THE PERMUTATION MODULES FOR

GL(n + 1, 𝔽_{q}) ACTING ON ℙ^{n}(𝔽_{q}) AND 𝔽_{q}^{n+1}.

Matthew Bardoe Peter Sin

Abstract. This paper studies the permutation representations of a finite general linear group, first on finite projective space and then on the set of vectors of its standard module. In both cases the submodule lattices of the permutation modules are determined. In the case of projective space, the result leads to the solution of certain incidence problems in finite projective geometry, generalizing the rank formula of Hamada. In the other case, the results yield as a corollary the submodule structure of certain symmetric powers of the standard module for the finite general linear group, from which one obtains the submodule structure of all symmetric powers of the standard module of the ambient algebraic group.

Introduction and Statements of Results

1. The permutation module on projective space.

Let P = ℙ^{n}(𝔽_{q}), n-dimensional projective space over the field of q = p^{t} elements. A certain class of problems in finite geometry concerns the determination of the 𝔽_{p}-ranks of the incidence relations between the set P of points and various other sets of geometric objects. The most significant result in this direction has been the formula of Hamada [Ha] for the rank of the incidence between P and the set of r-dimensional linear subspaces of P. This formula is valid for all possible choices of the parameters n, p, t and r. Hamada’s work was motivated by questions in coding theory; the incidence matrices are generator matrices of codes closely related to the Reed-Muller codes (see [AK], [MS]).

Problems of the type mentioned above can also be phrased as questions in group representation theory. The set P admits a natural action of the group G = GL(n + 1, q), making the vector space 𝔽_{p}^{P} with basis P into a permutation module. If Q is another set of subsets of P, permuted by G and 𝔽_{p}^{Q} its permutation module, then the incidence relation between P and Q defines an 𝔽_{p}G-module map

𝔽_{p}^{Q} → 𝔽_{p}^{P} , β ↦ ∑_{p∈β} p  (1)

Typeset by AMS-TEX

Supported in part by NSF grant DMS9701065

and the rank in question is the dimension of the image of this map, which is often called the code of the incidence relation. For technical reasons, it is more convenient to work over an algebraic closure k of 𝔽_{q} and to consider the map

k^{Q} → k^{P} .  (2)

Of course, the rank will be unaffected.

Informally, we may think of the image of the incidence map as the submodule of k^{P} generated by the objects in Q. Thus, incidence problems lead us to the study of the kGsubmodule lattice of k^{P}. In searching for a uniform method for treating all such incidence problems, one approach is to try to describe this lattice in enough detail that the submodule generated by a given element can be identified and its dimension read off. It is known that, in general, it is very difficult to determine the structures of modules for finite groups. However, the problem under consideration has two favorable special features which turn out to be decisive. First, the module k^{P} is multiplicity free, i.e. no two composition factors in a Jordan-Hölder series are isomorphic. This rather obvious fact (Lemma 2.1) has the heuristically important consequence that the submodule lattice is finite. Second, we may adopt the dual viewpoint and regard k^{P} as the space of functions on P, which is the set of 𝔽_{q}-rational points of ℙ^{n}(k). The homogeneous coordinate ring k[X_{0}, …, X_{n}] has a natural action of the algebraic group GL(n + 1, k) and restriction of functions defines a map of G-algebras from k[X_{0}, …, X_{n}]^{𝔽_{q}^{×}} onto k^{P}. This point of view enables us carry out many explicit computations in k^{P} and at the same time provides a useful point of contact with the representation theory of reductive groups.

The first principal result of this paper will be a full description of the kG-submodule lattice of k^{P}.

In k^{P}, viewed now as functions, we have a kG-decomposition

k^{P} = k.1 ⊕ Y_{P} ,  (3)

where k1 is the space of constant functions and Y_{P} is the kernel of the map k^{P} → k, f ↦ |P |^{−1} ∑_{p∈P} f(p), which splits the inclusion map of k.1 in k^{P}.

Thus, the essential point is to understand the structure of Y_{P}. We can now state our theorem.

Theorem A. Let H denote the set of t-tuples (s_{0}, …, s_{t−1}) of integers satisfying (for j = 0, …, t − 1)

(1) 1 ≤ s_{j} ≤ n;

(2) 0 ≤ ps_{j+1} − s_{j} ≤ (p − 1)(n + 1). (Subcripts mod t.) Let H be partially ordered in the natural way: (s′_{0}, …, s′_{t−1}) ≤ (s_{0}, …, s_{t−1}) if and only if s′_{j} ≤ s_{j} for all j.

Then

(a) The module k^{P} is multiplicity free and has composition factors L(s_{0}, …, s_{t−1}) parametrized by the set H ∪ {(0, …, 0)}.

(b) For (s_{0}, …, s_{t−1}) ∈ H, let λ_{j} = ps_{j+1}−s_{j}. Then the simple kG-module L(s_{0}, …, s_{t−1}) is isomorphic to the twisted tensor product

⨂_{j=0}^{t−1}(\overline{S}^{λ_{j}})^{(p^{j})},

where \overline{S}^{λ} denotes the component of degree λ in the truncated polynomial ring \overline{S} = k[X_{0}, …, X_{n}]/(X_{i}^{p})_{i=0}^{n} and the superscripts indicate twisting by powers of the Frobenius map.

(c) For each submodule M of Y_{P}, let H_{M} ⊆ H be the set of its composition factors. Then H_{M} is an ideal of the partially ordered set (H, ≤), i.e if (s_{0}, …, s_{t−1}) ∈ H_{M} and (s′_{0}, …, s′_{t−1}) ≤ (s_{0}, …, s_{t−1}), then (s′_{0}, …, s′_{t−1}) ∈ H_{M}.

(d) The mapping M ↦ H_{M} defines a lattice isomorphism between the submodule lattice of Y_{P} and the lattice of ideals, ordered by inclusion, of the partially ordered set (H, ≤)

Remarks.

1. The set H is essentially the same as the set of tuples introduced in [Ha]; much of the present work has been motivated by the desire to understand the algebraic structure behind his rank formula. In §8, we will give a new proof of this formula, as a corollary of one of the main steps in the proof of our theorem. Our proof is more conceptual, though less elementary, than the original and does not rely on the earlier result of K. Smith [Sm] on which [Ha] is based.

2. The twisted tensor product in (b) is an example of Steinberg’s Tensor Product Theorem [St]. A more detailed discussion of the composition factors is undertaken in §2.

3. (Stabilization of module structure.) It is interesting to note that condition (2) in the definition of H is automatically satisfied when t = 1, (i.e. q = p) or when p ≥ n. Thus, in both of these cases, the submodule lattice of Y_{P} is isomorphic to the lattice of ideals in the t-fold product of the integer interval [1, n]. In particular, it does not depend on p. Of course, in the case t = 1 the submodules of k^{P} are well understood by coding theorists, as they are generalizations of the Reed-Muller codes.

In order to apply the theorem, one needs to be able to read off the submodule generated by a given element. Now from parts (a) and (b) of Theorem A, we see that k^{P} has a basis of monomials coming from the composition factors L(s_{0}, …, s_{t−1}) and it is clear what is meant by the t-tuple in H ∪ {(0, …, 0)} of such a monomial. (The precise definitions are given in §§1, 2 and 3.)

For f ∈ k^{P}, let H_{f} ⊆ H ∪ {(0, …, 0)} denote the set of tuples of the basis monomials appearing with nonzero coefficients in the the expression for f.

Theorem B. The kG-submodule of k^{P} generated by f is the smallest submodule having all the L(s_{0}, …, s_{t−1}) for (s_{0}, …, s_{t−1}) ∈ H_{f} as composition factors.

In conjunction with Theorem A, this result enables one to read off the composition factors of the submodule generated by f. The dimensions, or, more precisely, the characters of the composition factors are given in §2.

2. The permutation module on vectors.

Let V (q) = 𝔽_{q}^{n+1} be the standard module for G. We may identify the permutation kGmodule on the set V (q) with the space of functions k[V (q)] = k[X_{0}, …, X_{n}]/(X_{i}^{q} − X_{i})_{i=0}^{n}. This kG-module deomposes as a direct sum of isotypic components with respect to the center Z(G) ≅ 𝔽_{q}^{×}. Thus,

k[V (q)] = ⨁_{[d]∈ℤ/(q−1)ℤ} A[d],  (1)

where [d] is the character t ↦ t^{d} of 𝔽_{q}^{×}. Explicitly, A(d) is the span of the images of monomials of degree congruent to d mod q − 1. Now the module A[0] is essentially the same as k^{P} (See 1.1 below), so its structure is given by Theorem A. The methods used to prove that theorem can be extended to give the structures of the summands A[d] for [d] ≠ [0]. Fix d with 0 < d < q − 1 and let its p-adic expression be

d = d_{0} + d_{1}p + ⋯ + d_{t−1}p^{t−1},  (0 ≤ d_{j} ≤ p − 1).  (2)

We have the following variation of Theorem A.

