BRANCHING RULES FOR MODULAR FUNDAMENTAL REPRESENTATIONS OF SYMPLECTIC GROUPS

A. A. BARANOV AND I. D. SUPRUNENKO

To our teacher A. E. Zalesskii on the occasion of his sixtieth birthday

Abstract. In this paper branching rules for the fundamental representations of the symplectic groups in positive characteristic are found. The submodule structure of the restrictions of the fundamental modules for the group Sp_{2n}(K) to the naturally embedded subgroup Sp_{2n−2}(K) is determined. As a corollary, inductive systems of fundamental representations for Sp_{∞}(K) are classified. The submodule structure of the fundamental Weyl modules is refined.

1. Introduction

The article is devoted to finding branching rules for the fundamental representations of the sympplectic groups in positive characteristic. The classical branching rules are concerned with the restrictions of representations of the classical algebraic and symmetric groups in characteristic 0 to naturally embedded subgroups of smaller ranks. For a group of rank n and its fixed irreducible representation φ they yield the composition factors of the restriction of φ to a naturally embedded subgroup of rank n − 1 and hence to similar subgroups of smaller ranks, at least algorithmically. These rules provide a basis for induction on rank and have found numerous applications. In positive characteristic one cannot expect to obtain complete branching rules in an explicit form in a near future since this problem is closely connected with that of finding the dimensions of arbitrary irreducible representations and the composition factors of the Weyl modules. So it is worth to investigate important particular cases where such rules can be found and to seek for asymptotic analogs of these rules. The notion of an inductive system of representations (see the definition below) introduced by Zalesskii in [11] yields an asymptotic version of the branching rules. It proved to be useful for the study of ideals in group algebras of locally finite groups as well, see, for instance, Zalesskii’s survey [12]. We classify the inductive systems of the fundamental representations for the infinite-dimensional symplectic group Sp_{∞}(K). This class of representations yields an example of representations of a simple form for which the branching rules in positive characteristic differ from the characteristic 0 case.

Let K ⊂ F be fields of characteristic p > 0, F̄ be the algebraic closure of F, and ℤ^{+} be the set of nonnegative integers. Let G_{n} = Sp_{2n}(K). Denote by ω_{i}^{n}, 0 ≤ i ≤ n, the ith fundamental module and representation of G_{n} over F where ω_{0}^{n} is the trivial one. Let W_{0}^{n} = ω_{0}^{n}, W_{1}^{n}, …, W_{n}^{n} be the corresponding Weyl modules. Set W_{i}^{n} = ω_{i}^{n} = 0 for i < 0 and for i > n. The labeling of the fundamental modules is standard, the fundamental and the Weyl modules for G_{n} are the F -modules affording the restrictions to G_{n} of the relevant representations of the group Sp_{2n}(F̄) (it is well known that these restrictions can be realized over F). For an integer z > 0 we denote by lp(z) the maximal i such that p^{i} | z. We have lp(z) = 0 if p |̸ z. Let M be a G_{n}-module. The restriction of M to G_{n−1} is denoted by M↓G_{n−1}. We shall write

1991 Mathematics Subject Classification. 20G05.

Both authors have been supported by the Institute of Mathematics of the National Academy of Sciences of Belarus in the framework of the program “Mathematical structures” and by INTAS (Project 93-183-Ext).

M ∼ N_{1} + ⋯ + N_{q}

if there is a series 0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{q} = M of submodules of M and a permutation σ such that N_{σ(i)} ≅ M_{i}/M_{i−1} for all i = 1, …, q. Moreover, if in addition M_{i}/M_{i−1} coincides with the socle of M/M_{i−1} for i = 1, …, q, then the sequence

N_{σ(1)} ≺_{s} N_{σ(2)} ≺_{s} … ≺_{s} N_{σ(q)}

is called the socle series of M.

Theorem 1.1 below describes the branching rules for the fundamental G_{n}-modules and the submodule structure of the restrictions of these modules to G_{n−1}.

Theorem 1.1. Let n ≥ 2 and 0 ≤ i ≤ n. Set d = lp(n − i + 1); ε = 0 if n − i + 1 ≡ −p^{d} (mod p^{d+1}) and ε = 1 otherwise. Then

(i) ω_{i}^{n}↓G_{n−1} ∼ ω_{i}^{n−1} + 2ω_{i−1}^{n−1} + (∑_{t=0}^{d−1} 2ω_{i−2p^{t}}^{n−1}) + εω_{i−2p^{d}}^{n−1} (the sum in the brackets is zero whenever d = 0);

(ii) ω_{i}^{n}↓G_{n−1} = ω_{i−1}^{n−1} ⊕ ω_{i−1}^{n−1} ⊕ D and the series

ω_{i−2}^{n−1} ≺_{s} ω_{i−2p}^{n−1} ≺_{s} ⋯ ≺_{s} ω_{i−2p^{d−1}}^{n−1} ≺_{s} ω ≺_{s} ω_{i−2p^{d−1}}^{n−1} ≺_{s} ⋯ ≺_{s} ω_{i−2p}^{n−1} ≺_{s} ω_{i−2}^{n−1}

with ω = ω_{i}^{n−1} ⊕εω_{i−2p^{d}}^{n−1} and ω_{j}^{n−1} omitted for j < 0 is the socle series of D. In particular, D = ω if i = 0, 1 or p |̸ n + 1 − i.

Corollary 1.2. For n ≥ 2 the restriction ω_{i}^{n}↓G_{n−1} is completely reducible if and only if i = 0, 1 or p |̸ n + 1 − i.

The proof of Theorem 1.1 is based on the description of the composition factors of the fundamental Weyl modules (Premet and Suprunenko [6, Theorem 2] for p > 2 and independently Adamovich [1, Theorem 2 and its Corollary 1] for arbitrary p) and Adamovich’s results [2] on the submodule structure of these Weyl modules. In Section 2 these results are refined (Theorem 2.13). In particular, a new irreducibility criterion for the fundamental Weyl modules is obtained (Corollary 2.14) and it is proved that their socles are always simple (Corollary 2.15).

