THE SUBMODULE STRUCTURE OF SPECHT MODULES

ALAIN REUTER

Abstract. In this paper, we will give a proof of the complete submodule structure of Specht modules corresponding to 2–part partitions for the general linear group GL(n, q) in characteristic p coprime to q (in non–defining characteristic).

The multiplicities in the Specht module S^{(n−l,l)} being at most 1, we introduce a partial order on the set of composition factors. Let e be the lowest integer such that p | 1 + q + … + q^{e−1}. We explicitly construct the j, such that D^{(n−j,j)} is a composition factor of S^{(n−l,l)}, by looking at well–defined sets I of exponents of p whose coefficients in the p–adic expansion of ⌊\frac{l−j}{e} ⌋ are relevant. We thus parametrise μ_{I}(l) = (n − j, j). The family of such sets I then forms a partially ordered set under inclusion. We show that this is isomorphic to the poset of composition factors in S^{(n−l,l)}.

The results in this paper are not to be confused with the results for 2–column partition Specht modules in the defining characteristic of the general linear group, obtained in [2] and discussed in [7].

If we were to take q ≡ 1 mod p, we would find the same submodule structure as in the corresponding Specht modules of the symmetric group (cf. [7]).

1. The Non–Defining Characteristic Representations of the General Linear Group

Let GL(n, q) be the group of n by n invertible matrices over the field K ≅ 𝔽_{q} of q elements, where q is a prime power. We are interested in the representations of GL(n, q) in non–defining characteristic, i.e. over a field F of characteristic p coprime to q. The main reference for this section is [6]. We will concentrate on presenting the objects we

Date: January 14, 2003. The research presented in this paper constitutes a part of the author’s PhD–

thesis. This research has been partly supported by an EPSRC EU award and the Aide Financière of the Luxembourg Government. The author also wishes to thank his supervisor, Gordon James, for the constant encouragement and comments provided throughout.

will be dealing with in a way that will enable us to work with them combinatorially at a later stage.

Let V = K^{n} be the underlying vector space of dimension n on which GL(n, q) acts naturally.

Definition 1.1. For λ = (λ_{1}, …, λ_{h}) an h–part partition of n, λ ⊢ n, a λ–flag is a sequence of subspaces of V such that V = V_{0} ≥ V_{1} ≥ … ≥ V_{h} = 0, with dim V_{i−1}/V_{i} = λ_{i}, ∀i = 1, …, h. Then GL(n, q) acts naturally on V and so it acts as well on λ–flags. We define the permutation module M^{λ} to be the F –space generated by λ–flags. The F –linear extension of the action of GL(n, q) on V makes M^{λ} into an F GL(n, q)–module. Now assume that 1 ≤ d ≤ h − 1 and 0 ≤ t ≤ λ_{d+1}. Let μ = (λ_{1}, …, λ_{d−1}, λ_{d} + λ_{d+1} − t, t, λ_{d+2}, …, λ_{h}). We define the homomorphism ψ_{d,t} : M^{λ} → M^{μ} by

V = V_{0} ≥ V_{1} ≥ … ≥ V_{d−1} ≥ V_{d} ≥ V_{d+1} ≥ V_{h} = 0

↦ ∑_{dim W_{d}/V_{d+1}=t^{W_{d}≤V_{d}}}{V = V_{0} ≥ V_{1} ≥ … ≥ V_{d−1} ≥ W_{d} ≥ V_{d+1} ≥ V_{h} = 0}.

We define the Specht module corresponding to λ to be:

S^{λ} = ⋂_{d=1}^{h−1} ⋂_{t=0}^{λ_{d}−1} Ker ψ_{d,t} ≤ M^{λ}.

Usually we would define S^{λ} as a right ideal of F GL(n, q). In that case much more work would be needed to actually prove the Kernel– Intersection property of S^{λ}.

We define the symmetric non–singular GL(n, q)–invariant bilinear form on M^{λ} by ⟨f_{i}, f_{j}⟩ = δ_{ij}, for λ–flags f_{i} and f_{j}. We may now give the Submodule Theorem for the general linear group.

Theorem 1.2 (James, [6]). If U is a submodule of M^{λ}, then either S^{λ} ≤ U or U ≤ S^{λ⊥}. D^{λ} = S^{λ}/(S^{λ} ∩ S^{λ⊥}) is self–dual and absolutely irreducible and the multiplicity of D^{λ} in M^{λ} is one.

It can be shown that the D^{μ}, where μ runs over all partitions, is a complete set of pairwise non–isomorphic irreducible unipotent F GL(n, q)– modules. Furthermore, it can be shown that the only composition factors of M^{μ} are of the form D^{λ}, where λ⊵μ. Here ⊵ denotes the dominance order on partitions of n: λ ⊵ μ if and only if ∑_{i=1}^{j} λ_{i} ≥ ∑_{i=1}^{j} μ_{i}, for all j.

2. Connecting F and K and Gaussian Polynomials

Here we define and give some properties of certain ‘numbers’ (some of these are actually polynomials in q) that will be used extensively in the study of 2–part partition modules.

Definition 2.1. For n ∈ ℕ and 0 ≠ q ∈ F, we define [0]_{q} = 0, [1]_{q} = 1 and [n]_{q} = 1+q+…+q^{n−1}. We further define [n]_{q}! = [n]_{q}[n−1]_{q} … [1]_{q}

and [\begin{matrix}n \\ m \end{matrix}]_{q} = \frac{[n]_{q}!}{[m]_{q}! [n−m]_{q}!}.

Let e be the least positive integer such that p divides [e]_{q}, if such an integer exists. If no such integer exists, we may take e to be equal to infinity. For b ∈ ℕ, we define b_{e}^{∗} and b′_{e} by b = b_{e}^{∗}e + b′_{e}, where 0 ≤ b′ < e.

We write as well [n] for [n]_{q}, b^{∗} for b_{e}^{∗} and b′ for b′_{e} if no confusion can

arise. Note that [\begin{matrix}n \\ m \end{matrix}]_{q} ∈ ℤ[q], the ring of integer coefficient polynomi-

als in q, and is called a Gaussian polynomial, although we sometimes consider it just to be an integer or an element of F. The number b_{e}^{∗} is the integer part of b/e, sometimes denoted by ⌊\frac{b}{e}⌋. From the definition of e above, we know immediately that e ≥ 2.

There is a relationship between e and p in the following sense. If p = e then p | [p]_{q}, so p | (q^{p} − 1) and q^{p} ≡ 1 mod p. By Fermat’s Little Theorem we have q^{p−1} ≡ 1 mod p, so we get q ≡ 1 mod p or p | (q − 1). The converse is obvious. So e = p if and only if p | q − 1. If however p ∤ (q − 1), then p | (q^{e} − 1)/(q − 1) = [e]_{q} and q^{e} ≡ 1 mod p, so e divides the order of the multiplicative group 𝔽_{p}^{∗}: e | (p − 1). In particular we always have 2 ≤ e ≤ p (cf. [4]).

When is a Gaussian Polynomial Divisible by p? In order to answer this question, we use lemma 19.2 of [6] which states that for an odd prime p, for 1 ≠ q ∈ ℕ and for e, t ∈ ℕ, where p | (q^{e} − 1) = [e](q − 1), we have ν_{p} (\frac{q^{te}−1}{q^{e}−1}) = ν_{p}(t), where ν_{p}(a), for a ∈ ℕ, denotes

the p–valuation of a, i.e. the largest exponent of p such that p^{ν_{p}(a)} | a. For a ∈ ℕ and b ∈ ℕ^{∗}, we may define ν_{p}(a/b) = ν_{p}(a) − ν_{p}(b). We then have, for e, s, t ∈ ℕ:

ν_{p} (\frac{q^{te} − 1}{q^{se} − 1}) = ν_{p}(\frac{t}{s}).

We have the following property of natural numbers:

A set of i consecutive natural numbers contains either i^{∗} or i^{∗} + 1 numbers that are divisible by e.

It can be shown, by simple enumeration, that for a ∈ ℕ, the set {a − i + 1, …, a} contains exactly i^{∗} numbers divisible by e if a′ = i′ = 0 or if a′ ≠ 0 and a′ + i′ ≤ e. Otherwise this set contains exactly i^{∗} + 1 numbers divisible by e.

We are then able to reduce the problem of when a Gaussian polynomial is divisible by the prime p to a similar problem in terms of binomial coefficients. For these the answer is known by lemma 22.4 in [5]: for a, b ∈ ℕ with p–adic expansions a = ∑_{j≥0} a_{j}p^{j} and b = ∑_{j≥0} b_{j}p^{j}, we have

(1)  (\begin{matrix}a \\ b \end{matrix}) ≡ ∏_{j≥0} (\begin{matrix}a_{j} \\ b_{j} \end{matrix}) mod p.

In particular p | (\begin{matrix}a \\ b \end{matrix}) if and only if there is a j ≥ 0 such that a_{j} < b_{j}.

The proof of the next theorem relies on a variation of an argument given in the proof of theorem 19.5 in [6].

Theorem 2.2. Let a, i ∈ ℕ and let e ∈ ℕ be minimal such that p | [e].

If a′ = i′ = 0 or if a′ ≠ 0 and a′ + i′ ≤ e then ν_{p} ([\begin{matrix}a \\ i \end{matrix}]) = ν_{p} ((\begin{matrix}a^{∗} \\ i^{∗} \end{matrix})).

Otherwise ν_{p} ([\begin{matrix}a \\ i \end{matrix}]) > 0.

Proof. We first examine the case for an odd prime p. By definition,

we have [\begin{matrix}a \\ i \end{matrix}] = \frac{(q^{a}−1)...(q^{a−i+1}−1)}{(q^{i}−1)...(q−1)}. Now if a′ = i′ = 0 or if a′ ≠ 0 and

a′ + i′ ≤ e, the set {a − i + 1, …, a} contains exactly i^{∗} numbers divisible by e and we use the property of ν_{p} discussed above to deduce

that ν_{p} ([\begin{matrix}a \\ i \end{matrix}]) = ν_{p} ((\begin{matrix}a^{∗} \\ i^{∗} \end{matrix})). If a′ ≠ 0 and a′ + i′ > e or if a′ = 0 and

i′ ≠ 0 however, then [\begin{matrix}a \\ i \end{matrix}] is a multiple of [e] = 1 + … + q^{e−1}, since the

set {a − i + 1, …, a} contains exactly i^{∗} + 1 numbers divisible by e.

But p | [e] and therefore ν_{p} ([\begin{matrix}a \\ i \end{matrix}]) > 0.

If p = 2, then e = p = 2, in which case every Gaussian polynomial is actually congruent to the corresponding binomial coefficient modulo p and the statement of the theorem still holds. □

3. Two–Part Partitions and Homomorphisms

We now consider some properties of 2–part partition modules in more detail. We will think of the 2–part partition permutation module M^{(n−l,l)} as the vector space with the l–dimensional subspaces of V as a basis.

3.1. The Kernel–Intersection Theorems. We discuss some properties of the following F GL(n, q)–homomorphisms.

Definition 3.1. For 0 ≤ j ≤ k ≤ l, let Θ_{j}^{k,l} : M^{(n−k,k)} → M^{(n−l,l)} be the homomorphism given by

U ↦ ∑{W ≤ V : dim W = l and dim(W ∩ U) = j}.

For 0 ≤ i ≤ l, let ψ_{i} : M^{(n−l,l)} → M^{(n−i,i)} be the homomorphism given by

W ↦ ∑{U ≤ W : dim U = i}.

We also write Θ_{j} or Θ_{j}^{.,l} for Θ_{j}^{k,l} when no confusion can occur. We will be mostly interested in Θ_{k}^{k,l} and Θ_{0}^{k,l}, called the semistandard and the reverse semistandard homomorphism from M^{(n−k,k)} → M^{(n−l,l)} respectively, by analogy to the symmetric group case. A particular property of Θ_{0}^{k,l} is the following, which can be easily deduced from theorems 3.1 and 19.5 of [6].

Theorem 3.2. For k, r ≤ l ≤ n/2, we have Im Θ_{0}^{k,l} ≤ ⋂_{i=r}^{l−1} Ker ψ_{i} if and only if e | n − k − l + 1 and (n − k − l + 1)^{∗} ≡ 0 mod p^{ℓ_{p}((l−r)^{∗})}.

In particular we have Im Θ_{0}^{k,l} ≤ S^{(n−l,l)} if and only if e | n − k − l + 1 and n − k − l + 1 ≡ 0 mod p^{ℓ_{p}(l^{∗})}. Furthermore, it can be shown (cf. [11]) by elementary methods that the reverse semistandard homomorphism preserves the kernel–intersection property of the Specht module S^{(n−k,k)}:

Theorem 3.3. For 0 ≤ m ≤ k − 1 we have

(⋂_{i=0}^{m} Ker ψ_{i}) Θ_{0}^{k,l} ≤ ⋂_{i=0}^{m} Ker ψ_{i}.

