BRANCHING RULES FOR MODULAR REPRESENTATIONS OF SYMMETRIC GROUPS, IV

ALEXANDER KLESHCHEV

Introduction

Let Σ_{n} be the symmetric group on n letters, and D be an irreducible representation of Σ_{n} over a ground field F of charactersitic p > 0. The main results of this paper are:

(1) We describe the algebra End_{Σ_{n−1}}(D ↓_{Σ_{n−1}});

(2) we obtain a criterion for the semisimplicity of the induced module D↑^{Σ_{n+1}}, and in the case of semisimplicity describe its composition factors.

We note that for characteristic p = 2 the criterion for the semisimplicity we get was conjectured by Benson [1, Conjecture 5]. Thus we generalize to an arbitrary p and prove the last unproved conjecture made by Benson at the Arcata Conference. Actually, (2) easily follows from (1) using a Mackey type theorem together with our previous results on D↑^{Σ_{n+1}} [15].

We recall the necessary definitions and give the exact formulations of the main results. The list of the notions defined in [15] and used here is as follows.

λ = (λ_{1} ≥ λ_{2} ≥ …) is a (fixed) p-regular partition of n; λ = {(i, j) ∈ ℕ × ℕ | j ≤ λ_{i}} is its Young diagram (we do not distinguish

between partitions and their Young diagrams); D^{λ} is the irreducible F Σ_{n}-module corresponding to λ [9]; (i, j) ∈ ℕ × ℕ is called a node; (i, λ_{i}) ∈ λ is called a removable node (of λ) if λ_{i} > λ_{i+1}; (i, λ_{i} + 1) is called an addable node (for λ) if i = 1 or i > 1 and λ_{i} < λ_{i−1}; λ_{A} = λ \ {A} = (λ_{1}, …, λ_{i−1}, λ_{i} − 1, λ_{i+1}, …) is a partition of n − 1 obtained by

removing a removable node A = (i, λ_{i}) from λ;

λ^{B} = λ ∪ {B} = (λ_{1}, …, λ_{i−1}, λ_{i} + 1, λ_{i+1}, …) is a partition of n + 1 obtained by adding an addable node B = (i, λ_{i} + 1) to λ;

res A = j − i (mod p) is the (p)-residue of a node A = (i, j).

1991 Mathematics Subject Classification. 20C05, 20G05.

Research supported by the National Science Foundation.

A removable node A of λ is called normal if for every addable node B above A with res B = res A there exists a removable node C(B) strictly between A and B with res C(B) = res A, and B ≠ B′ implies C(B) ≠ C(B′). A removable node is called good if it is the lowest among the normal nodes of a fixed residue. An addable node B is called good if it is good as a removable node of λ^{B}.

The first main result of the paper is

Theorem A.

(i) The dimension of End_{Σ_{n−1}}(D^{λ} ↓_{Σ_{n−1}}) is equal to the number of normal nodes;

(ii) the dimension of End_{Σ_{n+1}}(D^{λ} ↑^{Σ_{n+1}}) is one more than the number of normal nodes.

Actually we can describe the algebra End_{Σ_{n−1}}(D^{λ} ↓_{Σ_{n−1}}) explicitly. Assume that the normal nodes of λ have residues α_{1}, α_{2}, …, α_{s} ∈ ℤ/pℤ (there may be several or no normal nodes of a fixed residue), and let A_{1}, A_{2}, …, A_{s} be the good nodes with the corresponding residues. It is proved in [15] that

D^{λ} ↓_{Σ_{n−1}} ≅ I_{1} ⊕ I_{2} ⊕ ⋯ ⊕ I_{s}

where I_{1}, I_{2}, … I_{s} are indecomposable, belong to distinct blocks, and the socle of I_{l} is D^{λ_{A_{l}}} , l = 1, 2, …, s. Hence

End_{Σ_{n−1}}(D^{λ} ↓_{Σ_{n−1}}) ≅ End_{Σ_{n−1}}(I_{1}) ⊕ End_{Σ_{n−1}}(I_{2}) ⊕ ⋯ ⊕ End_{Σ_{n−1}}(I_{s}).

Now the algebras End_{Σ_{n−1}}(I_{l}), l = 1, 2, …, s, are described in the following

Theorem B. Let 1 ≤ l ≤ s. Suppose there are m(l) normal nodes of residue α_{l}. There exists an endomorphism φ_{l} ∈ End_{Σ_{n−1}}(I_{l}) such that φ_{l}^{m(l)} = 0 and

{id_{I_{l}}, φ_{l}, φ_{l}^{2}, …, φ_{l}^{m(l)−1}}

is a linear basis of End_{Σ_{n−1}}(I_{l}). In particular, End_{Σ_{n−1}}(I_{l}) is isomorphic to the algebra of truncated polynomials F [x]/(x^{m(l)}).

