Notes on Codes Invariant Under the Affine Group

Peter Sin^{∗} Department of Mathematics, University of Florida, 358, Little Hall, PO Box 118105 Gainesville, FL 32611–8105, USA

Abstract

Let k[V] be the space of functions from a finite vector space into the algebraically closure of its field of scalars. This paper describes the lattice of subspaces of k[V] which are invariant under the affine group AGL(V). The description provides a simple method for finding the submodule generated by any set of functions given as polynomials in the standard coordinates.

1 Introduction

This paper concerns the vector space k[V] of functions from a finite vector space to an algebraically closed field of the same characteristic and its kAGL(V)-submodules, the subspaces which are invariant under the natural action of the affine group AGL(V). Previously, this problem has been investigated as a topic in coding theory. The subspaces in question are precisely the codes of length |V | which are AGL(V)-invariant. In the case n = 1, all such codes were determined in [9]. Later, in [3], [5] the results of [9] were reformulated, giving a combinatorial description of the lattice of AGL(V)-invariant subcodes of k[V]. The general case was treated in the important paper [6], yielding a characterization of the AGL(V)-invariant subcodes of k[V]. The purpose of this paper is to give an explicit description of the kAGL(V)-submodules of k[V] (Theorem 4.3), using the standard coordinates on V, as well as a combinatorial description of the lattice of submodules (Theorem 4.1). In [1], a similar study was carried out for the kGL(V)-submodules of k[V] and some results from that work will be used here.

∗Supported by NSF grant DMS0071060

Much of the motivation for this work has been provided by two related topics outside coding theory.

First, there is quite a large body of research in representation theory, dating back to the origins of the subject, with the basic theme of studying invariant subspaces of polynomial rings under the action of classical groups and algebras. As a small sample we mention [7], [10], [11] and [12]; more can be found in the references of these papers. The present paper follows in this tradition, particularly from a technical standpoint. For example, the facts about simple modules in Section 3 below are standard in this theory.

A second connection is to doubly transitive permutation groups. The paper [14] raised the problem of describing the submodule structure of the doubly transitive permutation modules over all fields. The present paper supplies the answer for the permutation module in the natural characteristic for AGL(V) acting on V.

2 Notation and background

Let q = p^{t} be a prime power and let V = 𝔽_{q}^{n} be the n-dimensional affine space over 𝔽_{q}. Our principal object of study is the space k[V] of functions from V to an algebraically closed field k of characteristic p. The group GL(V) of linear automorphisms of V and the group AGL(V) = V ⋊ GL(V) of invertible affine transformations act on the basis V of k[V], so the latter is a permutation module for these groups. If we choose coordinates for V, then k[V] may be viewed as the quotient k[x_{1}, …, x_{n}]/(x_{i}^{q} − x_{i})_{i=1}^{n} of the polynomial ring, with the groups GL(V) and AGL(V) acting through homogeneous and inhomogeneous linear substitutions respectively.

2.1 Twisted degrees and types of monomials

The images in k[V] of the monomials ∏_{i=1}^{n} x_{i}^{a_{i}} with 0 ≤ a_{i} ≤ q − 1 form a basis of k[V]. We will refer to these elements of k[V] as basic monomials. It is clear what is meant by the degree of a basic monomial, namely the number a_{1} + … + a_{n}. Now the Galois group of 𝔽_{q}, a cyclic group of order t, acts on k[V] and in fact permutes the set of basic monomials; a generator σ raises each basic monomial to its p-th power, with the understanding that x_{i}^{q} is replaced by x_{i}. Given a basic monomial m = ∏_{i=1}^{n} x_{i}^{a_{i}}, we express each a_{i} p-adically as a_{i} = ∑_{j=0}^{t−1} b_{ij}p^{j} and rewrite m as ∏_{j=0}^{t−1}[∏_{i=1}^{n} x_{i}^{b_{ij}}]^{p^{j}}. Let λ_{j} = ∑_{i=1}^{n} b_{ij}. Then the degree of m is ∑_{j} λ_{j}p^{j}, and the degree of its image under σ^{−e} is ∑_{j} λ_{j}p^{(j−e)}, where the exponent (j − e) is the representative of the mod t congruence class of j − e in the range from 0 to t − 1.