For n − p + 2 ≤ i ≤ n Gow [5] has given an explicit construction of the modules ω_{i}^{n} and has described the submodule structure of the restrictions ω_{i}^{n}↓G_{1} × G_{n−1} (the natural embedding) ([5, Theorem 2.2]). This implies our Theorem 1.1 for these modules. In [5] a certain explicitly determined operator δ on the exterior algebra ΛV of the natural G_{n}-module V is considered and it is proved that for n − p + 2 ≤ i ≤ n the module ω_{i}^{n} can be realized as the quotient ker δ ∩ Λ^{i}V/δ^{p−1}(Λ^{i+2p−2}V) ([5, Corollary 2.4]). This nice construction gives a realization for an important class of modules without complicated representation-theoretic machinery. However, it cannot be extended to other fundamental modules since according to [5, Theorem 4.2], the quotient above is zero for i < n − p + 2.

In [7] Sheth has found the branching rules for modular representations of symmetric groups corresponding to two part partitions. The composition factors occurring in the relevant restrictions are similar to those of the module D in Theorem 1.1(ii). We conjecture that the submodule structure of these restrictions is also similar to that of D. The authors plan to consider this question as well as the similar one for representations of special linear groups with highest weights ω_{i} + ω_{j} in a subsequent paper.

In Section 4 Theorem 1.1 is applied to classify the inductive systems of fundamental F -representations for Sp_{∞}(K). Let

H_{1} ⊂ H_{2} ⊂ … ⊂ H_{n} ⊂ …  (1)

be a sequence of groups, and Ψ_{n}, n = 1, 2, …, be a nonempty finite set of (inequivalent) irreducible representations of H_{n} over a fixed field. The system Ψ = {Ψ_{n} | n = 1, 2, … } is called an inductive system (of representations) for the group H = ⋃_{n=1}^{∞} H_{n} if each Ψ_{n} coincides with the union of the sets of composition factors (up to equivalence) of the restrictions π↓H_{n} where π runs over Ψ_{n+1}. In this article (1) is the sequence of the naturally embedded groups G_{n} = Sp_{2n}(K), so ⋃_{n=1}^{∞} G_{n} = Sp_{∞}(K). Set

\begin{aligned}F_{n} & = & \{ω_{i}^{n} | 0 ≤ i ≤ n\}, & F & = & \{F_{n} | n = 1, 2, … \}; \\ L_{n}^{s} & = & \{ω_{i}^{n} | 0 ≤ i ≤ s\}, & L^{s} & = & \{L_{n}^{s} | n = 1, 2, … \}; \\ R_{n}^{u} & = & \{ω_{i}^{n} | n + 1 − u ≤ i ≤ n\}, & R^{u} & = & \{R_{n}^{u} | n = 1, 2, … \}. \end{aligned}

Theorem 1.3. The inductive systems of fundamental representations over F for Sp_{∞}(K) are exhausted by the systems F, L^{s}, R^{p^{t}−1}, and L^{s} ∪ R^{p^{t}−1} (s ≥ 0, t ≥ 1).

It is clear that L^{0} (which consists of the trivial representations) and R^{p−1} are minimal inductive systems. However, the question on the minimal inductive systems for Sp_{∞}(K) is far from solution. For p > 2 Zalesskii and Suprunenko [10] have described the inductive system Φ = {Φ_{n} | n = 1, 2, … } where for each n the set Φ_{n} consists of two irreducible representations with highest weights ω_{n−1} + \frac{1}{2}(p − 3)ω_{n} and \frac{1}{2}(p − 1)ω_{n}. The system Φ coincides with R^{2} for p = 3 and yields another example of a minimal inductive system for p > 3.

For other classical groups the questions investigated in this paper present no problems since the situation is the same as in characteristic 0.

The authors [3] have found the minimal and the minimal nontrivial inductive systems for the group SL_{∞}(K̄). For this group the system consisting of the trivial representation is the only minimal inductive system, and the minimal nontrivial ones are exhausted by the systems L^{j} = {L_{n}^{j} | n = 1, 2, … } and R^{j} = {R_{n}^{j} | n = 1, 2, … } where L_{n}^{j} consists of two irreducible representations of SL_{n+1}(K̄) with highest weights 0 and p^{j}ω_{1} and R_{n}^{j} of those with highest weights 0 and p^{j}ω_{n}. The picture is similar for the groups SL_{∞} and SU_{∞} over locally finite fields.

Until Proposition 4.2 we assume that K = F = F̄. At the end Proposition 4.2 transfers the results to arbitrary fields.

2. The structure of the fundamental Weyl modules

In this section we refine the results of [6], [1], and [2] on the structure of the fundamental Weyl modules for G_{n}.

Throughout the paper we set π_{i}^{n} = ω_{n+1−i}^{n} and V_{i}^{n} = W_{n+1−i}^{n}. We denote by [a, b] the set of all j ∈ ℤ^{+} with a ≤ j ≤ b. For an integer k ∈ ℤ^{+} write its p-adic expansion k = k_{0} + k_{1}p + ⋯ + k_{s}p^{s} with 0 ≤ k_{i} < p and set k_{i} = 0 for all such i ∈ ℤ^{+} that p^{i} > k. We shall write k = (k_{0}, k_{1}, …, k_{s}). We say that an integer m contains k to base p and write k ⊂_{p} m if and only if for each i either k_{i} = 0, or k_{i} = m_{i}. Set d_{k}^{m} = 1 if k ⊂_{p} m, and d_{k}^{m} = 0 otherwise.

Theorem 2.1. [6, Theorem 2] Let p > 2. Then W_{i}^{n} ∼ ∑_{k=0}^{∞} d_{k}^{n+1−i+2k}ω_{i−2k}^{n}.

We need some more notation to state Adamovich’s results. For λ ∈ ℤ^{+} define maps s′_{λ} : ℤ^{+} → ℤ^{+} and s_{λ} : ℤ^{+} → ℤ^{+} setting

s′_{λ}(l) = l + 2k′ where l + 1 = a′p^{λ} − k′, a′ ∈ ℤ^{+}, 0 ≤ k′ < p^{λ};

s_{λ}(l) = l + 2k where l = ap^{λ} − k, a ∈ ℤ^{+}, 0 ≤ k < p^{λ}. We say that the reflection s′_{λ} or s_{λ} is l-admissible if k′ ≠ 0 and p |̸ a′ or k ≠ 0 and p |̸ a, respectively. We denote by S(l) the set of all m > l that can be written in the form m = s_{λ_{u}} … s_{λ_{1}}(l) where λ_{u} < ⋯ < λ_{1} and for each i = 0, 1, …, u − 1 the reflection s_{λ_{i+1}} is s_{λ_{i}} … s_{λ_{1}}(l)-admissible. Similarly we define S′(l) (writing s′_{λ_{i}} instead of s_{λ_{i}}).