Theorem C. Let H[d] denote the set of t-tuples (r_{0}, …, r_{t−1}) of integers satisfying (for j = 0, …, t − 1)

(1) 0 ≤ r_{j} ≤ n;

(2) 0 ≤ d_{j} + pr_{j+1} − r_{j} ≤ (p − 1)(n + 1). (Subcripts mod t.) Let H[d] be partially ordered in the natural way: (r′_{0}, …, r′_{t−1}) ≤ (r_{0}, …, r_{t−1}) if and only if r′_{j} ≤ r_{j} for all j.

Then

(a) The module A[d] is multiplicity free and has composition factors L[d](r_{0}, …, r_{t−1}) parametrized by the set H[d].

(b) For (r_{0}, …, r_{t−1}) ∈ H[d], let λ_{j} = d_{j} + pr_{j+1} − r_{j}. Then the simple kG-module L[d](r_{0}, …, r_{t−1}) is isomorphic to the twisted tensor product

⨂_{j=0}^{t−1}(\overline{S}^{λ_{j}})^{(p^{j})}.

(c) For each submodule M of A[d], let H[d]_{M} ⊆ H[d] be the set of its composition factors. Then H[d]_{M} is an ideal of the partially ordered set (H[d], ≤).

(d) The mapping M ↦ H[d]_{M} defines a lattice isomorphism between the submodule lattice of A[d] and the lattice of ideals, ordered by inclusion, of the partially ordered set (H[d], ≤)

Remarks. 1. Let us compare Theorem C with Theorem A. By Theorem C, the module A[d] has a simple socle and a simple head, since the set H[d] has unique minimal and maximal elements (0, …, 0) and (n, …, n) respectively. The module A[0] on the other hand, is decomposable and it is the nontrivial summand Y_{P} which has simple head and socle. The submodule lattice of Y_{P} is given in terms of the set H of which the unique maximal element is (n, …, n), as for H[d], but the unique minimal element is (1, …, 1), not (0, …, 0). Thus, in a certain sense, Theorem C is the generic case and Theorem A the degenerate case. The proof of Theorem C requires only slight modifications to the arguments used to prove Theorem A. The details are given in §9.

2. Let G_{α} be a maximal parabolic subgroup stabilizing a one-dimensional subspace in V (q). Then the module A[d] can be viewed as the kG-module induced from the one-dimensional kG_{α}-module whose restriction to Z(G) affords [d]. The simplicity of the heads and socles of Y_{P} and A[d] can also be proved as a special case of more general results about such induced modules for finite groups of Lie type, due to Curtis [Cu]. 3.(Structure of symmetric powers.) In §10, we will describe in detail the following application of Theorem C in the study of polynomial representations of the algebraic group GL(n+1, k) Let S^{d} denote the space of homogeneous polynomials of degree d in the variables X_{0}, …,X_{n}. This space admits a natural action of GL(n + 1, k). Since d < q − 1, we have an embedding of kG-modules S^{d} ↪ A[d]. The image has a very simple description in terms of H[d] so Theorem C gives the submodule structure of S^{d} for the finite groups GL(n + 1, q). It is then shown that the submodule lattice is the same with respect to all the finite groups GL(n+1, p^{N}) with p^{N} −1 > d, hence equal to the submodule lattice for the algebraic group GL(n + 1, k). Thus, Theorem C generalizes the work of Doty [Do] and Krop [Kr] in which the latter structure was first described.

We thank Burt Totaro for sharing with the second author his ideas about the structure of modules induced from parabolic subgroups in finite groups of Lie type. Observations and questions from him over the last few years have been very helpful.

§1. Preliminaries

1.1. The following notation will be used throughout. Let q = p^{t} be a prime power, k an algebraic closure of 𝔽_{q}, V (q) = 𝔽_{q}^{n+1}, V = k^{n+1}, G = GL(n + 1, q) P = ℙ^{n}(𝔽_{q}). Our main object of study is the space k^{P} of functions from P to k. Let X_{0}, …,X_{n} be coordinates of V. Restriction of functions defines a surjective ring homomorphism

k[X_{0}, …, X_{n}] → k[V (q)]  (1)

from the polynomial ring onto the ring of functions on V (q), with kernel generated by the elements X_{i}^{q} − X_{i}, i = 0, …,n.

Further restriction to V (q) \ {0} gives a surjective ring homomorphism

k[V (q)] → k[V (q) \ {0}]  (2)

with kernel spanned by the characteristic function δ_{0} of {0}, which is the image of ∏_{i=0}^{n}(1− X_{i}^{q−1}). Now the inclusion map kδ_{0} ⊂ k[V (q)] has a G-splitting given by the evaluation map f ↦ f(0), so it follows that the map (2) is G-split and we have an isomorphism of kG-modules

k[X_{0}, …, X_{n}]/(X_{i}^{q} − X_{i})_{i=0}^{n} ≅ kδ_{0} ⊕ k[V (q) \ {0}].  (3)

The functions on V (q) \ {0} which descend to P are precisely those which are invariant under scalar multiplication by 𝔽_{q}^{×}, so we have:

k^{P} ≅ (k[X_{0}, …, X_{n}]/(X_{i}^{q} − X_{i}, 0 ≤ i ≤ n, ∏_{i=0}^{n}(1 − X_{i}^{q−1})))^{𝔽_{q}^{×}} .  (4)

Moreover, since 𝔽_{q}^{×} acts semisimply on k[X_{0}, …, X_{n}], the subring k[X_{0}, …, X_{n}]^{𝔽_{q}^{×}}, consisting of polynomials with total degree divisible by q − 1, maps onto k^{P}.

Next, consider the inclusion map k1 → k^{P} of the constant functions. This map has a G-splitting given by f ↦ ∑_{p∈P} f(p), since |P | ≡ 1 (mod p). Thus, we have

k^{P} = k1 ⊕ Y_{P} ,  (5)

where Y_{P} is the kernel of the latter map.

1.2. We now consider bases for the various modules. It is clear that the images under (1) of the monomials ∏_{i=0}^{n} X_{i}^{b_{i}} with 0 ≤ b_{i} ≤ q − 1 form a basis of k[V (q)], and that among these, those with ∑_{i} b_{i} ≡ 0 (mod q − 1) form a basis of k[V (q)]^{𝔽_{q}^{×}}. Then, because of the extra relation given by δ_{0}, we see that the images in k^{P} of the latter set of monomials will form a basis of k^{P} if we exclude the monomial ∏_{i=0}^{n} X_{i}^{q−1}. Let x_{i} be the image of X_{i} in k^{P}. By the monomial basis of k^{P} we shall mean the set

{∏_{i=0}^{n} x_{i}^{b_{i}} | 0 ≤ b_{i} ≤ q − 1, ∑_{i} b_{i} ≡ 0 (mod q − 1), (b_{0}, …, b_{n}) ≠ (q − 1, …, q − 1)} (6)

and refer to its elements as basis monomials.

In all cases, we will refer to the number ∑_{i} b_{i} as the degree of the basis monomial .

Lemma. The nonconstant basis monomials form a basis of Y_{p}.

Proof. Consider the basis monomial x = ∏_{i=0}^{n} x_{i}^{b_{i}} ≠ 1 Suppose first that all of the b_{i} are either 0 or q − 1. Then x takes the value 0 on the union of a nonempty subset of the coordinate hyperplanes of P and the value 1 on its complement. The complement has cardinality divisible by q, since x does not involve all n + 1 variables, so x ∈ Y_{P}. Now suppose some b_{i}, say b_{0} is neither 0 nor q − 1. Then

∑_{a∈P} x(a) = ∑_{μ∈𝔽_{q}^{×}} μ^{b_{0}} ∑_{a∈P} ∏_{i=1}^{n} x_{i}(a)^{b_{i}} = 0,

since the first factor on the right is 0. Therefore, x ∈ Y_{P}.

We shall call this the monomial basis of Y_{P}.

§2. Composition factors, formal characters and truncated polynomial rings

2.1. We recall that a kG-module is multiplicity-free if no two of its composition factors are isomorphic. The following result is well known and follows immediately from the fact that G has a cyclic subgroup which acts transitively on V (q) \ {0}.

Lemma. k[V (q) \ {0}] is a multiplicity-free kG-module.

It follows that k^{P} is multiplicity free. With regard to module structure, the most important consequence of this result is that the set of submodules of a multiplicity-free module is finite. Moreover, a submodule is completely determined by the isomorphism type of its head (maximal semisimple quotient).

2.2. In this section, we compute the composition factors of the modules k[V (q)] and k^{P}. This is done by comparing these modules with modules for the algebraic group GL(n+1, k), for which composition factors are determined by the action of a maximal torus.