Based on these observations we will now asociate with each basic monomial two t-tuples of non-negative integers which will be called its type and its twisted degree tuple. These data will play a fundamental role throughout the rest of this paper.

Definition 2.1 Given a basic monomial

m = ∏_{j=0}^{t−1}[∏_{i=1}^{n} x_{i}^{b_{ij}}]^{p^{j}}  (1)

with 0 ≤ b_{ij} ≤ p − 1, let λ_{j} = ∑_{i=1}^{n} b_{ij}. We define the type of m to be the t-tuple (λ_{0}, …, λ_{t−1}).

The e-th twisted degree of m is defined to be the degree of σ^{−e}(m), namely ∑_{j} λ_{j}p^{(j−e)} and the twisted degree tuple of m to be the t-tuple (d_{0}, …, d_{t−1}), where d_{e} is the e-th twisted degree.

For example, consider the basic monomial x_{i}^{p^{j}}. Its type is the j-th standard basis vector e_{j} (0 ≤ j ≤ t − 1) of Z^{t} and its twisted degree tuple is

v_{j} = (p^{j}, …, p, 1, …, p^{(j+1)}).  (2)

The t-tuples e_{j} and v_{j} are related by the formula

pv_{j−1} − v_{j} = (q − 1)e_{j}, (j = 0, …, t − 1).  (3)

As another example, if we set

1 = (1, 1, …, 1)  (4)

then the basic monomial ∏_{i=1}^{n} x_{i}^{q−1} has type n(p − 1)1 and twisted degree tuple n(q − 1)1.

The galois action induced on these tuples is easy to describe; the type and twisted degree tuple of σ(m) are obtained from those for m by shifting the entries cyclically one place to the right.

It is clear that the twisted degree tuple of a basic monomial depends only on its type. Let T be the set of all types of basic monomials and D the set of all their twisted degree tuples. Then

T = {λ = (λ_{0}, …, λ_{t}) ∈ ℕ^{t} | 0 ≤ λ_{i} ≤ n(p − 1)}.  (5)

Let Φ : T → D be the map assigning to a type λ its tuple of twisted degrees. It is additive in the sense that if a type is the sum of two types, then its twisted degree tuple is the sum of the those of its summands.

Since every type can be written as a nonnegative integral combination of the e_{j}, with coefficients between 0 and n(p − 1), we have

D = {∑_{j=0}^{t−1} c_{j}v_{j} | 0 ≤ c_{j} ≤ n(p − 1)}  (6)

The formula (3) shows that the map Φ is injective and that its image D lies in the sublattice

M = {(z_{0}, …, z_{t−1}) ∈ ℤ^{t} | pz_{j} − z_{j+1} ∈ (q − 1)ℤ^{t}, j = 0, …, t − 1} (7)

of ℤ^{t}. Moreover, it follows from (3) that M is equal to the sublattice of ℤ^{t} generated by D.

Definition 2.2 We give D the structure of a poset by taking the partial order ≤ to be the one induced by the standard partial order on ℤ^{t}, i.e. z ≤ z′ if and only if z_{j} ≤ z′_{j} for j = 0, …, t − 1.

3 Filtrations, and composition factors

3.1 Multiplicity-free modules

We recall that a module having a Jordan-Hölder series is called multiplicityfree if no two composition factors are isomorphic.

Lemma 3.1 (a) k[V] has a unique maximal kAGL(V) submodule J and k[V]/J ≅ k is a trivial module. J has as basis the set of all basic monomials of type ≠ n(p − 1)1.

(b) The space of constant functions on V is the unique minimal submodule of k[V].

(c) J is multiplicity-free as a kAGL(V)-module (or as a kGL(V)-module).

Proof: AGL(V) acts transitively on the basis V of k[V] and the stablizer of the origin is GL(V). If S is a simple kAGL(V)-module, then it is also simple for GL(V) (since the normal p-subgroup V must act trivially) and by Frobenius reciprocity, we have

Hom_{kAGL(V)}(k[V], S) ≅ Hom_{kGL(V)}(k, S).  (8)

The first part of (a) is immediate from this. The unique basic monomial of type n(p−1)1 is ∏_{i=1}^{n} x_{i}^{q−1}. Since all of the other basic monomials have lower degree, it follows that the subspace they span is a kAGL(V)-submodule and that the quotient is a one-dimensional trivial module. Therefore, by the uniqueness just proved, this maximal submodule is J. The fact that k[V] has a unique minimal submodule follows from (a) by the self-duality of the permutation module k[V]; and it is clear that the constant functions form a one-dimensional submodule, so (b) holds. Part (c) is a very well known and follows from the fact that GL(V) has a cyclic subgroup of order q^{n} −1 which acts regularly (simply transitively) on V \{0}. (In coding theory terms, this expresses the cyclicity of the GL(V)-invariant subcodes of k[V \ {0}].)