3.2. An Identity Connecting Θ_{j}^{k,l} and Θ_{k}^{k,l}. A very important result in the representation theory of 𝔖_{n} is that the semistandard homomorphisms form a basis of Hom_{F 𝔖_{n}}(S^{λ}, M^{μ}), for λ ▷ μ, (unless char F = 2 and λ is 2–singular). In particular therefore we have that Θ_{j}^{k,l} and Θ_{k}^{k,l} are multiples of each other in this setting. We establish a similar situation in the context of F GL(n, q).

Sinead Lyle ([9]) studied the following homomorphisms and was able to conjecture the identities in expressions (2) and (3). For 0 ≤ j ≤ k, we consider the homomorphism T_{j}^{k,l} = ∑_{i=j}^{k} Θ_{i}^{k,l}. So for 0 ≤ j < k, we have Θ_{j}^{k,l} = T_{j}^{k,l} − T_{j+1}^{k,l} and the semistandard homomorphism can be expressed by Θ_{k}^{k,l} = T_{k}^{k,l}. It can be shown by induction (cf. [11]) that, for 0 ≤ j ≤ k ≤ l, we have

(2) T_{j}^{k,l} ↓_{S^{(n−k,k)}}= (−1)^{k−j}q^{(k−j)(k−j+1)/2} [\begin{matrix}k − 1 \\ j − 1 \end{matrix}] T_{k}^{k,l} ↓_{S^{(n−k,k)}}

and thus, for 0 ≤ j ≤ k ≤ l,

(3)  Θ_{j}^{k,l} ↓_{S^{(n−k,k)}}= (−1)^{k−j}q^{(k−j)(k−j−1)/2} [\begin{matrix}k \\ j \end{matrix}] Θ_{k}^{k,l} ↓_{S^{(n−k,k)}} .

In particular we have Θ_{0}^{k,l} ↓_{S^{(n−k,k)}}= (−1)^{k}q^{k(k−1)/2}Θ_{k}^{k,l} ↓_{S^{(n−k,k)}}.

We are mostly interested in the restriction to the Specht module S^{(n−k,k)} of compositions of reverse semistandard homomorphisms. We know how Θ_{j} restricts to S^{(n−k,k)}, by (3) above, and thus theorem 3.3.(ii) in [6] gives us the next result: for A ∈ S^{(n−k,k)},

A\overline{Θ_{0}^{k,m}}\overline{Θ_{0}^{m,l}} = (−1)^{k+m}q^{\frac{k(k−1)+m(m−1)}{2}} [\begin{matrix}l − k \\ m − k \end{matrix}] A\overline{Θ_{k}^{k,l}}
= (−1)^{m}q^{\frac{m(m−1)}{2}} [\begin{matrix}l − k \\ m − k \end{matrix}] A\overline{Θ_{0}^{k,l}},

where \overline{Θ}_{0} denotes the restriction of the homomorphism Θ_{0} to the corresponding Specht module.

4. Multiplicity–Free Modules

Let M be a module of finite lenght. We denote by F(M) the multiset of composition factors of M and for A ∈ F(M), we denote by [M : A] the multiplicity of A in M, i.e. the number of times A appears in F(M). A module M is multiplicity–free if all the composition factors appearing in a composition series appear once. Thus F(M) is a set if M is multiplicity–free.

A Partial Order on The Composition Factors. We introduce the following partial order for multiplicity–free modules, as in [1] and [7].

Definition 4.1. Let M be a finite–dimensional multiplicity–free module. For A, B ∈ F(M), we say that A ≺_{M} B or that A ≺ B in M, if for all submodules N ≤ M we have B ∈ F(N) implies A ∈ F(N).

For B ∈ F(M), define M_{B} to be the minimal submodule with B as a composition factor: M_{B} = ⋂_{B∈F(N)^{N<M}} N and F(M_{B}) = {A ∈ F(M) :

A ≺_{M} B} ∪ {B}. Then (F(M), ≺_{M}) forms a partial order.

We have the following property of ≺_{M}:

A ≺_{M} B ⇔ M_{A} < M_{B}.

If M is multiplicity–free then the partial order (F(M), ≺_{M}) determines the submodule structure of M: if N_{1}, N_{2} ≤ M, then N_{1} < N_{2} if and only if F(N_{1}) ⊂ F(N_{2}) and N_{1} = N_{2} if and only if F(N_{1}) = F(N_{2}). Furthermore F(N_{1} + N_{2}) = F(N_{1}) ∪ F(N_{2}). Since every submodule of M is a sum of modules of the form M_{B}, we obtain the complete submodule structure of M by (F(M), ≺_{M}).

We have the following facts about the multiplicity–free modules M. These can be shown by elementary considerations on the submodules and subfactors of M.

Lemma 4.2. Suppose that A, B, C ∈ F(M) and that [M : A] = [M : B] = [M : C] = 1.

(1) Suppose that M ≥ V ≥ U ≥ 0. Then F(V/U) ⊆ F(M). Suppose furthermore that A ≺_{V/U} B. Then A ≺_{M} B.

(2) If A ≺_{M} B ≺_{M} C then there is no subquotient of M with composition factors A and C but not B.

(3) Let M = M_{0} ≥ M_{1} ≥ … ≥ M_{k} = 0 be a filtration of M and let A ∈ F(M_{i}/M_{i+1}) and B ∈ F(M_{j}/M_{j+1}), for some i, j ∈ ℕ. If i < j then A ⊀_{M} B.

5. The Generalisation of a Theorem by Gwendolen Murphy

In [10], Gwendolen Murphy gave a theorem (theorem 4.6) describing some submodules of S^{(n−l,l)} in the symmetric group setting. In particular this theorem showed that S^{(n−l,l)} has a unique minimal submodule. We now look at how this theorem can be presented in the general linear group case. By theorem 3.2, we have (4)

M^{(n−i,i)}Θ_{0}^{i,l} ≤ ⋂_{j=r}^{l−1} Ker ψ_{j} ⇔ \begin{cases}e | n − i − l + 1 and \\ (n − i − l + 1)^{∗} ≡ 0 mod p^{ℓ_{p}((l−r)^{∗})} \end{cases} ,

where Θ_{0}^{i,l} is the reverse semistandard homomorphism from M^{(n−i,i)} to M^{(n−l,l)} as in definition 3.1. Now let r = i and restrict Θ_{0}^{i,l} to the Specht module S^{(n−i,i)} ≅ ⋂_{j=0}^{i−1} Ker ψ_{j}. Then, since Θ_{0}^{i,l} preserves the kernel–intersection (theorem 3.3), we get S^{(n−i,i)}Θ_{0}^{i,l} ≤ S^{(n−l,l)} if and only if

e | n − l − i + 1 and (n − i − l + 1)^{∗} ≡ 0 mod p^{ℓ_{p}((l−i)^{∗})}.

Similarly S^{(n−i,i)}Θ_{i}^{i,l} ≤ S^{(n−l,l)} if and only if e | n − l − i + 1 and (n − i − l + 1)^{∗} ≡ 0 mod p^{ℓ_{p}((l−i)^{∗})}, by expression (3). We now give the equivalent of theorem 4.6 in [10] for the general linear group case.

Theorem 5.1. For 0 ≤ i_{2} < i_{1} ≤ l, we have

0 ≠ S^{(n−i_{2},i_{2})}Θ_{i_{2}}^{i_{2},l} < S^{(n−i_{1},i_{1})}Θ_{i_{1}}^{i_{1},l} ≤ S^{(n−l,l)}

if and only if e | (n − i_{2} − l + 1), e | (n − i_{1} − l + 1), (n − i_{2} − l + 1)^{∗} ≡ 0 mod p^{ℓ_{p}((l−i_{2})^{∗})} and (n − i_{1} − l + 1)^{∗} ≡ 0 mod p^{ℓ_{p}((l−i_{1})^{∗})} with ℓ_{p}((l − i_{1})^{∗}) < ℓ_{p}((l − i_{2})^{∗}).

Proof. Suppose e | (n−i_{2}−l+1) and (n−i_{2}−l+1)^{∗} ≡ 0 mod p^{ℓ_{p}((l−i_{2})^{∗})}. Then S^{(n−i_{2},i_{2})}Θ_{i_{2}}^{i_{2},l} ≤ S^{(n−l,l)}. Furthermore, suppose that e | (n−i_{1}−l+ 1) and (n−i_{1}−l+1)^{∗} ≡ 0 mod p^{ℓ_{p}((l−i_{1})^{∗})}. Then S^{(n−i_{1},i_{1})}Θ_{i_{1}}^{i_{1},l} ≤ S^{(n−l,l)}.

We now study Θ_{i_{2}}^{i_{2},i_{1}}Θ_{i_{1}}^{i_{1},l} = [\begin{matrix}l − i_{2} \\ i_{1} − i_{2} \end{matrix}] Θ_{i_{2}}^{i_{2},l} and investigate the Gaussian

polynomial in this expression. We have e | (n − l − i_{2} + 1) − (n − l − i_{1} + 1) = i_{1} − i_{2} and therefore

(5)  (i_{1} − i_{2})′ = 0.

Then (l − i_{2})′ + (i_{1} − i_{2})′ ≤ e, and by theorem 2.2 we have that

ν_{p} ([\begin{matrix}l − i_{2} \\ i_{1} − i_{2} \end{matrix}]) = ν_{p} ((\begin{matrix}(l − i_{2})^{∗} \\ (i_{1} − i_{2})^{∗} \end{matrix})). But by (5) we know that (i_{1} −

i_{2})^{∗} = (i_{1} − i_{2})/e = ((l − i_{2}) − (l − i_{1}))/e. Furthermore, ℓ_{p}((l − i_{1})^{∗}) < ℓ_{p}((l−i_{2})^{∗}) and therefore (i_{1} −i_{2})^{∗} = (n−l−i_{2} +1)^{∗} −(n−l−i_{1} +1)^{∗} = ((n−l−i_{2} +1)−(n−l−i_{1} +1))/e ≡ 0 mod p^{ℓ_{p}((l−i_{1})^{∗})}. Thus the p–adic expansion of (i_{1} − i_{2})^{∗} is that of (l − i_{2})^{∗} with some part removed. But

then (\begin{matrix}(l − i_{2})^{∗} \\ (i_{1} − i_{2})^{∗} \end{matrix}) ≢ 0 mod p by (1), so p ∤ [\begin{matrix}l − i_{2} \\ i_{1} − i_{2} \end{matrix}]. Therefore Θ_{i_{1}}^{i_{1},l}

maps the (proper) submodule M^{(n−i_{2},i_{2})}Θ_{i_{2}}^{i_{2},i_{1}} of M^{(n−i_{1},i_{1})} to

0 ≠ M^{(n−i_{2},i_{2})}Θ_{i_{2}}^{i_{2},l}Θ_{i_{1}}^{i_{1},l} ≤ M^{(n−i_{1},i_{1})}Θ_{i_{1}}^{i_{1},l}.

Now we restrict both of these modules to ⋂_{i=0}^{i_{2}−1} Ker ψ_{i} to get

0 ≠ S^{(n−i_{2},i_{2})}Θ_{i_{2}}^{i_{2},l} ≤ (M^{(n−i_{1},i_{1})} ∩ ⋂_{i=0}^{i_{2}−1} Ker ψ_{i}) Θ_{i_{1}}^{i_{1},l}.

Again restrict these two modules to ⋂_{i=i_{2}}^{i_{1}−1} Ker ψ_{i}. Note that we have S^{(n−i_{2},i_{2})}Θ_{i_{2}}^{i_{2},l} ≤ ⋂_{i=i_{2}}^{i_{1}−1} Ker ψ_{i}. Therefore we get

0 ≠ S^{(n−i_{2},i_{2})}Θ_{i_{2}}^{i_{2},l} ≤ S^{(n−i_{1},i_{1})}Θ_{i_{1}}^{i_{1},l} ≤ S^{(n−l,l)}.