Let A = k[V (q)] = k[X_{0}, …, X_{n}]/(X_{i}^{q} − X_{i})_{i=0}^{n} and A_{0} = k[X_{0}, …, X_{n}]/(X_{i}^{q})_{i=0}^{n}.

The natural filtration F_{r} = {f ∈ k[X_{0}, …, X_{n}], deg(f) ≤ r}, (r ≥ 0) is a filtration by GL(n + 1, k)-modules

This filtration induces filtrations on A_{0} and A, the first being one of GL(n+1, k)-modules, the second one of kG-modules. We consider the associated graded modules gr A_{0} and gr A. We use the same notation [f], to denote the symbol in the corresponding graded ring of an element f of either A_{0} or A.

Lemma. The graded modules gr A_{0} and gr A are isomorphic kG-modules.

Proof. Let y_{i} and z_{i} denote the images of X_{i} in A_{0} and A respectively. Then the symbols [∏_{i} y_{i}^{b_{i}}] and [∏_{i} z_{i}^{b_{i}}] with 0 ≤ b_{i} ≤ q − 1 and ∑_{i} b_{i} = r form bases for the degree r components of the respective graded rings, and the action of G on these rings is induced from the action on k[X_{0}, …, X_{n}]. Let m=∏_{i} X_{i}^{b_{i}} ∈ k[X_{0}, …, X_{n}] be of degree r with each b_{i} ≤ q − 1. For g ∈ G we may write

gm = ∑_{β} c_{β}X^{β} + f,

where the multi-indices β = (β_{0}, …, β_{n}) satisfy β_{i} ≤ q−1 and f is the sum of the remaining terms in gm. Since gm is homogeneous of degree r, each monomial in f contains at least one variable with exponent at least q. Thus, both in A_{0} and in A, the monomials in f have images in the images of F_{r−1}. Let m_{0} and m_{1} be the images of m in A_{0} and A. It follows that the map [∏_{i} y_{i}^{b_{i}}] ↦ [∏_{i} z_{i}^{b_{i}}] is a kG-isomorphism of the graded modules.

The lemma implies that A_{0} and A have the same kG-composition factors. The point is that the kG-module structure on A_{0} is the restriction of a GL(n + 1, k)-module structure. The composition factors are therefore determined by the weights with respect to the diagonal subgroup T (a maximal torus) of GL(n + 1, k).

2.3. Let \overline{S} = ⨁_{λ=0}^{(n+1)(p−1)} \overline{S}^{λ} denote the truncated polynomial ring S(V^{∗})/(V^{∗(p)}) ≅

k[X_{0}, …, X_{n}]/(X_{i}^{p})_{i=0}^{n} and let \overline{S}^{(p^{j})} denote the same ring but with the variables X_{i} replaced by their p^{j}-th powers. These graded algebras all have the strucure of modules for

GL(n + 1, k), with \overline{S}^{(p^{j})} being isomorphic to the j-th Frobenius twist of \overline{S}. Further, it is well known and easy to check by direct computation that the homogeneous components \overline{S}^{λ} are simple GL(n + 1, k)-modules which remain irreducible for SL(n + 1, p). To be precise, let ω_{i}, 1 ≤ i ≤ n denote the fundamental weights for SL(n + 1, k), chosen so that ω_{i} is the highest weight of the ith exterior power of the natural module V. Write λ = (p − 1)r + b, with 0 ≤ r ≤ n + 1, 0 ≤ b < p − 1. Then the highest weight of \overline{S}^{λ} is

\{ \begin{matrix}bω_{n}  if r = 0 \\ (p − 1 − b)ω_{n+1−r} + bω_{n−r}  if 1 ≤ r ≤ n − 1 \\ (p − 1)ω_{1}  if r = n \\ 0  if r = n + 1 \end{matrix}  (1)

In particular, the coefficients of the fundamental weights in the above expressions for the highest weights are at most p − 1, so the \overline{S}^{λ} are restricted simple modules for SL(n + 1, k)

Therefore, by Steinberg’s Tensor Product Theroem [St, Theorems 1.1, 1.3], the modules

S(λ_{0}, …, λ_{t−1}) = ⨂_{j=0}^{t−1}(\overline{S}^{λ_{j}})^{(p^{j})}  (2)

are simple SL(n + 1, k)-modules which remain simple for SL(n + 1, p^{t}). So they are also simple as GL(n + 1, k)-modules and as kG-modules.

Note that \overline{S}^{(p^{j})} has a basis consisting of the images of the monomials ∏_{i} X_{i}^{a_{i}p^{j}} with a_{i} ≤ p − 1. We use the same notation for the images. It is natural to refer to p^{j} ∑_{i} a_{ij} as the degree of such a basis element, with p^{j}a_{j} being the degree in the variable X_{i}. This leads

to a gradation of the module ⊗_{j=0}^{t−1}\overline{S}^{p^{j}}.

Lemma. The map ⨂_{j=0}^{t−1} \overline{S}^{p^{j}} → gr A_{0} given by

∏_{i} X_{i}^{a_{i0}} ⊗ ∏_{i} X_{i}^{a_{i1}p} ⊗ ⋯ ⊗ ∏_{i} X_{i}^{a_{i(t−1)}p^{t−1}}⊗ ↦ [∏_{i} y_{i}^{∑_{j} a_{ij}p^{j}}]

is an isomorphism of (graded) T -modules.

Proof. This is clear, since a basis is mapped to a basis, with corresponding elements having the same degree in each variable.

The lemma gives a T -isomorphism between two GL(n + 1, k) modules, which means that these modules have the same GL(n + 1, k) composition factors. Thus, we have found the GL(n + 1, k) composition factors of A_{0}, hence also the G-composition factors of A, by Lemma 2.2. Since 𝔽_{q}^{×} acts semisimply, the composition factors of A^{𝔽_{q}^{×}} are precisely those S(λ_{0}, …, λ_{t−1}), for which q − 1 divides the total degree λ_{0} + λ_{1}p + ⋯ + λ_{t−1}p^{t−1}. The composition factors for k^{P} and Y_{P} are then obtained by removing the trivial module, twice for Y_{P}, by 1.1, (3) and (5). We summarize this discussion.

Theorem.

(a) The composition factors of A_{0} and A are the modules S(λ_{0}, …, λ_{t−1}) for 0 ≤ λ_{j} ≤ (n + 1)(p − 1).

(b) The composition factors of k^{P} (resp. Y_{P}) are the modules S(λ_{0}, …, λ_{t−1}) for 0 ≤ λ_{j} ≤ (n + 1)(p − 1) satisfying the conditions:

(1) ∑_{j=0}^{t−1} λ_{j}p^{j} ≡ 0 (mod q − 1);

(2) (λ_{0}, …, λ_{t−1}) ≠ ((n + 1)(p − 1), …, (n + 1)(p − 1)) (resp. nor (0, …, 0)).

2.4. We shall derive a formula for the characters of the modules \overline{S}^{λ} in terms of ordinary symmetric powers and exterior powers.

Lemma. Let 0 ≤ λ ≤ (n + 1)(p − 1). Then in the Grothendieck group of GL(n + 1, k)- modules we have

\overline{S}^{λ} = ∑_{i=0}^{⌊ \frac{λ}{p} ⌋}(−1)^{i} ∧^{i} (V^{∗(p)}) ⊗ S^{λ−ip}(V^{∗}).

(Here, ⌊ . ⌋ signifies the integer part.)

Proof. The sequence X_{0}^{p}, …, X_{n}^{p} is a regular sequence ([M, p127) in the polynomial ring S(V^{∗}). Hence the Koszul complex (See [CE, VIII.4.3], [M,p127]) K_{∗}(X_{0}^{p}, …, X_{n}^{p}) gives us a resolution

⋯ → K_{1}(X_{0}^{p}, …, X_{n}^{p}) → K_{0}(X_{0}^{p}, …, X_{n}^{p}) → \overline{S} → 0.  (1)

We may identify K_{i}(X_{0}^{p}, …, X_{n}^{p}) with ∧^{i}(V^{∗(p)}) ⊗_{k} S(V^{∗}) so that (1) becomes an exact sequence of GL(n + 1, k)-modules. The scalar matrices act on homogeneous elements according to their total degrees, so for each λ, the components K_{i}^{λ} of degree λ in each term of the complex form a resolution of \overline{S}^{λ}. We have

K_{i}^{λ} = ∧^{i}(V^{∗(p)}) ⊗ S^{λ−ip}(V^{∗}).

Therefore, the character of \overline{S}^{λ} is the same as the Euler (alternating sum) character of K_{∗}^{λ}. Clearly, the resolution K_{∗}^{λ} runs out when i exceeds ⌊\frac{λ}{p}⌋, so the lemma is proved.

Corollary.

dim S(λ_{0}, …, λ_{t−1}) = ∏_{j=0}^{t−1}[ ∑_{i=0}^{⌊ \frac{λ_{j}}{p} ⌋}(−1)^{i}\binom{n + 1}{i}\binom{n + λ_{j} − ip}{n}].

