^{q^{|I^{<i}|}}).

We also have

∏_{l=1}^{n} ∏_{i∈I^{<l}} γ(t^{q^{|I^{<i}|}}) = ∏_{i∈I} ∏_{l=i+1}^{n} γ(t^{q^{|I^{<i}|}}) = ∏_{i∈I} γ(t^{q^{|I^{<i}|}})^{n−i}.

Returning to our expression for PS(T (I)), i.e. restoring the first product, we have

PS(T (I)) = ∏_{i∈I} \frac{γ(t^{|I^{<i}|})^{n−i}}{1 − t^{|I^{<i}|}} = ∏_{i∈I} ∑_{a_{i}≥0} κ_{a_{i}}^{(n−i)}t^{|I^{<i}|a_{i}},

by the definition of κ (8.5), completing the proof. □

9. Assembly

In this section we state and prove the key ingredients (9.1, 9.4, 9.5) for the proof by induction on |I| of Theorem 10.1. Here we use the Trace Lemma (16.1): the key application is in Proposition 9.4. This lemma is the only result from Sections 11 through 16 that we need to prove our Structure Theorem (10.1).

Proposition 9.1. Let X_{1}, …, X_{r} be kU_{I}-submodules of T (I) that are obtained via propagation (7.16), i.e. there are finite-dimensional kU_{I}-submodules X̄_{1}, …, X̄_{r} and subsets D_{i} of d_{I} such that X_{i} is the propagation k[D_{i}] · X̄_{i} ≅ k[D_{i}] ⊗ X̄_{i}. Suppose also that

(1) supp X_{i} ∩ supp X_{j} = ∅ if i ≠ j and

(2) there is a polynomial F ∈ ℤ[t] such that F + PS(X_{1}) + ⋯ + PS(X_{r}) = PS(T (I)). Then there is a kU_{I}-submodule B ⊂ T (I) such that

(1) PS(B) = F,

(2) supp B is the complement of ∪_{i} supp X_{i} in supp(T (I)) and

(3) T (I) is the direct sum of the submodules B, X_{1}, …, X_{r}.

Proof. Since F is a polynomial, its coefficients are zero in sufficiently large degrees. Therefore, the coefficients of t^{a} in ∑_{i=1}^{r} PS(X_{i}) and in PS(T (I)) are the same for sufficiently large powers of t. By 8.3, it follows that T (I) is the direct sum X_{1} ⊕ ⋯ ⊕ X_{r} in large degrees.

By 7.18, the X_{i} are individually summands in all degrees, with complements Y_{i} such that supp X_{i} ∩ supp Y_{i} = ∅. We can therefore apply 7.19 to see that the X_{i} are simultaneously summands, i.e. their direct sum is a summand. We write B for the complement of ⊕X_{i} in T (I) and note that conclusion 2 of 7.19 implies that B satisfies conclusion 2 of the present proposition. From hypothesis 2 it is immediate that B satisfies conclusion 1. By construction, B satisfies conclusion 3, so we have completed the proof. □

The least element of K will always be denoted by k_{1}.

Definition 9.2. [G(I, J), G(J)] For J ⫋ I let

G(I, J) = ∏_{k_{1}<i≤n, i∉J} \frac{θ(x_{i}; I^{<i})}{θ(x_{i}; J^{<i})}.

We will often write just G(J) when I is understood.

Note that G(I, J) is a polynomial, since θ(x_{i}; J^{<i}) divides θ(x_{i}; I^{<i}) from the definition.

Definition 9.3. [Tr] As usual, if G is a finite group, H < G and M is a kG-module then Tr_{G/H} : M^{H} → M^{G} is the map m ↦ ∑_{g∈G/H} gm. If H = 1 we write just Tr_{G}.

The proof of the next proposition depends on the Trace Lemma, which will eventually be proved in Section 16. (We ignore the sign in equation (1).)

First we define h_{i} = θ(x_{i}; J^{<i}), so S^{U_{J}} = k[h_{1}, …, h_{n}], by Proposition 4.6.

Lemma (Trace Lemma, 16.1). If a⃗ ∈ P (J) is S_{J}(J)-k_{1}-compatible then

(1)  Tr_{U_{I}/U_{J}} [G(I, J) · ha⃗ ; J] = ±d_{K}^{λ(J,a⃗)}da⃗ ; J + x,

where x ∈ H(d_{K}^{λ(J,a⃗)}da⃗ ; J, k[d_{I}]). Furthermore,

(2)  Tr_{U_{I}/U_{J}} [G(I, J) · H(ha⃗ ; J, k[h_{J}])] ⊂ H(d_{K}^{λ(J,a⃗)}da⃗ ; J, k[d_{I}]).

Proposition 9.4. Let T (I) be the kU_{I}-submodule of the polynomial ring S defined in Section 6, and suppose that B is a U_{J}-submodule of T (J) with supp B = S_{J}(J). Then there are U_{I}-submodules X̄_{J}(I) and X_{J}(I) of T (I) such that

(1) X̄_{J}(I) = ⨁_{g∈U_{I}/U_{J}} g · G(I, J) · B, which is isomorphic to Ind_{U_{J}}^{U_{I}} B (but with a change of grading),

(2) X_{J}(I) ≅ k[d_{K}] ⊗ X̄_{J}(I),

(3) supp X̄_{J}(I) = S̄_{J}(I) and supp X_{J}(I) = S_{J}(I), and

(4) PS(X_{J}(I)) = PS(B) · [U_{I}: U_{J}] · ∏_{k∈K}(\frac{1}{1−t^{deg d_{k}}}) · t^{deg G(I,J)}.

Notice that, since B is a U_{J}-submodule, supp B is calculated with respect to k[h_{1}, …, h_{n}], but supp X̄_{J}(I) and supp X_{J}(I) are calculated with respect to k[d_{1}, … d_{n}].

Proof. We define a map ρ: Ind_{U_{J}}^{U_{I}} B → T (I) using the inclusion T (J) ⊆ T (I), (6.6); if g ⊗ b ∈ Ind_{U_{J}}^{U_{I}} B = kU_{I} ⊗_{kU_{J}} B then we set ρ(g ⊗ b) = g · (G(I, J) · b). (Note that the first dot is the group action and the second dot multiplication of polynomials.) This map is well defined since G(I, J) is U_{J}-invariant.

We claim that ρ is injective; by 7.10, it suffices to show that ρ is injective on the socle and soc Ind_{U_{J}}^{U_{I}} B = (Ind_{U_{J}}^{U_{I}} B)^{U_{I}} = {∑_{g∈U_{I}/U_{J}} g ⊗ b | b ∈ B^{U_{J}} } ≅ soc B, by 7.11. We need to show that the map that takes (∑_{g∈U_{I}/U_{J}} g) ⊗ b to Tr_{U_{I}/U_{J}} [G(I, J) · b] is injective. By hypothesis, supp B = S_{J}(J), so there is a basis {b(a⃗) | a⃗ ∈ S_{J}(J)} for soc B with LM(b(a⃗)) = ha⃗ ; J and hence a basis {(∑_{g∈U_{I}/U_{J}} g) ⊗ b(a⃗) | a⃗ ∈ S_{J}(J)} for soc Ind_{U_{J}}^{U_{I}} B with

LM(b(a⃗)) = ha⃗ ; J.

By the Trace Lemma (16.1), the least monomial of

ρ( ∑_{g∈U_{I}/U_{J}} g ⊗ b(a⃗)) = Tr_{U_{I}/U_{J}} [G(I, J) · b(a⃗)] is d_{K}^{λ⃗(J,a⃗)}da⃗ ; J.

In particular, for different a⃗ the least monomials are distinct so, by 7.13, we have shown that the given basis of soc Ind_{U_{J}}^{U_{I}} B is mapped to a linearly independent set in T (I). It follows that ρ is injective and we have therefore produced a module X̄_{J}(I) = ρ(Ind_{U_{J}}^{U_{I}} B) ⊂ T (I), with the socle having a basis with least terms as specified above.

By the construction of X̄_{J}(I) it follows that

supp X̄_{J}(I) = {d_{K}^{λ⃗(J,a⃗)}da⃗ ; J | a⃗ ∈ S_{J}(J)}

and, by 5.7 this is equal to {da⃗ ; I | a⃗ ∈ S̄_{J}(I)}.

Consider the multiplication map k[d_{K}]⊗X̄_{J}(I) → T (I) and denote its image by X_{J}(I). Clearly

supp X_{J}(I) = {db⃗ ; Kda⃗ ; I | b⃗ ∈ P (K), a⃗ ∈ S̄_{J}(I)},

so supp X_{J}(I) = S_{J}(I) by 5.6. Also soc(k[d_{K}] ⊗ X̄_{J}(I)) = k[d_{K}] ⊗ soc X̄_{J}(I), which has a basis consisting of elements m ⊗ b, where m is a monomial in k[d_{K}] and b is from our basis for B. The images of these have distinct least monomials, so the multiplication map is injective by the same argument as was used for ρ.

To see that (4) holds, note that PS(X̄_{J}(I)) = t^{deg G(I,J)}[U_{I} : U_{J}] · P S(B), and that the extra factor in condition (4) comes from the propagation of X̄_{J}(I) by k[d_{K}] using 8.2. □

Corollary 9.5. With the hypotheses of Proposition 9.4, if we have in addition

PS(B) = ∑_{a⃗∈S_{J} (J)} ∏_{j∈J} κ_{a_{j}}^{(n−j)}t^{q^{|J^{<j} |}a_{j}},

then we may refine conclusion (4) to state that

PS(X_{J}(I)) = ∑_{a⃗∈S_{J} (I)} ∏_{i∈I} κ_{a_{i}}^{(n−i)}t^{q^{|I^{<i}|}a_{i}}.

To give the proof of Corollary 9.5 we require several lemmas, which we state now.

Definition 9.6. [ν(J, j, k)] Let

ν(J, j, k) = \begin{cases}(q − 1)q^{|(J^{<j})^{>k}|} if k < j \\ 0 otherwise. \end{cases}

Lemma 9.7. We have ψ(J^{>k}, a⃗) = ∑_{j∈J} ν(J, j, k)a_{j}.

Proof. By definition of ψ (5.3), ψ(J^{>k}, a⃗) = (q − 1) ∑_{j∈J^{>k}} q^{|(J^{>k})^{<j}|}. □

Lemma 9.8. We have deg h_{j} = deg d_{j} − ∑_{k∈K} deg d_{k}^{ν(J,j,k)}.