But clearly S^{(n−i_{2},i_{2})}Θ_{0}^{i_{2},l} ≠ S^{(n−i_{1},i_{1})}Θ_{0}^{i_{1},l}, since the unique top composition factors of these modules are not isomorphic. Conversely, let 0 ≠ S^{(n−i_{2},i_{2})}Θ_{i_{2}}^{i_{2},l} < S^{(n−i_{1},i_{1})}Θ_{i_{1}}^{i_{1},l} ≤ S^{(n−l,l)} with i_{2} < i_{1}. Then e | (n − l − i_{2} + 1), e | (n − l − i_{1} + 1), (n − l − i_{2} + 1)^{∗} ≡ 0 mod p^{l_{p}((l−i_{2})^{∗})} and (n − l − i_{1} + 1)^{∗} ≡ 0 mod p^{l_{p}((l−i_{1})^{∗})}. Now assume that ℓ_{p}((l − i_{1})^{∗}) = ℓ_{p}((l − i_{2})^{∗}). We get e | (n − l − i_{2} + 1) − (n − l − i_{1} + 1) = i_{1} − i_{2} and (i_{1} − i_{2})^{∗} ≡ 0 mod p^{l_{p}((l−i_{2})^{∗})}. But since (i_{1} − i_{2})^{∗} = (i_{1} − i_{2})/e = ((l − i_{2}) − (l − i_{1}))/e we get (i_{1} − i_{2})/e = 0 and i_{1} = i_{2}, a contradiction. □

Given l and i as in (4), there is usually more than one choice possible for r. We therefore define (6) r(l, i) = min{r : e | n − l − i + 1 and (n − l − i + 1)^{∗} ≡ 0 mod p^{ℓ_{p}((l−r)^{∗})}}

and

(7)  x(l, i) = ℓ_{p}((l − r(l, i))^{∗}) such that r(l, i) is as in (6).

Consequences for Specht Modules of GL(n, q). We are now able to draw a few preliminary conclusions about the occurrence of some composition factors of the Specht module S^{(n−l,l)}.

Lemma 5.2. Suppose that M^{(n−k,k)}Θ_{0}^{k,l} ≤ ⋂_{i=r}^{l−1} Ker ψ_{i} with D^{(n−u,u)} ≺ D^{(n−v,v)} in S^{(n−k,k)}, with u, v > r. Then D^{(n−v,v)} ≺ D^{(n−u,u)} in S^{(n−l,l)}.

Proof. Since Im Θ_{0}^{k,l} ≤ ⋂_{i=r}^{l−1} Ker ψ_{i}, we may restrict to ⋂_{i=0}^{r−1} Ker ψ_{i} to get Im Θ_{0}^{k,l} ∩ ⋂_{i=0}^{r−1} Ker ψ_{i} ≤ S^{(n−l,l)}. Since M^{(n−l,l)} has a Specht filtration, the composition factors of M^{(n−l,l)}/ ⋂_{i=0}^{r−1} Ker ψ_{i} are from the set ⋃_{i=0}^{r−1} F(S^{(n−i,i)}) (with multiplicities). Therefore, for all j = r + 1, …, l − 1, we have that D^{(n−j,j)} is not a composition factor of Im Θ_{0}^{k,l}/ (Im Θ_{0}^{k,l} ∩ ⋂_{i=0}^{r−1} Ker ψ_{i}) and consequently, for u, v > r, we

know that Im Θ_{0}^{k,l} contains D^{(n−u,u)} and D^{(n−v,v)} at most once. Furthermore, since Ker Θ_{0}^{k,l} ≤ S^{(n−k,k)⊥} and M^{(n−k,k)}/S^{(n−k,k)⊥} ≅ (S^{(n−k,k)})′, we know that Im Θ_{0}^{k,l} contains D^{(n−u,u)} and D^{(n−v,v)} at least once. In the quotient M^{(n−k,k)}/S^{(n−k,k)⊥}, we have as well D^{(n−v,v)} ≺ D^{(n−u,u)} and we therefore have D^{(n−v,v)} ≺ D^{(n−u,u)} in Im Θ_{0}^{k,l}, since Im Θ_{0}^{k,l} ≅ M^{(n−k,k)}/ Ker Θ_{0}^{k,l} and by lemma 4.2. But D^{(n−u,u)} and D^{(n−v,v)} occurring in Im Θ_{0}^{k,l} actually occur inside S^{(n−l,l)} and the result follows. □

Lemma 5.3. Suppose that 0 ≠ S^{(n−k,k)}Θ_{0}^{k,i} < S^{(n−l,l)}Θ_{0}^{l,i} in S^{(n−i,i)}. Then ∀ b such that D^{(n−k,k)} ≺ D^{(n−b,b)} ≺ D^{(n−l,l)} in S^{(n−l,l)}, we have D^{(n−k,k)} ≺ D^{(n−b,b)} ≺ D^{(n−l,l)} in S^{(n−i,i)}.

Proof. Since 0 ≠ S^{(n−k,k)}Θ ≤ S^{(n−l,l)}Θ in S^{(n−i,i)}, we know that D^{(n−k,k)} and D^{(n−l,l)} are composition factors of S^{(n−i,i)}. In S^{(n−l,l)}Θ_{0}^{l,i}, we have D^{(n−k,k)} ≺ D^{(n−l,l)}. Now, since D^{(n−k,k)} ≺ D^{(n−b,b)} ≺ D^{(n−l,l)} in S^{(n−l,l)}, we know that no subquotient of S^{(n−l,l)} has the factors D^{(n−k,k)} and D^{(n−l,l)} without having D^{(n−b,b)} as a factor, by lemma 4.2. But S^{(n−l,l)}Θ_{0}^{l,i} ≤ S^{(n−i,i)} is isomorphic to a subquotient of S^{(n−l,l)} with D^{(n−k,k)} and D^{(n−l,l)} as factors, so it must have D^{(n−b,b)} as a composition factor. □

Lemma 5.4. Let Θ ∈ Hom(S^{(n−i,i)}, M^{μ}), for (n − i, i) ⊵ μ. Suppose that D^{(n−j,j)} ≺ D^{(n−k,k)} in S^{(n−i,i)} and that D^{(n−j,j)} ∈ F(S^{(n−i,i)}Θ). Then D^{(n−k,k)} is a composition factor of S^{(n−i,i)}Θ.

Proof. Assume that the theorem does not hold. Then D^{(n−k,k)} ∈ F(Ker Θ) and D^{(n−j,j)} ∈ F(S^{(n−i,i)}/ Ker Θ). But then we cannot have D^{(n−j,j)} ≺ D^{(n−k,k)}, a contradiction. □

6. The Composition Factors of S^{(n−l,l)}

We will only study the case where e < p here. For the case where e = p, or equivalently, where q ≡ 1 mod p, we could go through the results discussed here, adjusting the notation if required.

Let l ∈ ℕ, l ≤ n/2 and λ = (n − l, l). For a, b ∈ ℕ with p–adic expansions a = ∑_{i≥0} a_{i}p^{i} and b = ∑_{i≥0} b_{i}p^{i}, we say that a contains b to base p, denoted by a ⊃_{p} b, if ℓ_{p}(b) < ℓ_{p}(a) and if for all i ∈ ℕ we have b_{i} = a_{i} or b_{i} = 0 (cf. [5]), where ℓ_{p}(m), for m ∈ ℕ is given by 0 ≤ p^{ℓ_{p}(m)−1} ≤ m < p^{ℓ_{p}(m)}. Recall that b_{e}^{∗} = b^{∗} denotes the integer part of b/e, where b, e ∈ ℕ and e is minimal such that p | [e] = 1 + q + … + q^{e−1}. In the case e < p, we usually write b^{∗} instead of b_{e}^{∗}. Note also that we have p ≠ 2 here, since 1 ≠ e < p. Now, given a, b ∈ ℕ, we define

f_{p,e}(a, b) = \begin{cases}1 if b = 0 \\ 1 if (a + 1)^{∗} ⊃_{p} b^{∗} and (e | b or e | (a + 1 − b)) \\ 0 otherwise \end{cases} .

The composition multiplicity of D^{(n−j,j)} in S^{(n−l,l)} is then given by theorem 20.6 in [6], Theorem 6.1 (James). [S^{(n−l,l)} : D^{(n−j,j)}] = f_{p,e}(n − 2j, l − j).

and is either 1 or 0.

We now give a Kleshchev–Sheth type description of the j such that the composition factors D^{(n−j,j)} occur in S^{(n−l,l)} (cf. [7]).

Definition 6.2. For λ = (n − l, l), with 0 ≤ l ≤ n/2 and ℓ_{p}(l) = s + 1, let a_{λ} = n − 2l + 1 = a_{λ}^{∗}e + a′_{λ}. Let

a_{λ}^{∗} = (n − 2l + 1)^{∗} = ∑_{i≥0} b_{i}p^{i}.

Define

B_{λ}^{∗,−} = \begin{cases}\{0 ≤ i ≤ ℓ_{p}(l) : b_{i} ≠ 0\}, if a′_{λ} = 0 \\ \{0 ≤ i ≤ ℓ_{p}(l) : b_{i} ≠ 0\} ∪ \{−1\} otherwise \end{cases} ,

and

B_{λ}^{∗,+} = \begin{cases}\{ℓ_{p}(l)\}, if b_{i} = p − 1, ∀ i = 0, …, ℓ_{p}(l) \\ \{0 ≤ i ≤ ℓ_{p}(l) : b_{i} ≠ p − 1\}, otherwise \end{cases} .

Let A_{λ}^{∗} be the family consisting of the empty set and all sets I^{∗} of the form

(8)  I^{∗} = [j_{1}, j_{2}[∪[j_{3}, j_{4}[∪ … ∪ [j_{2t−1}, j_{2t}[,

with j_{1} < j_{2} < … < j_{2t} and such that ∀ k = 1, …, t, we have j_{2k−1} ∈ B_{λ}^{∗,−} and j_{2k} ∈ B_{λ}^{∗,+} as defined above. Define δ_{∅} = 0 and for any ∅ ≠ I^{∗} ∈ A_{λ}^{∗} as above, define

(9)  δ_{I^{∗}} = ∑_{i≠−1^{i∈I^{∗}}}(p − 1 − b_{i})p^{i} + ∑_{j_{1}≠−1^{k=1}}^{t} p^{j_{2k−1}}.

For all I^{∗} ∈ A_{λ}^{∗}, we define

μ_{I^{∗}}(l) = \begin{cases}(n − l + δ_{I^{∗}}e + e − a′_{λ}, l − (δ_{I^{∗}}e + e − a′_{λ})), if − 1 ∈ I^{∗} \\ (n − l + δ_{I^{∗}}e, l − δ_{I^{∗}}e), otherwise \end{cases} .

Furthermore, we define A_{λ}^{∗} as follows

A_{λ}^{∗} = {I^{∗} ∈ A_{λ}^{∗} : −1 ∉ I^{∗} and δ_{I^{∗}}e ≤ l}
∪{I^{∗} ∈ A_{λ}^{∗} : −1 ∈ I^{∗}and δ_{I^{∗}}e + e − a′_{λ} ≤ l},

and let i_{1} = min{i ∈ B_{λ}^{∗,−}} and i_{2T} = max{i ∈ B_{λ}^{∗,−}}, such that [i_{1}, i_{2T} [∈ A_{λ}^{∗}. For i ∈ B_{λ}^{∗,−}, we define î = min{j ∈ B_{λ}^{∗,+} : j > i}, if it exists, and for i ∈ B_{λ}^{∗,+}, we define ĩ = max{j ∈ B_{λ}^{∗,−} : j < i}, if it exists.

Note that i_{1} = −1 if a′_{λ} > 0. The composition factors of S^{(n−l,l)} are then given by the following theorem.

Theorem 6.3. F(S^{(n−l,l)}) = {D^{μ_{I^{∗}}(l)} : I^{∗} ∈ A_{λ}^{∗}}.

Proof. For D^{(n−j,j)} to be a composition factor of S^{(n−l,l)}, we need e | (l − j) or e | (n − l − j + 1). If a′_{λ} = 0, then e | (l − j) if and only if e | (n − l − j + 1) = n − 2l + 1 + (l − j) and for all I^{∗} ∈ A_{λ}^{∗} we have −1 ∉ I^{∗}. We may then take (l − j)^{∗} = δ_{I^{∗}} and the result follows by the calculation of Sheth in [12] for the symmetric group case. Now suppose that a′_{λ} ≠ 0 and e | (l − j). Again we may take (l − j)^{∗} = δ_{I^{∗}}, since (l − j)′ = 0, i.e. −1 ∉ I^{∗}. Then (n − 2j + 1)^{∗} = (n − 2l + 1 + 2(l − j))^{∗}, since (l − j)′ = 0, and

(n − 2j + 1)^{∗} = (n − 2l + 1)^{∗} + 2δ_{I^{∗}}
= δ_{I^{∗}} + ∑_{k=1}^{t}(b_{j_{2k}} + 1)p^{j_{2k}} + ∑_{i≠j_{2},...,j_{2t}^{i∉I^{∗}}} b_{i}p^{i}

⊃_{p} δ_{I^{∗}}

and the result follows. Next suppose that a′_{λ} ≠ 0 and e | (n − l − j + 1). Then e ∤ (l − j) and so −1 ∈ I^{∗}. Since n − l − j + 1 = n − 2l + 1 + l − j, we have (l − j)′ = e − (n − 2l + 1)′ = e − a′_{λ} ≡ −a′_{λ} mod e. Here we take again (l − j)^{∗} = δ_{I^{∗}}, by the following.