2.5. We record here some statements about duality in \overline{S}.

Lemma.

(a) dim \overline{S}^{(n+1)(p−1)} = 1 and multiplication defines nonzero GL(n + 1, k)-bilinear maps

\overline{S}^{λ} × \overline{S}^{(n+1)(p−1)−λ} → \overline{S}^{(n+1)(p−1)}.

(b) \overline{S}^{λ} and \overline{S}^{(n+1)(p−1)−λ} are mutually dual as SL(n + 1, k)-modules. In particular they have the same dimension.

(c) S(λ_{0}, …, λ_{t−1}) and S((n + 1)(p − 1) − λ_{0}, …, (n + 1)(p − 1) − λ_{t−1}) are dual as kG-modules.

Proof. (a) is immediate and (b) follows from the simplicity of the modules \overline{S}^{λ}. Part (c) follows from (a), the Tensor Product theorem and the fact that G acts trivially on S((n + 1)(p − 1), …, (n + 1)(p − 1)).

§3. Twisted degrees, filtrations and Hamada’s parametrization

The purpose of this section is to give a second parametrization of the composition factors of k^{P}, by a set H, with a natural partial order which will be seen later to describe the submodule structure.

3.1. Types of monomials. Let X = ∏_{i=0}^{n} X_{i}^{b_{i}} ∈ k[X_{0}, …, X_{n}], with 0 ≤ b_{i} ≤ q − 1. Write b_{i} = ∑_{j=0}^{t−1} a_{ij}p^{j}, 0 ≤ a_{ij} ≤ p − 1 and set λ_{j} = ∑_{i=0}^{n} a_{ij}, so that

X = ∏_{j=1}^{t−1}(∏_{i=0}^{n} X_{i}^{a_{ij}})^{p^{j}}  (1)

We say that X is of type (λ_{0}, …, λ_{t−1}). The basis monomials in A_{0} and A and of their graded modules are each the image of exactly one of the above monomials, so we may define their types in the same way. For k^{P}, in order to make the preimage of a basis monomial unique, we exclude the monomial X_{0}^{q−1} ⋯ X_{n}^{q−1}, i.e the type of 1 is (0, …, 0).

The types of basis monomials of k^{P} (resp. Y_{P}) consist of all those (λ_{0}, …, λ_{t−1}) with 0 ≤ λ_{j} ≤ (n + 1)(p − 1) which satisfy the conditions

(1) ∑_{j=0}^{t−1} λ_{j}p^{j} ≡ 0 (mod q − 1);

(2) (λ_{0}, …, λ_{t−1}) ≠ ((n + 1)(p − 1), …, (n + 1)(p − 1)) (resp. nor (0, …, 0).)

3.2. Twisted degrees and the set H. The filtration {F_{s}} of k[X_{0}, …, X_{n}] by degree induces a filtration {F_{r}} on k^{P} given by

F_{r} = span of basis monomials x = ∏_{i} x_{i}^{b_{i}} with ∑_{i} b_{i} = s(q − 1) for some s ≤ r. (1)

This filtration is not stable under the action of the Galois group Gal(𝔽_{q}/𝔽_{p}) so we can define conjugate filtrations {F_{r}^{(e)}}, for 0 ≤ e ≤ t − 1 by setting

F_{r}^{(e)} = σ^{e}(F_{r}),  (2)

where σ is the generator of Gal(𝔽_{q}/𝔽_{p}) whose action on A sends f to f^{p}. To see what this means in terms of basis monomials let

x = ∏_{j=1}^{t−1}(∏_{i=0}^{n} x_{i}^{a_{ij}})^{p^{j}}  (3)

be a basis monomial of k^{P} and let its type be (λ_{0}, …, λ_{t−1}). Then

σ^{−e}(x) = ∏_{j=1}^{t−1}(∏_{i=0}^{n} x_{i}^{a_{ij}})^{p^{[j−e]}},

where “[ ]” indicates that the exponents of p are taken mod t. Thus, σ^{−e}(x) is of type (λ_{e}, …, λ_{[t−1+e]}).

We define the twisted degrees of basis monomials of k^{P} by

deg^{e}(x) = deg(σ^{−e}(x)) = ∑_{j=0}^{t−1} λ_{j}p^{[j−e]}  (4)

for e = 0, …,t−1. Then deg^{e}(x) ≡ 0 mod q − 1 and F_{r}^{(e)} is spanned by all basis monomials x with \frac{1}{q−1}deg^{e}(x) ≤ r. It is clear that the twisted degrees of a monomial depend only on its type. Thus, to each type (λ_{0}, …, λ_{t−1}) of a basis monomial of k^{P}, we may assign the t-tuple (s_{0}, …, s_{t−1}), where

(q − 1)s_{e} = ∑_{j=0}^{t−1} λ_{j}p^{[j−e]}.  (5)

We define H to be the set of t-tuples of integers (s_{0}, …, s_{t−1}) such that the following conditions hold:

(1) 1 ≤ s_{j} ≤ n;

(2) 0 ≤ ps_{j+1} − s_{j} ≤ (n + 1)(p − 1). (Set s_{t} = s_{0}.)

The following follows easily from the definition of H and the description of the types of basis monomials of k^{P} at the end of 3.1.

Lemma. Formula (5) defines a bijection from the set of types of basis monomials of Y_{p} (resp. k^{P}) and the set H (resp. H ∪ {(0, …, 0)}). The inverse map is given by the formula λ_{j} = ps_{j+1} − s_{j}, j = 0, … t − 1.

Thus, we may also speak of the H-tuple of a basis monomial of Y_{P}.

3.3. By Lemma 3.2, the set H∪{(0, …, 0)} also parametrizes the set of composition factors of k^{P}. We set

L(s_{0}, …, s_{t−1}) = S(λ_{0}, …, λ_{t−1}),  (1)

for (s_{0}, …, s_{t−1}) corresponding to (λ_{0}, …, λ_{t−1}).

There is a natural partial order on H given by

(s′_{0}, …, s′_{t−1}) ≤ (s_{0}, …, s_{t−1}) if and only ifs′_{j} ≤ s_{j} for all j.  (2)

We now consider the submodules

Y_{s_{0},... ,s_{t−1}} = Y_{P} ∩ F_{s_{0}}^{(0)} ∩ ⋯ ∩ F_{s_{t−1}}^{(t−1)}  (3)

of Y_{P}. Then Y_{s′_{0},... ,s′_{t−1}} ⊆ Y_{s_{0},... ,s_{t−1}} if and only if (s′_{0}, …, s′_{t−1}) ≤ (s_{0}, …, s_{t−1}).

Lemma.

(a)

Y_{s_{0},... ,s_{t−1}}/ ∑_{(s′_{0},... ,s′_{t−1})⪇(s_{0},... ,s_{t−1})} Y_{s′_{0},... ,s′_{t−1}} ≅ L(s_{0}, …, s_{t−1}).

(b) The composition factors of Y_{s_{0},... ,s_{t−1}} are the simple modules L(s′_{0}, …, s′_{t−1}) for (s′_{0}, …, s′_{t−1}) ≤ (s_{0}, …, s_{t−1}).

Proof. Let \overline{Y} denote the quotient in (a). This module has a basis consisting of the images of all monomials in A of type (λ_{0}, …, λ_{t−1}) corresponding to (s_{0}, …, s_{t−1}). Let M be the kG-submodule of A generated by these monomials. The M ⊆ A^{𝔽_{q}^{×}} and we have kG-maps

\overline{Y} ← M → gr^{(q−1)s_{0}} A ≅ gr^{(q−1)s_{0}} A_{0}.  (1)

The monomials of A of type (λ_{0}, …, λ_{t−1}) map to the monomials of the same type in gr A_{0}. The latter are weight vectors in gr A_{0} under the action of the maximal torus T, and map to a basis of the simple GL(n + 1, k)-module S(λ_{0}, …, λ_{t−1}) under the T -isomorphism of Lemma 2.3. Therefore, the GL(n + 1, k-submodule of gr A_{0} generated by monomials of type (λ_{0}, …, λ_{t−1}) has a simple quotient L isomorphic to S(λ_{0}, …, λ_{t−1}) such that the images of the monomials in question form a basis. It is now clear that the map M → L induces the isomorphism in (a). Part(b) now follows easily.

§4. Combinatorics of H

4.1. For (s_{0}, …, s_{t−1}) ∈ H we set λ_{j} = ps_{j+1} − s_{j} as in Lemma 3.2.

Lemma. Let (s_{0}, …, s_{t−1}) ∈ H. Then (s_{0}, …, s_{j} − 1, …, s_{t−1}) ∈ H if and only if (i) λ_{j} < (n + 1)(p − 1) and (ii) λ_{j−1} ≥ p.