Because of Lemma 3.1, the submodule lattice of k[V] will be known once we know that of J. Further, since J is multiplicity-free, we know by general representation theory that its submodule lattice is isomorphic to the lattice of ideals in the poset in which the elements are the isomorphism classes of composition factors, partially ordered by the rule that one composition factor lies above another if and only if every submodule having the first composition factor also has the second.

3.2 The simple modules S(λ)

Next we shall define a set of simple kGL(V)-modules parametrized by the set T of types of basic monomials. These will turn out to be the composition factors of k[V]. For λ = (λ_{0}, … λ_{t−1}) ∈ T , the corresponding simple module, denoted S(λ), can be constructed in the following way. Let V_{k} = k ⊗_{𝔽_{q}} V. Then the algebraic group GL(V_{k}) acts (by homogeneous linear substitutions) on k[x_{1}, …, x_{n}] and hence also on k[x_{1}, …, x_{n}]/(x_{i}^{p})_{i=1}^{n}. For 0 ≤ λ ≤ n(p−1), let \overline{S}^{λ} = \overline{S}^{λ}(V_{k}) denote the degree λ component of the graded module k[x_{1}, …, x_{n}]/(x_{i}^{p})_{i=1}^{n}. Then we may form the twisted tensor product

S(λ) = \overline{S}^{λ_{0}} ⊗ (\overline{S}^{λ_{1}})^{(p)} ⊗ ⋯ ⊗ (\overline{S}^{λ_{t−1}})^{(p^{t−1})},  (9)

where the powers of p in the superscripts indicate Frobenius twists [8, I.9.10].

It is well known that the GL(V_{k})-modules obtained in this way are simple and remain simple when restricted to the finite group GL(V). Details and references can be found in [1, §2.3]. The dimensions and characters of the modules S(λ) are also well known ([1, §2.4]).

3.3 Twisted Filtrations and the modules Y_{d}

We now consider some natural filtrations of k[V] which will allow us to show that the composition factors of J are the modules S(λ) for λ ∈ T \{n(p−1)1}

For each natural number r the basic monomials of degree ≤ r span an AGL(V) submodule F_{r} of k[V]. Thus we have the degree filtration 0 ≤ F_{0} ≤ F_{1} ≤ ⋯ F_{n(q−1)} = k[V]. If we let σ also denote the Frobenius map on AGL(V), then for g ∈ AGL(V) we have

gσ^{−e}(m) = σ^{−e}(σ^{e}(g)m),  (10)

which shows that the e-th twisted degree of each basic monomial occuring in gm is no greater than the e-th twisted degree of the basic monomial m. Thus we obtain a total of t filtrations by twisted degrees of AGL(V)-modules 0 ≤ F_{0}^{(j)} ≤ F_{1}^{(j)} ≤ ⋯ F_{n(q−1)}^{(j)} = k[V], where F_{r}^{(j)} = σ^{−j}F_{r}.

For each d = (d_{0}, …, d_{t−1}) ∈ D define the submodule

Y_{d} = F_{d_{0}} ∩ F_{d_{1}}^{(1)} ∩ … ∩ F_{d_{t−1}}^{(t−1)}.  (11)

Let Y_{<d} be the subspace of Y_{d} spanned by all basic monomials with twisted degree tuples < d. This is a kAGL(V)-submodule.

Lemma 3.2 Let λ = Φ^{−1}(d). Then Y_{d}/Y_{<d} ≅ S(λ).

Proof: It is clear from the definitons that Y_{d}/Y_{<d} has as basis the images of all basic monomials of type λ. The action of GL(V) on this module is by homogeneous linear substitution of the monomials, followed by deletion of all monomials of type different from λ.