Theorem 2.2. [1] Let 0 ≤ l ≤ n. Then V_{l+1}^{n} ∼ π_{l+1}^{n} + ∑_{m∈S′(l)} π_{m+1}^{n}.

As s′_{λ}(x − 1) = s_{λ}(x) − 1, the following theorem yields an equivalent statement.

Theorem 2.3. Let 1 ≤ l ≤ n + 1. Then V_{l}^{n} ∼ π_{l}^{n} + ∑_{m∈S(l)} π_{m}^{n}.

Let us rewrite Theorem 2.1 in terms of π_{m}^{n} and V_{l}^{n} (without restrictions on p).

Theorem 2.4. Let 1 ≤ l ≤ n + 1. Then V_{l}^{n} ∼ ∑_{k=0}^{∞} d_{k}^{l+2k}π_{l+2k}^{n}.

Now our goal is to show that Theorems 2.3 and 2.4 are equivalent, so Theorem 2.4 (and 2.1) holds for p = 2. For this purpose we prove some technical facts on the triples k, l, m with k ⊂_{p} m = l + 2k and admissible reflections.

Until the end of the section l ≥ 1. For each m ∈ S(l) the tuple (λ_{1}; …; λ_{u}) is uniquely determined (see the comments before the Theorem in [2]). If u is odd for some m, set λ_{u+1} = lp(m). Then s_{λ_{u+1}}(m) = m and λ_{u+1} < λ_{u}. Now for every m ∈ S(l) we have a uniquely determined sequence of reflections s_{λ_{1}}, …, s_{λ_{2t}}. Such sequences will be called l-admissible. For an integer 0 ≤ a ≤ p − 1 set ā = p − 1 − a. The following lemma is straightforward.

Lemma 2.5. Set q = lp(l). The reflection s_{λ} is l-admissible if and only if λ > q and l_{λ} ≠ p − 1. In that case s_{λ}(l) = p^{q}(l̄_{q} + 1, l̄_{q+1}, … l̄_{λ−1}, l_{λ} + 1, l_{λ+1}, …).

Two consequent applications of Lemma 2.5 yield

Proposition 2.6. Let lp(l) ≤ μ < λ, m = s_{μ}s_{λ}(l), and k = (m − l)/2. The pair s_{λ}, s_{μ} is l-admissible if and only if l_{λ} ≠ p − 1 and l_{μ} ≠ 0. In that case

\begin{aligned}m & = & (l_{0}, & …, & l_{μ−1}, & l̄_{μ} + 1, & l̄_{μ+1}, & …, & l̄_{λ−1}, & l_{λ} + 1, & l_{λ+1}, & …), \\ k & = & (0, & …, & 0, & l̄_{μ} + 1, & l̄_{μ+1}, & …, & l̄_{λ−1}). \end{aligned}

In particular, k ⊂_{p} m = l + 2k.

We call a tuple σ = (λ_{1}; …; λ_{2t}) l-admissible if λ_{i} ∈ ℤ^{+}, λ_{1} > ⋯ > λ_{2t}, l_{λ_{2j−1}} ≠ p − 1 and l_{λ_{2j}} ≠ 0 for j = 1, …, t. For an l-admissible σ set

𝔔_{l}(σ) = ⋃_{j=1}^{t}[λ_{2j}, λ_{2j−1} − 1],

δ_{iσ} = 1 if i = λ_{j} for some j and δ_{iσ} = 0 otherwise. Define l^{σ} ∈ ℤ^{+} putting

l_{i}^{σ} = \{ \begin{matrix}l̄_{i} + δ_{iσ}, & i ∈ 𝔔_{l}(σ) \\ l_{i} + δ_{iσ}, & i ∉ 𝔔_{l}(σ). \end{matrix}

Proposition 2.6 yields the following corollary.

Corollary 2.7. A sequence s_{λ_{1}}, …, s_{λ_{2t}} is l-admissible if and only if the tuple σ = (λ_{1}; …; λ_{2t}) is l-admissible. In that case s_{λ_{2t}} … s_{λ_{1}}(l) = l^{σ}.

Proposition 2.8. An integer m ∈ S(l) if and only if m − l = 2k > 0 and k ⊂_{p} m.

Proof. Let σ = (λ_{1}; …; λ_{2t}) be an l-admissible tuple and m = s_{λ_{2t}} … s_{λ_{1}}(l). Set m^{0} = l, m^{j} = s_{λ_{2j}} … s_{λ_{1}}(l), and k^{j} = (m^{j} − m^{j−1})/2, 1 ≤ j ≤ t. Using Proposition 2.6, one deduces that k^{j} ⊂_{p} m^{j}; m^{j} = l^{σ^{j}} with σ^{j} = (λ_{1}; …; λ_{2j}); k_{i}^{j} = 0 and m_{i}^{j} = m_{i}^{j−1} for i ∉ [λ_{2j}, λ_{2j−1}]. Therefore k = k^{1} + … + k^{t} ⊂_{p} m.

Assume now that m − l = 2k > 0 and k ⊂_{p} m. Choose integers τ_{1}, τ_{2}, …, τ_{2t} as follows. Set τ_{0} = −1. Assume that τ_{2j} is chosen. If there is no i > τ_{2j} such that k_{i} = m_{i} ≠ 0, we set t = j and stop the process. Otherwise we choose for τ_{2j+1} minimal i > τ_{2j} with k_{i} = m_{i} ≠ 0. As m > 2k, there exists f > τ_{2j+1} with k_{f} ≠ m_{f} (observe that in this case k_{f} = 0). We choose minimal such f for τ_{2j+2}. Set λ_{q} = τ_{2t+1−q}. Since k ⊂_{p} m, using Corollary 2.7 and analyzing the p-adic expansions of k and m, one can conclude that the tuple σ = (λ_{1}; …; λ_{2t}) is l-admissible and m = l^{σ}.

Corollary 2.9. Theorems 2.3 and 2.4 are equivalent, so Theorems 2.1 and 2.4 are valid in characteristic 2 as well.

Now we rewrite Adamovich’s results [2] on the submodule structure of the Weyl modules in our terms. We fix n and write V_{l} and π_{m} instead of V_{l}^{n} and π_{m}^{n}. For m ∈ S(l) or m = l we denote by P_{l}(m) the smallest submodule of V_{l} that has a composition factor π_{m}. Since V_{l} is multiplicity-free, P_{l}(m) is correctly defined and each submodule of V_{l} is a sum of P_{l}(m) for some m. Hence the submodule structure of V_{l} is determined by the inclusion relations between the submodules P_{l}(m) (see also comments at the beginning of [2]). We shall write π_{m} ≺ π_{q} if P_{l}(m) ⊂ P_{l}(q). Let σ = (λ_{1}; …; λ_{2t}) be an l-admissible tuple. For m = l^{σ} set

𝔔_{l}(m) = 𝔔_{l}(σ) = ⋃_{j=1}^{t}[λ_{2j}, λ_{2j−1} − 1].