Now we turn to the criterion for D^{λ} ↑^{Σ_{n+1}} to be semisimple.

Theorem C. Let λ have r good addable nodes and s normal removable nodes. Then D^{λ} ↑^{Σ_{n+1}} is semisimple if and only if r = s + 1, in which case

D^{λ} ↑^{Σ_{n+1}} ≅ ⊕D^{λ^{B}},

the sum being taken over all good addable nodes B.

If p = 2, the condition in Theorem C is equivalent to the one conjectured by Benson [1, Conjecture 5]:

Corollary D. Let p = 2 and λ = (λ_{1} > λ_{2} > ⋯ > λ_{t} > 0) be a 2-regular partition. Then D^{λ} ↑^{Σ_{n+1}} is semisimple if and only if all λ_{i} are even. In this case

D^{λ} ↑^{Σ_{n+1}} ≅ D^{(λ_{1},...,λ_{t−1},λ_{t}+1)} ⊕ D^{(λ_{1},...,λ_{t},1)}.

As a by-product of our calculations with GL(n) (in Section 2 of this paper), we get a curious filtration of the indecomposables I_{l}. Let S^{λ} be a Specht module (see [9]). A quotient module T of S^{λ} is called a small quotient of S^{λ} if the following two conditions hold:

(1) T has no composition factors D^{μ} with μ obtained from λ by moving one box up;

(2) if λ is p-regular then T ≠ 0.

As before Theorem B, we fix a residue α_{l}, and let C(1), C(2), …, C(m(l)) be all the normal nodes of residue α_{l} listed from bottom to top (1 ≤ l ≤ s). Consider the set of pairs

{(j, k) | 1 ≤ j ≤ k ≤ m(l)}

totally ordered as follows:

(j, k) ≺ (t, u) ⇔ 2k − j < 2u − t or 2k − j = 2u − t and j < t.

We denote by (j, k)′ the pair immediately preceding (j, k) in this total order. (Of course (1, 1)′ is not defined—below we interpret Ψ_{(1,1)′} as 0.)

Theorem E. Let 1 ≤ l ≤ s. There exists a filtration {Ψ_{(j,k)} | 1 ≤ j ≤ k ≤ m(l)} of I_{l} with Ψ_{(j,k)} ⊆ Ψ_{(t,u)} if and only if (j, k) ≺ (t, u) such that

(1) Ψ_{(1,m(l))} = I_{l};

(2) every consecutive quotient F_{(j,k)} = Ψ_{(j,k)}/Ψ_{(j,k)′} is a small quotient of S^{λ_{C(j)}};

(3) Let k_{2} > k_{1} ≥ j. Then F_{(j,k_{1})} is a quotient of F_{(j,k_{2})}. Moreover,

φ_{l}^{k_{2}−k_{1}}(Ψ_{(j,k_{2})}) ⊆ Ψ_{(j,k_{1})}, φ_{l}^{k_{2}−k_{1}}(Ψ_{(j,k_{2})′}) ⊆ Ψ_{(j,k_{1})′},

and in this way φ_{l}^{k_{2}−k_{1}} induces a surjective homomorphism from F_{(j,k_{2})} to F_{(j,k_{1})}.

Example. Let m(l) = 3. Then we have a filtration

(0) ⊆ Ψ_{(1,1)} ⊆ Ψ_{(2,2)} ⊆ Ψ_{(1,2)} ⊆ Ψ_{(3,3)} ⊆ Ψ_{(2,3)} ⊆ Ψ_{(1,3)} = I_{l},

and Ψ_{(1,1)}, Ψ_{(2,2)}/Ψ_{(1,1)}, Ψ_{(1,2)}/Ψ_{(2,2)}, Ψ_{(3,3)}/Ψ_{(1,2)}, Ψ_{(2,3)}/Ψ_{(3,3)}, Ψ_{(1,3)}/Ψ_{(2,3)} are small quotients of S^{λ_{C(1)}}, S^{λ_{C(2)}}, S^{λ_{C(1)}}, S^{λ_{C(3)}}, S^{λ_{C(2)}}, S^{λ_{C(1)}}, respectively. Moreover, Ψ_{(1,1)} is a quotient of Ψ_{(1,2)}/Ψ_{(2,2)}, which is in turn a quotient of Ψ_{(1,3)}/Ψ_{(2,3)}, and Ψ_{(2,2)}/Ψ_{(1,1)} is a quotient of Ψ_{(2,3)}/Ψ_{(3,3)}. (Actually, it is known that Ψ_{(1,1)} is always equal to D^{λ_{C(1)}}).