Consider the k-vector space isomorphism from the tensor product (9) to Y_{d}/Y_{<d} given by

⊗_{j=0}^{t−1}(∏_{i} x_{i}^{b_{ij}})^{(p^{j})} ↦ ∏_{i} x_{i}^{∑_{j} b_{ij}p^{j}}.  (12)

(Here we have kept the same notation for the images of basic monomials in Y_{d}/Y_{<d}.)

The action of GL(V) on each tensor factor in (9) is by homogeneous linear substitution of the variables followed by deletion of terms involving a p-th power or higher in any variable. Since we have an explicit description of the GL(V)-action on both spaces, it is now elementary to verify that (12) defines a kGL(V)-isomorphism. In checking this, two things to keep in mind are the definitions of scalar multiplication in the twisted tensor factors and the fact that in characteristic p, the p-th power of a sum is the sum of the p-th powers.

It is immediate from Lemma 3.2 that each S(λ) with λ ∈ T is a composition factor of k[V]. They are in fact all of the composition factors since S(λ) has as basis (the images of) all basic monomials of type λ, while k[V] is spanned by monomials of all types. More generally, the same argument proves the following corollary.

Corollary 3.3 For d ∈ D composition factors of Y_{d} are the simple modules S(λ′) such that Φ(λ′) ≤ d.

Since J = Y_{<n(p−1)1}, its composition factors are the S(λ) for λ ≠ n(p − 1)1. Lemma 3.1(c) therefore implies that these simple modules are mutually non-isomorphic, while Lemma 3.1(a) and (b) show that S(n(p − 1)1) ≅ k ≅ S(01).

3.4 The poset (T , ⪯)

Since the set of composition factors of J is in bijection with T \ {n(p − 1)1}, the partial ordering on the composition factors of J induces a partial order on this set.

Definition 3.4 The partial order ⪯ on T is the extension of the above partial order on T \ {n(p − 1)1}, in which n(p − 1)1 is declared to be the unique maximal element.

By virtue of Lemma 3.1, the lattice of ideals in (T , ⪯) is isomorphic to the lattice of kAGL(V)-submodules of k[V], ordered by inclusion.

The bijection Φ : T → D allows us to ordered sets (T , ⪯) and (D, ≤). Since the space Y_{Φ(λ)} is a kAGL(V)-submodule of k[V] having as composition factors all S(λ′) with Φ(λ′) ≤ Φ(λ), it follows that Φ is orderpreserving.

4 The submodule lattice of k[V] with respect to AGL(V)

We are now ready to state our main theorems.

Theorem 4.1 The map Φ is an isomorphism of partially ordered sets.

Remark 4.2 Theorem 4.1 should be viewed as a reformulation of the characterization of affine-invariant codes in [6]. The results of [6] are not used in its proof.

Theorem 4.3 (a) The submodule of k[V] generated by a basic monomial m of type λ is equal to Y_{Φ(λ)}.

(b) The submodule Y_{<Φ(λ)} is the unique maximal submodule of Y_{Φ(λ)} and Y_{Φ(λ)}/Y_{<Φ(λ)} ≅ S(λ). The composition factors of Y_{Φ(λ)}, each occurring with multiplicity one, are the simple modules S(λ′), for all λ′ such that Φ(λ′) ≤ Φ(λ).

(c) For λ ≠ n(p − 1)1 any submodule which has S(λ) as a composition factor contains Y_{Φ(λ)}.

(d) The submodule of k[V] generated by an element ∑ α_{m}m, expressed as a linear combination of basic monomials, is the sum of the submodules Y_{Φ(λ)} as λ runs over the types of the basic monomials with non-zero coefficients.

Remark 4.4 The case t = 1 of Theorem 4.3 which is a fundamental result in the theory of generalized Reed-Muller codes, was worked out in detail in [6].

The theorems will be proved in 4.2 below, with the aid of the lemmas in the next subsection.

4.1

Lemma 4.5 Let d, d′ ∈ D with d′ ≤ d and d − d′ ∈ (q − 1)ℤ^{t}. Then Φ^{−1}(d′) ⪯ Φ^{−1}(d) in T .