Proof. Observe that deg h_{j} = q^{|J^{<j}|} and deg d_{j} = q^{|I^{<j}|}. By the definition of ν (9.6), we have

∑_{k∈K} deg d_{k}^{ν(J,j,k)} = ∑_{k∈K^{<j}} q^{|I^{<k}|}(q − 1)q^{|(J^{<j})^{>k}|}.

It is easily verified that, for k < j, I^{<k} ⨿ (J^{<j})^{>k} = J^{<j} ⨿ K^{<k} and it follows that

∑_{k∈K} deg d_{k}^{ν(J,j,k)} = ∑_{k∈K^{<j}} q^{|J^{<j}∪K^{<k}|}(q − 1) = ∑_{u=|J^{<j}|}^{|I^{<j}|−1} q^{u}(q − 1) = q^{|I^{<j}|} − q^{|J^{<j}|}.

□

Lemma 9.9. We have deg G(I, J) = ∏_{k∈K} deg d_{k}^{μ(J,k)}.

Proof. A direct proof is possible, but it is easier to apply the Trace Lemma (16.1) with a⃗ = 0 and note that λ(J, 0, k) = μ(J, k) (see 5.4). Thus the degree of Tr_{U_{I}/U_{J}} G(I, J) is the right hand side of the equation of the lemma. Since G(I, J) is homogeneous, this must be deg G(I, J). □

Proof of Corollary 9.5. To simplify this proof, instead of powers of t in the Poincaré series we will write invariants, where each invariant d is meant to stand for t^{deg d}.

Using conclusion (4) of Proposition 9.4 and the expression for PS(B) in the hypothesis, we obtain:

P S(X_{J}(I)) = ∑_{a⃗∈S_{J} (J)} [( ∏_{j∈J} κ_{a_{j}}^{(n−j)}) · ha⃗ ; J · [U_{I}: U_{J}] · ( ∏_{k∈K} \frac{1}{1 − d_{k}} ) · t^{deg G(I,J)}].

Using 9.9 we can replace t^{deg G(I,J)} by ∏_{k∈K} d_{k}^{μ(J,k)} and by 9.8 we can replace h_{j} by d_{j} ∏_{k∈K} d_{k}^{−ν(J,j,k)}. (All that this means is that the degrees are the same.) We know from the definition of the groups that [U_{I} : U_{J}] = |U_{K}| = ∏_{k∈K} q^{n−k}. We can also write

∏_{k∈K} \frac{1}{1−d_{k}} = ∑_{b⃗∈P (K)} db⃗ ; K. Therefore we have:

P S(X_{J}(I)) = ∑_{a⃗∈S_{J} (J)} [ ∏_{j∈J} κ_{a_{j}}^{(n−j)} · ∏_{j∈J}(d_{j} ∏_{k∈K} d_{k}^{−ν(J,j,k)})^{a_{j}} · ∏_{k∈K} q^{n−k} · ∑_{b⃗∈P (K)} db⃗ ; K · ∏_{k∈K} d_{k}^{μ(J,k)}].

But ∏_{j∈J} d_{k}^{−ν(J,j,k)a_{j}} = d_{k}^{∑_{j∈J} −ν(J,j,k)a_{j}} = d_{k}^{−ψ(J^{>k},a⃗)}, by 9.7, and combining this with d_{k}^{μ(J,k)} and d_{k}^{b_{k}} we obtain d_{k}^{b_{k}+λ(J,a⃗,k)}.

Now set α_{k} = b_{k} + λ(J, a⃗, k) for k ∈ K. We combine this with a⃗ to form a new vector α⃗ ∈ P (I). The condition b_{k} ≥ 0 for k ∈ K now becomes α_{k} ≥ λ(J, α⃗, k), which is IE_{J,k} for α⃗, and the condition a⃗ ∈ S_{J}(J) yields IE_{J,j} for j ∈ J.

Thus ∑_{a⃗∈S_{J} (J)} ∑_{b⃗∈P (K)} can be replaced by ∑_{α⃗∈S_{J} (I)} and we obtain:

PS(X_{J}(I) = ∑_{α⃗∈S_{J} (I)} [( ∏_{j∈J} κ_{a_{j}}^{(n−j)}) · ( ∏_{k∈K} q^{n−k}) · dα⃗ ; I].

But, since α⃗ ∈ S_{J}(I), we can apply 8.7 to replace ∏_{k∈K} q^{n−k} by ∏_{k∈K} κ_{α_{k}}^{(n−k)} and combine it with the other κ-term to obtain ∏_{i∈I} κ_{α_{i}}^{(n−i)}.

All that remains is to replace the d_{i} by powers of t. □

10. The Main Theorem

We can now put all of this together to prove the promised Structure Theorem.

Theorem 10.1. For all I ⊆ {1, …, n − 1} there is a direct sum decomposition as graded kU_{I}-modules T (I) ≅ ⊕_{J⊆I}X_{J}(I), where

(1) X_{J}(I) ≅ k[d_{k} | k ∈ K] ⊗ X̄_{J}(I) by the multiplication map,

(2) X̄_{J}(I) = ⨁_{g∈U_{I}/U_{J}} g · G(I, J) · X_{J}(J), which is isomorphic to Ind_{U_{J}}^{U_{I}} X_{J}(J) (but with a change of grading),

(3) supp X̄_{J}(I) = S̄_{J}(I) and supp X_{J}(I) = S_{J}(I),

(4) PS(X_{J}(I)) = ∑_{a⃗∈S_{J} (I)} ∏_{i∈I} κ_{a_{i}}^{(n−i)}t^{q^{|I^{<i}|}a_{i}}.

Proof. We prove the theorem by induction on t = |I|, the number of rows. Let J ⫋ I, so that U_{J} is a group with fewer than t rows. By induction, we may apply the Theorem to U_{J} and obtain a direct sum decomposition of T (J) and, in particular, a finite-dimensional submodule X_{J}(J). Applying Proposition 9.4 and Corollary 9.5, we obtain a submodule X_{J}(I) ⊆ T (I) with the properties required by the statement of the Theorem. It remains to exhibit the module X_{I}(I).

By Proposition 5.10 the complement of ∪_{J⫋I}S_{J}(I) in P (I) is S_{I}(I) and since, by 8.8,

PS(T (I)) = ∑_{a⃗∈P (I)} ∏_{i∈I} κ_{a_{i}}^{(n−i)}t^{q^{|I^{<i}|}a_{i}},

it follows that

PS(T (I)) − ∑_{J⫋I}PS(X_{J}(I)) = ∑_{a⃗∈S_{I}(I)} κ_{a_{i}}^{(n−i)}t^{q^{|I^{<i}|}a_{i}}.

Now set

F = ∑_{a⃗∈S_{I}(I)} κ_{a_{i}}^{(n−i)}t^{q^{|I^{<i}|}a_{i}}

and apply Proposition 9.1 to obtain a module B. We set X_{I}(I) = B and note that, by construction, X_{I}(I) has the properties required, by the statement of the present theorem.

□

Combining this with 6.4 we obtain the Main Theorem of the Introduction 1.1.

11. Overview of the Trace Lemmas

Our aim in the next part of the paper is to prove the Trace Lemma (16.1), which will complete the proof of Theorem 10.1. In order to prove this lemma, we start with an elementary fact about finite fields (12.3), which can be considered as computing a trace over the additive group of k. Since we can obtain the group U_{K} over which we must compute a trace by repeated extensions by groups isomorphic to k^{+}, we can perform a proof by induction.

More explicitly, the group U_{K} can be written as a product U_{{k_{1}}} ·U_{{k_{2}}} ⋯ U_{{k_{|K|}}}, breaking it up row by row. We refer to the result for U_{{k_{i}}} as the “One-Row” Trace Lemma (15.3). In order to obtain this we need a result for groups with one row having only m positions with nonzero entries, which we call the “m-Step” Lemma (15.2). The key point in the proof of the m-Step Lemma is the induction step, or “One-Step” Lemma (14.1). This is where almost all of the bookkeeping is concentrated.

Once the proof of the One-Row Trace Lemma is complete, we must do a second induction to prove the “m-Row” Lemma (16.2). This lemma follows in a completely formal way from the One-Row Lemma, and the Trace Lemma (16.1), which is our ultimate goal, is a special case of the m-Row Lemma.

The reader should note that the only result in Sections 12 to 16 that is used in any other part of the paper is the Trace Lemma itself (16.1). Furthermore, the extra bookkeeping (starting in Lemma 12.18) required to get the sign in the Trace Lemma is not necessary for our application.

In principle, the traces we need can be calculated by the elegant method of [17], but the change of basis required seems difficult to formulate explicitly.

Now we fix the notation for the invariants of all the different groups that we will consider. We start with the invariants of U_{J}, which are denoted by h’s. As we move to larger groups, we move the letters used to denote the invariants down the alphabet, so that g’s denote invariants of U_{J∪{k_{1}}}, f’s denote invariants of U_{J∪K^{≤k_{m−1}}}, e’s denote invariants of U_{J∪K^{≤km}}, and d’s denote invariants of U_{I} (consistently with 4.5). To summarize:

Definition 11.1. [d_{i}, e_{i}, f_{i}, g_{i}, h_{i}] For given k_{m} ∈ K, let h_{i} = θ(x_{i}; J^{<i}), g_{i} = θ(x_{i}; (J ∪ {k_{1}})^{<i}), f_{i} = θ(x_{i}; (J ∪ K^{≤k_{m−1}})^{<i}), e_{i} = θ(x_{i}; (J ∪ K^{≤k_{m}})^{<i}), d_{i} = θ(x_{i}; I^{<i}).

By Proposition 4.6, the d_{i}, e_{i}, f_{i}, g_{i}, and h_{i} are algebraically independent and generate the invariant rings of the groups U_{I}, U_{J∪K^{≤km}}, U_{J∪K^{≤k_{m−1}}}, U_{J∪{k_{1}}}, and U_{J}, i.e.:

S^{U_{I}}  = k[d_{1}, …, d_{n}]  S^{U_{J∪{k_{1}}}} = k[g_{1}, …, g_{n}]
S^{U_{J∪K^{≤km}}} = k[e_{1}, …, e_{n}]  S^{U_{J}}  = k[h_{1}, …, h_{n}]
S^{U_{J∪K^{≤k_{m−1}}}} = k[f_{1}, …, f_{n}].

We also need some basic subgroups, from which the other groups are built.

Notation 11.2. [A(x_{i}, x_{j})] A(x_{i}, x_{j}) is the subgroup of U_{n} consisting of matrices with just one non-zero off-diagonal entry in position (i, j). (Mnemonic: Add x_{i} to x_{j}.)

Thus A(x_{i}, x_{j}) = {a_{ij}(λ) | λ ∈ k} (see §4).