Note that 𝔔_{l}(m) = ⋃_{j=1}^{t}[τ_{2j−1}, τ_{2j} − 1] where τ_{i} are as in the proof of Proposition 2.8. For instance, for p = 3,

m = (0, 1, \underline{2, 2, 0}, 1, 0, \underline{2, 1, 0, 0}, 2, 1),
k = (0, 0, \underline{2, 2, 0}, 0, 0, \underline{2, 1, 0, 0}, 0, 0),

we have 𝔔_{m−2k}(m) = [2, 4] ∪ [7, 10]. Put also 𝔔_{l}(l) = ∅.

Theorem 2.10. [2] For 1 ≤ m, q ≤ n + 1 and m, q ∈ S(l) ∪ {l} the module π_{m} ≺ π_{q} (as composition factors of V_{l}) if and only if 𝔔_{l}(q) ⊂ 𝔔_{l}(m).

Remark 2.11. Actually the sets 𝔓_{l}(m) which are considered in [2] differ slightly from 𝔔_{l}(m). For m ∈ S(l) one has 𝔓_{l}(m) = ∪_{j=1}^{t}[μ_{2j} + 1, μ_{2j−1}] where μ_{s} = λ_{s} for s < 2t, μ_{2t} = λ_{2t} if m ≠ s_{λ_{2t−1}} … s_{λ_{1}}(l), and μ_{2t} = 0 otherwise. However, Lemma 2.5 enables one to deduce that 𝔓_{l}(m) ⊂ 𝔓_{l}(q) if and only if 𝔔_{l}(m) ⊂ 𝔔_{l}(q). The crucial point is that lp(l) = lp(m) for m ∈ S(l).

For l-admissible tuples σ = (λ_{1}; …; λ_{2t}) and σ′ = (λ′_{1}; …; λ′_{2s}) we say that σ ≤ σ′ if there exists f ≤ 2t, 2s such that λ_{i} = λ′_{i} for 1 ≤ i ≤ f and either f = 2t, or f < 2t, 2s and λ_{f+1} < λ′_{f+1}. It is convenient to assume that the empty tuple ∅ is l-admissible, ∅ ≤ σ for all σ, l^{∅} = l, and 𝔔_{l}(∅) = ∅. The following is obvious.

Lemma 2.12. Let σ and σ′ be l-admissible tuples. Then l^{σ} ≤ l^{σ′} if and only if σ ≤ σ′.

Set n′ = n + 1. Construct an l-admissible tuple σ^{max} = (μ_{1}; …; μ_{2t}) as follows. Put μ_{0} = +∞. Assume that μ_{2j} is chosen. Set μ = μ_{2j} − 1. If there is no l-admissible tuple (α; β) such that μ ≥ α > β and

(l_{0}, …, l_{μ})^{(α;β)} = (l_{0}, …, l̄_{β} + 1, l̄_{β+1}, …, l̄_{α−1}, l_{α} + 1, …, l_{μ}) ≤ (n′_{0}, …, n′_{μ}),

we stop the process and set t = j (σ^{max} = ∅ if t = 0). Otherwise we choose maximal such pair (α; β) (with respect to ≤); set μ_{2j+1} = α and μ_{2j+2} = β; and if

(l̄_{β} + 1, l̄_{β+1}, …, l̄_{α−1}, l_{α} + 1) < (n′_{β}, …, n′_{α}),

we stop the process and determine (μ_{2j+3}; …; μ_{2t}) as the maximal l-admissible tuple with μ_{2j+3} < β. Obviously, l^{σ^{max}} is the maximal integer m such that π_{m}^{n} is a composition factor of V_{l}^{n}.

For l-admissible tuples σ and σ′ we write σ ≺ σ′ if and only if 𝔔_{l}(σ) ⊃ 𝔔_{l}(σ′). Using Corollary 2.7, Theorem 2.10, and Lemma 2.12, we get our main result on the structure of fundamental Weyl modules.

Theorem 2.13. The map σ ↦ π_{l^{σ}}^{n} is a poset isomorphism between the l-admissible tuples σ ≤ σ^{max} and the composition factors of V_{l}^{n} with the partial orders ≺.

If l < n′, we denote by v the maximal integer such that l_{v} ≠ n′_{v}. If l_{v} + 1 = n′_{v}, we denote by u the maximal integer < v such that l̄_{u} ≠ n′_{u} setting u = −1 if l̄_{i} = n′_{i} for all i < v. Put s = lp(l).

Corollary 2.14. Let n′ = n + 1 and 1 ≤ l ≤ n′. Then V_{l}^{n} is irreducible (i.e. σ^{max} = ∅) if and only if one of the following holds.

(1) l = n′;

(2) l < n′ and s ≥ v;

(3) l < n′, s < v, l_{v} + 1 = n′_{v}, l̄_{s} ≥ n′_{s}; l_{i} = p − 1 and n′_{i} = 0 for s < i < v.

Proof. This follows from Proposition 2.6 and Corollary 2.7.

Corollary 2.15. Let n′ = n + 1 and 1 ≤ l ≤ n′. The socle of V_{l}^{n} is always simple. For reducible V_{l}^{n} it has the form π_{l^{γ}} with γ = (t; s) and t as follows.

(1) t = v if s < v and either l_{v} + 1 < n′_{v}, or l̄_{u} < n′_{u};

(2) t = w if l_{v} + 1 = n′_{v}; u = −1 or l̄_{u} > n′_{u}; s < w < v; l_{w} ≠ p − 1; and l_{j} = p − 1 for w < j < v.

Proof. Applying Results 2.6, 2.7, and 2.13, we conclude that π_{l^{γ}} is a composition factor of V_{l}^{n} and for each l-admissible tuple τ ≤ σ^{max} the set 𝔔_{l}(τ) ⊂ [s, t − 1], so π_{l^{γ}} ≺ π_{l^{τ}}.

3. Branching rules and the submodule structure of the restrictions

In this section the main results of the article are proved. We shall need the following simple lemma.

Lemma 3.1. Assume that d_{k}^{l+2k} = 1 (i.e. k ⊂_{p} l + 2k).