Proof: In view of Lemma 3.1(a) and (b), we can assume d, d′ ≠ (0, … 0), (n(q − 1), …, n(q − 1)). The hypothesis on d − d′ implies that S(Φ^{−1}(d)) and S(Φ^{−1}(d′)) are composition factors in the same direct summand of the kGL(V)-module k[V]. The submodule structure of these summands is given in [1, Theorems A and C]. (We note that there are slight notational differences; for example the tuples H in [1, Theorems A] should be multiplied by q − 1 to give the corresponding twisted degree tuples.) It follows from these results that any kGL(V)-submodule of k[V] having S(Φ^{−1}(d)) as a composition factor also has S(Φ^{−1}(d′)). Hence, the same holds for kAGL(V)-submodules. See also [6].

Lemma 4.6 Let λ = (λ_{0}, λ_{1}, …, λ_{t−1}) and λ′ = (λ′_{0}, λ′_{1}, …, λ′_{t−1}). Suppose that for some j, we have λ′_{j} = λ_{j} + 1 and λ′_{k} = λ_{k} for k ≠ j. Then λ ⪯ λ′.

Proof: By Lemma 3.1, we can assume λ, λ′ ≠ (0, …, 0), n(p−1)1. We may also assume without loss of generality that j = 0. Let Φ(λ) = d and Φ(λ′) = d′. Set E = Y_{d′}/(∑_{d′′} Y_{d′′}), where the sum is over all d′′ ≤ d′ except for d′ and d. Now there is no element of D betweeen d and d′, so E has a simple submodule U ≅ S(λ) with quotient E/U ≅ S(λ′). To prove the lemma, we must show that U is the unique simple submodule of E. If q = 2 we are in the case of the classical Reed-Muller codes, This is much simpler than the general case and it is well known and straightforward to show that k[V] is uniserial, from which the lemma follows easily. We will not repeat this case here and assume from now on that q > 2. (An independent proof of the uniseriality of k[V] in the more general case q = p is given in Theorem 5.1 below.) Now the k-span of basic monomial m is invariant under the subgroup of scalar matrices 𝔽_{q}^{×}I and the character afforded is αI ↦ α^{deg(m)}. Since the degrees of λ and λ′ differ by 1 so are incongruent modulo q − 1, monomials of these two types afford distinct characters. Linear independence of characters then shows that any k𝔽_{q}^{×}I-submodule of E contains both the λ-component and λ′-component of each of its elements. By hypothesis λ′_{0} ≠ 0, so there existis a basic monomial m = x_{1}^{a_{1}}g of type λ′, in which the degree of x_{1} has p-adic expression a_{1} = b_{10} + b_{11}p + ⋯ + b_{1(t−1)}p^{t−1}, with b_{10} ≠ 0. Let \overline{m} ∈ E/U be its image. Suppose that a kAGL(V)-submodule of E is not contained in U. Then since E/U ≅ S(λ′), it follows that the submodule contains an element mapping to \overline{m} and so by linear independence of characters the submodule contains the image in E of m. We are therefore reduced to proving that the kAGL(V)-submodule of E generated by the image of m contains an element with a nonzero λ-component. We apply the substitution τ : x_{1} ↦ x_{1} + 1, with all other variables left fixed. Then τ(m) = m + b_{10}x_{1}^{a−1}g, of which the last term is of type λ, so the proof is complete.

Remark 4.7 Let ≤ denote the ordering on T induced by the standard ordering of ℤ^{t}. By Lemma 4.6 λ ≤ λ′ implies λ ⪯ λ′, while Lemma 4.5 shows for example, in the case t = 2 that (0, 1) ⪯ (p, 0).

Recall the definition (7) of the lattice M. As noted in Remark 4.7 the ordering on T is a refinement of the standard ordering. Theorem 4.1 predicts that this should be reflected in D by the presence of positive elements of M which are not non-negative combinations of the v_{j}. Equation (3) provides some examples of these positive elements. The following lemma can be interpreted as saying that (3) accounts for the existence of all such positive elements.

Lemma 4.8 Each element y of M ∩ ℤ_{+} can be written in the form

y = ∑_{j=0}^{t−1} a_{j}(q − 1)e_{j} + b_{j}v_{j},  (13)

with a_{j} ≥ 0 and 0 ≤ b_{j} ≤ p − 1.