12. Trace Lemmas

Many of the results in this section are well known.

Notation 12.1. [θ(y, x)] We write θ(y, x) for ∏_{λ∈k}(y+λx). (This is just a slight variation on the definition of θ in 4.4.)

Lemma 12.2. θ has the following properties:

(1) θ(y, x) = y^{q} − yx^{q−1},

(2) θ(v; I) = 0 if v is a linear combination of the x_{i} for i ∈ I,

(3) θ(t; I) is linear in t over k,

(4) θ(t; I) = θ(θ(t; I − {i}), θ(x_{i}; I − {i})) for i ∈ I.

Proof. (1) The expression y^{q}−yx^{q−1} vanishes whenever y = λx, λ ∈ k, so must be divisible by θ(y, x). But both expressions have the same degree and the same coefficient of y^{q}, so must be equal.

(2) This is clear from the definition.

We prove (3) and (4) together by induction on the size of I. If I = {i} then θ(t; I) = θ(t, x_{i}), so (3) is true by (1) and (4) is clear from the definitions. If |I| > 1 then it follows from the induction hypothesis that θ(θ(t; I − {i}), θ(x_{i}; I − {i})) is linear in t. Suppose that t = v + λx_{i}, where v is a linear combination of {x_{j} | j ∈ I − {i}}. Then

θ(θ(t; I−{i}), θ(x_{i}; I−{i})) = θ(θ(v; I−{i}), θ(x_{i}; I−{i}))+λθ(θ(x_{i}; I−{i}), θ(x_{i}; I−{i})),

and we see that the right hand side is zero using property (2). So θ(θ(t; I − {i}), θ(x_{i}; I − {i})) is divisible by all expressions t − v − λx_{i}, hence by θ(t; I). Since both expressions have the same degree and the same coefficient of t^{q^{|I|}}, they must be equal. □

Lemma 12.3. If i > 0, then

∑_{λ∈k} λ^{i} = \begin{cases}0 & i ≢ 0 mod q − 1, \\ −1 & i ≡ 0 mod q − 1. \end{cases}

Proof. Let ω be a generator of the cyclic multiplicative group 𝔽_{q}^{×}, so ∑_{λ∈k} λ^{i} = ∑_{j=1}^{q−1} ω^{ij}. If i ≡ 0 mod q − 1, then ω^{i} = 1 so this sum is −1. Otherwise we sum the geometric series to obtain ω^{i}(1 − ω^{i(q−1)})(1 − ω^{i})^{−1} = 0. □

Definition 12.4. [Tr_{A(x,y)}] Let R be a commutative k-algebra without zero-divisors, and let Tr_{A(x,y)} : R[x, y] → R[x, y] be the function defined by f ↦ ∑_{λ∈k} f(x, y + λx) for f ∈ R[x, y]. This extends to the field of fractions of R[x, y].

Observe that this definition is consistent with Definition 9.3 and Notation 11.2 if we regard A(x, y) as the additive group of k and let it act on R[x, y] in such a way that λ ∈ k sends x ↦ x and y ↦ y + λx.

Lemma 12.5. Let f ∈ R[x, y] and write

f(x, y + h) = a_{0}(x, y) + a_{1}(x, y)h + a_{2}(x, y)h^{2} + ⋯

in R[x, y, h], where a_{i}(x, y) ∈ R[x, y]. Then

Tr_{A(x,y)} f = −[a_{q−1}(x, y)x^{q−1} + a_{2(q−1)}(x, y)x^{2(q−1)} + a_{3(q−1)}(x, y)x^{3(q−1)} + ⋯]

Proof. Note that Tr_{A(x,y)} f = qa_{0}(x, y) + a_{1}(x, y)(∑_{λ∈k} λ)x + a_{2}(x, y)(∑_{λ∈k} λ^{2})x^{2} + ⋯, and apply 12.3. □

Lemma 12.6. We have

Tr_{A(x,y)} y^{i} = \begin{cases}0, & i < q − 1 \\ −x^{q−1}, & i = q − 1 \end{cases}

Proof. Use the binomial expansion for (y + h)^{i} and 12.5. □

Lemma 12.7. Let ⟨u, v⟩ ⊆ R[u, v] denote the k-vector space spanned by u and v. If f ∈ R[y], ϕ : ⟨u, v⟩ → R is linear over k and Tr_{A(x,y)} f = g(x, y) then

Tr_{A(u,v)} f(ϕ(v)) = g(ϕ(u), ϕ(v)).

Proof. Tr_{A(u,v)} f(ϕ(v)) = ∑_{λ∈k} f(ϕ(v + λu)) = ∑_{λ∈k} f(ϕ(v) + λϕ(u)) = g(ϕ(u), ϕ(v)). □

Lemma 12.8. Tr_{A(x,y)} θ(y; I)^{i} = 0 if i < q − 1, while Tr_{A(x,y)} θ(y; I)^{q−1} = −θ(x; I)^{q−1}.

Proof. Apply 12.6 and 12.7 with θ(·; I) as ϕ: k⟨x, y⟩ → R. □

Lemma 12.9. In the field of fractions of k[x, y],

Tr_{A(x,y)} \frac{1}{y} = \frac{−x^{q−1}}{θ(y, x)}.

Proof. Note that Tr_{A(x,y)} \frac{1}{y} = \frac{1}{θ(y, x)} Tr_{A(x,y)} \frac{θ(y, x)}{y} = Tr_{A(x,y)}(y^{q−1} − x^{q−1}) and use 12.6.

□

Lemma 12.10. If l, t ∉ I, l ≠ t then in the field of fractions of k[x_{1}, …, x_{n}] we have

Tr_{A(x_{t},x_{l})} \frac{1}{θ(x_{l}; I)} = \frac{−θ(x_{t}; I)^{q−1}}{θ(x_{l}; I ∪ \{t\})}.

Proof. Use Lemmas 12.9 and 12.7 with θ(·; I) as ϕ: k⟨x_{t}, x_{l}⟩ → R to obtain

Tr_{A(x_{t},x_{l})} \frac{1}{θ(x_{l}; I)} = \frac{−θ(x_{t}; I)^{q−1}}{θ(θ(x_{l}; I); θ(x_{t}; I))}.

By 12.2(4), this is equal to the form claimed. □

Lemma 12.11. Write B for the group of automorphisms of k[x, y] given by y ↦ μy +λx, x ↦ x, μ, λ ∈ k, μ ≠ 0. Then the invariant ring k[x, y]^{B} is k[x, θ(y, x)^{q−1}].

Proof. First note that B = A ⋊ C, where C ≅ k^{×} is the subgroup for which λ = 0 and A is the subgroup with μ = 1. Now notice that k[x, y]^{A} = k[x, θ(y, x)] has a basis {x^{i}θ(y, x)^{j} | i ≥ 0, j ≥ 0} consisting of eigenvectors for the action of C. The invariant ring k[x, θ(y, x)]^{C} is the +1 eigenspace, which is just k[x, θ(y, x)^{q−1}]. Alternatively, use the method of 4.6 or of [7] 3.7.5. □

Lemma 12.12. Tr_{A(x,y)}((x^{q−1} − y^{q−1})^{a}) = x^{(q−1)a} + θ(y, x)^{q−1}P (x, θ(y, x)^{q−1}), for some polynomial P in two variables.

Proof. Notice that y^{q−1} is already invariant under y ↦ μy for μ ∈ k, μ ≠ 0. So, in the notation of 12.11, Tr_{A}((x^{q−1} − y^{q−1})^{a}) is B-invariant and lies in k[x, θ(y, x)^{q−1}].

Regarding Tr_{A(x,y)}((x^{q−1}−y^{q−1})^{a}) as a polynomial in x and θ(y, x)^{q−1} it suffices to check that the coefficient of x^{(q−1)a} is 1. Since we are checking a polynomial identity, we can set y = 0, so θ(y, x) = 0 also. We have

[Tr_{A}((x^{q−1} − y^{q−1})^{a})]|_{y=0} = ∑_{λ∈k}(x^{q−1} − (λx)^{q−1})^{a}

= x^{(q−1)a} + ∑_{λ∈k^{×}}(x^{q−1} − λ^{q−1}x^{q−1})^{a}

= x^{(q−1)a},

and the coefficient of x^{(q−1)a} is indeed 1. □

Lemma 12.13. If l, t ∉ I then, for some polynomial P,

Tr_{A(x_{t},x_{l})} [θ(x_{t}; I ∪ {l})^{a}] = θ(x_{t}; I)^{qa}+θ(x_{l}; I∪{t})^{q−1}·θ(x_{t}; I)^{a}·P (θ(x_{t}; I), θ(x_{l}; I∪{t})^{q−1}).

THE MODULE STRUCTURE OF A GROUP ACTION ON A POLYNOMIAL RING 24

Proof. First observe that, by Lemma 12.2 parts (1) and (4), θ(x_{t}; I∪{l})^{a} = θ(x_{t}; I)^{a}[θ(x_{t}; I)^{q−1}− θ(x_{l}; I)^{q−1}]^{a}. Now θ(x_{t}; I) is invariant under A(x_{t}, x_{l}) so, by 12.12 and 12.7,

Tr_{A(x_{t},x_{l})} [θ(x_{t}; I ∪ {l})^{a}] =

θ(x_{t}; I)^{a}[θ(x_{t}; I)^{(q−1)a} + θ(θ(x_{l}; I), θ(x_{t}; I))^{q−1} · P (θ(x_{t}; I), θ(θ(x_{l}; I), θ(x_{t}; I))^{q−1})]. But θ(θ(x_{l}; I), θ(x_{t}; I)) = θ(x_{l}, I ∪ {t}), by 12.2(4). □ Lemma 12.14. Let 1 ≤ k, l ≤ n − 1 but k, l ∉ I. Then θ(x_{l}; I ∪ {k}) · θ(x_{k}; I) = −θ(x_{l}; I) · θ(x_{k}; I ∪ {l}). Proof.

θ(x_{l}; I ∪ {k}) · θ(x_{k}; I) = [θ(x_{l}; I)^{q} − θ(x_{k}; I)^{q−1}θ(x_{l}; I)] · θ(x_{k}; I) by Lemma 12.2(4)
= θ(x_{l}; I) · [θ(x_{l}; I)^{q−1}θ(x_{k}; I) − θ(x_{k}; I)^{q}]
= −θ(x_{l}; I)θ(x_{k}; I ∪ {l})  by Lemma 12.2(4).
□

Recall from 11.1 that h_{i} = θ(x_{i}; J^{<i}), g_{i} = θ(x_{i}; (J ∪ {k_{1}})^{<i}). Lemma 12.15. If l ∉ J then

Tr_{A(x_{k_{1}},x_{l})} \frac{1}{h_{l}} = \frac{−θ(x_{k_{1}}; J^{<(l+1)})^{q−1}}{g_{l}} .