Remark. In general, ≺ is not the only total order on {(j, k) | 1 ≤ j ≤ k ≤ m(l)} for which Theorem E holds.

The following result, first proved in [16], is a special case of Theorem E (but it was used in the proof of this theorem).

Corollary F. [16, 1.4] Let A be a removable node of λ, and λ_{A} be p-regular. Then the multiplicity [D^{λ} ↓_{Σ_{n−1}} : D^{λ_{A}}] is non-zero if and only if A is normal. Moreover, for A normal we have

[D^{λ} ↓_{Σ_{n−1}} : D^{λ_{A}}] = 1 + #{normal B above A with res B = res A}

Using the now standard technique, we translate our problems from Σ_{n} to GL(n) = GL_{n}(F) (we assume F to be algebraically closed and all modules to be rational when we work with the algebraic group GL(n)). Then we apply the lowering operators introduced in [14] to do with the arising problems for GL(n). We use various properties of these operators proved in [14] and [2]. These properties are gathered in Section 1. In the same section we also deal with Murphy operators introduced in [2]. They are closely related with a “Casimir operator” which ultimately provides us with the endomorphisms we are looking for. The main results about GL(n) are proved in Section 2. They are translated into “Σ_{n}-language” in Section 3. The Benson’s Conjecture is proved in Section 4.

We complete this introduction with the following conjecture generalizing one of the main results of the paper.

Conjecture. Denote by L_{n}(λ) (resp., Δ_{n}(λ)) the irreducible (resp., Weyl) module with high weight λ. Following [2], define a dominant GL(n − 1)-weight μ to be λ-normal if

Hom_{GL(n−1)}(Δ_{n−1}(μ), L_{n}(λ) ↓_{GL(n−1)}) ≠ 0.

We conjecture that dim End_{GL(n−1)}(L_{n}(λ) ↓_{GL(n−1)}) is equal to the number of λnormal weights.

1. Preliminaries on GL(n)

The category of finite dimensional rational F GL(n)-modules is equivalent to the category of finite dimensional modules over the hyperalgebra U(n) of GL(n). We are going to work in the latter one. The main reference here is [17].

To define U(n) we first consider the Lie algebra 𝔤𝔩_{n}(ℂ) with the basis consisting of the matrix units {e_{ij} | 1 ≤ i, j ≤ n}. The elements

\{\frac{e_{ij}^{k}}{k!} , ( \begin{matrix}e_{ii} \\ k \end{matrix} ) | 1 ≤ i ≠ j ≤ n, k ≥ 0\}

generate a ℤ-subalgebra U(n, ℤ) in the universal enveloping U(𝔤𝔩_{n}(ℂ)). Now U(n) is U(n, ℤ) ⊗_{ℤ} F. We use special names for certain elements of U(n):

E_{ij}^{(k)} = \frac{e_{ij}^{k}}{k!} ⊗ 1, F_{ij}^{(k)} = \frac{e_{ji}^{k}}{k!} ⊗ 1, 1 ≤ i < j ≤ n;

( \begin{matrix}H_{i} \\ k \end{matrix} ) = ( \begin{matrix}e_{ii} \\ k \end{matrix} ) ⊗ 1, 1 ≤ i ≤ n.

We also write E_{ij} for E_{ij}^{(1)}, H_{i} for ( \begin{matrix}H_{i} \\ 1 \end{matrix} ), F_{ij} for F_{ij}^{(1)}. The subalgebras of U(n)

generated by

{E_{ij}^{(k)} | 1 ≤ i < j ≤ n, k ≥ 0},

\{( \begin{matrix}H_{i} \\ k \end{matrix} ) | 1 ≤ i ≤ n, k ≥ 0\},

{F_{ij}^{(k)} | 1 ≤ i < j ≤ n, k ≥ 0}

are denoted by U(n)^{+}, U(n)^{0}, U(n)^{−}, respectively. By a version of PBW-theorem, proved in [17, Theorem 2],

{F_{1,2}^{(N_{1,2})}F_{1,3}^{(N_{1,3})}F_{2,3}^{(N_{2,3})} … F_{1,n}^{(N_{1,n})}F_{2,n}^{(N_{2,n})} … F_{n−1,n}^{(N_{n−1,n})} | N_{ij} ≥ 0}
is a basis of U(n)^{−}.