Proof: It follows from (3) that (q −1)ℤ^{t} ≤ M and that M/(q −1)ℤ^{t} is cyclic of order q − 1, generated by the image of any v_{i}. Therefore, the element y is congruent mod (q − 1)ℤ^{t} to bv_{0}, with 0 ≤ b ≤ q − 2. Let b = ∑_{j=0}^{t−1} b_{j}p^{j} with 0 ≤ b_{j} ≤ p − 1. From (3) we obtain that p^{j}v_{0} = v_{j} mod (q − 1)ℤ^{t}. Thus, we may write

y = ∑_{j=0}^{t−1} a_{j}(q − 1)e_{j} + b_{j}v_{j}.  (14)

Then since y ∈ ℤ_{+}^{t}, by considering the entries in each coordinate of y in (14) we see that a_{j} ≥ 0 for all j.

Lemma 4.9 Let Φ(λ) = d and Φ(λ′) = d′. If d ≤ d′ then λ ⪯ λ′.

Proof: By Lemmas 4.8 and 4.5 and proceeding inductively, we are reduced to the case d′ = d + v_{j}. This means that if λ = (λ_{0}, …, λ_{j}, …, λ_{t−1}) then λ′ = (λ_{0}, …, λ_{j} + 1, …, λ_{t−1}). So Lemma 4.6 applies.

4.2 Proofs of theorems

Theorem 4.1 is immediate from Lemma 4.9 and the order-preserving property of the bijection Φ. We turn to Theorem 4.3. Let m be a basic monomial of type λ. Then m ∈ Y_{Φ(λ)} and has non-zero image in Y_{Φ(λ)}/Y_{<Φ(λ)}. Therfore the submodule generated by m has S(λ) as a composition factor. Therefore it also has as a composition factor all S(λ′) with λ′ ⪯ λ, which by Theorem 4.1 are all of the composition factors of Y_{Φ(λ)}. This proves (a). Part (b) follows from Theorem 4.1 and Corollary 3.3. We prove (c) by contradiction. Suppose a submodule N of k[V] has S(λ) as a composition factor, but N ∩Y_{Φ(λ)} ⪇ Y_{Φ(λ)}. Then by (b), S(λ) is not a composition factor of N ∩ Y_{Φ(λ)} and must therefore be a composition factor of both summands of (N/N ∩ Y_{Φ(λ)}) ⊕ (Y_{Φ(λ)}/N ∩ Y_{Φ(λ)}) ≅ (N + Y_{Φ(λ)})/N ∩ Y_{Φ(λ)}, contrary to the fact that J is multiplicity-free.

To prove (d), let

y = ∑ α_{m}m α_{m} ≠ 0  (15)

be a linear combination of basic monomials. Let T_{y} ⊂ T be the set of types in the expression (15) and let T_{y}^{∗} be the maximal members of this set with respect to ⪯. Next consider the kAGL(V)-submodule Y generated by y. We have

Y ≤ ∑_{μ∈T_{y}} Y_{Φ(μ)} = ∑_{μ∈T_{y}^{∗}} Y_{Φ(μ)}.  (16)

Let μ ∈ T_{y}^{∗}. Then maximality of μ implies that there is a nonzero homomorphism from ∑_{μ∈T_{y}^{∗}} Y_{Φ(μ)} to S(μ) such that every basic monomial of different type from μ mapped to zero, while the images of basic monomials of type μ form a basis of S(μ). In particular, the image of y is not zero. Therefore S(μ) is a composition factor of Y. Then (b) implies that Y_{Φ(μ)} ≤ Y. Since this holds for each μ ∈ T_{y}^{∗}, we have equality in (16).

5 Supplementary results

In this final section we include several observations which are closely related to our main discussion but not central to it. They are based on very well known material and do not require the main theorems.

5.1 The radical and socle series of k[V]

The socle of a module M, denoted soc(M), is the sum of all simple submodules or, equivalently, the maximal semisimple submodule. The radical, rad(M), is the intersection of all maximal submodules or, equivalently, the smallest submodule by which the quotient module is semisimple. The higher radicals and socles are defined recursively in the usual way: soc^{i}(M) is the full preimage in M of soc(M/soc^{i−1}(M)) and rad^{i}(M) = rad(rad^{i−1}(M)). The radical length ℓ(M) is the smallest index i such that rad^{i}(M) = 0. Thus, the socle series is an ascending chain of submodules with semisimple quotients and the radical series is a descending chain with semisimple quotients. The radical series and socle series are said to be equal if they are the same as sets of subspaces, namely rad^{ℓ(M)−i}M = soc^{i}M.