Proof. Apply Lemma 12.10. □ Notation 12.16. [J̃] Let J̃ denote J^{>k_{1}}. Lemma 12.17 (Theta Lemma). θ(x_{k_{1}}; J^{<l}) is an element of k[h_{k_{1}}, hJ̃^{<l}]. As an element of k[h_{1}, …, h_{n}], its least term is h_{k_{1}}^{q^{|J̃^{<l}|}}. Proof. We prove the result by induction on |J̃^{<l}|. If |J̃^{<l}| = 0 then θ(x_{k_{1}}; J^{<l}) = h_{k_{1}}, so the initial case is clear. To prove the induction step, let j be the largest number in J̃^{<l} and apply Lemma 12.2 parts (1) and (4) to get the equation

θ(x_{k_{1}}; J^{<l}) = θ(x_{k_{1}}; J^{<l} − {j})^{q} − θ(x_{j}; J^{<l} − {j})^{q−1} · θ(x_{k_{1}}; J^{<l} − {j}). By the induction hypothesis, we have an expression θ(x_{k_{1}}; J^{<l} −{j}) = h_{k_{1}}^{q^{|J̃^{<l}|−1}} +f, where f ∈ k[h_{k_{1}}, h_{J̃^{<l}−{j}}] is a polynomial with deg_{h_{k_{1}}} f < q^{|J̃^{<l}|−1}. In any case, θ(x_{j}; J^{<l}−{j}) = h_{j}. Thus we can write θ(x_{k_{1}}; J^{<l}) = (h_{k_{1}}^{q^{|J̃^{<l}|−1}} + f)^{q} − h_{j}^{q−1}(h_{k_{1}}^{q^{|J̃^{<l}|−1}} + f) = h_{k_{1}}^{q^{|J̃^{<l}|}} + f^{q} − h_{j}^{q−1}h_{k_{1}}^{q^{|J̃^{<l}|−1}} − h_{j}^{q−1}f.

Because of the condition on deg_{h_{k_{1}}} f, we see that h_{k_{1}}^{q^{|J̃^{<l}|}} is the least term of the last expression, and this proves the lemma. □ Lemma 12.18 ((a, b)-Lemma). Let l ∈ J̃. If a < b then

(1) Tr_{A(x_{k_{1}},x_{l})} [h_{l}^{a}θ(x_{k_{1}}; J^{<l+1})^{b}] = (−1)^{a}g_{l}^{a}θ(x_{k_{1}}; J^{<l})^{a+q(b−a)} mod g_{l}^{a+1}k[g_{k_{1}}, hJ̃^{<l}, g_{l}].

If, instead, a ≥ b then, for some polynomial Q,

(2)  Tr_{A(x_{k_{1}},x_{l})} [h_{l}^{a}θ(x_{k_{1}}; J^{<l+1})^{b}] = g_{l}^{b}θ(x_{k_{1}}; J^{<l})^{b} · Q(g_{l}, θ(x_{k_{1}}; J^{<l})).

Proof. Since l > k_{1} we have, by Lemma 12.14, that h_{l} · θ(x_{k_{1}}; J^{<(l+1)}) = −g_{l} · θ(x_{k_{1}}; J^{<l}). Case 1: a < b. From the observation above, we have

Tr_{A(x_{k_{1}},x_{l})} [h_{l}^{a}θ(x_{k_{1}}; J^{<(l+1)})^{b}] = (−1)^{a}g_{l}^{a}θ(x_{k_{1}}; J^{<l})^{a} Tr_{A(x_{k_{1}},x_{l})} [θ(x_{k_{1}}; J^{<(l+1)})^{b−a}] ,

and now we use Lemma 12.13 to obtain

(−1)^{a}g_{l}^{a}θ(x_{k_{1}}; J^{<l})^{a}[θ(x_{k_{1}}; J^{<l})^{q(b−a)} + g_{l}^{q−1}θ(x_{k_{1}}; J^{<l})^{b−a}P (θ(x_{k_{1}}; J^{<l}), g_{l}^{q−1})].

We know that θ(x_{k_{1}}; J^{<l}) ∈ k[h_{k_{1}}, hJ̃^{<l}] by 12.17 and g_{k_{1}} = h_{k_{1}} from the definitions. Case 2: a ≥ b. From the observation again, we get

\begin{aligned}Tr_{A(x_{k_{1}},x_{l})} [h_{l}^{a}θ(x_{k_{1}}; J^{<(l+1)})^{b}] & = (−1)^{b}g_{l}^{b}θ(x_{k_{1}}; J^{<l})^{b} Tr_{A(x_{k_{1}},x_{l})} [h_{l}^{a−b}] \\ & = g_{l}^{b}θ(x_{k_{1}}; J^{<l})^{b}Q(g_{l}, θ(x_{k_{1}}; J^{<l})). \end{aligned}

The last step perhaps requires a little explanation, even though the reason it works is quite general. Consider the action of A(x_{k_{1}}, x_{l}) on k[h_{l}, θ(x_{k_{1}}; J^{<l})]. This has the effect h_{l} ↦ h_{l} +λθ(x_{k_{1}}; J^{<l}), where λ ∈ k. The invariant subring under this action is k[g_{l}, θ(x_{k_{1}}; J^{<l})], and so Tr_{A(x_{k_{1}},x_{l})}[h_{l}^{a−b}] = Q(g_{l}, θ(x_{k_{1}}; J^{<l})) for some polynomial Q (which is possibly 0). □

13. Preparations for the One-Step Lemma

The purpose of this section is to set up some notation that will be used for the One-Step Lemma (14.1) and establish some simple properties of this notation.

Definition 13.1. [G(I, J)_{l}] For l = k_{1} + 1, …, n + 1, let

G(I, J)_{l} = ∏_{k_{1}<i<l i∉J} \frac{θ(x_{i}; I^{<i})}{θ(x_{i}; J^{<i})} ∏_{l≤i≤n i∉J} \frac{θ(x_{i}; I^{<i})}{θ(x_{i}; J^{<i} ∪ \{k_{1}\})}.

We will usually write just G(J) for short, since I will not vary.

Lemma 13.2. G(J)_{n+1} = G(J) and G(J)_{k_{1}+1} = G(J ∪ {k_{1}})

Lemma 13.3. For l > k_{1} both G(J)_{l} and g_{l} are A(x_{k_{1}}, x_{l})-invariant.

Notation 13.4. [w_{l,i}, w_{l,X}] Let w_{l,i} = h_{i} if i < l and w_{l,i} = g_{i} if i ≥ l.

For X ⊆ {1, …, n}, we write w_{l,X} for the set {w_{l,i} | i ∈ X} and for X ⊆ Y and

a⃗ ∈ P (Y) we write wa⃗ ; l,X for ∏_{i∈X} w_{l,i}^{a_{i}}.

Lemma 13.5. If l ∉ J then

(1) θ(x_{k_{1}}; J^{<l+1}) = θ(x_{k_{1}}; J^{<l}) is A(x_{k_{1}}, x_{l})-invariant (3) G(J)_{l+1} = G(J)_{l} \frac{g_{l}}{h_{l}}
(2) wa⃗ ; l+1,J̃ = wa⃗ ; l,J̃ is A(x_{k_{1}}, x_{l})-invariant  (4) J̃^{≥l} = J̃^{≥l+1} and J̃^{<l} = J̃^{<l+1}

Lemma 13.6. If l ∈ J then

(1) wa⃗ ; l+1,J̃ = wa⃗ ; l,J̃(\frac{h_{l}}{g_{l}})^{a_{l}}  (3) G(J)_{l+1} = G(J)_{l}

(2) wa⃗ ; l,J̃ is A(x_{k_{1}}, x_{l})-invariant Lemma 13.7. If l ∉ J, l > k_{1} then

G(J)_{l+1}θ(x_{k_{1}}; J^{<l+1})^{f(l+1)}wa⃗ ; l+1,J̃ = G(J)_{l}g_{l}wa⃗ ; l,J̃ · \frac{θ(x_{k_{1}}; J^{<l+1})^{f(l+1)}}{h_{l}} ,

and everything on the right hand side before the dot is A(x_{k_{1}}, x_{l})-invariant.

Proof. Apply Lemmas 13.5 and 13.3. □

Lemma 13.8. If l ∈ J, l > k_{1} then

G(J)_{l+1}θ(x_{k_{1}}; J^{<l+1})^{f(l+1)}wa⃗ ; l+1,J̃ = G(J)_{l}wa⃗ ; l,J̃\frac{1}{g_{l}^{a_{l}}} · h_{l}^{a_{l}}θ(x_{k_{1}}; J^{<l+1})^{f(l+1)},

and everything on the right hand side before the dot is A(x_{k_{1}}, x_{l})-invariant.

Proof. Apply Lemmas 13.6 and 13.3. □

The One-Step Lemma will compute certain traces up to an error term which lies in a certain ideal R_{l}(a⃗) over which we have some control. We now define this ideal.

Remark 13.9. In reading the following definition, recall that h_{k_{1}} = g_{k_{1}} and also the ordering on elements of P (I) (7.2).

If a⃗ ∈ P (J) and J′ ⊂ J we will write a_{J′} ∈ P (J′) for the obvious restriction, and sometimes just a⃗ when no confusion is likely to arise. We will also on occasion extend a⃗ by 0 to give a vector a⃗_{0} ∈ P (J ∪ {k_{1}}).

Definition 13.10. Given k_{1},J̃, a⃗ ∈ P (J̃) and l such that k_{1} < l ≤ n + 1 and f(l) ≥ 0, we write R_{l}(a⃗) for the k-subspace of k[g_{k_{1}}, w_{l,J̃}] spanned by elements of the form

(1)  g_{k_{1}}^{c}wv⃗ ; l,J̃ where v⃗ ∈ P (J̃), v⃗J̃^{≥l} > a⃗J̃^{≥l}, and c ≥ 0,

and elements of the form

(2)  g_{k_{1}}^{d}θ(x_{k_{1}}; J^{<l})^{f(l)}wv⃗ ; l,J̃ where v⃗J̃^{≥l} = a⃗J̃^{≥l}, v⃗J̃^{<l} > a⃗J̃^{<l}, and d ≥ 0.

Thus R_{l}(a⃗) depends implicitly on k_{1} and J̃, although they do not appear in the notation.

Remark 13.11. The extreme cases l = n + 1 and l = k_{1} + 1 are of special interest in the above definition.