(n − 2j + 1)^{∗} = (n − 2l + 1 + 2(l − j))^{∗}
= (n − 2l + 1)^{∗} + 2(l − j)^{∗} + (a′_{λ} + 2(l − j)′)^{∗}
= ∑_{i≥0} b_{i}p^{i} + 2δ_{I^{∗}} + 1,

since a′_{λ} + 2(l − j)′ = a′_{λ} + 2e − 2a′_{λ} = 2e − a′_{λ} and (2e − a′_{λ})^{∗} = 1. Then

(n − 2j + 1)^{∗} = ∑_{0≤i∉I^{∗}} b_{i}p^{i} + ∑_{−1≠i∈I^{∗}}(p − 1 − b_{i})p^{i} + ∑_{k=2}^{t} p^{j_{2k−1}}

+ ∑_{−1≠i∈I^{∗}}(p − 1)p^{i} + ∑_{k=2}^{t} p^{j_{2k−1}} + 1

= ∑_{i≠j_{2},...,j_{2t}^{0≤i∉I^{∗}}} b_{i}p^{i} + δ_{I^{∗}} − 1 + ∑_{k=1}^{t}(b_{j_{2k}} + 1)p^{j_{2k}} + 1

⊃_{p} δ_{I^{∗}},

since j_{1} = −1 and so ∑_{−1≠i∈I^{∗}}(p − 1)p^{i} + ∑_{k=2}^{t} p^{j_{2k−1}} = −p^{j_{1}+1} + ∑_{k=1}^{t} p^{j_{2k}} = −1 + ∑_{k=1}^{t} p^{j_{2k}}. So in all the cases examined, we may take (l − j)^{∗} = δ_{I^{∗}} and either (l − j)′ = e − a′_{λ} or (l − j)′ = 0, depending on whether −1 ∈ I^{∗} or not (respectively). □

We give an example to illustrate the notation.

Example 6.4 (n=62, e=2, p=3, l=21). We examine the composition factors for the Specht module S^{(41,21)} for GL(62, q), over the field F such that char F = 3 and where q ≡ −1 mod 3.

\begin{matrix}j & l − j & n − 2j + 1 & (l − j)_{2}^{∗} & (n − 2j + 1)_{2}^{∗} & I^{∗} \\ 5 & 16 & 53 & (2, 2) & (2, 2, 2) & [0, 2[ \\ 6 & 15 & 51 & (1, 2) & (1, 2, 2) & [−1, 2[ \\ 17 & 4 & 29 & (2, 0) & (2, 1, 1) & [0, 1[ \\ 18 & 3 & 27 & (1) & (1, 1, 1) & [−1, 1[ \\ 20 & 1 & 23 & (0) & (2, 0, 1) & [−1, 0[ \\ 21 & 0 & 21 & (0) & (1, 0, 1) & ∅ \end{matrix}

Table 1. (n − 2j + 1)_{2}^{∗} ⊃_{3} (21 − j)_{2}^{∗} for S^{(41,21)} and q ≡ −1 mod 3

Here a_{λ} = n − 2l + 1 = 21 and so a_{λ}^{∗} = 10 = 1 + 9 = (1, 0, 1) and a′_{λ} = 1. Therefore b_{0} = 1, b_{1} = 0, b_{2} = 1 and B_{λ}^{∗,−} = {−1, 0, 2} and B_{λ}^{∗,+} = {0, 1, 2}. We now form all possible sets I^{∗} ∈ A_{λ}^{∗} (last column in table 1) and calculate (l − j)^{∗}. Then we get l − j by multiplying (l − j)^{∗} by e, and adding e − a′_{λ} = 1 if −1 ∈ I^{∗}. For example for I^{∗} = [−1, 2[ we have (l − j)^{∗} = (p − 1 − b_{0}) + (p − 1 − b_{1})p = 1 + 6 = 7 and l − j = 14 + 1 = 15. The 3–adic expansions of (n − 2j + 1)^{∗} are also shown in table 1. The entries in table 1 are shown by increasing j, but they really should be in the order in which we obtained I^{∗}.

The following result is a corollary of theorem 6.3 (cf. corollaries 2.14 and 2.15 in [3]) and is given here to illustrate the notation.

Corollary 6.5.

(1) S^{(n−l,l)} is irreducible if and only if A_{(n−l,l)} = {∅}

(2) The socle of S^{(n−l,l)} is irreducible and isomorphic to D^{μ_{[i_{1},i_{2T} [}(l)}.

7. The Submodule Structure of S^{(n−l,l)}

We now want to prove that the submodule structure of the 2–part partition Specht modules in F GL(n, q) is poset–isomorphic to the incidence structure in A_{λ}^{∗}. In order to prove this, we assume that we know the submodule structure of all the Specht modules S^{μ}, for μ▷(n−l, l), and deduce the submodule structure of S^{(n−l,l)}. We examine the occurrence of the composition factor D^{μ_{I^{∗}}(l)} in S^{(n−l,l)} according to the form of the set I^{∗} ∈ A_{λ}^{∗}.

7.1. Sets With a Common Extremum. We first examine the case where two sets have a common maximum.

Lemma 7.1. Suppose that i ∈ B_{λ}^{∗,+}. For all I^{∗} ∈ A_{λ}^{∗} with max I^{∗} = i − 1, the composition factor D^{μ_{I^{∗}}(l)} of S^{(n−l,l)} is as well a composition

factor of S^{(n−f,f)}, where f = \begin{cases}l − (δ_{[i_{1},i[}e + e − a′_{λ}), if a′_{λ} > 0 \\ l − δ_{[i_{1},i[}e, otherwise \end{cases} .

Proof. We compute

n − 2f + 1 = \begin{cases}n − 2l + 1 + 2δ_{[i_{1},i[}e + 2e − 2a′_{λ}, if a′_{λ} > 0 \\ n − 2l + 1 + 2δ_{[i_{1},i[}e, otherwise \end{cases} .

If a′_{λ} > 0, we have i_{1} = −1. In this case we have n − 2f + 1 = n − 2l + 1 + 2δ_{[−1,i[}e + 2e − 2a′_{λ}, and

(n − 2f + 1)^{∗} = a_{λ}^{∗} + 2δ_{[−1,i[} + (2e − a′_{λ})^{∗}
= a_{λ}^{∗} + 2δ_{[−1,i[} + 1
= ∑_{j≥0} b_{j}p^{j} + 2 ∑_{j=0}^{i−1}(p − 1 − b_{j})p^{j} + 1

= ∑_{j=0}^{i−1}(p − 1 − b_{j})p^{j} + ∑_{j=0}^{i−1}(p − 1)p^{j} + ∑_{j≥i} b_{j}p^{j} + 1

= δ_{[−1,i[} + (b_{i} + 1)p^{i} + ∑_{j>i} b_{j}p^{j},

since ∑_{j=0}^{i−1}(p − 1)p^{j} = −1 + p^{i}. In the case where a′_{λ} = 0, we obviously get (n − 2f + 1)^{∗} = a_{λ}^{∗} + 2δ_{[i_{1},i[}. Thus

(10) (n − 2f + 1)^{∗} = \begin{cases}δ_{[−1,i[} + (b_{i} + 1)p^{i} + ∑_{j>i} b_{j}p^{j}, if a′_{λ} > 0 \\ δ_{[i_{1},i[} + (b_{i} + 1)p^{i} + ∑_{j∉[i_{1},i]} b_{j}p^{j}, otherwise \end{cases} .

THE SUBMODULE STRUCTURE OF SPECHT MODULES 15

Now let I^{∗} be as in equation (8), but with max I^{∗} = j_{2t} − 1 = i − 1,

and let h = \begin{cases}l − (δ_{I^{∗}}e + e − a′_{λ}), if − 1 ∈ I^{∗} \\ l − δ_{I^{∗}}e, otherwise \end{cases} . We then get f − h =

\begin{cases}l − (δ_{[i_{1},i[}e + e − a′_{λ}) − l + (δ_{I^{∗}}e + e − a′_{λ}), if i_{1} = −1 and − 1 ∈ I^{∗} \\ l − (δ_{[i_{1},i[}e + e − a′_{λ}) − l + δ_{I^{∗}}e, if i_{1} = −1 and − 1 ∉ I^{∗} \\ l − δ_{[i_{1},i[}e − l + (δ_{I^{∗}}e + e − a′_{λ}), if i_{1} ≠ −1 and − 1 ∈ I^{∗} \\ l − δ_{[i_{1},i[}e − l + δ_{I^{∗}}e, if i_{1} ≠ −1 and − 1 ∉ I^{∗} \end{cases}.

But then

f − h = \begin{cases}(δ_{I^{∗}} − δ_{[i_{1},i[})e, if i_{1} = −1 and − 1 ∈ I^{∗} \\ (δ_{I^{∗}} − δ_{[i_{1},i[})e − e + a′_{λ}, if i_{1} = −1 and − 1 ∉ I^{∗} \\ (δ_{I^{∗}} − δ_{[i_{1},i[})e + e − a′_{λ}, if i_{1} ≠ −1 and − 1 ∈ I^{∗} \\ (δ_{I^{∗}} − δ_{[i_{1},i[})e, if i_{1} ≠ −1 and − 1 ∉ I^{∗} \end{cases} .

So, if i_{1} = −1 and −1 ∈ I^{∗}, we have

(f − h)^{∗} = − ∑_{j<i^{j∉I^{∗}}}(p − 1 − b_{j})p^{j} + ∑_{r=2}^{t} p^{j_{2r−1}}
(11)  = ∑_{r=1}^{t−1}(b_{j_{2r}} + 1)p^{j_{2r}} + ∑_{j<i^{{j∉I^{∗} \atop j≠j_{2},...,j_{2}t}}} b_{j}p^{j},

since − ∑_{j=j_{2r}}^{j_{2r+1}−1}(p−1)p^{j} = p^{j_{2r}}−p^{j_{2r+1}}, for all r = 1 …, t−1. Similarly, if i_{1} = −1 and −1 ∉ I^{∗}, we have

(f − h)^{∗} = δ_{I^{∗}} − δ_{[i_{1},i[} − 1
= − ∑_{j=0}^{j_{1}}(p − 1 − b_{j})p^{j} + p^{j_{1}} − ∑_{j_{1}<j<i^{j∉I^{∗}}}(p − 1 − b_{j})p^{j}

+ ∑_{r=2}^{t} p^{j_{2r−1}} − 1

(12)  = ∑_{j=0}^{j_{1}−1} b_{j}p^{j} + ∑_{r=1}^{t−1}(b_{j_{2r}} + 1)p^{j_{2r}} + ∑_{j<i^{{j∉I^{∗} \atop j≠j_{2},...,j_{2}t}}} b_{j}p^{j},

by the calculation above and since we have − ∑_{j=0}^{j_{1}−1}(p − 1)p^{j} = 1 − p^{j_{1}}. If i_{1} ≠ −1 and −1 ∈ I^{∗}, we proceed similarly:

(f − h)^{∗} = δ_{I^{∗}} − δ_{[i_{1},i[}
= ∑_{j=0}^{i_{1}−1}(p − 1 − b_{j})p^{j} − p^{i_{1}} − ∑_{i_{1}<j<i^{j∉I^{∗}}}(p − 1 − b_{j})p^{j} + ∑_{r=2}^{t} p^{j_{2r−1}}

(13)  = ∑_{j=0}^{i_{1}−1}(p − 1 − b_{j})p^{j} + ∑_{r=1}^{t−1}(b_{j_{2r}} + 1)p^{j_{2r}} + ∑_{j<i^{{j∉I^{∗} \atop j≠j_{2},...,j_{2}t}}} b_{j}p^{j},

by the above. Finally, if i_{1} ≠ −1 and −1 ∉ I^{∗}, we have

(f − h)^{∗} = ∑_{j=j_{1}}^{i_{1}−1}(p − 1 − b_{j})p^{j} − p^{i_{1}} − ∑_{i_{1}<j<i^{j∉I^{∗}}}(p − 1 − b_{j})p^{j} + ∑_{r=2}^{t} p^{j_{2r−1}}

(14)  = ∑_{j=j_{1}}^{i_{1}−1}(p − 1 − b_{j})p^{j} + ∑_{r=1}^{t−1}(b_{j_{2r}} + 1)p^{j_{2r}} + ∑_{j<i^{{j∉I^{∗} \atop j≠j_{2},...,j_{2}t}}} b_{j}p^{j},

where the sum ∑_{j=j_{1}}^{i_{1}−1}(p − 1 − b_{j})p^{j} is empty if i_{1} ≤ j_{1}. In all these cases, we get that (f − h)^{∗}, in expressions (11), (12), (13) and(14), is of a valid form for D^{(n−h,h)} to be a composition factor of S^{(n−f,f)}, by theorem 6.3. □

Remark 7.2. Note that if J^{∗} ∈ A_{λ}^{∗} and max J^{∗} < i − 1, then clearly D^{μ_{J^{∗}}(l)} ∉ F(M^{μ_{[i_{1},i[}(l)}), since δ_{[i_{1},i[} > δ_{J^{∗}} and thus (l − δ_{[i_{1},i[}e)^{∗} < (l − δ_{J^{∗}}e)^{∗}. Note also that if I^{∗}, J^{∗} ∈ A_{λ}^{∗} such that max I^{∗} = max J^{∗} = i−1 and D^{μ_{I^{∗}}(l)} ≺ D^{μ_{J^{∗}}(l)} in S^{(n−f,f)}, then D^{μ_{J^{∗}}(l)} ≺ D^{μ_{I^{∗}}(l)} in S^{(n−l,l)}, by lemma 5.2. The pictorial argument for the case where i = i_{2T} and (n − m, m) = μ_{[i_{1},i_{2T} [}(l) is given in figure 1.