(1)

There are natural embeddings of U(n−1), U(n−1)^{−}, etc., into U(n) corresponding to the embedding of GL(n − 1) into the top left hand corner of GL(n).

If λ = (λ_{1}, …, λ_{n}) is an n-tuple of integers, we can identify it with the weight

λ : U(n)^{0} → F, ( \begin{matrix}H_{i} \\ k \end{matrix} ) ↦ ( \begin{matrix}λ_{i} \\ k \end{matrix} ) .

A weight λ = (λ_{1}, …, λ_{n}) is called dominant if λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{n}. The set of all weights is denoted by X(n) and the set of all dominant weights is denoted by X_{+}(n).

If λ, μ ∈ X(n) we say λ dominates μ, written λ ≥ μ, if ∑_{i=1}^{j} λ_{i} ≥ ∑_{i=1}^{j} μ_{i} for all j = 1, 2, …, n.

It is convenient to use the weights

ε_{i} = (0, …, 0, 1, 0, …, 0),

with 1 on the ith position, i = 1, 2, …, n. Then

α(i, j) := ε_{i} − ε_{j}, 1 ≤ i < j ≤ n,

are the positive roots, and

α(i, i + 1), i = 1, 2, …, n − 1,

are the simple roots.

If μ is a weight we denote by U(n)_{−μ}^{−} the vector subspace of U(n)^{−} spanned by all monomials of the basis in (1) such that

N_{1,2}α(1, 2) + N_{1,3}α(1, 3) + ⋯ + N_{n−1,n}α(n − 1, n) = μ.

U(n)_{μ}^{+} is defined similarly.

Finally, for λ = (λ_{1}, …, λ_{n}) ∈ X(n) we denote by λ ↓ the weight (λ_{1}, …, λ_{n−1}) ∈ X(n − 1).

We use [i, j], (i, j], etc., to denote the intervals of integers, for example [i, j) = {k ∈ ℤ | i ≤ k < j}.

We write a ≡ b for a ≡ b (mod p).

From now on we fix a dominant weight λ ∈ X_{+}(n). For 1 ≤ i ≤ j < n we define

B(i, j) = λ_{i} − λ_{j+1} + j − i,
C(i, j) = λ_{i} − λ_{j} + j − i.

For 1 ≤ i ≤ j ≤ n put

B(i, j) = {k ∈ [i, j) | B(i, k) ≡ 0},
C(i, j) = {k ∈ (i, j) | C(i, k) ≡ 0}.

For two sets C and B of integers, a weakly decreasing (resp., increasing) injection from B to C is an injective map φ : B → C such that φ(b) ≤ b (resp., φ(b) ≥ b) for any b ∈ B. We say C compensates B if there exists a weakly decreasing injection from B to C.

Let 1 ≤ i < j ≤ n. We say i is normal in [i, j) if C(i, j) compensates B(i, j). We say i is normal if it is normal in [i, n). It follows from the definitions that

(2) if i is normal then λ_{i} − λ_{i+1} ≢ 0.

The following combinatorial lemma was proved in [16, 3.4].

Lemma 1.1. Let i be normal in [i, j), and |B(i, j)| < |C(i, j)|. Then there exists c ∈ C(i, j) which is normal in [c, j).

We denote by L_{n}(λ) and Δ_{n}(λ) the irreducible and the Weyl module, respectively, over U(n) with high weight λ. Let v_{+} be a high weight vector of L_{n}(λ).

Let w be a weight vector of a U(n)-module V. Then the vector w is called (U(n)-)primitive if E_{ij}^{(k)}w = 0 for all 1 ≤ i < j ≤ n, k > 0. It is called U(n − 1)- primitive if E_{ij}^{(k)}w = 0 for all 1 ≤ i < j ≤ n − 1, k > 0. Thus, a weight vector is U(n − 1)-primitive if it is a “high weight vector” for U(n − 1) as embedded earlier.

If V is a U(n)-module and μ is a weight we denote by V_{μ} the corresponding weight space of V.

We now recall some properties of lowering operators proved in [14]. We note that Brundan [3, 4] found easier proofs of these properties. Observe that our T_{ij}(M) corresponds to T_{i,j−1}(M) from [14] and to S_{ij}((i, j) \ M) from [3, 4, 2]. (Actually our T_{ij}(M) is obtained by evaluating Brundan’s S_{ij}((i, j) \ M) at weight λ.)

Let 1 ≤ i < j ≤ n. For U = {u_{1} < u_{2} < ⋯ < u_{a}} ⊆ (i, j) we set

F_{ij}(U) = F_{i,u_{1}}F_{u_{1},u_{2}} … F_{u_{a},j} ∈ U(n)^{−} (F_{ij}(∅) = F_{ij}).