(i) If p^{s} | l + 2k, then p^{s} | k and p^{s} | l.

(ii) If p^{s} | l, then p^{s} | k and p^{s} | l + 2k.

Proof. One can assume that s ≥ 1.

(i) Let p^{s} | l + 2k. Since k ⊂_{p} l + 2k, we have p^{s} | k. This implies that p^{s} | l. (ii) Let p^{s} | l. Then l_{0} = ⋯ = l_{s−1} = 0. Let r = lp(k). Assume that r < s. Since k ⊂_{p} l + 2k, we have k_{r} = (2k)_{r} ≠ 0, which is impossible. Therefore r ≥ s, so p^{s} | k and p^{s} | l + 2k.

As in Section 2 , we shall omit the superscript n in our notation for modules when it is known which group is considered. Replacing ω_{i} by π_{n+1−i} and W_{i} by V_{n+1−i}, one immediately concludes that Theorem 1.1(i) is equivalent to the following

Theorem 3.2. Let 1 ≤ i ≤ n + 1 and d = lp(i). Then

π_{i}^{n}↓G_{n−1} ∼ π_{i−1} + 2π_{i} + (∑_{t=0}^{d−1} 2π_{i−1+2p^{t}}) + επ_{i−1+2p^{d}}

where ε = 0 if i ≡ −p^{d} (mod p^{d+1}) and ε = 1 otherwise.

Proof. One can rewrite the formula in Theorem 3.2 as follows.

π_{i}^{n}↓G_{n−1} ∼ π_{i−1} + 2π_{i} + ∑_{t=0}^{∞} b_{t}^{i}π_{i−1+2p^{t}}  (2)

where

b_{t}^{i} = \{ \begin{matrix}2, & i ≡ 0 (mod p^{t+1}), \\ 1, & i ≡ ap^{t} (mod p^{t+1}) and a ≢ 0, −1 (mod p), \\ 0, & i ≢ 0 (mod p^{t}) or i ≡ −p^{t} (mod p^{t+1}). \end{matrix}  (3)

Recall that by convention π_{i}^{n} = 0 for all i > n + 1, and π_{n+1}^{n} is the trivial one-dimensional G_{n}-module. So (2) holds for i ≥ n + 1. Assume now that 1 ≤ l < n + 1 and (2) is valid for all i > l. We shall prove it for i = l. Then the theorem will follow by induction.

It follows from [4, Proposition 3.3.2 and Theorem 4.3.1] that V_{l}↓G_{n−1} has a filtration by Weyl modules for G_{n−1}. Then the classical branching rules for characteristic 0 [13] and Theorem 2.4 imply

V_{l}^{n}↓G_{n−1} ∼ V_{l−1} + 2V_{l} + V_{l+1} ∼ 2V + ∑_{t=0}^{∞} f_{t}^{l+2t−1}π_{l+2t−1}  (4)

where V = ∑_{k=0}^{∞} d_{k}^{l+2k}π_{l+2k}, f_{0}^{l−1} = d_{0}^{l−1}, and

f_{t}^{l+2t−1} = d_{t}^{l+2t−1} + d_{t−1}^{l+2t−1} for t ≥ 1.

On the other hand, by Theorem 2.4, V_{l}↓G_{n−1} ∼ ∑_{k=0}^{∞} d_{k}^{l+2k}(π_{l+2k}↓G_{n−1}). Since d_{0}^{l} = 1 and the branching rules for π_{i} with i > l are assumed to satisfy (2), one can determine the branching of π_{l}. Therefore it suffices to check that the right part of (4) is equal to ∑_{k=0}^{∞} d_{k}^{l+2k}U_{l+2k} where U_{i} is the right part of (2). The latter sum can be rewritten as follows:

2V + ∑_{k=0}^{∞} d_{k}^{l+2k}(π_{l+2k−1} + ∑_{s=0}^{∞} b_{s}^{l+2k}π_{l+2k+2p^{s}−1}) = 2V + ∑_{t=0}^{∞} e_{t}^{l+2t−1}π_{l+2t−1}

where

e_{t}^{l+2t−1} = d_{t}^{l+2t} + ∑_{k,s≥0, k+p^{s}=t} d_{k}^{l+2k}b_{s}^{l+2k}.  (5)

We have to show that e_{t}^{l+2t−1} = f_{t}^{l+2t−1} for all t ≥ 0. Note that f_{t}^{l+2t−1} ≤ 2. We proceed by steps.

Step 1. At most one summand in (5) is nonzero. In particular, e_{t}^{l+2t−1} ≤ 2.

Assume that d_{k}^{l+2k}b_{s}^{l+2k} ≠ 0 and d_{k′}^{l+2k′}b_{s′}^{l+2k′} ≠ 0 with t = k + p^{s} = k′ + p^{s′} and s > s′. Since b_{s}^{l+2k} ≠ 0, we have p^{s} | l + 2k. As d_{k}^{l+2k}, d_{k′}^{l+2k′} ≠ 0, by Lemma 3.1, p^{s} divides k, l, and k′. Hence p^{s} | k + p^{s} − k′ = p^{s′}, which yields a contradiction.

Now assume that d_{t}^{l+2t} ≠ 0 and d_{k}^{l+2k}b_{s}^{l+2k} ≠ 0 with k + p^{s} = t. As above, we get that p^{s} divides k, l, and t. Let r = lp(l). Then by Lemma 3.1 (ii), p^{r} | k and p^{r} | t. Since k + p^{s} = t, we have r = s, so l_{s} ≠ 0. Consider the following cases.

Case 1. k ≢ 0 and t ≢ 0 (mod p^{s+1}). Then l + 2t ≡ t and l + 2k ≡ k (mod p^{s+1}), so p^{s} = t − k ≡ 2(t − k) (mod p^{s+1}), which is impossible.

Case 2. k ≡ 0 (mod p^{s+1}). Then t_{s} = 1. Since l + 2t ≡ t (mod p^{s+1}), we have l_{s} = p − 1, so (l + 2k)_{s} = p − 1. This implies that b_{s}^{l+2k} = 0 and yields a contradiction.

Case 3. t ≡ 0 (mod p^{s+1}). Then k_{s} = p − 1. Therefore (l + 2k)_{s} = k_{s} = p − 1, so as above, b_{s}^{l+2k} = 0.

Step 2. e_{t}^{l+2t−1} = 2 if and only if f_{t}^{l+2t−1} = 2 (equivalently, d_{t}^{l+2t−1} = d_{t−1}^{l+2t−1} = 1).