We now look at what happens when two sets have the same minimum. Recall that for j ∈ ℕ ∪ {−1}, we define ĵ to be min{i ∈ B_{λ}^{∗,+} : i > j}.

Lemma 7.3. Let j ∈ B_{λ}^{∗,−}. Then for all I^{∗} ∈ A_{λ}^{∗} such that min I^{∗} = j, the composition factor D^{μ_{I^{∗}}(l)} of S^{(n−l,l)} is also a composition factor of

S^{(n−g,g)}, where g = \begin{cases}l − (δ_{[j,ĵ[}e + e − a′_{λ}), if j = −1 \\ l − δ_{[j,ĵ[}e, otherwise \end{cases} .

Proof. We proceed as in the previous lemma. If j = −1, we have n − 2g + 1 = n − 2l + 1 + 2δ_{[j,ĵ[}e + 2e − 2a′_{λ}, and if j ≠ −1, we have

[figure: M^{(n−m,m)}; F(S ∩ S^{⊥}); S^{(n−m,m)}; D^{(n−m,m)}; S^{⊥} = Ker Θ_{0}^{m,l}; S ∩ S^{⊥}; F(S ∩ S^{⊥}); 0; Θ_{0}^{m,l}; S^{(n−l,l)}; D^{(n−l,l)}; F(S ∩ S^{⊥}); D^{(n−m,m)}; 0]
Figure 1. Θ_{0}^{m,l} : M^{(n−m,m)} ↪ S^{(n−l,l)}, its image and kernel

n − 2g + 1 = n − 2l + 1 + 2δ_{[j,ĵ[}e. Thus

(n − 2g + 1)^{∗} = \begin{cases}δ_{[−1, ˆ−1[} + (b ˆ−1 + 1)p ˆ−1 + ∑_{i> ˆ−1} b_{i}p^{i}, if j = −1 \\ δ_{[j,ĵ[} + (bĵ + 1)pĵ + ∑_{i∉[j,ĵ]} b_{i}p^{i}, otherwise \end{cases} .

Now let I^{∗} be as in equation (8), but with min I^{∗} = j. Let

h = \begin{cases}l − (δ_{I^{∗}}e + e − a′_{λ}), if j = −1 \\ l − δ_{I^{∗}}e, otherwise \end{cases} .

Then we get g − h = (δ_{I^{∗}} − δ_{[j,ĵ[})e, regardless of whether j = −1 or not. Therefore

(g − h)^{∗} = ∑_{i∈I\[j,ĵ[}(p − 1 − a_{i})p^{i} + ∑_{r=2}^{t} p^{j_{2r−1}}
= ∑_{i∈I\[j,ĵ]}(p − 1 − a_{i})p^{i} + (p − 1 − (aĵ + 1))pĵ + pĵ

+ ∑_{r=2}^{t} p^{j_{2r−1}}

= ∑_{i∈I\[j,ĵ[}(p − 1 − g_{i})p^{i} + ∑_{r=2}^{t} p^{j_{2r−1}} + pĵ

is of a valid form for D^{(n−h,h)} to be a composition factor of S^{(n−g,g)}. □

The following lemma gives us some information about the order of composition factors corresponding to partitions as described in the above lemma. It tells us that their order in S^{(n−l,l)} is the same as that in S^{(n−g,g)}.

Lemma 7.4. Suppose that j ∈ B_{λ}^{∗,−} and that j ≠ i_{1} and j ≠ i_{2T}. We then have D^{μ_{[j,i_{2T} [}(l)} = S^{μ_{[j,i_{2T} [}(l)}Θ_{0}^{.,h} ≤ S^{μ_{[j,ĵ[}(l)}Θ_{0}^{.,h} ≤ S^{(n−l,l)}Θ_{0}^{l,h} in

S^{(n−h,h)}, where h = \begin{cases}l + ∑_{i=0}^{j−1} b_{i}p^{i}e + a′_{λ}, if a′_{λ} > 0 \\ l + ∑_{i=i_{1}}^{j−1} b_{i}p^{i}e, otherwise \end{cases} .

Proof. We look at x(h, l − δ_{[j,i_{2T} [}e), x(h, l − δ_{[j,ĵ[}e) and x(h, l) as defined in expression (7), provided they exist. Since

n − h − (l − δ_{[j,i_{2T} [}e) + 1

= \begin{cases}n − l − ∑_{i=0}^{j−1} b_{i}p^{i}e − a′_{λ} − l + δ_{[j,i_{2T} [}e + 1, if a′_{λ} > 0 \\ n − l − ∑_{i=i_{1}}^{j−1} b_{i}p^{i}e − l + δ_{[j,i_{2T} [}e + 1, otherwise \end{cases}

= \begin{cases}n − 2l + 1 + δ_{[j,i_{2T} [}e − ∑_{i=0}^{j−1} b_{i}p^{i}e − a′_{λ}, if a′_{λ} > 0 \\ n − 2l + 1 + δ_{[j,i_{2T} [}e − ∑_{i=i_{1}}^{j−1} b_{i}p^{i}e, otherwise \end{cases}

= \begin{cases}∑_{i≥0} b_{i}p^{i}e + a′_{λ} + δ_{[j,i_{2T} [}e − ∑_{i=0}^{j−1} b_{i}p^{i}e − a′_{λ}, if a′_{λ} > 0 \\ ∑_{i≥0} b_{i}p^{i}e + δ_{[j,i_{2T} [}e − ∑_{i=i_{1}}^{j−1} b_{i}p^{i}e, otherwise \end{cases}

= pp^{j}e + ∑_{i=j+1}^{i_{2T} −1}(p − 1)p^{i}e + ∑_{i≥i_{2T}} b_{i}p^{i}e,

we have e | (n − h − (l − δ_{[j,i_{2T} [}e) + 1). We see as well that (n − h − (l − δ_{[j,i_{2T} [}e) + 1)^{∗} ≡ 0 mod p^{i_{2T}}, and so x(h, l − δ_{[j,i_{2T} [}e) ≥ i_{2T} and D^{μ_{[j,i_{2T} [}(l)} = S^{μ_{[j,i_{2T} [}(l)}Θ_{0}^{.,h}. Similarly, we get e | (n − h − (l − δ_{[j,ĵ[}e) + 1) and (n − h − (l − δ_{[j,ĵ[}e) + 1)^{∗} ≡ 0 mod pĵ, so that x(h, l − δ_{[j,ĵ[}e) ≥ ĵ, and we get e | (n − h − l + 1) and (n − h − l + 1)^{∗} ≡ 0 mod p^{j}, so that x(h, l) ≥ j. The result then follows by theorem 5.1. □

7.2. The Comparison Result. We are now able to explore how the composition factors corresponding to comparable sets I^{∗}, J^{∗} ∈ A_{λ}^{∗} occur in S^{(n−l,l)}.

Theorem 7.5. Let I^{∗}, J^{∗} ∈ A_{λ}^{∗}. If J^{∗} ⊂ I^{∗}, then D^{μ_{I^{∗}}(l)} ≺ D^{μ_{J^{∗}}(l)} in S^{(n−l,l)}.

Proof. The submodule structure of S^{(n)} and S^{(n−1,1)}, together with the incidence structure of A_{(n)} and A_{(n−1,1)} provide the base step for the induction. Lemma 7.1 and induction show that the theorem holds for sets I^{∗} and J^{∗} such that max I^{∗} = max J^{∗}. The order of the composition factors is reversed compared to that of the respective preimage under Θ_{0}^{.,l}, by lemma 5.2. Lemmas 7.3 and 7.4 and induction show that the theorem holds for sets I^{∗} and J^{∗} such that min I^{∗} = min J^{∗}. Here the order of the composition factors is the same as that in the preimage under Θ_{0}^{.,l}, by lemma 5.3. Now, for general I^{∗}, J^{∗} ≠ ∅, suppose that J^{∗} ⊂ I^{∗}. Then we construct a set K^{∗} ∈ A_{λ}^{∗} such that

J^{∗} ⊂ K^{∗} ⊂ I^{∗}

with min K^{∗} = min I^{∗} and max K^{∗} = max J^{∗} to get D^{μ_{K^{∗}}(l)} ≺ D^{μ_{J^{∗}}(l)} and D^{μ_{I^{∗}}(l)} ≺ D^{μ_{K^{∗}}(l)} in S^{(n−l,l)}. Therefore D^{μ_{I^{∗}}(l)} ≺ D^{μ_{J^{∗}}(l)} in S^{(n−l,l)} and the result holds. □

7.3. Some Kernels in the Specht Module. We now prove a nice theorem that explains which composition factors belong to the kernel of the restriction to the Specht module of the reverse semistandard homomorphism.

Theorem 7.6. Let I^{∗}, J^{∗} ∈ A_{λ}^{∗} and let i_{3} = min I^{∗} and i_{5} = min J^{∗}

with i_{1} ≤ i_{3} < i_{5} < i_{2T}. Let h = \begin{cases}l + ∑_{i=0}^{i_{5}−1} b_{i}p^{i}e + a′_{λ}, if a′_{λ} > 0 \\ l + ∑_{i=i_{1}}^{i_{5}−1} b_{i}p^{i}e, otherwise \end{cases} .

Then D^{μ_{I^{∗}}(l)} ∈ F(S^{(n−l,l)} ∩ Ker Θ_{0}^{l,h}) and D^{μ_{J^{∗}}(l)} ∈ F(S^{(n−l,l)}Θ_{0}^{l,h}).

Proof. We first examine the case where we have i_{1} = i_{3}. We know that 0 ≠ S^{μ_{[i_{1},î_{1}[}(l)}Θ_{0}^{.,l} ≤ S^{(n−l,l)}, since

\begin{cases}n − l − (l − (δ_{[i_{1},î_{1}[}e + e − a′_{λ})) + 1, if a′_{λ} > 0 \\ n − l − (l − δ_{[i_{1},î_{1}[}e) + 1, otherwise \end{cases}

= \begin{cases}n − 2l + 1 + δ_{[−1, ˆ−1[}e + e − a′_{λ}, if a′_{λ} > 0 \\ n − 2l + 1 + δ_{[i_{1},î_{1}[}e, otherwise \end{cases}

= \begin{cases}∑ ˆ−1 ; i=0(p − 1)p^{i}e + e + ∑_{i≥ ˆ−1} b_{i}p^{i}, if a′_{λ} > 0 \\ pe + ∑î_{1} ; i=1(p − 1)p^{i}e + ∑_{i≥î_{1}} b_{i}p^{i}, otherwise \end{cases}
≡ 0 mod pî_{1}.