By (1), {F_{ij}(U) | U ⊆ (i, j)} is a basis of U(n)_{−α(i,j)}^{−}. It is shown in [14] that there exists another basis

{T_{ij}(M) | M ⊆ (i, j)}

of U(n)_{−α(i,j)}^{−} with certain remarkable properties. Some of these properties are listed below for future reference.

Lemma 1.2. [14, 2.17]

T_{ij}(M) = ( ∏_{k∈(i,j)\M} C(i, k))F_{ij} + [a linear combination of F_{ij}(U) with U ≠ ∅].

Lemma 1.3. [14, 2.14] Let 1 ≤ i < k < j ≤ n, M ⊆ (i, k), N ⊆ (k, j), m = max M interpreted as i if M = ∅. Then

T_{ik}(M)T_{kj}(N) =T_{ij}(M ∪ N) − T_{ij}(M \ {m} ∪ {k} ∪ N)
− C(m, k)T_{ij}(M ∪ {k} ∪ N),

with the second term omitted if M = ∅.

The following result tells how the E’s corresponding to all but one simple roots act on the lowering operators.

Lemma 1.4. [14, 2.13] Let k ∈ [1, n) \ {j − 1}. Then

E_{k,k+1}T_{ij}(M)v_{+}

= \begin{cases}−T_{ik}(M ∩ (i, k))T_{k+1,j}(M ∩ (k + 1, j))v_{+} & if k ∉ M but k + 1 ∈ M; \\ 0 & otherwise. \end{cases}

The next lemma provides a criterion for T_{ij}(M)v_{+} to be zero.

Lemma 1.5. [14, 3.2] T_{ij}(M)v_{+} = 0 if and only if there exists k ∈ (i, j] \ M and a weakly increasing injection φ : {i} ∪ (M ∩ (i, k)) → [i, k) such that B(m, φ(m)) ≡ 0 for any m ∈ {i} ∪ (M ∩ (i, k)).

Definition 1.6. Let 1 ≤ i < j ≤ n. We say that F_{ij}v_{+} can be expressed through non-trivial subdivisions, if F_{ij}v_{+} is a linear combination of F_{ij}(U)v_{+} with U ≠ ∅ (in particular, if F_{ij}v_{+} = 0).

The following result tells us exactly when F_{ij}v_{+} can be expressed through nontrivial subdivisions (for a generalization see [2, 5.2], [5, 3.3]).

Lemma 1.7. [14, 4.6,4.2] F_{ij}v_{+} can be expressed through non-trivial subdivisions if and only if i is not normal in [i, j).

We recall the definition and some properties of the Murphy operators introduced in [2]. Set

L_{j} = ∑_{i=1}^{j−1} F_{ij}E_{ij}, j = 2, 3, …, n. (3)

These are Murphy operators. It is easy to check that they commute. A remarkable fact first observed by Brundan is that Murphy operators can be used to define lowering operators T_{ij}(M):

Lemma 1.8. [2, Section 3.3]

T_{ij}(M)v_{+} = ∏_{k∈(i,j)\M}(C(i, k) − L_{i+1} − L_{i+2} − ⋯ − L_{k})F_{ij}v_{+}

(one does not need to specify the order in the product since the factors commute).

Another result proved in [2] is useful in calculations:

Lemma 1.9. [2, 3.2(ii)] Let i < k < j. Then

(C(i, k) − L_{i+1} − L_{i+2} − ⋯ − L_{k})L_{k}F_{ij}v_{+} = 0.

We prove a couple of further results concerning the L_{k}.

Lemma 1.10. Assume 1 ≤ i < j < k ≤ n. Then L_{k}F_{ij} = F_{ij}L_{k}.

Proof. We have

L_{k}F_{ij} = ∑_{l=1}^{k−1} F_{lk}E_{lk}F_{ij} = ∑_{l=1}^{k−1} F_{lk}F_{ij}E_{lk} + F_{ik}[E_{ik}, F_{ij}]

= ∑_{l=1}^{k−1} F_{ij}F_{lk}E_{lk} + [F_{jk}, F_{ij}]E_{jk} + F_{ik}[E_{ik}, F_{ij}].

It remains to use [F_{jk}, F_{ij}] = F_{ik}, [E_{ik}, F_{ij}] = −E_{jk}.