Assume that e_{t}^{l+2t−1} = 2. By Step 1, this is equivalent to the following: there exist k, s ≥ 0 with k + p^{s} = t such that b_{s}^{l+2k} = 2 and d_{k}^{l+2k} = 1. Hence p^{s+1} | l + 2k and k ⊂_{p} l+2k. By Lemma 3.1 (i), p^{s+1} divides l and k. Note that l+2t−1 = l+2k+p^{s}+(p^{s}−1).

Therefore t = k+p^{s} ⊂_{p} l+2t−1 and t−1 = k+(p^{s}−1) ⊂_{p} l+2t−1, so d_{t}^{l+2t−1} = d_{t−1}^{l+2t−1} = 1, as required.

Assume now that d_{t}^{l+2t−1} = d_{t−1}^{l+2t−1} = 1. Let s = lp(t). Then t_{s} ≠ 0 and (t−1)_{s} = t_{s}−1. Since both t and t − 1 are contained in l + 2t − 1, we have t_{s} = 1. Moreover, we have (l −1)_{0} = ⋯ = (l −1)_{s−1} = (t−1)_{0} = ⋯ = (t−1)_{s−1} = p−1. Since (l +2t−1)_{s} = t_{s} = 1, we get (l − 1)_{s} = p − 1. Hence p^{s+1} | l. Set k = t − p^{s}. Then p^{s+1} | k, so p^{s+1} | l + 2k and b_{s}^{l+2k} = 2. It remains to observe that k = t − p^{s} ⊂_{p} (l + 2t − 1) − p^{s} − (p^{s} − 1), so d_{k}^{l+2k} = 1 and e_{t}^{l+2t−1} = d_{k}^{l+2k}b_{s}^{l+2k} = 2.

Step 3. If e_{t}^{l+2t−1} ≠ 0, then f_{t}^{l+2t−1} ≠ 0, i.e. t ⊂_{p} l + 2t − 1 or t − 1 ⊂_{p} l + 2t − 1.

Let r = lp(l). First assume that d_{t}^{l+2t} = 1 (see (5)), i.e. t ⊂_{p} l + 2t. Then by Lemma 3.1 (ii), p^{r} | t. One easily checks that if p^{r+1} |̸ t, then t − 1 ⊂_{p} l + 2t − 1, and if p^{r+1} | t, then t ⊂_{p} l + 2t − 1, as required.

Assume now that there exist k, s ≥ 0 with k + p^{s} = t such that d_{k}^{l+2k} = 1 and b_{s}^{l+2k} ≠ 0. By Lemma 3.1, p^{r} | k and lp(l + 2k) = r. If b_{s}^{l+2k} = 2, then by Step 2, f_{t}^{l+2t−1} = 2 ≠ 0. Hence we can assume that b_{s}^{l+2k} = 1. By (3), p^{s} | l + 2k and (l + 2k)_{s} ≠ 0, p − 1, so r = s. Assume that k_{s} = 0. Then (t−1)_{s} = 0 and t−1 = k+p^{s}−1 ⊂_{p} l+2k+p^{s}+(p^{s}−1) = l+2t− 1. If k_{s} ≠ 0, we have k_{s} = (l+2k)_{s} ≠ p−1, so t = k+p^{s} ⊂_{p} l+2k+p^{s}+(p^{s}−1) = l+2t−1, as required.

Step 4. If f_{t}^{l+2t−1} ≠ 0, then e_{t}^{l+2t−1} ≠ 0.

If t = 0, then e_{0}^{l−1} = 1, so assume that t ≥ 1. We have either t ⊂_{p} l + 2t − 1, or t − 1 ⊂_{p} l + 2t − 1. One needs to show that either t ⊂_{p} l + 2t (i.e. d_{t}^{l+2t} = 1), or there exist k, s ≥ 0 with k + p^{s} = t such that k ⊂_{p} l + 2k, p^{s} | l + 2k, and (l + 2k)_{s} ≠ p − 1 (i.e. d_{k}^{l+2k} = 1 and b_{s}^{l+2k} ≠ 0). First assume that t ⊂_{p} l + 2t − 1. Let s = lp(t). We have (l + 2t − 1)_{s} = t_{s} ≠ 0. If p^{s} | l + 2t, then k = t − p^{s} ⊂_{p} l + 2t − 1 − p^{s} − (p^{s} − 1) = l + 2k, p^{s} | l + 2t − 2p^{s} = l + 2k, and (l + 2k)_{s} = (l + 2t − 1)_{s} − 1 ≠ p − 1, as required. If p^{s} |̸ l + 2t, one gets t ⊂_{p} (l + 2t − 1) + 1 = l + 2t, as desired.

Assume now that t − 1 ⊂_{p} l + 2t − 1. Consider the following cases.

Case 1. (t − 1)_{0} = 0. If (l + 2t − 1)_{0} = 0, then t ⊂_{p} l + 2t. Assume that (l + 2t − 1)_{0} ≠ 0. Set s = 0, k = t − 1. Then k = t − 1 ⊂_{p} (l + 2t − 1) − 1 = l + 2k and (l + 2k)_{0} = (l + 2t − 1)_{0} − 1 ≠ p − 1, as required.

Case 2. (t − 1)_{0} ≠ 0, p − 1. Then (l + 2t − 1)_{0} = (t − 1)_{0} ≠ p − 1, so t ⊂_{p} l + 2t.

Case 3. p | t. Let s = lp(t). Since t − 1 ⊂_{p} l + 2t − 1, we have p^{s} | l + 2t, so p^{s} | l. As p^{s+1} |̸ t, the integer (t − 1)_{s} ≠ p − 1. If (t − 1)_{s} ≠ 0, then (l + 2t − 1)_{s} = (t − 1)_{s} ≠ p − 1, so t ⊂_{p} l + 2t. Assume now that (t − 1)_{s} = 0. This implies t_{s} = 1. If (l + 2t − 1)_{s} = 0, then t ⊂_{p} l + 2t. Therefore one can suppose that (l + 2t − 1)_{s} ≥ 1. Then k = t − p^{s} ⊂_{p} (l+2t−1)−p^{s}−(p^{s}−1) = l+2k, p^{s} | l+2t−2p^{s} = l+2k, and (l+2k)_{s} = (l+2t−1)_{s}−1 ≠ p−1, as required.