Now suppose that h = \begin{cases}l + ∑_{i=0}^{i_{5}−1} b_{i}p^{i}e + a′_{λ}, if a′_{λ} > 0 \\ l + ∑_{i=i_{1}}^{i_{5}−1} b_{i}p^{i}e, otherwise \end{cases} . Then, by

lemma 7.4, we know that 0 ≠ S^{μ_{[i_{5},î_{5}[}(l)}Θ_{0}^{.,h} ≤ S^{(n−l,l)}Θ_{0}^{l,h} ≤ S^{(n−h,h)}. We now want to examine S^{μ_{[i_{1},î_{1}[}(l)}Θ_{0}^{.,l}Θ_{0}^{l,h}. We consider the cases a′_{λ} > 0 and a′_{λ} = 0 separately. For a′_{λ} > 0, we have

[\begin{matrix}h − (l − (δ_{[i_{1},î_{1}[}e + e − a′_{λ})) \\ l − (l − (δ_{[i_{1},î_{1}[}e + e − a′_{λ})) \end{matrix}] =

[\begin{matrix}∑_{i=0}^{i_{5}−1} b_{i}p^{i}e + δ_{[−1, ˆ−1[}e + e \\ δ_{[−1, ˆ−1[}e + e − a′_{λ} \end{matrix}]
= [\begin{matrix}e + (b ˆ−1 + 1)p ˆ−1e + ∑_{i= ˆ−1+1}^{i_{5}−1} b_{i}p^{i}e \\ δ_{[−1, ˆ−1[}e + e − a′_{λ} \end{matrix}]
≡ 0 mod p,

by theorem 2.2. For a′_{λ} = 0, we similarly have

[\begin{matrix}h − (l − δ_{[i_{1},î_{1}[}e) \\ l − (l − δ_{[i_{1},î_{1}[}e) \end{matrix}] ≡ (\begin{matrix}∑_{i=i_{1}}^{i_{3}−1} b_{i}p^{i} + δ_{[i_{1},î_{1}[} \\ δ_{[i_{1},î_{1}[} \end{matrix})
≡ (\begin{matrix}(bî_{1} + 1)pî_{1} + ∑_{i=î_{1}+1}^{i_{3}−1} b_{i}p^{i} \\ δ_{[i_{1},î_{1}[} \end{matrix})
≡ 0 mod p,

by theorem 2.2. Thus S^{μ_{[i_{1},î_{1}[}(l)}Θ_{0}^{.,l}Θ_{0}^{l,h} = 0. Furthermore we have S^{μ_{[i_{1},î_{1}[}(l)}Θ_{0}^{.,l} ≤ Ker Θ_{0}^{l,h}∩S^{(n−l,l)} and for all L^{∗} ∈ A_{λ}^{∗} such that min L^{∗} = i_{1}, so in particular for I^{∗}, we have D^{μ_{L^{∗}}(l)} ≺ D^{μ_{[i_{1},î_{1}[}(l)} in S^{(n−l,l)}, by theorem 7.5, so that D^{μ_{L^{∗}}(l)} ∈ F(Ker Θ_{0}^{l,h} ∩ S^{(n−l,l)}). By theorem 7.5, we know as well that D^{μ_{[i_{5},i_{2T} [}(l)} ≺ D^{μ_{J^{∗}}(l)} in S^{(n−l,l)}, and by lemma 7.4 we know that D^{μ_{[i_{5},i_{2T} [}(l)} is the unique bottom composition factor of S^{(n−h,h)}. Therefore D^{μ_{J^{∗}}(l)} ∈ F(S^{(n−l,l)}Θ_{0}^{l,h}). We now examine the case where i_{1} < i_{3} here. Note that in the case i_{1} = i_{3} examined above, i_{5} plays the role that i_{3} plays here. Furthermore, we assume that there is no K^{∗} ∈ A_{λ}^{∗} such that i_{1} < min K^{∗} < i_{3} or i_{3} < min K^{∗} < i_{5}. If this were the case, we would just have to apply the following argument first for min K^{∗}, then for i_{5}. Therefore we have that S^{μ_{[i_{3},î_{3}[}(l)}Θ_{0}^{.,k} is isomorphic to a subfactor of S^{(n−l,l)}, or S^{μ_{[i_{3},î_{3}[}(l)}Θ_{0}^{.,k} ≤ S^{(n−l,l)}/(Ker Θ_{0}^{l,k} ∩ S^{(n−l,l)}), where

k = \begin{cases}l + ∑_{i=0}^{i_{3}−1} b_{i}p^{i}e + a′_{λ}, if a′_{λ} > 0 \\ l + ∑_{i=i_{1}}^{i_{3}−1} b_{i}p^{i}e, otherwise \end{cases}

plays the role of h in the case above. We now look at S^{(n−l,l)}Θ_{0}^{l,h}. Then, by lemma 7.4, we know again that 0 ≠ S^{μ_{[i_{5},î_{5}[}(l)}Θ_{0}^{.,h} ≤ S^{(n−l,l)}Θ_{0}^{l,h} ≤ S^{(n−h,h)}. But, for j = i_{1} or j = i_{3}, we have e | (n − h − (l − δ_{[j,ĵ[}e) + 1) and (n − h − (l − δ_{[j,ĵ[}e) + 1)^{∗} ≡ 0 mod p^{j}, whereas e | (n − h − l + 1) and (n − h − l + 1)^{∗} ≡ 0 mod p^{i_{5}}, by the calculation in the proof of lemma 7.4. Thus S^{μ_{[j,ĵ[}(l)}Θ_{0}^{.,h} ≰ S^{(n−h,h)} by theorem 5.1: if we had S^{μ_{[j,ĵ[}(l)}Θ_{0}^{.,h} ≤ S^{(n−h,h)}, then we would have S^{μ_{[j,ĵ[}(l)}Θ_{0}^{.,h} ≤ S^{(n−l,l)}Θ_{0}^{l,h} ≤ S^{(n−h,h)} and we would need j > i_{5}, which contradicts j = i_{1} or j = i_{3}. But 0 ≠ S^{μ_{[i_{1},î_{1}[}(l)}Θ_{0}^{.,l} ≤ S^{(n−l,l)}, so

S^{μ_{[i_{1},î_{1}[}(l)}Θ_{0}^{.,l}Θ_{0}^{l,h} = \begin{cases}either S^{μ_{[i_{1},î_{1}[}(l)}Θ_{0}^{.,h} \\ or 0 \end{cases} .

Since 0 ≠ S^{(n−l,l)}Θ_{0}^{l,h} ≤ S^{(n−h,h)}, we have necessarily S^{μ_{[i_{1},î_{1}[}(l)}Θ_{0}^{.,l} ≤ Ker Θ_{0}^{l,h} ∩ S^{(n−l,l)}. Furthermore, for all L^{∗} ∈ A_{λ}^{∗} such that min L^{∗} = i_{1}, we have D^{μ_{L^{∗}}(l)} ≺ D^{μ_{[i_{1},î_{1}[}(l)} in S^{(n−l,l)}, by theorem 7.5 so that D^{μ_{L^{∗}}(l)} ∈ F(Ker Θ_{0}^{l,h} ∩ S^{(n−l,l)}). Similarly 0 ≠ S^{μ_{[i_{3},î_{3}[}(l)} ≤ S^{(n−l,l)}/(S^{μ_{[i_{1},î_{1}[}(l)}Θ_{0}^{.,l}) and

S^{μ_{[i_{3},î_{3}[}(l)}Θ_{0}^{.,l}Θ_{0}^{l,h} = \begin{cases}either S^{μ_{[i_{3},î_{3}[}(l)}Θ_{0}^{.,h} \\ or 0 \end{cases} ,

since S^{μ_{[i_{1},î_{1}[}(l)}Θ_{0}^{.,l} ≤ Ker Θ_{0}^{l,h} ∩ S^{(n−l,l)}. Thus we have S^{μ_{[i_{3},î_{3}[}(l)}Θ_{0}^{.,k} ≤ Ker Θ_{0}^{l,h} ∩ (S^{(n−l,l)}/(S^{μ_{[i_{1},î_{1}[}(l)}Θ_{0}^{.,l})). Then for all L^{∗} ∈ A_{λ}^{∗} such that

min L^{∗} = i_{3} —so in particular for I^{∗}— we have D^{μ_{L^{∗}}(l)} ≺ D^{μ_{[i_{3},î_{3}[}(l)} in S^{(n−l,l)}, by theorem 7.5, so that D^{μ_{L^{∗}}(l)} ∈ F (Ker Θ_{0}^{l,h} ∩ S^{(n−l,l)}). But

since we know, by lemma 7.4, that D^{μ_{[i_{5},i_{2T} [}(l)} is the unique bottom composition factor of S^{(n−h,h)}, we have D^{μ_{J^{∗}}(l)} ∈ F(S^{(n−l,l)}Θ_{0}^{l,h}), as before. □

7.4. Properties of Some Incomparable Sets. We now examine the origin of some composition factors corresponding to sets of A_{λ}^{∗} that are incomparable in the incidence order.

Lemma 7.7. Let λ = (n−l, l) and let I^{∗}, J^{∗} ∈ A_{λ}^{∗}, such that I^{∗}, J^{∗} ≠ ∅ and such that neither I^{∗} ⊂ J^{∗} nor J^{∗} ⊂ I^{∗}. Assume further that I^{∗} and J^{∗} are as follows

I^{∗} = [j_{1}, j_{2}[∪ … ∪ [j_{2t−1}, j_{2t}[
(15)  J^{∗} = [h, k[,

with j_{1} < … < j_{2t} and h < k < j_{2t}. We have the following properties

(1) if I^{∗} ∩ J^{∗} = ∅ then the composition factor D^{μ_{I^{∗}}(l)} of S^{(n−l,l)} is not a composition factor of S^{μ_{J^{∗}}(l)};

(2) if there are s ≤ u with s, u = 1, …, t − 1, such that j_{2s−1} ≤ h < j_{2s} ≤ k ≤ j_{2u+2} then the composition factor D^{μ_{I^{∗}}(l)} of S^{(n−l,l)} is a composition factor of S^{μ_{J^{∗}}(l)} if and only if j_{2s−1} = h and j_{2u+1} < k ≤ j_{2u+2};

(3) if there is an s = 1, …, t − 1 such that j_{2s} ≤ h < j_{2s+1} < k ≤ j_{2s+2}, then the composition factor D^{μ_{I^{∗}}(l)} of S^{(n−l,l)} is a composition factor of S^{μ_{J^{∗}}(l)}.

Proof. Let g = \begin{cases}l − (δ_{J^{∗}}e + e − a′_{λ}), if h = −1 \\ l − δ_{J^{∗}}e, otherwise \end{cases} and let γ = (n −

g, g). We compute a_{γ}^{∗} = (n − 2g + 1)^{∗} = δ_{[h,k[} + (b_{k} + 1)p^{k} + ∑_{i∉[h,k]} b_{i}p^{i} as we did for (n − 2f + 1)^{∗} in lemma 7.1. Then we put

(16)  a_{γ}^{∗} = ∑_{i≥i_{1}} g_{i}p^{i},

by equating the coefficient g_{i} in (16) with the coefficient of p^{i} in the p–adic expansion of a_{γ}^{∗} above. For

j = \begin{cases}l − (δ_{I^{∗}}e + e − a′_{λ}), if − 1 ∈ I^{∗} \\ l − δ_{I^{∗}}e, otherwise \end{cases} ,

we then calculate (g − j)^{∗} in the different cases considered. For sets of the form (1), if h = −1, we have g −j = δ_{I^{∗}}e−δ_{J^{∗}}e−e+a′_{λ}, whereas if h ≠ −1, we get

(17)  g − j = \begin{cases}δ_{I^{∗}}e − δ_{J^{∗}}e + e − a′_{λ}, if − 1 ∈ J^{∗} \\ δ_{I^{∗}}e − δ_{J^{∗}}e, otherwise \end{cases} .

Thus we get (taking h = 0 in the calculation below to cover the case h = −1):

(g − j)^{∗} = δ_{I^{∗}} − δ_{J^{∗}}
= δ_{I^{∗}} − ∑_{i∈J^{∗}}(p − 1 − b_{i})p^{i} − p^{h}

= δ_{I^{∗}} + ∑_{i∈J^{∗}} b_{i}p^{i} − (∑_{i∈J^{∗}}(p − 1)p^{i} + p^{h})
(18)  = δ_{I^{∗}} + ∑_{i∈J^{∗}} b_{i}p^{i} − p^{k}.