Lemma 1.11. Let k ∈ (1, n] \ {j}. Then

L_{k}T_{ij}(M)v_{+} = \begin{cases}−T_{ik}(M ∩ (i, k))T_{kj}(M ∩ (k, j))v_{+} & if k ∈ M; \\ 0 & otherwise. \end{cases}

Proof. If k ≤ i or k > j then any E_{lk} annihilates T_{ij}(M)v_{+} since the weight of E_{lk}T_{ij}(M)v_{+} is λ − α(i, j) + α(l, k) which is not dominated by λ in this case. So we may assume that k ∈ (i, j). If k ∈ (i, j) \ M then L_{k}T_{ij}(M)v_{+} = 0 by Lemmas 1.8 and 1.9.

Finally, assume k ∈ M. Let m = max(M ∩(i, k)) interpreted as i if M ∩(i, k) = ∅. Then, again using Lemmas 1.8, 1.9 and what have already been proved, we get

L_{k}T_{ij}(M)v_{+}
= −(C(i, k) − L_{i+1} − ⋯ − L_{k})T_{ij}(M)v_{+}
+ (C(i, k) − L_{i+1} − ⋯ − L_{k−1})T_{ij}(M)v_{+}
= −T_{ij}(M \ {k})v_{+}
+ (C(i, m) − L_{i+1} − ⋯ − L_{m} + C(m, k) − L_{m+1} − ⋯ − L_{k−1})T_{ij}(M)v_{+}
= −T_{ij}(M \ {k})v_{+} + T_{ij}(M \ {m})v_{+} + C(m, k)T_{ij}(M)v_{+},

with the second term omitted if M ∩ (i, k) = ∅. But the last expression equals

−T_{ik}(M ∩ (i, k))T_{kj}(M ∩ (k, j))v_{+}

by virtue of Lemma 1.3.

We now need a “Casimir element” z_{n−1} ∈ U(n − 1) which is the Carter-Lusztig’s central element C_{2}, see [6, 2.2]. Set

z_{n−1} := ∑_{1≤i<j≤n−1}((H_{i} − i)(H_{j} − j) − F_{ij}E_{ij}) = θ_{n−1} − L_{2} − L_{3} − ⋯ − L_{n−1}, (4)

where

θ_{n−1} = ∑_{1≤i<j≤n−1}(H_{i} − i)(H_{j} − j) ∈ U(n − 1)^{0} ⊂ U(n)^{0}.

It follows from [6, Corollary 2.2] that

(5) z_{n−1} is central in U(n − 1).

Lemma 1.12. Assume 1 ≤ i_{1} < i_{2} < n. Then

(λ − α(i_{1}, n))(θ_{n−1}) − (λ − α(i_{2}, n))(θ_{n−1}) = C(i_{1}, i_{2}). (6)

Proof. We have for l = 1, 2:

λ − α(i_{l}, n) = (λ_{1}, …, λ_{i_{l}−1}, λ_{i_{l}} − 1, λ_{i_{l}+1}, …, λ_{n−1}, λ_{n} + 1).

So

(λ − α(i_{l}, n))(θ_{n−1}) = ∑_{{1≤j<k≤n−1 \atop j,k≠i_{l}}}(λ_{j} − j)(λ_{k} − k) + ∑_{j=1}^{i_{l}−1}(λ_{j} − j)(λ_{i_{l}} − 1 − i_{l})

+ ∑_{k=i_{l}+1}^{n−1}(λ_{i_{l}} − 1 − i_{l})(λ_{k} − k).

Now a simple calculation proves (6).

2. A filtration of the first level

The representation

det^{c} : GL(n) → GL(1), g ↦ (det g)^{c}

has high weight (c, c, …, c), c ∈ ℤ. It follows that

L_{n}(λ_{1}, λ_{2}, …, λ_{n}) ⊗ det^{c} ≅ L_{n}(λ_{1} + c, λ_{2} + c, …, λ_{n} + c).

Since all interesting properties of L_{n}(λ) are invariant with respect to such an operation we may assume that λ_{n} = 0. Thus, from now on,

λ is an arbitrary dominant weight of the form λ = (λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{n}), with λ_{n} = 0

This is convenient because now we can consider the Young diagram of λ. If (i, j) is a node of λ we define its p-residue to be (i − j) (mod p). (Observe that when we work with the symmetric group the residue is defined to be (j − i) (mod p). The reason for the difference comes from the fact that Schur functors transpose diagrams). A node (i, λ_{i}) is called normal if and only if i is normal as defined in Section 1. It follows from (2) that normal nodes are removable. (Note again that the normality just defined is equivalent to the normality in the introduction but up to transposing Young diagrams.)