Moreover if (s_{0}, … s_{t−1}) ≠ (1, …, 1), there exists some j with (s_{0}, …, s_{j}−1, …, s_{t−1}) ∈ H.

Proof. The necessary and sufficient conditions are immediate from the definition of H. To prove the last statement, suppose s_{e} > 1. Then ∑_{j} λ_{j}p^{[j−e]} = (q−1)s_{e} > q−1, which forces λ_{j_{1}} ≥ p for some j_{1}. Also, since s_{j} ≤ n for all j, there is some j_{2} with λ_{j_{2}} < (n + 1)(p − 1). So either there exists j with λ_{j−1} = (n + 1)(p − 1) ≥ p and λ_{j} < (n + 1)(p − 1), in which case, this j satisfies (i) and (ii), or else for all j we have λ_{j} < (n + 1)(p − 1), in which case j_{1} + 1 satisfies (i) and (ii).

4.2.

Lemma. Let (s_{0}, …, s_{t−1}), (s′_{0}, …, s′_{t−1}) ∈ H with (s′_{0}, …, s′_{t−1}) ⪇ (s_{0}, …, s_{t−1}). Then for some j we have

(s′_{0}, …, s′_{t−1}) ≤ (s_{0}, …, s_{j} − 1, …, s_{t−1}).

Proof. Let s_{j} = s′_{j} + a_{j}. We must show that the conditions of Lemma 4.1 hold for some j with a_{j} ≠ 0. Let I = {j | a_{j} ≠ 0}. By assumption I ≠ ∅. Suppose for a contradiction that for every j ∈ I, either (i) or (ii) in Lemma 4.1 fails. Assume first that there exists j ∈ I such that λ_{j} = (n + 1)(p − 1). Then

\begin{aligned}(n + 1)(p − 1) = λ_{j} & = ps_{j+1} − s_{j} \\ & = ps′_{j+1} + pa_{j+1} − s′_{j} − a_{j} \\ & = λ′_{j} + pa_{j+1} − a_{j}. \end{aligned}  (1)

Since λ′_{j} ≤ (n + 1)(p − 1) and a_{j} ≠ 0, it follows that a_{j+1} ≠ 0, so j + 1 ∈ I. Hence either (i) or (ii) fails for j + 1. But λ_{(j+1)−1} = λ_{j} = (n + 1)(p − 1) ≥ p so it must be (i) which fails for j + 1, namely λ_{j+1} = (n + 1)(p − 1). Repeating the argument, we eventually deduce that I = {0, …, t − 1}, contradicting the last sentence of Lemma 4.1. We may therefore assume for every j ∈ I, that λ_{j} < (n + 1)(p − 1) and hence that condition (ii) fails for all such j, which is to say that λ_{j} < p for every j ∈ I. Then

0 ≤ λ′_{j−1} = λ_{j−1} − pa_{j} + a_{j−1}.  (2)

Since a_{j} ≠ 0, we must have a_{j−1} ≠ 0, whence j − 1 ∈ I. This leads us eventually to the same contradiction as before, that I = {0, …, t − 1}.

By induction, we obtain the following.

Corollary. Under the hypotheses of the lemma, there is an descending chain from (s_{0}, … s_{t−1}) to (s′_{0}, … s′_{t−1}) in which successive terms are obtained by subtracting 1 from a suitable entry of the t-tuple.

§5. Submodules generated by monomials

The main aim of this section is to show that a basis monomial with H-tuple (s_{0}, … s_{t−1}) generates Y_{s_{0},... ,s_{t−1}} as a kG-module.

5.1. For 0 ≤ r ≠ s ≤ n let g_{rs} denote the linear substitution in k[X_{0}, …, X_{n}] given by

g_{rs} : X_{r} ↦ X_{s},  X_{l} ↦ X_{l}  (l ≠ r)  (1)

To simplify notation, we take r = 0 and s = 1. Let

X = ∏_{i=0}^{n} X_{i}^{b_{i}} = ∏_{j=1}^{t−1}(∏_{i=0}^{n} X_{i}^{a_{ij}})^{p^{j}}  (2)

be a monomial with b_{i} ≤ q − 1. We consider ther following two expressions for g_{01}X.

g_{01}X = (∑_{u=0}^{b_{0}} \binom{b_{0}}{u}X_{0}^{(b_{0}−u)}X_{1}^{b_{1}+u}) ∏_{i=2}^{n} X_{i}^{b_{i}}  (3)

g_{01}X = ∏_{j=0}^{t−1} (( ∑_{u_{j}=0}^{a_{0j}} \binom{a_{0j}}{u_{j}}X_{0}^{a_{0j}−u_{j}}X_{1}^{a_{1j}+u_{j}}) ∏_{i=2}^{n} X_{i}^{a_{ij}} )^{p^{j}}  (4)

From (4) we see that there are ∏_{j=0}^{t−1}(a_{0j} + 1) distinct monomials appearing with nonzero coefficients in g_{01}X. From (3) we see that all monomials in g_{01}X afford distinct characters of T (q) (the subgroup of diagonal matrices in G), except when b_{0} = q − 1, in which case the monomials X and X_{1}^{b_{1}+q−1}X_{2}^{b_{2}} ⋯ X_{n}^{b_{n}} are the only two which afford the same character, for this is the only case where exponents of X_{0} in the monomials in (3) are congruent mod q − 1.

Now suppose

x = ∏_{i=0}^{n} x_{i}^{b_{i}} = ∏_{j=1}^{t−1}(∏_{i=0}^{n} x_{i}^{a_{ij}})^{p^{j}}  (5)

is a basis monomial of Y_{P}. From (3), we obtain

g_{01}x = (∑_{u=0}^{b_{0}} \binom{b_{0}}{u}x_{0}^{(b_{0}−u)}x_{1}^{\{b_{1}+u\}}) ∏_{i=2}^{n} x_{i}^{b_{i}},  (6)

where “{c}” indicates that q − 1 should be subtracted if c ≥ q.

Since the map k[X_{0}, …, X_{n}]^{𝔽_{q}^{×}} → k^{P} is a kG-map, the only basis monomials in (6) affording the same character of T (q) will be x and x′ = x_{1}^{b_{1}+q−1}x_{2}^{b_{2}} ⋯ x_{n}^{b_{n}}. Since any kGmodule is the direct sum of its T (q)-isotypic components, it follows that the submodule kGx generated by x contains all of the basis monomials which appear in (6) with nonzero coefficients (including x′, since x + x′ is a T (q)-isotypic component of g_{01}x, so both x and x + x′ belong to kGx). We have proved the following result.

Lemma. Let x be a basis monomial of Y_{p}. Then when g_{rs}x is expressed as a linear combination of basis monomials, each basis monomial which appears with a nonzero coefficient lies in the kG-submodule generated by x. If x is given by (5) (with 0 and 1 replaced by r and s), then these monomials are precisely the images of the ∏_{j=0}^{t−1}(a_{rj} + 1) monomials which occur in (4).

5.2.

Lemma. If x is a basis monomial of Y_{P} of type (λ_{0}, …, λ_{t−1}), then the submodule kGx contains every basis monomial of type (λ_{0}, …, λ_{t−1})

Proof. Write

x = ∏_{j=1}^{t−1}(∏_{i=0}^{n} x_{i}^{a_{ij}})^{p^{j}} = (∏_{i=0}^{n} x_{i}^{a_{i0}})x^{∗}  (1)

Suppose a_{r0} ≠ 0 and a_{s0} < p − 1. Then g_{rs}x will involve the basis monomial

x_{r}^{a_{r0}−1}x_{s}^{a_{s0}+1}( ∏_{i≠r,s} x_{i}^{a_{i0}})x^{∗}  (2)

with coefficient a_{r0} ≠ 0, so by Lemma 5.2, this monomial belongs to the submodule kGx. It is clear that monomial (2) is also of type (λ_{0}, …, λ_{t−1}). By repeating this procedure for different choices of r and s, we see that kGx contains all basis monomials of the form (∏_{i=0^{n}} x_{i}^{c_{i}})x∗, for any values of c_{i} with 0 ≤ c_{i} ≤ p − 1 and ∑_{i} c_{i} = λ_{0}. Then we can carry out the whole process with a different value of j fixed instead of 0 in (1). It follows that all monomials of type (λ_{0}, …, λ_{t−1}) belong to kGx.

5.3.