This is not of a valid form for D^{(n−j,j)} to be a composition factor of S^{(n−g,g)}, because of the term −p^{k}. For sets I^{∗} and J^{∗} of the form (2), we may take u = s, since the reasoning will be the same as for s < u, but the notation will be vastly improved. Let us suppose first that j_{2s−1} < h and that k < j_{2s+1}. Then h ≠ −1 and g − j is as given in (17) above. It can then be shown that

(19)  (g − j)^{∗} = ∑_{i∈I^{∗}\[j_{2s−1},j_{2s}[}(p − 1 − g_{i})p^{i} + ∑_{r≠s^{r=1}}^{t} p^{j_{2r−1}}

+ ∑_{i∈[j_{2s−1},h[}(p − 1 − g_{i})p^{i} + p^{j_{2s−1}} − p^{h}

+ ∑_{i∈[j_{2s},k[}(p − 1 − g_{i})p^{i} + p^{j_{2s}} − p^{k},

by a similar argument as above. Since p − 1 − g_{h} = b_{h} − 1 ≠ −1 and p − 1 − g_{k} = p − 2 − b_{k} ≠ −1, this is not of the required form for D^{(n−j,j)} to be a composition factor of S^{(n−g,g)}. Now if we take j_{2s−1} = h and k < j_{2s+1}, the sum ∑_{i∈[j_{2s−1},h[}(p − 1 − g_{i})p^{i} + p^{j_{2s−1}} − p^{h} in the penultimate line of (19) is empty, but (g − j)^{∗} is still not of a valid form for D^{(n−j,j)} to be a composition factor of S^{(n−g,g)}, because of the last line of (19). If we take j_{2s−1} < h and j_{2s+1} < k ≤ j_{2s+2}, the sum in the penultimate line of (19) is not empty, and (g − j)^{∗} is not of a valid form for D^{(n−j,j)} to be a composition factor of S^{(n−g,g)}. If we take j_{2s−1} = h and j_{2s+1} < k ≤ j_{2s+2}, then g − j = δ_{I^{∗}}e − δ_{J^{∗}}e and

(g − j)^{∗} = ∑_{i∈I^{∗}\([j_{2s−1},j_{2s}[∪[j_{2s+1},j_{2s+2}[)}(p − 1 − g_{i})p^{i} + ∑_{r≠s,s+1^{r=1}}^{t} p^{j_{2r−1}}

+ ∑_{i∈[j_{2s},j_{2s+1}[}(p − 1 − g_{i})p^{i} + p^{j_{2s}}

+ ∑_{i∈[k,j_{2s+2}[}(p − 1 − g_{i})p^{i} + p^{k},

by a similar argument as above. This is of a valid form for D^{(n−j,j)} to be a composition factor of S^{(n−g,g)}. The only remaining case to investigate here is for −1 = j_{1} < h and j_{2s+1} < k ≤ j_{2s+2}. Without loss of generality, we may take s = 1 and I^{∗} = [−1, j_{2}[∪[j_{3}, j_{4}[, since we are only interested in what happens in–between −1 and h. We have in this case: g − j = l − δ_{[h,k[}e − l + δ_{I^{∗}}e + e − a′_{λ}, so that

(20)  (g − j)^{∗} = ∑_{i=0}^{h−1}(p − 1 − g_{i})p^{i} − p^{h} + ∑_{i=j_{2}}^{j_{3}−1} g_{i}p^{i} + p^{j_{2}}

+ ∑_{i=k}^{j_{4}}(p − 1 − g_{i})p^{i} + p^{k}.

This is not of the required form for D^{(n−j,j)} to be a composition factor of S^{(n−g,g)}, because of the term −p^{h} appearing in (20). For sets I^{∗} an J^{∗} of the form (3), we necessarily have h > −1 and so g − j is as in (17). Therefore, it can be shown that

(g − j)^{∗} = ∑_{i∈I^{∗}\[j_{2s+1},j_{2s+2}[}(p − 1 − g_{i})p^{i} + ∑_{r≠s+1^{r=1}}^{t} p^{j_{2r−1}}

+ ∑_{i∈[h,j_{2s+1}[}(p − 1 − g_{i})p^{i} + p^{h} + ∑_{i∈[k,j_{2s+2}[}(p − 1 − g_{i})p^{i}

+p^{k}. (21)

This is of a valid form for D^{(n−j,j)} to be a composition factor of S^{(n−g,g)}. □

We now give two more lemmas, that discuss composition factors appearing or not appearing in the image of certain homomorphisms. Recall that for b ∈ ℕ, we denote max{j ∈ B_{λ}^{−} : j < b} by b̃, if it exists.

Lemma 7.8. Let J^{∗} = [a, b[∈ A_{λ}^{∗}, for some a ∈ B_{λ}^{∗,−} and b ∈ B_{λ}^{∗,+}, with i_{1} < a < b ≤ ℓ_{p}(l). Let J_{1}^{∗} = [b̃, b[∈ A_{λ}^{∗} and let J_{0}^{∗} = [i_{1}, b[∈ A_{λ}^{∗}. Let Θ_{0}^{.,l} be the reverse semistandard homomorphism: M^{μ_{J^{∗}_{0}}(l)} → M^{(n−l,l)} and let

(22)  U = ⋂_{k=0}^{l−δ_{J^{∗}_{1}}e−1} Ker ψ_{k} ∩ M^{μ_{J^{∗}_{0}}(l)}

(23)  V = ⋂_{k=0}^{l−δ_{J^{∗}}e−1} Ker ψ_{k} ∩ M^{μ_{J^{∗}_{0}}(l)}.

Then V Θ_{0}^{.,l} ≤ UΘ_{0}^{.,l} < S^{(n−l,l)} and D^{μ_{J^{∗}}(l)} ∈ F(V Θ_{0}^{.,l}).

Proof. Please refer to figure 2 for a pictorial description of M^{μ_{J^{∗}_{0}}(l)}. Since (n−l−(l−δ_{J_{0}^{∗}})+1)/e = (n−l−(l−δ_{J_{0}})+1)^{∗} ≡ 0 mod p^{b} by the proof of lemma 7.4, we know that UΘ_{0}^{.,l} ≤ S^{(n−l,l)} by (4) and theorem

[figure: 0; V ∩ (U ∩ Ker Θ_{0}^{.,l}); V; U ∩ Ker Θ_{0}^{.,l}; V + (U ∩ Ker Θ_{0}^{.,l}); U; U + Ker Θ_{0}^{.,l}; Ker Θ_{0}^{.,l}; M^{μ_{J^{∗}_{0}}(l)}]

Figure 2. Diagram of M^{μ_{J^{∗}_{0}}(l)}, U, V and Ker Θ_{0}^{.,l}

5.1. Therefore V Θ_{0}^{.,l} ≤ UΘ_{0}^{.,l} ≤ S^{(n−l,l)}, since V ≤ U ≤ M^{μ_{J^{∗}_{0}}(l)}. Suppose that the multiplicity of D^{μ_{J^{∗}}(l)} in V is y:

(24)  [V : D^{μ_{J^{∗}}(l)}] = y.

Then so is the multiplicity of D^{μ_{J^{∗}}(l)} in M^{μ_{J^{∗}_{0}}(l)}, since M^{μ_{J^{∗}_{0}}(l)}/V has a Specht filtration with factors S^{(n)}, S^{(n−1,1)}, …, S^{α}, where α is the 2–part partition equal to min{μ : μ ▷ μ_{J^{∗}}(l)} (recall that the dominance order for partitions is a total order if we only consider 2–part partitions). But V ≤ U ≤ M^{μ_{J^{∗}_{0}}(l)} and so

(25)  [U : D^{μ_{J^{∗}}(l)}] = y.

Now D^{μ_{J^{∗}_{1}}(l)} ∈ F(UΘ_{0}^{.,l}), since D^{μ_{J^{∗}_{1}}(l)} is the top composition factor of U, and therefore of M^{μ_{J^{∗}_{0}}(l)}Θ_{0}^{.,l} ∩ S^{(n−l,l)}, by the maximality of b̃, theorem 7.5 and remark 7.2. But D^{μ_{J^{∗}}(l)} ≺ D^{μ_{J^{∗}_{1}}(l)} in S^{(n−l,l)}, since J_{1}^{∗} ⊂ J^{∗}, so D^{μ_{J^{∗}}(l)} ∈ F(UΘ_{0}^{.,l}) and

(26)  [UΘ_{0}^{.,l} : D^{μ_{J^{∗}}(l)}] = 1,

since UΘ_{0}^{.,l} ≤ S^{(n−l,l)}. By (24) and (25) we know that D^{μ_{J^{∗}}(l)} ∉ F(U/V), therefore

(27)  D^{μ_{J^{∗}}(l)} ∉ F (U/(V + (U ∩ Ker Θ_{0}^{.,l}))) .

Now, by (26) and (27), we know that D^{μ_{J^{∗}}(l)} is a composition factor (once) of (V + (U ∩ Ker Θ_{0}^{.,l}))/(U ∩ Ker Θ_{0}^{.,l}) ≅ V/(V ∩ Ker Θ_{0}^{.,l}) ≅

V Θ_{0}^{.,l}, so D^{μ_{J^{∗}}(l)} ∈ F(V Θ_{0}^{.,l}). □

Lemma 7.9. Let I^{∗}, J^{∗} ∈ A_{λ}^{∗} be of the form

I^{∗} = [j_{1}, j_{2}[∪ … ∪ [j_{2t−1}, j_{2t}[
J^{∗} = [h, k[,

with j_{1} < … < j_{2t} and h < k < j_{2t}. Let J_{1}^{∗} = [k̃, k[∈ A_{λ}^{∗}, let J_{0}^{∗} = [i_{1}, k[∈ A_{λ}^{∗} and let Θ_{0}^{.,l} be the reverse semistandard homomorphism: M^{μ_{J^{∗}_{0}}(l)} → M^{(n−l,l)}. If I^{∗} ∩ J_{1}^{∗} = ∅, then D^{μ_{I^{∗}}(l)} ∉ F(M^{μ_{J^{∗}_{0}}(l)}Θ_{0}^{.,l} ∩ S^{(n−l,l)}).

Proof. Let d_{1} = k̃, d_{2} = d̃_{1}, …, d_{r} = i_{1} and for all s = 1, …, r, let J_{s}^{∗} = [d_{s}, k[. Note that with our notation we have J_{r}^{∗} = J_{0}^{∗}. Now, since I^{∗} ∩ J_{1}^{∗} = ∅, we have that for all s = 1, …, r, the set J_{s}^{∗} is one of the forms (1) or (2) described for J^{∗} in lemma 7.7, but such that we do not have j_{2u+1} < k ≤ j_{2u+2}, for any u = 1, …, t − 1. Hence D^{μ_{I^{∗}}(l)} ∉ F(S^{μ_{J^{∗}_{s}}(l)}), for any s = 1, …, r. But then D^{μ_{I^{∗}}(l)} ∉ F(M^{μ_{J^{∗}_{0}}(l)}Θ_{0}^{.,l}∩S^{(n−l,l)}), since M^{μ_{J^{∗}_{0}}(l)}Θ_{0}^{.,l}∩S^{(n−l,l)} has a Specht filtration with factors S^{μ_{J^{∗}_{1}}(l)}, …, S^{μ_{J^{∗}_{r}}(l)}. □

7.5. The Incomparability Result. We are now able to give our main result.

Theorem 7.10. The posets (A_{λ}^{∗}, ⊃) and (F(S^{(n−l,l)}), ≺) are poset– isomorphic.

Proof. Theorem 7.5 shows that if I^{∗}, J^{∗} ∈ A_{λ}^{∗} and I^{∗} ⊂ J^{∗}, then D^{μ_{I^{∗}}(l)} ≺ D^{μ_{J^{∗}}(l)} in S^{(n−l,l)}. For incomparable I^{∗}, J^{∗} ∈ A_{λ}^{∗} we proceed as follows. If max J^{∗} = max I^{∗} or min J^{∗} = min I^{∗}, the composition factors D^{μ_{I^{∗}}(l)} and D^{μ_{J^{∗}}(l)} are incomparable by induction and lemmas 7.1 to 7.4. Otherwise, assume that J^{∗} ⊄ I^{∗}. Then we will prove that D^{μ_{I^{∗}}(l)} ⊀ D^{μ_{J^{∗}}(l)} in S^{(n−l,l)}. We have one of the following cases

(1) min J^{∗} < min I^{∗}

(2) max J^{∗} > max I^{∗}

(3) min J^{∗} > min I^{∗} and max J^{∗} < max I^{∗}.

Case (1): Let min J^{∗} < min I^{∗}. If min J^{∗} = i_{1}, then D^{μ_{J^{∗}}(l)∈F(S^{μ_{[i_{1},î_{1}[}(l)})}

and D^{μ_{I^{∗}}(l)∉F(S^{μ_{[i_{1},î_{1}[}(l)})} and we are done. Thus we may take min J^{∗} > i_{1}.

We then look at Θ_{0}^{l,h} : M^{(n−l,l)} → M^{(n−h,h)}, where

h = \begin{cases}l + ∑_{i=0}^{min I^{∗}−1} b_{i}p^{i}e + a′_{λ}, if a′_{λ} > 0 \\ l + ∑_{i=i_{1}}^{min I^{∗}−1} b_{i}p^{i}e, otherwise \end{cases} .