Fix a residue α ∈ ℤ/pℤ. Let

(7) Γ_{α} = {i_{1}^{(α)} < i_{2}^{(α)} < ⋯ < i_{m_{α}}^{(α)}} = {i | i is normal and res(i, λ_{i}) = α}.

Since α is fixed we often omit α everywhere and write Γ = {i_{1} < i_{2} < ⋯ < i_{m}}. Of course Γ might be empty, which will turn out to be an easy case. So we now concentrate on the case where Γ ≠ ∅. Note that C(i, j) = λ_{i} − λ_{j} + j − i is just the difference between the residues of (j, λ_{j}) and (i, λ_{i}) so

C(i_{j}, i_{k}) ≡ 0 for all 1 ≤ j ≤ k ≤ m,

and

C(min(i_{j}, l), max(i_{j}, l)) ≢ 0

for any normal l ∉ {i_{1}, i_{2}, …, i_{m}} and any j ∈ {1, 2, …, m}. Set

B_{j} = B(i_{j}, i_{j+1}), C_{j} = C(i_{j}, i_{j+1}), C̄_{j} = {i_{j}} ⊔ C_{j}, j = 1, 2, …, m.

(we always interpret i_{m+1} as n). Also put

B_{jk} = ∪_{j≤l≤k}B_{l}, C_{jk} = ∪_{j≤l≤k}C_{l}, C̄_{jk} = ∪_{j≤l≤k}C̄_{l}, 1 ≤ j, k ≤ m.

Note that

B(i_{j}, i_{j+1} − 1) = C(i_{j}, i_{j+1}) − 1 ≡ −1 ≢ 0, j = 1, 2, …, m − 1.

This together with

B(i_{j}, i_{j}) = λ_{i_{j}} − λ_{i_{j}+1} ≢ 0

(see (2)) implies

B_{j} ⊆ (i_{j}, i_{j+1} − 2], j = 1, 2, …, m − 1. (8)

Lemma 2.1. Let 1 ≤ j ≤ m. Then (i) |C_{j}| = |B_{j}|; (ii) C_{j} compensates B_{j}; (iii) C(i_{j}, n) = C_{j} ⊔ C̄_{j+1} ⊔ ⋯ ⊔ C̄_{m}; (iv) B(i_{j}, n) = B_{j} ⊔ B_{j+1} ⊔ ⋯ ⊔ B_{m}.

Proof. By definition, the unions (iii) and (iv) are disjoint. To prove (iv), note that

B(i_{j}, i_{k}) = C(i_{j}, i_{k}) + (λ_{i_{k}} − λ_{i_{k}+1}) ≡ (λ_{i_{k}} − λ_{i_{k}+1}) ≢ 0

since i_{k} is normal, see (2). So i_{k} ∉ B(i_{j}, n). If l ∈ (i_{j}, n) \ {i_{j+1}, i_{j+2}, …, i_{m}}, we have l ∈ (i_{k}, i_{k+1}) for some k ≥ j. Then

B(i_{j}, l) = C(i_{j}, i_{k}) + B(i_{k}, l) ≡ B(i_{k}, l),

which implies (iv). The proof of (iii) is similar.

Recall that i_{j} is normal by definition. This implies that i_{j} is normal in [i_{j}, i_{j+1}), i.e. C_{j} compensates B_{j}, which proves (ii). In particular, |C_{j}| ≥ |B_{j}|. If |C_{j}| > |B_{j}|, then, by Lemma 1.1, there exists c ∈ C_{j}, normal in [c, i_{j+1}). Observe that

{l | i_{j+1} ≤ l < n, B(c, l) ≡ 0} = B_{j+1} ⊔ ⋯ ⊔ B_{m},

which is compensated by C_{j+1}⊔⋯⊔C_{m}. This, together with the fact that c is normal in [c, i_{j+1}), implies that c is normal in [c, n). But C(i_{j}, c) ≡ 0 and i_{j} < c < i_{j+1}, which contradicts the choice of i_{1}, i_{2}, …, i_{m}.

Now we define a direct summand of L_{n}(λ) ↓_{U(n−1)}, called the first level, which is of crucial importance for the symmetric groups. Informally, the first level consists of the weight spaces corresponding to the weights obtained from λ by subtracting the last simple root α(n − 1, n) and then subtracting an arbitrary linear combination of the other simple roots.

Definition 2.2. The first level L^{(1)} is

L^{(1)} = ⊕L_{n}(λ)_{μ}

where the sum is over all weights μ such that

λ − μ = ∑_{j=1}^{n−2} k_{j}α(j, j + 1) + α(n − 1, n)

for some k_{1}, …, k_{n−2} ∈ ℤ_{≥0}.