Theorem 5.1 The radical and socle series of k[V] with respect to kAGL(V) are equal. The socle length is N = nt(p − 1) + 1 and the i-th socle layer is isomorphic to

⨁_{|λ|=i−1} S(λ),  (17)

where |λ| = λ_{0} + ⋯ + λ_{t−1}.

Since Theorem 4.1 completely describes the submodule lattice of k[V], the socle and radical series of any submodule can be found directly from this result. Thus, Theorem 5.1 is just one example. However, Theorem 5.1 does not require the full strength of Theorem 4.1 and it may be instructive to see how it can be deduced from the classical prime field case t = 1.

The restriction of k[V] to the subgroup V is isomorphic to the regular module for V. Let soc_{V}^{i} (k[V]) denote its i-th socle and rad_{V}^{i} (k[V]) its i-th radical.

Assume now that we are in the prime field case. For 0 ≤ r ≤ n(p−1), the r-th order p-ary Generalized Reed-Muller code is simply the subspace F_{r} in the degree filtration. We set F_{−1} = 0. It is a well known fact, first proved in [2], that this family of subcodes of k[V] is equal to the socle and radical series with respect to V. (See [4] for the history, a proof and generalizations, [15, 7.2] for further discussion and references.) We present here a self-contained proof of this fact.

Lemma 5.2 For 0 ≤ i ≤ n(p − 1) + 1 we have

soc_{V}^{i} (k[V]) = F_{i−1} = rad_{V}^{n(p−1)+1−i}(k[V]).  (18)

Proof: The action of V on k[V] is by translation of functions. In coordinates, let f = f[x_{1}, …, x_{n}] be written in terms of the monomial basis, let d be its degree and let u = −(a_{1}, …, a_{n}). Then we have a “Taylor expansion”

(uf)(x_{1}, …, x_{n}) = f(x_{1} + a_{1}, …, x_{n} + a_{n})

= f(x_{1}, …, x_{n}) + ∑_{i} a_{i} \frac{∂f}{∂x_{i}} + terms of degree ≤ d − 2 ,

(19)

It is immediate from (19) that for 0 ≤ r ≤ n(p − 1), the subspace F_{r} is a kV -submodule of k[V] and that V acts trivially on each quotient F_{r}/F_{r−1}. In order to prove the first equality of the lemma, it suffices to show that the module F_{r}/F_{r−1} is the whole socle, that is the set of all fixed points, of the module k[V]/F_{r−1}. In other words, we must prove that if f ∈ k[V] has degree d ≥ r + 1 then there exists u ∈ V such that uf − f ≠ 0 mod F_{r−1}. It will be enough if uf − f ≠ 0 mod F_{d−2}. Without loss of generality, we can assume the variable x_{1} occurs in f. The desired conclusion is then read off from (19) by setting u = (1, 0, …, 0) and recalling that the partial degree of f in x_{1} is ≤ p − 1.

The argument to prove the second equality is dual in some sense. It suffices to prove that the elements uf − f for f ∈ F_{r} and u ∈ V generate F_{r−1}. This will follow if we show that each basis monomial of degree ≤ r −1 is of the form uf − f for some element f ∈ F_{r} and some u ∈ V. Since r − 1 < n(p − 1), at least one variable in such a monomial has exponent < p − 1 and so by (19) we obtain f by partial integration.

Proof of Theorem 5.1

AGL(V) acts on the kV -socle series and from Lemma 5.2 we see that soc_{V}^{i} (k[V])/soc_{V}^{i−1}(k[V]) is isomorphic to the simple kAGL(V)-module \overline{S}^{i−1}(V ⊗_{𝔽_{p}} k). Since V acts trivially on any semisimple kAGL(V)-module, it follows that the kV -socle series is also equal (as sets of subspaces) to the kAGL(V)- socle series and radical series (and incidentally that k[V] is a uniserial kAGL(V)-module). Thus, Theorem 5.1 holds in the case t = 1.