Now we investigate the submodule structure of the restriction π_{i}^{n}↓G_{n−1}. Let n > 1 and 1 ≤ i ≤ n. As π_{i}^{n} is the top composition factor of V_{i}^{n}, it follows from (4) that π_{i}^{n}↓G_{n−1} is a quotient of the G_{n−1}-module V_{i}^{n}↓G_{n−1} ∼ V_{i−1} + 2V_{i} + V_{i+1}. Applying Smith’s theorem [8] both to V_{i} and π_{i}, we conclude that V_{i}^{n}↓G_{n−1} = V_{i} ⊕V_{i} ⊕V where V ∼ V_{i−1} +V_{i+1}, and π_{i}^{n}↓G_{n−1} = π_{i} ⊕π_{i} ⊕D where D is a quotient of V. Now Theorem 1.1(ii) and Corollary 1.2 follow immediately from

Theorem 3.3. Let d = lp i. Set ε = 0 if i ≡ −p^{d} (mod p^{d+1}) and ε = 1 otherwise; j_{q} = i − 1 + 2p^{q}. Choose minimal t ∈ ℤ^{+} such that j_{t} > n. Put d′ = min{d, t}. Then

π_{j_{0}} ≺_{s} π_{j_{1}} ≺_{s} ⋯ ≺_{s} π_{j_{d′−1}} ≺_{s} π_{i−1} ⊕ επ_{j_{d}} ≺_{s} π_{j_{d′−1}} ≺_{s} ⋯ ≺_{s} π_{j_{1}} ≺_{s} π_{j_{0}}

is the socle series of D. In particular, D = π_{i−1} ⊕ επ_{j_{d}} if d′ = 0.

Proof. By Theorem 3.2, D ∼ π_{i−1} +2π_{j_{0}} +⋯+2π_{j_{d′−1}} +επ_{j_{d}}. It follows from Theorem 2.4 that the factors π_{i−1}, π_{j_{0}}, …, π_{j_{d′−1}} come from V_{i−1} and the factors π_{j_{0}}, …, π_{j_{d′−1}}, and επ_{j_{d}} if nonzero come from V_{i+1}. Note that

j_{k} = i − 1 + 2p^{k} = i + p^{k} + (p^{k} − 1) = i + (\underbrace{p − 1, …, p − 1}_{k}, 1),

so 𝔔^{i−1}(j_{k}) = [k, d − 1] for all 0 ≤ k ≤ d − 1. Therefore by Theorem 2.10,

π_{j_{0}} ≺ π_{j_{1}} ≺ ⋯ ≺ π_{j_{d′−1}} ≺ π_{i−1} in V_{i−1}.  (6)

Similarly, we get 𝔔^{i+1}(j_{k}) = [0, k − 1] for 1 ≤ k < d (and for k = d if ε ≠ 0 and d > 0). Hence

επ_{j_{d}} ≺ π_{j_{d′−1}} ≺ ⋯ ≺ π_{j_{1}} ≺ π_{j_{0}} in V_{i+1}.  (7)

(Here the symbol ≺ is extended to the zero module in the natural way.) Since π_{i}^{n} is selfdual, D is selfdual also. Let D_{1} ≺_{s} ⋯ ≺_{s} D_{m} be the socle series of D. Recall that D has a filtration by quotients of V_{i−1} and V_{i+1}. As the factor π_{i−1} has multiplicity 1 and D is selfdual, (6) implies that π_{i−1} is a factor of D_{q} with d′ + 1 ≤ q ≤ m − d′, so m ≥ 2d′ + 1. If επ_{j_{d}} = 0, then m = 2d′ + 1 is the composition length of D and the theorem follows from (6) and (7). Assume that επ_{j_{d}} ≠ 0. As above, by the selfduality of D and (7), π_{j_{d}} is a factor of D_{q′} with d′ + 1 ≤ q′ ≤ m − d′. Assume that q′ ≠ q. Then m = 2d′ + 2, so D is uniserial, which contradicts the selfduality of D. Hence q′ = q and the theorem follows from (6) and (7).

Remark 3.4. Obviously, if επ_{j_{d}} = 0 (i.e. d′ < d or i ≡ −p^{d} (mod p^{d+1})), then the module D is uniserial, so has exactly 2d′ + 2 different submodules. Since D is selfdual, one can easily observe that D has exactly 2d + 4 different submodules in the case where επ_{j_{d}} ≠ 0 (i.e. d = d′ and i ≢ −p^{d} (mod p^{d+1})).

4. Inductive systems and the transfer to an arbitrary field

In this section the inductive systems of fundamental representations for Sp_{∞}(K) are classified and the results of the paper are transferred to an arbitrary field.

Proposition 4.1. (i) Let Ψ be an inductive system and R^{p^{t−1}} ⊂ Ψ. Then R^{p^{t}−1} ⊂ Ψ. (ii) R^{k} is an inductive system if and only if k = p^{t} − 1, t ≥ 1.

Proof. (i) Assume that R^{p^{t}−1} ⊄ Ψ. Choose maximal l such that R^{l} ⊂ Ψ. Then p^{t−1} ≤ l < p^{t} − 1. Take minimal s such that l_{s} ≠ p − 1 and set i = p^{s}(l_{s}, l_{s+1}, …). Then i > 0 and R^{i} ⊂ Ψ. One has lp(i) ≥ s. Moreover, if lp(i) = s, then i ≢ −p^{s} (mod p^{s+1}). Therefore Theorem 3.2 implies that π_{i−1+2p^{s}}^{n−1} is a composition factor of π_{i}^{n}↓G_{n−1} for n ≥ i − 1 + 2p^{s}, and R^{i−1+2p^{s}} ⊂ Ψ. As i − 1 + 2p^{s} > l, we get a contradiction.

(ii) In view of (i), it suffices to verify that R^{p^{t}−1} is an inductive system. By Theorem 3.2, we need only to check that if i < p^{t} −1 and s = lp(i), then i−1+2p^{s} ≤ p^{t} −1 if i_{s} ≠ p−1, and i−1+2p^{s−1} ≤ p^{t} −1 if i_{s} = p−1 and s > 0. But this is clear since i ≤ (p−2)p^{s} +(p− 1)p^{s+1} + … + (p − 1)p^{t−1} in the first case and i ≤ (p − 1)p^{s} + (p − 1)p^{s+1} + … + (p − 1)p^{t−1} in the second one.

Proof of Theorem 1.3. Theorem 1.1(i) and Proposition 4.1 yield that F, L^{s}, R^{p^{t}−1}, and L^{s} ∪ R^{p^{t}−1}, s ≥ 0, t ≥ 1, are inductive systems for Sp_{∞}(K).