Lemma. Suppose (s_{0}, …, s_{t−1}), (s_{0}, …, s_{j} − 1, …, s_{t−1}) ∈ H. Then their exists a basis monomial x with H-tuple (s_{0}, …, s_{t−1}) such that the submodule kGx contains a basis monomial with H-tuple (s_{0}, …, s_{j} − 1 …, s_{t−1})

Proof. Let (λ_{0}, …, λ_{t−1}) correspond to (s_{0}, …, s_{t−1}). Without loss, we may assume j = 0. Since (s_{0} − 1, …, s_{t−1}) ∈ H, we have λ_{t−1} ≥ p and λ_{0} < (n + 1)(p − 1). Therefore there exists a monomial of type (λ_{0}, …, λ_{t−1})

x = ∏_{j=1}^{t−1}(∏_{i=0}^{n} x_{i}^{a_{ij}})^{p^{j}}  (1)

such that a_{0,t−1} = p − 1, a_{1,t−1} ≠ 0 and a_{0,0} < p − 1. Write x = ∏_{j=0}^{t−1} m_{j}^{p^{j}} and let

X = ∏_{j=0}^{t−1} M_{j}^{p^{j}} be the corresponding monomial in k[X_{0}, …, X_{n}]. By 5.1 (4), g_{10}X involves all of the monomials in

∏_{j≠t−1} M_{j}^{p^{j}}(g_{01}M_{t−1})^{p^{t−1}} = ∏_{j≠t−1} M_{j}^{p^{j}}(∏_{i=2}^{n} X_{i}^{a_{ij}} ∑_{l=0}^{a_{1,t−1}} \binom{a_{1,t−1}}{l}X_{0}^{p−1+l}X_{1}^{a_{1,t−1}−l})^{p^{t−1}} (2)

The term for l = 1 is

a_{1,t−1} ∏_{j≠t−1} M_{j}^{p^{j}}(∏_{i=2}^{n} X_{i}^{a_{ij}}X_{0}^{p}X_{1}^{a_{1,t−1}−1})^{p^{t−1}}  (3)

When this is mapped into Y_{P}, the X^{q} from the last factor maps to x_{0}^{q} = x_{0}, so the image of (3) is

a_{1,t−1}(x_{0}^{a_{00}+1} ∏_{i=1}^{n} x_{i}^{a_{i0}}) ∏_{j≠0,t−1} M_{j}^{p^{j}}(∏_{i=2}^{n} X_{i}^{a_{ij}}X_{1}^{a_{1,t−1}−1})^{p^{t−1}}.  (4)

By Lemma 5.2, the monomial in (4) lies in kGx, since a_{1,t−1} ≠ 0. Its type is (λ_{0} + 1, …, λ_{t−1} − p), which corresponds to the H-tuple (s_{0} − 1, …, s_{t−1}), by Lemma 3.2.

Combining Lemmas 5.2 and 5.3 yields the main result of this section.

Theorem. Any basis monomial with H-tuple (s_{0}, …, s_{t−1}) generates Y_{s_{0},... ,s_{t−1}} as a kGmodule.

§6. Non-split module extensions

6.1. Suppose (s_{0}, …, s_{t−1}), (s_{0}, …, s_{j} − 1, …, s_{t−1}) ∈ H. Assume for simplicity that j = 0. Let

E = Y_{s_{0},... ,s_{t−1}}/ ∑_{\begin{matrix}(s′_{0},... ,s′_{t−1})⪇(s_{0},... ,s_{t−1}) \\ (s′_{0},... ,s′_{t−1})≠(s_{0}−1,... ,s_{t−1}) \end{matrix}} Y_{s′_{0},... ,s′_{t−1}} ≅ L(s_{0}, …, s_{t−1}).  (1)

Then we have a short exact sequence

0 ⟶ L(s_{0} − 1, …, s_{t−1}) ⟶ E ⟶ L(s_{0}, …, s_{t−1}) ⟶ 0.  (2)

Theorem. The sequence (2) does not split.

Proof. We shall adopt the following notational conventions. For a basis monomial x of Y_{s_{0},... ,s_{t−1}} we shall denote its image in E by the same symbol and use \overline{x} for its image in the quotient L(s_{0}, …, s_{t−1}). Also, for I ⊆ {0, …, n}, we will write X_{I}^{q−1} for the monomial ∏_{i∈I} X_{i}^{q−1} in k[X_{0}, …, X_{n}], with corresponding notations for monomials in Y_{P} or E or L(s_{0}, …, s_{t−1}).

Let (λ_{0}, …, λ_{t−1}) be the type corresponding to (s_{0}, …, s_{t−1}). For each j write λ_{j} = (p − 1)k_{j} + r_{j}, with 0 ≤ r_{j} < p − 1, set

M_{j} = X_{0}^{p−1} ⋯ X_{k_{j}−1}^{p−1}X_{k_{j}}^{r_{j}}  (3)

and let m_{j} and \overline{m}_{j} denote its images in E and L(s_{0}, …, s_{t−1}) respectively.

Let M = ∏_{j} M_{j}^{p^{j}}, with images m and \overline{m}. We can write in the forms

m = ∏_{i=0}^{n} x_{i}^{b_{i}} = x_{I}^{q−1}x′,  (4)

where I = {0, …, min{k_{j}} − 1} and x′ is the remaining factor.

In order to prove that (4) does not split, it will be sufficient to show that every submodule E’ of E which contains a preimage m̂ of \overline{m} actually contains m, since it follows from Theorem 5.3 that m generates E. Suppose such an E′ and m̂ are given. Since every kG-module is the direct sum of its T (q)-isotypic components, we may assume that km̂ is T (q)-stable (and therefore affords the same character of T (q) as m). Therefore, we know that m̂ has the form

m̂ = x_{I}^{q−1}x′ + ∑_{J} a_{J}x_{J}^{q−1}x′,  (5)

where a_{j} ∈ k and the monomials x_{J}^{q−1}x′ have H-tuples (s_{0} − 1, …, s_{t−1}). In particular, the subsets J have size |I| − 1. Also, there no variable x_{i} with i ∈ J occurs in x′. The main idea now is to define linear endomorphisms of E which preserve E′ and which eliminate the error terms x_{J}^{q−1}x′ from (5).

By definition of H, we have I ≠ ∅, {0, …, n}. Fix r and s with 0 ≤ r, s ≤ n, s ∈ I and r ∉ I Let T_{i}(q) ≅ 𝔽_{q}^{×} denote the subgroup of T (q) with any nonzero entry in the i − th diagonal position and 1 in all other diagonal positions. We denote by the ordered pair (μ, ν) of integers mod q − 1 the character of the subgroup T_{r}(q) × T_{s}(q) of T (q) taking (γ, θ) ∈ T_{r}(q) × T_{s}(q) to γ^{μ}θ^{ν} ∈ k. Let

π_{sr} = projection onto the (q − 2, b_{r} + 1)-component for T_{r}(q) × T_{s}(q),
π_{s} = projection onto the trivial component for T_{s}(q)

(6)

Of course, when q = 2, the maps in (6) are the identity, so we first give the proof under the assumption q > 2. The linear map π_{s}g_{rs}π_{sr}g_{sr} of E maps every kG-submodule into itself; we now consider its effect on the monomials in (5). (See 5.1 for action of g_{rs}) Write x′ = x_{r}^{b_{r}}x′′. Under π_{sr}g_{sr}, the monomial m = x_{I}^{q−1}x′ = x_{I}^{q−1}x_{r}^{b_{r}}x′′ is mapped to

m′ = x_{I\{s}}^{q−1}x_{s}^{q−2}x_{r}^{b_{r}+1}x′′.  (7)

Then π_{s}g_{rs} maps m′ back to m.

Next we consider what happens to the other monomials x_{J}^{q−1}x′.

If s ∈ J, r ∉ J, then the computation is similar to the one for m: x_{J}^{q−1}x′ is mapped under π_{sr}g_{sr} to x_{J\{s}}^{q−1}x_{s}^{q−2}x_{r}^{b_{r}+1}x′′ and π_{s}g_{rs} maps the latter back to x_{J}^{q−1}x′.

If If s ∉ J, then g_{sr} fixes x_{J}^{q−1}x′ and π_{sr} annihilates it.

Finally, if s, r ∈ J, then g_{sr} maps x_{J}^{q−1}x′ to

x_{J}^{q−1}x′ + x_{J\\{s,r\}}^{q−1}x′ ∑_{l=1}^{q−1} \binom{q − 1}{l}x_{s}^{q−1−l}x_{b}^{l} = x_{J}^{q−1}x′  (8)

since the degree of the other terms is q − 1 lower than the degree of x_{J}^{q−1}x′, so they are zero in E. Then π_{s}r annihilates x_{J}^{q−1}x′. Thus, applying π_{s}g_{rs}π_{sr}g_{sr} to (5) yields

π_{s}g_{rs}π_{sr}g_{sr}m̂ = x_{I}^{q−1}x′ + ∑_{J}′a_{J}x_{J}^{q−1}x′,  (9)

where ∑′ indicates that the summation now runs only over those subsets J in (5) for which s ∈ J and r ∉ J. Since the sets J are smaller than I, the application of the operators π_{s}g_{rs}π_{sr}g_{sr} for all possible s ∈ I will kill off all the terms x_{J}^{q−1}x′, leaving us to conclude that m ∈ E′. This completes the proof of the theorem for q > 2. For q = 2, the same proof works if we use (g_{rs} − 1)(g_{sr} − 1) instead of π_{s}g_{rs}π_{sr}g_{sr}.