Returning now to the general case, we can place ourselves in the prime case if we forget the 𝔽_{q}-structure of V and regard it as an nt-dimensional 𝔽_{p}-vector space, which we denote by V_{𝔽_{p}} when necessary. Then we apply the prime field case to the nt-dimensional affine group AGL(V_{𝔽_{p}}). Thus, we have a kAGL(V_{𝔽_{p}})-isomorphism

soc_{V}^{i} (k[V])/soc_{V}^{i−1}(k[V]) ≅ \overline{S}^{i−1}(V ⊗_{𝔽_{p}} k), (1 ≤ i ≤ nt(p − 1) + 1). (20)

We next compute the restriction of this module to AGL(V). Since, as kAGL(V)-modules, we have

V ⊗_{𝔽_{p}} k ≅ ⨁_{j=0}^{t−1} V_{k}^{(p^{j})}  (21)

it follows from the standard properties of the truncated symmetric powers with respect to direct sums that

\overline{S}^{i−1}(V ⊗_{𝔽_{p}} k) ≅ ⨁_{\begin{matrix}0≤λ_{j}≤n(p−1) \\ λ_{0}+···+λ_{t−1}=i−1 \end{matrix}} \overline{S}^{λ_{0}}(V_{k}) ⊗ \overline{S}^{λ_{1}}(V_{k}^{(p)}) ⊗ ⋯ \overline{S}^{λ_{t−1}}(V_{k}^{(p^{t−1})}).

(22)

In particular, this shows that soc_{V}^{i} (k[V])/soc_{V}^{i−1}(k[V]) is a semsisimple kAGL(V)- module, which implies that the socle and radical series of k[V] with respect to kAGL(V) are the same as the series with respect to kV. Since (22) also gives the composition factors of the socle layers, the proof of Theorem 5.1 is complete.

Our proof of Theorem 5.1 contains the following additional information.

Corollary 5.3 The radical and socle series of the module k[V] are equal to each other and the same (as sets of k-subspaces) whether k[V] is regarded as a kAGL(V)-module or as the regular kV -module. In particular, the radical and socle series are independent of the field over which V is taken to be a vector space.

5.2 Change of scalars

For completeness, we will give the general formulation for the change of fields involved in the preceding subsection. Let u be a divisor of t and set v = t/u. When necessary, we will use the notation V_{𝔽_{p^{u}}} when we wish to consider V as an nv-dimensional vector space over 𝔽_{p^{u}}. The types of the composition factors of k[V] with respect to AGL(V_{𝔽_{p^{u}}}) (or, equivalently, GL(V_{𝔽_{p^{u}}}) since V acts trivially) are u-tuples ρ = (ρ_{0}, …, ρ_{u−1}) with 0 ≤ ρ_{ℓ} ≤ nv(p − 1). We will modify the notation of section 3.2 slightly by calling these 𝔽_{p^{u}}-types and denoting the corresponding simple modules by S_{𝔽_{p^{u}}}(ρ). We keep the previous notation for the case u = t.

Our aim is to describe the restriction of S_{𝔽_{p^{u}}}(ρ_{0}, …, ρ_{u−1}) to GL(V). Given a 𝔽_{p^{u}}-type ρ = (ρ_{0}, …, ρ_{t−1}) we consider the set the set of all u × v matrices with entries λ_{ℓm}, (0 ≤ ℓ ≤ u − 1, 0 ≤ m ≤ v − 1), satisfying:

1. 0 ≤ λ_{ℓm} ≤ n(p − 1);

2. For each ℓ, the sum of the entries in the ℓ-th row equals ρ_{ℓ}.

We obtain an 𝔽_{q}-type from each such matrix by listing the t matrix entries starting from the top left and moving down successive columns. Thus λ_{ℓm} is the um + ℓ entry of the 𝔽_{q}-type. Let T (ρ) be the set of types obtained in this way.

Theorem 5.4 As kGL(V)-modules, we have

S_{𝔽_{p^{u}}}(ρ_{0}, …, ρ_{u−1}) ≅ ⨁_{λ∈T (ρ)} S(λ).  (23)

Proof: The result follows from the definitions and the additive properties of tensor products, together with the following straightforward generalization of (22). For 0 ≤ ρ ≤ nv(p − 1), we have

\overline{S}^{ρ}(V ⊗_{𝔽_{p^{u}}} k) ≅ ⨁_{\begin{matrix}0≤λ_{j}≤n(p−1) \\ λ_{0}+···+λ_{v−1}=ρ \end{matrix}} \overline{S}^{λ_{0}}(V_{k}) ⊗ \overline{S}^{λ_{1}}(V_{k}^{(p^{u})}) ⊗ ⋯ \overline{S}^{λ_{v−1}}(V_{k}^{(p^{(t−u)})}).

(24)

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