Let Ψ = {Ψ_{i}, i = 1, 2, … } be an inductive system of fundamental representations. It is clear that either for every s, u ∈ ℤ^{+} there exist n and l such that ω_{l}^{n} ∈ Ψ_{n}, l > s, and n + 1 − l > u, or Ψ ⊂ L^{s} ∪ R^{u} for some s and u. In the first case we claim that Ψ = F. Indeed, fix m and l, 0 ≤ l ≤ m. Then one can choose k and n such that ω_{k}^{n} ∈ Ψ_{n}, k ≥ l, and n − k ≥ m − l. Since Ψ is an inductive system, Theorem 1.1(i) implies that ω_{l}^{n+l−k} ∈ Ψ_{n+l−k} and ω_{l}^{m} ∈ Ψ_{m}. Hence Ψ = F.

Next, suppose that Ψ ⊂ L^{s} ∪ R^{u}. Choose minimal s and u with this property assuming that s = −1 if Ψ ⊂ R^{u} and u = 0 if Ψ ⊂ L^{s}. (Observe that for all s and u, (L^{s} ∩ R^{u})_{n} = ∅ for n large enough.) We shall prove that Ψ = L^{s} ∪ R^{u} and u = p^{t} − 1 with t ∈ ℤ^{+} (in particular, Ψ = R^{u} for s = −1 and Ψ = L^{s} for u = 0).

First let u > 0. We claim that R^{u} ⊂ Ψ and u = p^{t} − 1. As Ψ ⊄ L^{s} and Ψ and L^{s} are inductive systems, Ψ_{n} ∩ R_{n}^{u} ≠ ∅ for infinitely many integers n. So there exists v ≤ u such that π_{v}^{n} ∈ Ψ_{n} for infinitely many n. Choose maximal such v. Theorem 3.2 yields that π_{v}^{n} ∈ Ψ_{n} for all n ≥ v − 1 and R^{v} ⊂ Ψ. Now Proposition 4.1 and the choice of v imply that v = p^{t} − 1 and R^{v} is an inductive system. It remains to show that v = u. Suppose this is not the case. As Ψ ⊄ L^{s} ∪ R^{v}, there exist l and t such that v < l ≤ u, t > s + l − 1, and π_{l}^{t} ∈ Ψ_{t}. Since Ψ, L^{s}, and R^{v} are inductive systems, this implies that for every k > t there exists π_{m_{k}}^{k} ∈ Ψ_{k} with v < m_{k} ≤ u which contradicts the choice of v. Hence v = u = p^{t} − 1 and R^{u} ⊂ Ψ.

Now we show that L^{s} ⊂ Ψ if s ≥ 0. As Ψ ⊄ L^{s−1} ∪ R^{u}, for some n > s + u − 1 we have ω_{s}^{n} ∈ Ψ_{n}. Since Ψ, R^{u}, and L^{s−1} for s ≥ 1 are inductive systems, this forces ω_{s}^{n} ∈ Ψ_{n} for all n ≥ s. Now Theorem 1.1 yields that L^{s} ⊂ Ψ, as desired.

Proposition 4.2. All theorems of the paper hold for arbitrary F ⊃ K.

Proof. Since the restrictions of the fundamental representations of a semisimple algebraic group over an algebraically closed field to relevant Chevalley groups over arbitrary subfields remain irreducible and can be realized over these subfields, only Theorems 1.1(ii) (or 3.3), 2.10, and 2.13 require some analysis. Let M be the G_{n}-module ω_{i}^{n+1}↓G_{n} or W_{i}^{n}.

First assume that F = F̄ is algebraically closed. Set H = Sp_{2n}(F). Let L be the Lie algebra of H. For a root α of H and t ∈ F denote by x_{α}(t) ∈ H and X_{α} ∈ L the root elements in H and L associated with α. It is well known that x_{α}(t)(m) = (1 + tX_{α})m for m ∈ M and long α (see, for instance, [6, Lemma 1]). For g ∈ G_{n} set x_{α}^{g}(t) = gx_{α}(t)g^{−1} and X_{α}^{g} = gX_{α}g^{−1}. It is clear that X_{α}^{g} ∈ L. It suffices to show that each G_{n}-submodule N ⊂ M is an H-submodule. Obviously, x_{α}^{g}(t)N = N and X_{α}^{g}N ⊂ N for all long roots α, g ∈ G_{n}, and t ∈ K. But this forces x_{α}^{g}(t)N = N for all t ∈ F. However, using the commutator relations for the Chevalley groups of type C (see, for instance, [9, Lemma 15]), one can deduce that the subgroup generated by all x_{α}^{g}(t) with g ∈ G_{n}, t ∈ F, and long α coincides with H. (Here, in fact, it suffices to make computations within subgroups of type C_{2} and show that our subgroup contains all short root subgroups). Hence N is an H-module, as desired.

Now let F ⊃ K be arbitrary. For a finite dimensional F G_{n}-module S set S̄ = S ⊗_{F} F̄ and denote the socle of S by soc(S). Since dim Hom_{F G_{n}}(E, S) = dim Hom_{F̄G_{n}}(Ē, S̄) for any F G_{n}-module E, we have soc(S̄) = \overline{soc(S)} if all composition factors of S are absolutely irreducible. The same holds for other members of the socle series of S. If M = W_{i}^{n}, then M and M̄ are multiplicity-free and their submodules are completely determined by the sets of composition factors. Therefore the arguments on socles allow us to conclude that each submodule of M̄ has the form S̄ for some submodule S ⊂ M.

Let M = ω_{i}^{n+1}↓G_{n} with i, n ≥ 1. Then the socle of M contains a submodule V ≅ ω_{i−1}^{n}⊕ ω_{i−1}^{n}. Since M and V are selfdual and M has only two composition factors isomorphic to ω_{i−1}^{n}, there exists a submodule D of M such that M = V ⊕ D and D̄ is the unique submodule in M̄ with M̄ = V̄ ⊕D̄. Now one can see that the socle series of D is determined by that of D̄ and is described by Theorem 1.1(ii).

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Institute of Mathematics, National Academy of Sciences of Belarus, Surganova 11, Minsk, 220072, Belarus.

E-mail address: baranov@im.bas-net.by

Institute of Mathematics, National Academy of Sciences of Belarus, Surganova 11, Minsk, 220072, Belarus.

E-mail address: suprunenko@im.bas-net.by