By definition, R_{n+1}(a⃗) is spanned by elements of types 1 and 2. If l = n + 1 then J̃^{≥l} is empty and thus v⃗J̃^{≥l} = a⃗J̃^{≥l}. In particular, v⃗J̃^{≥l} > a⃗J̃^{≥l} is impossible and so there are no elements of type 1. As for the elements of type 2, f(n + 1) = 0 (5.24) and g_{k_{1}} = h_{k_{1}} and so R_{n+1}(a⃗) is the ideal in k[h_{k_{1}}, hJ̃] generated by monomials with exponent vector greater than a⃗_{0}. In other words, R_{n+1}(a⃗) = H(a⃗_{0}, k[h_{k_{1}}, hJ̃]), (see 7.4).

If l = k_{1} + 1 then J̃^{<k_{1}+1} is empty and v⃗J̃^{<k_{1}+1} > a⃗J̃^{<k_{1}+1} is impossible. It follows that there are no elements of type 2. Similarly, this means that R_{k_{1}+1}(a⃗) is the ideal in k[g_{k_{1}}, gJ̃] generated by monomials with exponent vector greater than a⃗J̃. That is, R_{k_{1}+1}(a⃗) = H(a⃗_{0}, k[g_{k_{1}}, gJ̃]).

Lemma 13.12. R_{l}(a⃗) is an ideal in k[g_{k_{1}}, w_{l,J̃}].

Proof. In order to prove that a sub-vector-space of k[g_{k_{1}}, w_{l,J̃}] is an ideal, it is enough to show that the subspace is closed under multiplication by g_{k_{1}} and w_{l,i} for i ∈ J̃.

First we show that the span of the elements of type 1 forms an ideal. Let g_{k_{1}}^{c}wv⃗ ; l,J̃ be an element of type 1 in R_{l}(a⃗), so v⃗J̃^{≥l} > a⃗J̃^{≥l}. Consider the product of this element with each of the following:

(1) g_{k_{1}}: The product is again an element of type 1 in R_{l}(a⃗).

(2) h_{i}, i < l: Write v′ for the exponent vector of wv⃗ ; l,J̃ · h_{i}. Then v′ ; J̃^{≥l} = v⃗J̃^{≥l} > a⃗J̃^{≥l} and so the product is still an element of type 1 in R_{l}(a⃗).

(3) g_{i}, i ≥ l: Write v′ for the exponent vector of wv⃗ ; l,J̃· g_{i}. Then v′ ; J̃^{≥l} > v⃗J̃^{≥l} > a⃗J̃^{≥l} and so the product is still an element of type 1 in R_{l}(a⃗).

Thus we have shown that the elements of type 1 form an ideal in k[g_{k_{1}}, w_{l,J̃}].

Now let g_{k_{1}}^{d}θ(x_{k_{1}}; J^{<l})^{f(l)}wv⃗ ; l,J̃ be an element of type 2. We have the conditions v⃗J̃^{≥l} = a⃗J̃^{≥l}, v⃗J̃^{<l} > a⃗J̃^{<l}. Consider the product of this element with each of the following:

(1) g_{k_{1}}: The product is again an element of type 2 in R_{l}(a⃗).

(2) h_{i}, i < l: Write v′ for the exponent vector of wv⃗ ; l,J̃ · h_{i}. Then v′ ; J̃^{≥l} = v⃗J̃^{≥l} = v⃗J̃^{≥l} and v′ ; J̃^{<l} > v⃗J̃^{<l} > a⃗J̃^{≥l} so the product is still an element of type 2.

(3) g_{i}, i ≥ l: This is the only tricky part. Write v′ for the exponent vector of wv⃗ ; l,J̃ · g_{i}. Then v′ ; J̃^{≥l} > v⃗J̃^{≥l} = a⃗J̃^{≥l} and so wv⃗ ; l,J̃ · g_{i} is an element of type 1 in R_{l}(a⃗). We have already shown that the span of the elements of type 1 forms an ideal in k[g_{k_{1}}, w_{l,J̃}]. By the Theta Lemma (12.17), θ(x_{k_{1}}; J^{<l}) ∈ k[g_{k_{1}}, w_{l,J̃}]. Hence the product g_{k_{1}}^{d}θ(x_{k_{1}}; J^{<l})^{f(l)}w_{l,J̃}^{v′} is a linear combination of elements of type 1 and therefore lies in R_{l}(a⃗).

□

14. The One-Step Lemma

Lemma 14.1 (One-Step Lemma). Let a⃗ ∈ P (J̃) be S_{J}(J)-(l − 1)-compatible and k_{1} < l ≤ n. Then we have

(1) Tr_{A(x_{k_{1}},x_{l})} [G(J)_{l+1}θ(x_{k_{1}}; J^{<l+1})^{f(l+1)}wa⃗ ; l+1,J̃] = G(J)_{l} [(−1)^{σ}θ(x_{k_{1}}; J^{<l})^{f(l)}wa⃗ ; l,J̃ + x] ,

where x ∈ R_{l}(a⃗) and σ = 1 if l ∉ J, σ = a_{l} if l ∈ J. Furthermore,

(2)  Tr_{A(x_{k_{1}},x_{l})} [G(J)_{l+1}R_{l+1}(a⃗)] ⊆ G(J)_{l}R_{l}(a⃗).

Notice that both f(l) and f(l + 1) are non-negative, by 5.28.

Proof of the One-Step Lemma. Formula (1). In this part we compute a trace up to an element of R_{l}(a⃗). We divide the calculation into two cases, according to whether l ∈ J or not.

THE MODULE STRUCTURE OF A GROUP ACTION ON A POLYNOMIAL RING 28

Case l ∉ J. By Lemma 13.7, it is enough to compute:

Tr_{A(x_{k_{1}},x_{l})} \frac{θ(x_{k_{1}}; J^{<l+1})^{f(l+1)}}{h_{l}} = −\frac{θ(x_{k_{1}}; J^{<l+1})^{f(l+1)+q−1}}{g_{l}} by Lemma 12.15
= −\frac{θ(x_{k_{1}}; J^{<l})^{f(l)}}{g_{l}}  by Lemmas 5.24 and 13.5(1).

Thus Tr_{A(x_{k_{1}},x_{l})} [G(J)_{l+1}θ(x_{k_{1}}; J^{≤l+1})^{f(l+1)}wa⃗ ; l+1,J̃] = −G(J)_{l}θ(x_{k_{1}}; J^{<l})^{f(l)}wa⃗ ; l,J̃ exactly, with

no error term. Case l ∈ J. By Lemma 13.8, we need to compute Tr_{A(x_{k_{1}},x_{l})} [h_{l}^{a_{l}}θ(x_{k_{1}}; J^{<l+1})^{f(l+1)}] .

Since a⃗ is S_{J}(J)-(l − 1)-compatible, 5.27 applies and we know that f(l + 1) > a_{l}. So from the first part of the (a, b)-Lemma (12.18(1)), the required trace is equal to (−1)^{a_{l}}g_{l}^{a_{l}}θ(x_{k_{1}}; J^{<l})^{a_{l}+q(f(l+1)−a_{l})} + g_{l}^{a_{l}+1}X, where X ∈ k[g_{k_{1}}, w_{l,J̃}].

Now we apply 5.24 to simplify the exponent; we obtain (−1)^{a_{l}}g_{l}^{a_{l}}θ(x_{k_{1}}; J^{<l})^{f(l)} +g_{l}^{a_{l}+1}X and so

Tr_{A(x_{k_{1}},x_{l})} [G(J)_{l+1}θ(x_{k_{1}}; J^{<l+1})^{f(l+1)}wa⃗ ; l+1,J̃]

= G(J)_{l} [(−1)^{a_{l}}θ(x_{k_{1}}; J^{<l})^{f(l)}wa⃗ ; l,J̃ + wa⃗ ; l,J̃g_{l}X] .

To complete this part of the proof, we must show that wa⃗ ; l,J̃g_{l}X ∈ R_{l}(a⃗). Note that the power of g_{l} in the monomial wa⃗ ; l,J̃g_{l} is greater than a_{l}, so wa⃗ ; l,J̃g_{l} is a generating element of R_{l}(a⃗) of type 1. In particular, wa⃗ ; l,J̃g_{l} ∈ R_{l}(a⃗). Since X ∈ k[g_{k_{1}}, w_{l,J̃}] and R_{l}(a⃗) is an ideal in this ring (13.12), wa⃗ ; l,J̃g_{l}X ∈ R_{l}(a⃗).

This completes the proof of the first formula; we now turn to the second one. Formula (2). There are two types of traces, corresponding to the two types of generators of R_{l}(a⃗).

Type 1. The first type is Tr_{A(x_{k_{1}},x_{l})} [G(J)_{l+1}g_{k_{1}}^{c}wv⃗ ; l+1,J̃] for v⃗J̃^{≥l+1} > a⃗J̃^{≥l+1}.

The calculation splits into two cases, depending on whether l ∈ J or not. Type 1, l ∉ J. If l ∉ J we apply Lemmas 13.3 and 13.5 and obtain:

\begin{aligned}Tr_{A(x_{k_{1}},x_{l})} [G(J)_{l+1}g_{k_{1}}^{c}wv⃗ ; l+1,J̃] & = Tr_{A(x_{k_{1}},x_{l})} [G(J)_{l} \frac{g_{l}}{h_{l}} g_{k_{1}}^{c}wv⃗ ; l+1,J̃] \\ & = G(J)_{l}g_{l}g_{k_{1}}^{c}wv⃗ ; l,J̃ Tr_{A(x_{k_{1}},x_{l})} \frac{1}{h_{l}} . \end{aligned}

By 12.15, Tr_{A(x_{k_{1}},x_{l})} \frac{1}{h_{l}} = \frac{−θ(x_{k};J^{≤l})^{q−1}}{g_{l}}, so

Tr_{A(x_{k_{1}},x_{l})} [G(J)_{l} \frac{g_{l}}{h_{l}} g_{k_{1}}^{c}wv⃗ ; l+1,J̃] = −G(J)_{l}g_{k_{1}}^{c+q−1}wv⃗ ; l,J̃.

Since J̃^{≥l} = J̃^{≥l+1} we have v⃗J̃^{≥l} > a⃗J̃^{≥l} and therefore g_{k_{1}}^{c+q−1}wv⃗ ; l,J̃ ∈ R_{l}(a⃗).

THE MODULE STRUCTURE OF A GROUP ACTION ON A POLYNOMIAL RING 29

Type 1, l ∈ J. If l ∈ J we apply Lemma 13.6 and get:

Tr_{A(x_{k_{1}},x_{l})} [G(J)_{l+1}g_{k_{1}}^{c}wv⃗ ; l+1,J̃] = G(J)_{l}g_{k_{1}}^{c} \frac{wv⃗ ; l,J̃}{g_{l}^{v_{l}}} Tr_{A(x_{k_{1}},x_{l})} h_{l}^{v_{l}}.