We then have D^{μ_{J^{∗}}(l)} ∈ F(Ker Θ_{0}^{l,h}∩S^{(n−l,l)}) and D^{μ_{I^{∗}}(l)} ∉ F(Ker Θ_{0}^{l,h}∩ S^{(n−l,l)}), by theorem 7.6, so that D^{μ_{I^{∗}}(l)} ⊀ D^{μ_{J^{∗}}(l)} in S^{(n−l,l)}. Case (2): Let max J^{∗} > max I^{∗}. Let max J^{∗} = i − 1 for some i ∈ B_{λ}^{∗,+}. Let Θ_{0}^{.,l} be the reverse semistandard homomorphism M^{μ_{[i_{1},i[}(l)} → M^{(n−l,l)}. Then D^{μ_{J^{∗}}(l)} ∈ F(M^{μ_{[i_{1},i[}(l)}Θ_{0}^{.,l} ∩ S^{(n−l,l)}) by lemma 7.1 and D^{μ_{I^{∗}}(l)} ∉ F(M^{μ_{[i_{1},i[}(l)}Θ_{0}^{.,l} ∩ S^{(n−l,l)}), by the first note in remark 7.2, so that D^{μ_{I^{∗}}(l)} ⊀ D^{μ_{J^{∗}}(l)} in S^{(n−l,l)}. Case (3): Let min J^{∗} > min I^{∗} and max J^{∗} < max I^{∗}. This case needs to be examined in greater detail. Let I^{∗} and J^{∗} be as follows:

I^{∗} = [j_{1}, j_{2}[∪ … ∪ [j_{2t−1}, j_{2t}[,
J^{∗} = [k_{1}, k_{2}[∪ … ∪ [k_{2r−1}, k_{2r}[,

with i_{1} ≤ j_{1} < j_{2} < … < j_{2t} and i_{1} < k_{1} < … < k_{2r} < j_{2t}. Since J^{∗} ⊄ I^{∗}, we have J^{∗} \ (J^{∗} ∩ I^{∗}) ≠ ∅ and

(28)  J^{∗} \ (J^{∗} ∩ I^{∗}) is a union of intervals.

For the remainder of this proof, let Θ_{0}^{.,l} be the reverse semistandard homomorphism M^{μJ̃^{∗}_{0}(l)} → M^{(n−l,l)} and let

(29)  U = M^{μJ̃^{∗}_{0}(l)} ∩ ⋂_{i=0}^{l−δJ̃^{∗}−1} Ker ψ_{i},

for the different J̃^{∗} and J̃_{0}^{∗} to be described below. Then UΘ_{0}^{.,l} ≤ S^{(n−l,l)}, by lemma 7.8 and since J̃^{∗} ∈ A_{λ}^{∗}, as we will see below.

(a) Suppose first that there is an interval J̃^{∗} = [a, b[∈ A_{λ}^{∗} in the union (28). Let J̃_{0}^{∗} = [i_{1}, b[. Then J̃^{∗} ∩ I^{∗} = ∅ and D^{μ_{I^{∗}}(l)} ∉ F(UΘ_{0}^{.,l}), by lemma 7.9. But since J̃^{∗} ⊂ J^{∗}, we have D^{μ_{J^{∗}}(l)} ≺ D^{μJ̃^{∗}(l)}, by the comparability result (theorem 7.5), and so D^{μ_{J^{∗}}(l)} ∈ F(UΘ_{0}^{.,l}). Thus D^{μ_{I^{∗}}(l)} ⊀ D^{μ_{J^{∗}}(l)} in S^{(n−l,l)}.

(b) Next suppose that there is no interval of A_{λ}^{∗} in the union (28), but that [a, b[, such that a ∈ B_{λ}^{∗,−}, appears as an interval in this union. Then let J̃^{∗} = [a, â[∈ A_{λ}^{∗} and let J̃_{0}^{∗} = [i_{1}, â[. If J̃^{∗} ∩ I^{∗} = ∅ then D^{μ_{I^{∗}}(l)} ∉ F(UΘ_{0}^{.,l}), but D^{μ_{J^{∗}}(l)} ∈ F(UΘ_{0}^{.,l}) by the case (a) above. Otherwise, J̃^{∗} and I^{∗} are of the same form as J^{∗} and I^{∗} are respectively in point (3) of lemma 7.7 and D^{μ_{I^{∗}}(l)} ∈ F(S^{μJ̃^{∗}(l)}). By the proof of that point, we have as well

(30)  μ_{I^{∗}}(l) = μ_{(I^{∗}∪J̃^{∗})\(I^{∗}∩J̃^{∗})}(l − δJ̃^{∗})
(31)  μ_{I^{∗}∪J̃^{∗}}(l) = μ_{I^{∗}\(I^{∗}∩J̃^{∗})}(l − δJ̃^{∗}).

Now since I^{∗} \ (I^{∗} ∩ J̃^{∗}) ⊂ (I^{∗} ∪ J̃^{∗}) \ (I^{∗} ∩ J̃^{∗}), we have, by theorem 7.5, D^{μ_{(I^{∗}∪J̃^{∗})\(I^{∗}∩J̃^{∗})}(l−δJ̃^{∗})} ≺ D^{μ_{I^{∗}\(I^{∗}∩J̃^{∗})}(l−δJ̃^{∗})} in S^{μJ̃^{∗}(l)}, or, written differently:

(32)  D^{μ_{I^{∗}}(l)} ≺ D^{μ_{I^{∗}∪J̃^{∗}}(l)} in S^{μJ̃^{∗}(l)}.

Let V = U ∩ Ker ψ_{l−δJ̃^{∗}}. Since U/V ≅ S^{μJ̃^{∗}(l)}, we have D^{μ_{I^{∗}}(l)} ≺ D^{μ_{I^{∗}∪J̃^{∗}}(l)} in U/(V ∩ Ker Θ_{0}^{.,l}). But since J̃^{∗} ⊂ (I^{∗} ∪ J̃^{∗}), we have D^{μ_{I^{∗}∪J̃^{∗}}(l)} ≺ D^{μJ̃^{∗}(l)} in S^{(n−l,l)} and so D^{μ_{I^{∗}∪J̃^{∗}}(l)} ∈ F(UΘ_{0}^{.,l}). Furthermore, since UΘ_{0}^{.,l} ≅ U/(U ∩ Ker Θ_{0}^{.,l}) ≤ S^{(n−l,l)}, we have either

\begin{cases}D^{μ_{I^{∗}}(l)} ≺ D^{μ_{I^{∗}∪J̃^{∗}}(l)} in UΘ_{0}^{.,l} or \\ D^{μ_{I^{∗}}(l)} ∉ F(UΘ_{0}^{.,l}) \end{cases} .

But now I^{∗} ⊂ (I^{∗} ∪ J̃^{∗}) and so D^{μ_{I^{∗}∪J̃^{∗}}(l)} ≺ D^{μ_{I^{∗}}(l)} in S^{(n−l,l)}, by theorem 7.5. Therefore D^{μ_{I^{∗}}(l)} ∉ F(UΘ_{0}^{.,l}). But since J̃^{∗} ⊂ J^{∗}, we have D^{μ_{J^{∗}}(l)} ∈ F(UΘ_{0}^{.,l}), and thus D^{μ_{I^{∗}}(l)} ⊀ D^{μ_{J^{∗}}(l)} in S^{(n−l,l)}.

(c) Thirdly, we suppose that there are no intervals of A_{λ}^{∗}, nor any intervals K^{∗} with min K^{∗} ∈ B_{λ}^{∗,−} in the union (28), but that [a, b[, with b ∈ B_{λ}^{∗,+}, appears as an interval in this union. Then let J̃^{∗} = [b̃, b[∈ A_{λ}^{∗} and let J̃_{0}^{∗} = [i_{1}, b[. Then J̃^{∗} ∩ I^{∗} ≠ ∅, but J̃^{∗} and I^{∗} are of the same form as J^{∗} and I^{∗} are respectively in point (2) of lemma 7.7, without j_{2s+1} < b ≤ j_{2s+2}, for any s = 1, …, t − 1. Therefore we have D^{μ_{I^{∗}}(l)} ∉ F(UΘ_{0}^{.,l}), by the proof of lemma 7.9. But D^{μ_{J^{∗}}(l)} ∈ F(UΘ_{0}^{.,l}), since J̃^{∗} ⊂ J^{∗}. Thus D^{μ_{I^{∗}}(l)} ⊀ D^{μ_{J^{∗}}(l)} in S^{(n−l,l)}.

(d) Finally, let us suppose that none of the assumptions about the union (28) in cases (a), (b), and (c) apply. Then there is an interval in the union of J^{∗} \ (J^{∗} ∩ I^{∗}) of the form [a, b[ for a ∉ B_{λ}^{∗,−} and b ∉ B_{λ}^{∗,+}. Let J̃^{∗} = [ã, b̂[∈ A_{λ}^{∗} and let J̃_{0}^{∗} = [i_{1}, b̂[. Let d_{1} = ã, d_{2} = d̃_{1},…, d_{v} = i_{1}, and for u = 1, …, v, let J̃_{u}^{∗} = [d_{u}, b̂[. Note that with this notation we have J̃^{∗} = J̃_{1}^{∗} and J̃_{0}^{∗} = J̃_{v}^{∗}. Since J̃^{∗} ⊂ J^{∗}, we have D^{μ_{J^{∗}}(l)} ∈ F(UΘ_{0}^{.,l}). Now, for all u = 1, …, v, we have J̃_{u}^{∗} ∩ I^{∗} ≠ ∅ and J̃_{u}^{∗} and I^{∗} are of the same form as J^{∗} and I^{∗} are respectively in point (2) of lemma 7.7, but this time with j_{2s+1} < b̂ ≤ j_{2s+2}, for some s = 1, …, t−1. If d_{u} ≠ j_{2x−1} however, for any x = 1, …, s, then D^{μ_{I^{∗}}(l)} ∉ F(S^{μJ̃^{∗}_{u}(l)}) by that lemma. We therefore try all successive J̃_{1}^{∗}, J̃_{2}^{∗}, … up until we get a J̃_{u}^{∗} such that d_{u} = j_{2x−1} for some x = 1, …, s. Then D^{μ_{I^{∗}}(l)} ∈ F(S^{μJ̃^{∗}_{u}(l)}) by lemma 7.7 and since I^{∗} ⊂ (I^{∗} ∪ J̃_{u}^{∗}), we derive

(33)  D^{μ_{I^{∗}}(l)} ≺ D^{μ_{I^{∗}∪J̃^{∗}_{u}}(l)} in S^{μJ̃^{∗}_{u}(l)}

by using an similar argument as above for (30), (31) and (32), but with J̃_{u}^{∗} in the role of J̃^{∗}. Let V_{u} = U ∩ ⋂_{i=l−δJ̃^{∗}}^{l−δJ̃^{∗}_{u}−1} Ker ψ_{i} and let V_{u_{∗}} = V_{u}∩Ker ψ_{l−δJ̃^{∗}_{u}}. Then we know that D^{μ_{I^{∗}}(l)} ∉ F(U/V_{u}). Moreover, since

V_{u}/V_{u_{∗}} ≅ S^{μJ̃^{∗}_{u}(l)}, we have D^{μ_{I^{∗}}(l)} ≺ D^{μ_{I^{∗}∪J̃^{∗}_{u}}(l)} in V_{u}/(V_{u_{∗}} ∩ Ker Θ_{0}^{.,l}). We have D^{μ_{I^{∗}∪J̃^{∗}_{u}}(l)} ∈ F(V_{u}Θ_{0}^{.,l}) as well, since J̃_{u}^{∗} ⊂ (I^{∗} ∪ J̃_{u}^{∗}). But V_{u}Θ_{0}^{.,l} ≅ V_{u}/(V_{u} ∩ Ker Θ_{0}^{.,l}) ≤ S^{(n−l,l)}, so we have either

\begin{cases}D^{μ_{I^{∗}}(l)} ≺ D^{μ_{I^{∗}∪J̃^{∗}_{u}}(l)} in V_{u}Θ_{0}^{.,l} or \\ D^{μ_{I^{∗}}(l)} ∉ F(V_{u}Θ_{0}^{.,l}) \end{cases} .

But now I^{∗} ⊂ (I^{∗} ∪ J̃_{u}^{∗}), so we have D^{μ_{I^{∗}∪J̃^{∗}_{u}}(l)} ≺ D^{μ_{I^{∗}}(l)} in S^{(n−l,l)}, by theorem 7.5. Therefore D^{μ_{I^{∗}}(l)} ∉ F(V_{u}Θ_{0}^{.,l}). But D^{μ_{I^{∗}}(l)} ∉ F(U/V_{u}), so D^{μ_{I^{∗}}(l)} ∉ F((U/V_{u})Θ_{0}^{.,l}) and D^{μ_{I^{∗}}(l)} ∉ F(UΘ_{0}^{.,l}). Thus D^{μ_{I^{∗}}(l)} ⊀ D^{μ_{J^{∗}}(l)} in S^{(n−l,l)}. □

This concludes the study of the submodule structure of 2–part partition Specht modules in the non–defining characteristic of the general linear group.

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Dept. of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2BZ

E-mail address: alain.reuter@gmx.net