Corollary. Y_{s_{0},... ,s_{t−1}} is the smallest submodule of Y_{P} which has L(s_{0}, …, s_{t−1}) as a composition factor.

Proof. Let Z be the smallest submodule of Y_{P} which has L(s_{0}, …, s_{t−1}) as a composition factor and let P (s_{0}, …, s_{t−1}) denote the projective kG-module which has L(s_{0}, …, s_{t−1}) as its unique simple quotient (See [F],I.13.6). Since Y_{P} is multiplicity-free, the space Hom_{kG}(P (s_{0}, …, s_{t−1}), Y_{P}) is one-dimensional and any nonzero map in this space has Z as its image (cf. [F,I.13.9]). We proceed by induction on ∑_{j} s_{j}. Consider the maps

\begin{aligned}Y_{s_{0},... ,s_{t−1}} & ⟶^{γ} & E & ⟶^{τ} & L(s_{0}, …, s_{t−1}) \\ ↑ & π \\ & & & & P (s_{0}, …, s_{t−1}) \end{aligned}  (10)

By projectivity, the map π of P (s_{0}, …, s_{t−1}) onto its simple head may be lifted to a map P (s_{0}, …, s_{t−1}) ⟶^{π̃} E, so that τπ̃ = π. Theorem 6.1 shows that E has a simple head, so π̃ must be surjective. Again, projectivity yields a lifting P (s_{0}, …, s_{t−1}) ⟶^{π^{∗}} Y_{s_{0},... ,s_{t−1}}, with γπ^{∗} = π̃. Therefore, the image of π^{∗}, which must be equal to Z, has L(s_{0} − 1, …, s_{t−1}) as a composition factor. The same is true with 0 replaced by any j such that (s_{0}, …, s_{j} −1, …, s_{t−1}) ∈ H. The inductive hypothesis implies that Z contains all the modules Y_{s′_{0},... ,s′_{t−1}} for which (s′_{0}, …, s′_{t−1}) ≤ (s_{0}, …, s_{t−1}) and ∑_{j} s′_{j} = (∑_{j} s_{j}) − 1. By Lemma 4.2, it now follows that Z contains all Y_{s′_{0},... ,s′_{t−1}} with (s′_{0}, …, s′_{t−1}) ⪇ (s_{0}, …, s_{t−1}). Therefore, Z = Y_{s′_{0},... ,s′_{t−1}}, by Lemma 3.3(a).

§7. Proof of Theorems A and B

7.1. Proof of Theorem A. Parts (a) and (b) have been proved in Lemma 2.1. and Theorem 2.3. (See also 3.3.). To prove (c), note that Corollary 6.1 implies that each submodule of Y_{P} is a sum of the submodules Y_{s_{0},... ,s_{t−1}}. Since the H-tuples of the composition factors of Y_{s_{0},... ,s_{t−1}} form the ideal in H of the elements dominated by (s_{0}, …, s_{t−1}) and since taking the sum of submodules corresponds to taking the union of their sets of composition factors, (c) is proved. The mapping in (d) is injective, by (a). Surjectivity is also immediate; any ideal I of H corresponds to the submodule

∑_{(s_{0},... ,s_{t−1})∈I} Y_{s_{0},... ,s_{t−1}}.  (1)

This completes the proof of Theorem A.

7.2. Proof of Theorem B. Write f ∈ k^{P} as

f = ∑_{(s_{0},... ,s_{t−1})∈H_{f}} f_{(s_{0},... ,s_{t−1})},  (1)

where f_{(s_{0},... ,s_{t−1})} is a linear combination of basis monomials of k^{P} with tuple (s_{0}, …, s_{t−1}) ∈ H ∪ {0}.

If (0, …, 0) ∈ H_{f}, then f has a nonzero projection onto the space of constant functions k1. Since k^{P} is multiplicity-free, this implies that k1 ⊆ kGf. So from now on, we may assume that f ∈ Y_{P} and H_{f} ⊆ H. Then

kGf ⊆ ∑_{(s_{0},... ,s_{t−1})∈H_{f}} Y_{s_{0},... ,s_{t−1}}  (2)

and in the sum we may even replace the set H_{f} by its set H_{f}^{∗} of maximal elements. For each (s′_{0}, …, s′_{t−1}) ∈ H_{f}^{∗}, we have a nonzero kG-map

∑_{(s_{0},... ,s_{t−1})∈H_{f}^{∗}} Y_{s_{0},... ,s_{t−1}} → L(s′_{0}, …, s′_{t−1})  (3)

Therefore, kGf has L(s′_{0}, …, s′_{t−1}) as a quotient. By the theorem, we then have Y_{s′_{0},... ,s′_{t−1}} ⊂ kGf, so the inclusion (2) is actually equality. Since the right hand side of (2) is the smallest submodule of Y_{P} which has all L(s_{0}, …, s_{t−1}) with (s_{0}, …, s_{t−1}) ∈ H_{f} as composition factors, Theorem B is proved.

7.3. Socle and radical series.

Recall that the radical of a module is the smallest submodule with semisimple quotient (called the head). Iterating, we obtain the radical series. Dually, the maximal semisimple submodule is called the socle and we have the socle series.

Define G_{r} ⊂ Y_{P} to be the span of the basis monomials x whose H-tuples in H satisfy ∑_{j} s_{j} ≤ r. In other words,

G_{r} = ∑_{\begin{matrix}(s_{0},... ,s_{t−1}) \\ ∑_{j} s_{j}=r \end{matrix}} Y_{s_{0},... ,s_{t−1}}.  (1)

Then, since elements of H which the same value for ∑_{j} s_{j} are incomparable, we have semisimple layers

G_{r}/G_{r−1} ≅ ⨁_{\begin{matrix}(s_{0},... ,s_{t−1}) \\ ∑_{j} s_{j}=r \end{matrix}} L(s_{0}, …, s_{t−1}).  (2)

The following result now follows easily from Theorem A.

Corollary. The filtration G_{r} is equal (after suitable re-indexing) to the socle filtration and to the radical filtration in reverse.

§8. Hamada’s Formula

8.1. Let C_{r} ⊆ k^{P} be the subspace spanned by the characteristic functions of r-dimensional linear subspaces of P. Since G acts transitively on the set of these subspaces, C_{r} is equal to kGχ_{L}, where L is defined by the equations X_{i} = 0, i = r + 1 …, n. Its characteristic function can be written as

χ_{L} = ∏_{i=r+1}^{n}(1 − X_{i}^{q−1}) = ∑_{I⊆{r+1,... ,n}}(−1)^{|I|}X_{I}^{q−1}.  (1)

For I ≠ ∅ the monomial X_{I}^{q} − 1 has H-tuple (|I|, …, |I|), which lies below the H-tuple (n − r, …, n − r) of ∏_{i=r+1}^{n} X_{i}^{q−1}. Therefore, Theorems A and B yield

C_{r} = k1 ⊕ Y_{n−r,... ,n−r}.  (2)

Therefore by Lemma 3.3(b) and Corollary 2.4 we obtain:

dim C_{r} = 1 + ∑_{(s_{0},... ,s_{t−1})} ∏_{j=0}^{t−1}[ ∑_{i=0}^{⌊ \frac{ps_{j+1}−s_{j}}{p} ⌋}(−1)^{i}\binom{n + 1}{i}\binom{n + ps_{j+1} − s_{j} − ip}{n}],  (3)

where the sum extends over (s_{0}, …, s_{t−1}) ∈ H with 1 ≤ s_{j} ≤ n − r. Next, we note that in the definition of H (§3.2), if we change condition (1) to (1’)0 ≤ s_{j} ≤ n, the only extra tuple will be (0, … 0), so we could rewrite (3) by summing over H ∪ {(0, …, 0)} and omitting the term 1 from (3). Then, applying Lemma 2.5 (c) we obtain Hamada’s rank formula [Ha, Theorem 1].

dim C_{r} = ∑_{\begin{matrix}(s_{0},... ,s_{t−1}) \\ r+1≤s_{j}≤n+1 \\ 0≤ps_{j+1}−s_{j}≤(n+1)(p−1) \end{matrix}} ∏_{j=0}^{t−1}[ ∑_{i=0}^{⌊ \frac{ps_{j+1}−s_{j}}{p} ⌋}(−1)^{i}\binom{n + 1}{i}\binom{n + ps_{j+1} − s_{j} − ip}{n}]. (4)

Remark. Note that this formula could have been proved without the full strength of Theorems A and B. For (2) can be deduced from (1) using only Theorem 5.3 and induction on n − r.

§9. Structure of A.