Note that \frac{wv⃗ ; l,J̃}{g_{l}^{v_{l}}} is a polynomial (13.4), which contains gv⃗ ; J̃^{≥l+1} as a factor. Since g_{k_{1}}^{c}wv⃗ ; l+1,J̃ is of type 1 in R_{l+1}(a⃗) by hypothesis, it follows that v⃗J̃^{≥l+1} > a⃗J̃^{≥l+1} and hence that gv⃗ ; J̃^{≥l+1} is an element of R_{l}(a⃗) of type 1. Since R_{l}(a⃗) is an ideal in k[g_{k_{1}}, w_{l,J̃}] (13.12), \frac{wv⃗ ; l,J̃}{g_{l}^{v_{l}}} ∈ R_{l}(a⃗).

Now we have to study the trace of h_{l}^{v_{l}}. We apply the (a, b)-Lemma (12.18) with a = v_{l}, b = 0 to conclude that Tr_{A(x_{k_{1}},x_{l})} h_{l}^{v_{l}} lies in k[g_{l}, θ(x_{k_{1}}; J^{<l})], which in turn is contained in k[h_{k_{1}}, hJ̃^{<l}, g_{l}] by the Theta Lemma (12.17). Now k[h_{k_{1}}, hJ̃^{<l}, g_{l}] ⊂ k[g_{k_{1}}, w_{l,J̃}] and, since R_{l}(a⃗) is an ideal in k[g_{k_{1}}, w_{l,J̃}] (13.12), the product g_{k_{1}}^{c} \frac{wv⃗ ; l,J̃}{g_{l}^{v_{l}}} Tr_{A(x_{k_{1}},x_{l})} h_{l}^{v_{l}} lies in R_{l}(a⃗).

Type 2. The second type of trace is Tr_{A(x_{k_{1}},x_{l})} [G(J)_{l+1}g_{k_{1}}^{d}θ(x_{k_{1}}; J^{<l+1})^{f(l+1)}wv⃗ ; l+1,J̃], where

v⃗J̃^{≥l+1} = a⃗J̃^{≥l+1} but v⃗J̃^{<l+1} > a⃗J̃^{<l+1}. Type 2, l ∉ J. Since l ∉ J we can apply Lemma 13.5 and get

Tr_{A(x_{k_{1}},x_{l})} [G(J)_{l+1}g_{k_{1}}^{d}θ(x_{k_{1}}; J^{<l+1})^{f(l+1)}wv⃗ ; l+1,J̃]

= Tr_{A(x_{k_{1}},x_{l})} [G(J)_{l} \frac{g_{l}}{h_{l}} g_{k_{1}}^{d}θ(x_{k_{1}}; J^{<l})^{f(l+1)}wv⃗ ; l,J̃]

= G(J)_{l}g_{k_{1}}^{d}g_{l}wv⃗ ; l,J̃ Tr_{A(x_{k_{1}},x_{l})} \frac{θ(x_{k_{1}}; J^{<l})^{f(l+1)}}{h_{l}}  by Lemmas 13.3 and 13.5
= −G(J)_{l}g_{k_{1}}^{d}g_{l}wv⃗ ; l,J̃\frac{θ(x_{k_{1}}; J^{<l})^{f(l+1)+q−1}}{g_{l}}  by Lemma 12.15
= −G(J)_{l}g_{k_{1}}^{d}θ(x_{k_{1}}; J^{<l})^{f(l)}wv⃗ ; l,J̃  by Lemma 5.24.

Since we started with an element of type 2 in R_{l+1}(a⃗), v⃗J̃^{≥l+1} = a⃗J̃^{≥l+1} and v⃗J̃^{<l+1} > a⃗J̃^{<l+1}. In view of 13.5(4), this gives us the same two conditions on v⃗ and a⃗ but with l+1 replaced by l. Thus the trace lies in G(J)_{l}R_{l}(a⃗). Type 2, l ∈ J. Using Lemmas 13.3 and 13.6 we reduce the problem to computing

G(J)_{l}g_{k_{1}}^{d} \frac{wv⃗ ; l,J̃}{g_{l}^{v_{l}}} Tr_{A(x_{k_{1}},x_{l})} [h_{l}^{v_{l}}θ(x_{k_{1}}; J^{<l+1})^{f(l+1)}] .

Recall that \frac{wv⃗ ; l,J̃}{g_{l}^{v_{l}}} is a polynomial (13.4).

The problem now splits into two cases again, depending on whether v_{l} < f(l + 1) or not. Type 2, l ∈ J, v_{l} < f(l + 1). The computation here is analogous to that for Formula 1, l ∈ J.

THE MODULE STRUCTURE OF A GROUP ACTION ON A POLYNOMIAL RING 30

We need to compute

Tr_{A(x_{k_{1}},x_{l})} [h_{l}^{v_{l}}θ(x_{k_{1}}; J^{<l+1})^{f(l+1)}] .

Since f(l + 1) > v_{l}, the first part of the (a, b)-Lemma (12.18(1)) tells us that the required trace is

(−1)^{v_{l}} [g_{l}^{v_{l}}θ(x_{k_{1}}; J^{<l})^{v_{l}+q(f(l+1)−v_{l})} + g_{l}^{v_{l}+1}X] ,

where X ∈ k[g_{k_{1}}, w_{l,J̃}]. Returning to the original trace computation, we have shown that

Tr_{A(x_{k_{1}},x_{l})} [G(J)_{l+1}θ(x_{k_{1}}; J^{<l+1})^{f(l+1)}g_{k_{1}}^{d}wv⃗ ; l+1,J̃]

= ±G(J)_{l}g_{k_{1}}^{d} [wv⃗ ; l,J̃θ(x_{k_{1}}; J^{<l})^{v_{l}+q(f(l+1)−v_{l})} + wv⃗ ; l,J̃g_{l}X] .

It is sufficient to show that wv⃗ ; l,J̃θ(x_{k_{1}}; J^{<l})^{v_{l}+q(f(l+1)−v_{l})} and wv⃗ ; l,J̃g_{l}X lie in R_{l}(a⃗). We start with the easy case, wv⃗ ; l,J̃g_{l}X.

If v′ is the exponent vector of wv⃗ ; l,J̃g_{l}, we have immediately that v′_{l} > v_{l} ≥ a_{l} and hence that v′ ; J̃^{≥l} > a⃗J̃^{≥l}. This implies that wv⃗ ; l,J̃g_{l} is an element of R_{l}(a⃗) of type 1. Since X ∈ k[g_{k_{1}}, w_{l,J̃}] and R_{l}(a⃗) is an ideal in this ring (13.12), wv⃗ ; l,J̃g_{l}X ∈ R_{l}(a⃗).

To see that the term wv⃗ ; l,J̃θ(x_{k_{1}}; J^{<l})^{v_{l}+q(f(l+1)−v_{l})} lies in R_{l}(a⃗) is slightly trickier. There are two possibilities: first, that v_{l} > a_{l} and second, that v_{l} = a_{l} but also v⃗J̃^{<l} > a⃗J̃^{<l}.

If v_{l} > a_{l} then wv⃗ ; l,J̃ satisfies the definition of an element of type 1 in R_{l}(a⃗). By the Theta Lemma (12.17), θ(x_{k_{1}}; J^{<l}) ∈ k[g_{k_{1}}, w_{l,J̃}], and R_{l}(a⃗) is an ideal in this ring (13.12),

so wv⃗ ; l,J̃θ(x_{k_{1}}; J^{<l})^{v_{l}+q(f(l+1)−v_{l})} ∈ R_{l}(a⃗).

Now suppose that v_{l} = a_{l}, but also v⃗J̃^{<l} > a⃗J̃^{<l}. By 5.24, wv⃗ ; l,J̃θ(x_{k_{1}}; J^{<l})^{v_{l}+q(f(l+1)−v_{l})} =

wv⃗ ; l,J̃θ(x_{k_{1}}; J^{<l})^{f(l)}, which is an element of R_{l}(a⃗) of type 2, so we are done.

Type 2, l ∈ J, v_{l} ≥ f(l + 1). Just as in the case v_{l} < f(l + 1), we need to compute Tr_{A(x_{k_{1}},x_{l})} h_{l}^{v_{l}}θ(x_{k_{1}}; J^{<l+1})^{f(l+1)}. Since v_{l} ≥ f(l + 1), the second part of the (a, b)-Lemma (12.18(2)) shows us that the required trace is g_{l}^{f(l+1)}θ(x_{k_{1}}; J^{<l})^{f(l+1)}Q, where Q ∈ k[g_{k_{1}}, θ(x_{k_{1}}; J^{<l})], which is contained in k[g_{k_{1}}, w_{l,J̃}] by the Theta Lemma (12.17). Returning to the original trace computation we have shown that

Tr_{A(x_{k_{1}},x_{l})} [G(J)_{l+1}θ(x_{k_{1}}; J^{<l+1})^{f(l+1)}g_{k_{1}}^{d}wv⃗ ; l+1,J̃]

= G(J)_{l}g_{k_{1}}^{d} [\frac{wv⃗ ; l,J̃}{g_{l}^{v_{l}}}g_{l}^{f(l+1)}θ(x_{k_{1}}; J^{<l})^{f(l+1)}Q] .

Again, recall that \frac{wv⃗ ; l,J̃}{g_{l}^{v_{l}}} is a polynomial (13.4).

We must now show that \frac{wv⃗ ; l,J̃}{g_{l}^{v_{l}}}g_{l}^{f(l+1)}θ(x_{k_{1}}; J^{<l})^{f(l+1)}Q is an element of R_{l}(a⃗). First we

show that \frac{wv⃗ ; l,J̃}{g_{l}^{v_{l}}}g_{l}^{f(l+1)} ∈ R_{l}(a⃗).

By 5.27, f(l + 1) > a_{l}, so if we define v′ to be the same as v⃗ except that v_{l} is replaced

by f(l + 1), we can write w_{l,J̃}^{v′} = \frac{wv⃗ ; l,J̃}{g_{l}^{v_{l}}}g_{l}^{f(l+1)} and we have v′ ; J̃^{≥l} > a⃗J̃^{≥l}. This means that

\frac{wv⃗ ; l,J̃}{g_{l}^{v_{l}}}g_{l}^{f(l+1)} is an element of R_{l}(a⃗) of type 1.

Since θ(x_{k_{1}}; J^{<l})^{f(l+1)}Q ∈ k[g_{l}, w_{l,J̃}] and R_{l}(a⃗) is an ideal in this ring (13.12), the product

\frac{wv⃗ ; l,J̃}{g_{l}^{v_{l}}}g_{l}^{f(l+1)}θ(x_{k_{1}}; J^{<l})^{f(l+1)}Q lies in R_{l}(a⃗), as we have argued before in this proof.

□

15. The One-Row Trace Lemma

For groups A, B < C we write C = A · B to signify that C ≅ A × B as sets and that the bijection is given by multiplication. This happens if and only if |C| = |A| · |B| and A ∩ B = 1.

The induction argument in this section is based on the decomposition U_{{i}} = ∏_{j=i+1}^{n} A(x_{i}, x_{j}) (where ∏ signifies repeated ·).

The next lemma is easy and well known.

Lemma 15.1. If C = A · B then Tr_{C} = Tr_{A} Tr_{B} and also Tr_{C/B} = Tr_{A} on B-invariants.

Lemma 15.2 (m-Step Lemma). Let a⃗ ∈ P (J) be S_{J}(J)-(l − 1)-compatible and k_{1} < l ≤ n + 1. Then we have

(1) Tr_{A(x_{k_{1}},x_{l})} Tr_{A(x_{k_{1}},x_{l+1})} ⋯ Tr_{A(x_{k_{1}},x_{n})} [G(J) · ha⃗ ; J]

= G(J)_{l}ha⃗ ; J^{<k_{1}} [(−1)^{τ}θ(x_{k_{1}}; J^{<l})^{f(l)}wa⃗ ; l,J̃ + x] ,

where x ∈ R_{l}(a⃗) and τ = τ(l, J, a⃗) = n + l + 1 + ∑_{j∈J^{≥l}}(a_{j} + 1) (mod 2). Furthermore,

(2)  Tr_{A(x_{k_{1}},x_{l})} Tr_{A(x_{k_{1}},x_{l+1})} ⋯ Tr_{A(x_{k_{1}},x_{n})} [G(J) · R_{n+1}(a⃗)] ⊆ G(J)_{l}R_{l}(a⃗).

If l = n + 1 we interpret the left hand side of (1) as G(J) · ha⃗ ; J.

Proof. The result is proved by downward induction on l. In the initial case, l = n + 1, the left hand side is G(J) · ha⃗ ; J and, on the right hand side, G(J)_{n+1} = G(J) by 13.2, τ = 0, f(l) = 0 by 5.24 and wa⃗ ; n+1,J̃ = ha⃗ ; J̃ by definition (13.4). The latter combines with ha⃗ ; J^{<k_{1}} to give ha⃗ ; J, so the two sides are equal when x = 0.

The induction step is a formal application of the One-Step Lemma (14.1). □

Up until now we have always extended a⃗ ∈ P (J) to a vector in P (J ∪ {k_{1}}) by giving it the value 0 on k_{1}. But now we want it to take the value λ(J, a⃗, k_{1}) on k_{1}. The simplest way to express this is to use monomials and to write g_{k_{1}}^{λ(J,a⃗,k_{1})}ga⃗ ; J.

Lemma 15.3. (One-Row Trace Lemma) Let a⃗ ∈ P (J) be S_{J}(J)-k_{1}-compatible. Then we have

(1)  Tr_{U_{{k_{1}}}} [G(J) · ha⃗ ; J] = G(J ∪ {k_{1}}) [(−1)^{τ(k_{1}+1,J,a⃗)}g_{k_{1}}^{λ(J,a⃗,k_{1})}ga⃗ ; J + x] ,

where x ∈ H(g_{k_{1}}^{λ(J,a⃗,k_{1})}ga⃗ ; J, k[g_{k_{1}}, g_{J}]) and τ = τ(k_{1} + 1, J, a⃗) is as in Lemma 15.2. Furthermore,

(2)  Tr_{U_{{k_{1}}}} [G(J) · H(ha⃗ ; J, k[h_{k_{1}}, h_{J}])] ⊆ G(J ∪ {k_{1}})H(g_{k_{1}}^{λ(J,a⃗,k_{1})}ga⃗ ; J, k[g_{k_{1}}, g_{J}]).

Proof of the One-Row Trace Lemma. We just need to interpret the result of the m-Step Lemma (15.2) in the case when l = k_{1} + 1. We are calculating the correct trace by Lemma 15.1 and the fact that U_{{k_{1}}} = ∏_{j=k_{1}+1}^{n} A(x_{k_{1}}, x_{j}). Notice also that G(J)_{k_{1}+1} =

G(J ∪ {k_{1}}) (Lemma 13.2), ha⃗ ; J^{<k_{1}} = ga⃗ ; J^{<k_{1}}, θ(x_{k_{1}}; J^{<k_{1}+1}) = g_{k_{1}} and f(k_{1} + 1) = λ(J, a⃗, k_{1})

(by the definition of f (5.23)).

Also R_{k_{1}+1}(a⃗) = H((a⃗J̃)_{0}, k[g_{k_{1}}, gJ̃]) and R_{n+1}(a⃗) = H(a⃗_{0}, k[h_{k_{1}}, hJ̃]) (Remark 13.11). The old x (in 15.2) is in R_{k_{1}+1}(a⃗), so the new x (in 15.3) is in ha⃗ ; J^{<k_{1}}R_{k_{1}+1}(a⃗) = H(a⃗, k[g_{k_{1}}, g_{J}]) = H(ga⃗ ; J, k[g_{k_{1}}, g_{J}]). In fact, it is in H(g_{k_{1}}^{λ(J,a⃗,k_{1})}ga⃗ ; J, k[g_{k_{1}}, g_{J}]), since there is some j greater than k_{1} for which g_{j} occurs to a higher power than a_{j} and we are using the lexicographic ordering. □

The λ inside the H-terms is not necessary at this stage, but we want it for use later.

16. The Trace Lemma

In this section we will finally be able to prove the Trace Lemma (16.1), which is a partial calculation of Tr_{U_{I}/U_{J}} : S^{U_{J}} → S^{U_{I}}. Recall that the notation in the statement of the result was introduced in Section 7 (7.1, 7.4), in particular the definitions of the invariants d_{i}, e_{i}, f_{i}, g_{i}, and h_{i}. We will regard λ(J, a⃗, −) as a function on K (or some subset), denoted by λ⃗(J, a⃗).

Lemma 16.1 (Trace Lemma). Let a⃗ ∈ P (J) be S_{J}(J)-k_{1}-compatible. Then we have

(1)  Tr_{U_{I}/U_{J}} [G(J) · ha⃗ ; J] = (−1)τ̂d_{K}^{λ⃗(J,a⃗)}da⃗ ; J + x,

where x ∈ H(d_{K}^{λ⃗(J,a⃗)}da⃗ ; J, k[d_{I}]) and τ̂ = τ̂(I, J, a⃗) = ns + ∑_{k∈K} k + ∑_{j∈J} |K^{<j}|(a_{j} + 1).

Furthermore,

(2)  Tr_{U_{I}/U_{J}} [G(J) · H(ha⃗ ; J, k[h_{J}])] ⊆ H(d_{K}^{λ⃗(J,a⃗)}da⃗ ; J, k[d_{I}]).

The above result is the special case, m = |K|, of the following slightly more general lemma, which is a partial calculation of Tr_{U_{J∪K^{≤km}} /U_{J}} : S^{U_{J}} → S^{U_{J∪K^{≤km}}}.

The idea of the proof is that U_{I} = U_{K} · U_{J} and if we write K = ⟨k_{1}, …, k_{|K|}⟩ then U_{K} = U_{{k_{|K|}}} · U_{{k_{|K|−1}}} ⋯ U_{{k_{2}}} · U_{{k_{1}}}. So, using 15.1, we can decompose Tr_{U_{I}/U_{J}} = Tr_{U_{K}} as the composition Tr_{U_{{k_{|K|}}}} Tr_{U_{{k_{|K|−1}}}} … Tr_{U_{{k_{2}}}} Tr_{U_{{k_{1}}}} and compute the trace bit by bit.

Lemma 16.2. (m-Row Trace Lemma, P (m)) Let a⃗ be S_{J}(J)-k_{1}-compatible and let k_{m} ∈ K. Then we have

(1)  Tr_{U_{J∪K^{≤km}} /U_{J}} [G(J) · ha⃗ ; J] = G(J ∪ K^{≤k_{m}}) · [(−1)^{τ^{(m)}}e_{K^{≤km}}^{λ⃗(J,a⃗)}ea⃗ ; J + x] ,

where x ∈ H(e_{K^{≤km}}^{λ⃗(J,a⃗)}ea⃗ ; J, k[e_{K^{≤km}}, e_{J}]) and τ^{(m)} = ∑_{i=1}^{m} τ(k + 1, J, a⃗). Furthermore,

(2) Tr_{U_{J∪K^{≤km}} /U_{J}} [G(J) · H(ha⃗ ; J, k[h_{J}])] ⊂ G(J ∪ K^{≤k_{m}}) · H(e_{K^{≤km}}^{λ⃗(J,a⃗)}ea⃗ ; J, k[e_{K^{≤km}}, e_{J}]).

We prove the m-Row Trace Lemma by induction on m. Of course, all that we actually need to prove is that P (m − 1) ⇒ P (m), since P (1) is the One-Row Trace Lemma. Here we record what we need to show for the induction step of the proof of Lemma 16.2, which is a partial calculation of Tr_{U_{{km}}} = Tr_{U_{J∪K^{≤km}} /U_{J∪K^{≤k_{m−1}}}} : S^{U_{J∪K^{≤k_{m−1}}}} → S^{U_{J∪K^{≤km}}}.

Lemma 16.3. (P (m − 1) ⇒ P (m)) Let a⃗ be S_{J}(J)-k_{1}-compatible. Then

(1) Tr_{U_{{km}}} [G(J ∪ K^{≤k_{m−1}}) · f_{K^{≤k_{m−1}}}^{λ⃗(J,a⃗)}f a⃗ ; J] = G(J ∪K^{≤k_{m}})·[(−1)^{τ(k_{m}+1,J,a⃗)}e_{K^{≤km}}^{λ⃗(J,a⃗)}ea⃗ ; J + x] ,

where x ∈ H(e_{K^{≤km}}^{λ⃗(J,a⃗)}ea⃗ ; J, k[e_{K^{≤km}}, e_{J}]). Furthermore,

(2) Tr_{U_{{km}}} [G(J ∪ K^{≤k_{m−1}}) · H(f_{K^{≤k_{m−1}}}^{λ⃗(J,a⃗)}f a⃗ ; J, k[f_{K^{≤k_{m−1}}}, f_{J}])] ⊆

G(J ∪ K^{≤k_{m}}) · H(e_{K^{≤km}}^{λ⃗(J,a⃗)}ea⃗ ; J, k[e_{K^{≤km}}, e_{J}]).