As we mentioned in the introduction, we can extend our results on Y_{P} to obtain the submodule structure of A = k[X_{0}, …, X_{n}]/(X_{i}^{q} − x_{i})_{i=0}^{n}. Under the action of Z(G) ≅ 𝔽_{q}^{×}, we have the decomposition

A = ⨁_{[d]∈ℤ/(q−1)ℤ} A[d]  (1)

into isotypic components, where [d] is the character ϵ ↦ ϵ^{d}. The summand A[0] is isomorphic to k ⊕ k^{P} by 1.1(3), so we may restrict our attention to the other summands. From now on let d be a fixed integer with 0 < d < q − 1. Our aim is to prove Theorem C. The proof consists of generalizing the definitions we have made to study Y_{P} and then checking that the arguments used in proving Theorem A carry over with suitable modifications. We note that A[d] is multiplicity-free, by Lemma 2.1.

9.1. The purpose of this subsection is to set up the notation for Theorem C. The discussion runs parallel to that of §3. Recall that z_{i} denotes the image of X_{i} in A Then A[d] has a basis consisting of the monomials ∏_{i=0}^{n} z_{i}^{b_{i}}, 0 ≤ b_{i} ≤ q − 1 whose degrees are congruent to d mod q−1, which we shall call the monomial basis of A[d]. The types (§3.1) of the basis monomials are all the possible tuples (λ_{0}, …, λ_{t−1}) of integers satisfying 0 ≤ λ_{j} ≤ (n + 1)(p − 1) and ∑_{j} λ_{j}p^{j} ≡ d mod q −1. The degree filtration of k[X_{0}, …, X_{n}] induces the filtration {F_{r}[d]} on A[d] given by

F_{r}[d] = span of basis monomials z = ∏_{i} z_{i}^{b_{i}} with ∑_{i} b_{i} = d+s(q−1) for some s ≤ r. (1)

In its action on A, the generator σ of Gal(𝔽_{q}/𝔽_{p}) maps A[m] to A[pm]. Let d^{(e)} denote the smallest nonnegative integer in the congruence class p^{−e}[d]. Then can define for each e = 0, …, t − 1 a filtration {F_{r}^{(e)}[d]} by

F_{r}^{(e)}[d] = σ^{e}(F_{r}[d^{(e)}])  (2)

We define the twisted degree of a basis monomial z of type (λ_{0}, …, λ_{t−1}) to be

deg^{e}(z) = ∑_{j} λ_{j}p^{[j−e]}.  (3)

Then deg^{e}(z) ≅ d^{(e)} mod q−1 and F_{r}^{(e)}[d] is spanned by basis monomials z with \frac{1}{q−1}(deg^{e}(z)− d^{(e)}) ≤ r. Thus, may associate with each type (λ_{0}, …, λ_{t−1}) the t-tuple (r_{0}, …, r_{t−1}) of nonnegative integers by the formula

∑_{j} λ_{j}p^{[j−e]} = d^{(e)} + (q − 1)r_{e}, (e = 0, …, t − 1).  (4)

Let d = ∑ d_{j}p^{j}, 0 ≤ d_{j} ≤ p − 1 be the p-adic expression for d. The type λ can be recovered from its tuple by the formula

λ_{j} = d_{j} + pr_{j+1} − r_{j}, (r_{t} = r_{0}).  (5)

Define the set

H[d] = {(r_{0}, …, r_{t−1}) | 0 ≤ r_{j} ≤ n, 0 ≤ d_{j} + pr_{j+1} − r_{j} ≤ (n + 1)(p − 1)}.  (6)

Then (cf. Lemma 3.2) formulas (4) and (5) define inverse bijections between the set of types of basis monomials for A[d] and the set H[d]. It follows from Lemma 2.3, Lemma 2.2 and Theorem 2.3 that the set of types parametrizes the set of composition factors of A[d], so we can also parametrize them with H[d], setting

L[d](r_{0}, …, r_{t−1}) = S(λ_{0}, …, λ_{t−1}).  (7)

The set H[d] is partially ordered by the rule (r′_{0}, …, r′_{t−1}) ≤ (r_{0}, …, r_{t−1}) if and only if r′_{j} ≤ r_{j} for each j.

We consider now the submodules

F[d]_{r_{0},... ,r_{t−1}} = F_{r_{0}}^{(0)}[d] ∩ ⋯ ∩ F_{r_{t−1}}^{(t−1)}[d]  (8)

for (r_{0}, …, r_{t−1}) ∈ H[d]. These will play the same role as the modules Y_{s_{0},... ,s_{t−1}} in the discussion of Y_{P}. We have F[d]_{r′_{0},... ,r′_{t−1}} ⊆ F[d]_{r_{0},... ,r_{t−1}} if and only if (r′_{0}, …, r′_{t−1}) ≤ (r_{0}, …, r_{t−1}). Also, the analogue of Lemma 3.3 holds, so we have

F[d]_{r_{0},... ,r_{t−1}}/  ∑_{(r′_{0},... ,r′_{t−1})⪇(r_{0},... ,r_{t−1})} F[d]_{r′_{0},... ,r′_{t−1}} ≅ L[d](r_{0}, …, r_{t−1})  (9)

and the composition factors of F[d]_{r_{0},... ,r_{t−1}} are the simple modules L(r′_{0}, …, r′_{t−1}) for (r′_{0}, …, r′_{t−1}) ≤ (r_{0}, …, r_{t−1}).

9.2 Proof of Theorem C. In order to prove Theorem C it remains to point out the necessary changes to the statements and proofs in §§4-7, used in proving Theorem A.

§4. If we replace H by H[d], the results remain valid. In Lemma 4.1, the minimal element (1, …, 1) must of course be replaced by (0, …, 0). The inequality in the second line becomes ∑_{j} λ_{j}p^{[j−e]} = d^{(e)} +(q −1)r_{e} > q −1, which holds because d^{(e)} > 0. In Lemma 4.2, the relation λ_{j} = ps_{j+1} − s_{j} must be replaced by 9.1(5), but the equality between the first and last members of (1) still holds, as does equation (2) of that lemma.

§5. The very general discussion here makes no use of the congruence class of the degrees of the basis monomials, so all results remain valid with Y_{p}, Y_{s_{0},... ,s_{t−1}} and H replaced by A[d], F[d]_{r_{0},... ,r_{t−1}} and H[d].

§6. Again the arguments do not depend on the congruence class mod q −1, and all that is required is to substitute the relevant notations into both the statements and proofs. In fact since we are assuming 0 < d < q − 1, the proof of the analogue of Theorem 6.1 is simplified by not having to consider the case q = 2.

§7. Parts (a) and (b) of Theorem C have been proved in the discussion leading uo to 9.1 (7). The other parts are proved just as for Theorem A, using the anologue Corollary 6.1.

This completes our outline of the proof of Theorem C.

§10. Structure of Symmetric powers.

Let S^{d} ⊆ k[X_{0}, …, X_{n}] denote the space of homogeneous polynomials of degree d. It has a natural structure as a rational module for the algebraic group GL(n + 1, k). Since d < q − 1 the map S^{d} → A[d] is an embedding of kG-modules with image F_{0}[d]. Thus, S^{d} corresponds to the ideal

H[d]_{S^{d}} = {(r_{0}, …, r_{t−1}) ∈ H[d] | r_{0} = 0}.  (1)

This gives the submodule structure of S^{d} as a module for G = GL(n + 1, p^{t}).

We now examine what happens if we fix d and replace p^{t} by a higher power p^{N}. Let A[d](N), H[d](N), etc. denote the corresponding objects for G(N) = GL(n + 1, p^{N}). Then

H[d](N)_{S^{d}} = {(r_{0}, …, r_{N−1}) ∈ H[d](N) | r_{0} = 0}.

The p-adic expresion for d is unchanged, so that we have d_{j} = 0 for t ≤ j ≤ N − 1. Let (r_{0}, …, r_{N−1}) ∈ H[d](N)_{S^{d}}. Then from the definitions we have 0 ≤ d_{N−1} + pr_{0} − r_{N−1} = −r_{N−1}, which forces r_{N−1} = 0. Repeating this, we obtain r_{j} = 0 for t ≤ j ≤ N − 1. Moreover, the conditions on the entries r_{j} for 0 ≤ j ≤ t − 1 are exactly the conditions for the t-tuple (r_{0}, …, r_{t−1}) to belong to H[d]_{S^{d}}.

We have proved the following theorem.

Theorem. The submodule lattice of S^{d} is the same for all of the groups GL(n + 1, p^{t}) for p^{t} − 1 > d. Consequently, it is also the same for the algebraic group GL(n + 1, k). This lattice is isomorphic to the lattice of ideals in the partially ordered set H[d]_{S^{d}}

The GL(n + 1, k) submodule structure of S^{d} was first given in [Do] and independently in [Kr1,2].

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Department of Mathematics, University of Florida, Gainesville, FL 32611, USA e-mail: sin@math.ufl.edu