Finally,

(3) P (m − 1), 1, and 2 together imply P (m).

Proof of Lemma 16.2. Use induction on m: Lemma 15.3 is the initial case and Lemma 16.3 is the induction step. □

Proof of Lemma 16.3. For convenience, we write K′ = K^{≤k_{m−1}} and K′′ = K^{≤k_{m}}.

To prove part 1 we use the One-Row Trace Lemma (15.3), with J replaced by J ∪ K′. In order to apply this lemma we must verify that f_{K′}^{λ⃗(J,a⃗)}f a⃗ ; J is S_{J∪K′}(J ∪ K′)-compatible. This is true by Lemma 5.31. We may therefore conclude that

Tr_{U_{{km}}} [G(J ∪ K′) · f_{K′}^{λ⃗(J,a⃗)}f a⃗ ; J] =

G(J ∪ K′′) · [(−1)^{τ(k_{m}+1,J∪K′,α⃗)}e_{k_{m}}^{λ(J∪K′,α⃗,k_{m})}e_{K′}^{λ⃗(J,a⃗)}ea⃗ ; J + x] ,

where α⃗ ∈ P (J ∪ K′) is the exponent vector associated to the monomial f_{K′}^{λ⃗(J,a⃗)}f a⃗ ; J, and

x ∈ H(e_{k_{m}}^{λ(J,a⃗,k_{m})}e_{K′}^{λ⃗(J,a⃗)}ea⃗ ; J, k[e_{K′′}, e_{J}]).

Observe that λ(J ∪ K′, α⃗, k_{m}) = λ(J, α⃗, k_{m}) by Lemma 5.30. Also observe that τ(k_{m} + 1, J ∪ K′, α⃗) = τ(k_{m} + 1, J, a⃗) by definition of τ (15.2).

Up to a change of notation, we have now proved Part 1 of Lemma 16.3.

Part 2 of Lemma 16.3 is likewise an application of the One-Row Trace Lemma. We apply Lemma 15.3(2), with J replaced by J ∪ K′ and k_{1} replaced by k_{m}. This gives the desired equation.

The proof of part 3 is easy and therefore left to the reader. □

17. Consequences

Perhaps the most surprising corollary is the following result.

Theorem 17.1. For any finite group G acting on S, only a finite number of (isomorphism types of) indecomposable kG-modules can occur as a summand of S.

Proof. First we deal with the case of a p-group P, so we may assume that P ⊆ U_{n}. Taking the statement of the Main Theorem 10.1 and restricting it to P, we see that any indecomposable summand must be a summand of some Res_{P}^{U_{n}} Ind_{U_{J}}^{U_{n}} X_{J}(J). Since there are only finitely many possible J and each X_{J}(J) is finite, the result follows for P.

For general G, let P be a Sylow p-subgroup of G. Then, as kG-modules, S is a summand of Ind_{P}^{G} Res_{P}^{G} S and the result follows from the one for P. □

Given a finite group G, let A be an abelian group (often ℤ). We will consider an additive homomorphism χ : a_{k}(G) → A from the Green ring to A, that is a function on the isomorphism classes of finite dimensional kG-modules with values in A which is additive on direct sums.

For any ℤ-graded kG-module M, finite dimensional in each degree, the Poincaré series relative to χ is PS_{χ}(M, t) = ∑_{n∈ℤ} χ(M_{n})t^{n}.

Interesting possibilities for χ include the dimension of the invariants or coinvariants, dim H^{n}(G, −) or the multiplicity of a given indecomposable kG-module as a summand. We can even take the identity function id : a_{k}(G) → a_{k}(G), in which case PS_{id}(M, t) expresses the isomorphism class of M.

Theorem 17.2. For any p-group P acting on S, the Poincaré series relative to χ has the form

PS_{χ}(S, t) = \frac{f(t)}{∏_{i=1}^{n}(1 − t^{q^{i−1}})},

where f(t) is a polynomial of degree at most \frac{q^{n}−1}{q−1} − n.

If we can calculate PS_{χ}(S, t) in degrees up to and including \frac{q^{n}−1}{q−1} − n then we know it in all degrees.

Proof. We may assume that P ⊆ U_{n}. By the Main Theorem (10.1) with I = {1, …, n−1},

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

as kP -modules. Applying PS_{χ} we find that

PS_{χ}(S, t) = ∑_{J⊆I} \frac{PS_{χ}(X̄_{J}(I), t)}{∏_{i∈(I−J)∪\{n\}}(1 − t^{q^{i−1}})}.

If we put each term over the common denominator ∏_{i=1}^{n}(1 − t^{q^{i−1}}) then the numerator becomes PS_{χ}(X̄_{J}(I), t) ∏_{j∈J}(1 − t^{q^{j−1}}); this is bounded as required, by 5.21 and 5.22.

The last claim comes from the observation that f(t) = PS_{χ}(S, t) ∏_{i=1}^{n}(1 − t^{q^{i−1}}), so we can calculate f(t). □

We can reduce the bound on the degree at the cost of a more complicated formulation.

Theorem 17.3. If we know the decomposition into indecomposables of S as a kU_{I}-module in degrees less than or equal to q^{n−1} − n (or even just deg_{I}(p⃗)), then we know the decomposition into indecomposables of S in all degrees.

Proof. We show by induction on |I| that we know all the X_{J}(I) for J ⊆ I; this proves the claim and the case |I| = 0 is trivial.

Since we know S as a kU_{I}-module in the range of degrees given, we certainly know it as a kU_{J}-module for J ⊂ I and by induction we know X_{J}(J) up to isomorphism for J ⫋ I. But this tells us X_{J}(I), since the latter is constructed by induction and propagation from the former in the Main Theorem 10.1.

We can now identify X_{I}(I) up to isomorphism as the remainder since, according to Lemma 5.22, the degree of an element of X_{I}(I) is bounded by the two expressions given in the statement of the theorem. □

Describing the ring structure of the invariants is a classical problem.

Theorem 17.4. For any p-group P acting on S, the invariants S^{P} are generated as a ring by elements of degree less than or equal to \frac{q^{n}−1}{q−1} − n if n ≥ 3, (q if n = 2).

For any group G acting on S, the invariants S^{G} are generated as a ring by elements of degree less than or equal to \frac{q^{n}−1}{q−1}(nq − n − 1) if n ≥ 3, (2q^{2} − q − 2 if n = 2).

Proof. Taking invariants in formula A above we obtain S^{P} = ⨁_{J⊆I} k[d_{k}|k ∈ (I − J) ∪ {n}] · X̄_{J}(I)^{P}. Thus S^{P} is generated by the d_{i} and X̄_{J}(I)^{P}. The degrees of the former are bounded by q^{n−1}. The degrees of the latter are bounded by \frac{q^{n}−1}{q−1} − n, by Lemma 5.21. This is greater unless n = 2.

For general G, let P be a Sylow p-subgroup of G. Now S^{G} ⊆ S^{P}, and is, in fact, a summand as an S^{GL^{n}}-module. The splitting is given by |G : P |^{−1} Tr_{P}^{G} (where Tr_{P}^{G} s = ∑_{g∈G/P} gs).

Thus it suffices to show that S^{P} is finitely generated as an S^{GL^{n}}-module by elements of the claimed degree (since S^{GL^{n}} is generated by the Dickson invariants, which have degree at most q^{n} − 1).

But we have already seen that every s ∈ S^{P} can be expressed as s = ∑_{i} λ_{i}a_{i} with λ_{i} ∈ S^{U_{n}} and deg a_{i} ≤ \frac{q^{n}−1}{q−1} − n. To finish we note that it is shown in [6] that every λ_{i} ∈ S^{U_{n}} can be written as ∑_{j} μ_{i,j}b_{i,j} with μ_{i,j} ∈ S^{GL^{n}} and b_{i,j} ∈ S^{U_{n}} such that deg b_{i,j} ≤ nq^{n} − 2\frac{q^{n}−1}{q−1}. So s = ∑_{i,j} μ_{i,j}b_{i,j}a_{i} and S^{P} is generated over S^{GL^{n}} by elements of the form ba with deg a ≤ \frac{q^{n}−1}{q−1} − n and deg b ≤ nq^{n} − 2\frac{q^{n}−1}{q−1}. These have the degrees claimed. □

Of course, it is well known that S^{G} is finitely generated: it is the explicit bound on the degrees of the generators that is relevant here. For a long time no such a priori bound was known, but recently Derksen and Kemper [7], using forgotten work of Hermann [9], produced the bound

(B)  n(|G| − 1) + |G|^{n·2^{n−1}+1} · n^{2^{n−1}+1}.

It is curious that this bound depends only on the order of the group and not the field, while the bound in Theorem 17.4 depends only on the field and not the group.

For “large” groups the bound in Theorem 17.4 is very much smaller than Derksen and Kemper’s bound (B). For example, for U_{n}, where the best possible bound is q^{n−1}, it is of order q^{n−1} in q, while Derksen and Kemper’s bound is roughly of order (q^{n−1})^{n^{2}2^{n−2}}.

The bound in Theorem 17.4 can be improved if we know that P is contained in some U_{I}.

18. Index of Notation

Symbol  Definition Symbol Definition  Symbol  Definition
k  §4  p, q  §4  S  §4
U_{n}  §4  I, J, K  4.1  U_{I}  4.2
J^{<i}  4.3  θ  4.4,12.1  d_{i}  4.5
μ  5.1  P (I)  5.2  ψ  5.3
λ  5.4  S_{J}(I)  5.5  IE, E  5.5
b_{i}^{J}  5.15  deg_{I}  5.19  f(l)  5.23
T (I)  6.1  d_{I}, da⃗ ; I  7.1  H(a⃗, k[d_{I}]), H(a⃗)  7.4
LM  7.6  soc  7.9  supp  7.12
PS  8.1  γ  8.4  κ  8.5
G(I, J)  9.2  Tr  9.3, 12.4  ν  9.6
d_{i}, e_{i}, f_{i}, g_{i}, h_{i}  11.1  A(x_{i}, x_{j})  11.2  J̃  12.16
G(I, J)_{l}  13.1  w_{l,i}, w_{l,X}  13.4  R_{l}(a⃗)  13.10

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http://www.ma.umist.ac.uk/pas/preprints

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

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

School of Mathematics, University of Manchester, P.O. Box 88, Manchester M60 1QD, England

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