Since U(n − 1) is embedded “by the first n − 2 simple roots”, L^{(1)} is invariant with respect to U(n − 1), so it is a submodule, actually a direct summand, of L_{n}(λ) ↓_{U(n−1)}, see [13] for more details. We always consider L^{(1)} as a U(n − 1)- module.

To describe a filtration of L^{(1)} we define

v(j, k) = T_{i_{j},n} (C_{j} ∪ C̄_{j+1,k} ∪ C_{k+1,m}) v_{+}
= T_{i_{j},n}(C_{jm} ∪ {i_{j+1}, …, i_{k}})v_{+} ∈ L^{(1)}, 1 ≤ j ≤ k ≤ m,

and

V = {v(j, k) | 1 ≤ j ≤ k ≤ m}.

Remark. It was proved in [14, Theorems 4.2, 4.11] that v(j, j), j = 1, 2, …, m, are U(n − 1)-primitive vectors, and the module U(n − 1)v(j, j) is irreducible if and only if j = 1.

Set

Q_{α} = Q_{Γ_{α}} = Q_{Γ} = U(n − 1)⟨V ⟩, (9)

the U(n − 1)-submodule of L^{(1)} generated by V. Below we shall prove that

L^{(1)} = ⊕_{α∈ℤ/pℤ}Q_{α}

and that the non-zero Q_{α} are indecomposable (by definition, Q_{α} = 0 if and only if there are no normal nodes of residue α). For r = 1, 2, …, 2m − 1 we define sets V_{r} as follows:

V_{r} = {v(j, k) | 1 ≤ j ≤ k ≤ m, 2k − j = r}.

Then

V = V_{1} ⊔ V_{2} ⊔ ⋯ ⊔ V_{2m−1}.

12 ALEXANDER KLESHCHEV

Example. Let m = 5. Then V_{9} = {v(1, 5)}, V_{8} = {v(2, 5)}, V_{7} = {v(1, 4), v(3, 5)}, V_{6} = {v(2, 4), v(4, 5)}, V_{5} = {v(1, 3), v(3, 4), v(5, 5)}, V_{4} = {v(2, 3), v(4, 4)}, V_{3} = {v(1, 2), v(3, 3)}, V_{2} = {v(2, 2)}, V_{1} = {v(1, 1)}.

and v(1, 5) = T_{i_{1},n}(C_{1} ∪ C̄_{2} ∪ C̄_{3} ∪ C̄_{4} ∪ C̄_{5})v_{+}, v(2, 5) = T_{i_{2},n}(C_{2} ∪ C̄_{3} ∪ C̄_{4} ∪ C̄_{5})v_{+}, v(1, 4) = T_{i_{1},n}(C_{1} ∪ C̄_{2} ∪ C̄_{3} ∪ C̄_{4} ∪ C_{5})v_{+}, v(3, 5) = T_{i_{3},n}(C_{3} ∪ C̄_{4} ∪ C̄_{5})v_{+}, v(2, 4) = T_{i_{2},n}(C_{2} ∪ C̄_{3} ∪ C̄_{4} ∪ C_{5})v_{+}, v(4, 5) = T_{i_{4},n}(C_{4} ∪ C̄_{5})v_{+}, v(1, 3) = T_{i_{1},n}(C_{1} ∪ C̄_{2} ∪ C̄_{3} ∪ C_{4} ∪ C_{5})v_{+}, v(3, 4) = T_{i_{3},n}(C_{3} ∪ C̄_{4} ∪ C_{5})v_{+}, v(5, 5) = T_{i_{5},n}(C_{5})v_{+}, v(2, 3) = T_{i_{2},n}(C_{2} ∪ C̄_{3} ∪ C_{4} ∪ C_{5})v_{+}, v(4, 4) = T_{i_{4},n}(C_{4} ∪ C_{5})v_{+}, v(1, 2) = T_{i_{1},n}(C_{1} ∪ C̄_{2} ∪ C_{3} ∪ C_{4} ∪ C_{5})v_{+}, v(3, 3) = T_{i_{3},n}(C_{3} ∪ C_{4} ∪ C_{5})v_{+}, v(2, 2) = T_{i_{2},n}(C_{2} ∪ C_{3} ∪ C_{4} ∪ C_{5})v_{+}, v(1, 1) = T_{i_{1},n}(C_{1} ∪ C_{2} ∪ C_{3} ∪ C_{4} ∪ C_{5})v_{+}.

We now define two important filtrations (F) and (Φ) of Q_{Γ}. Set

(F) (0) = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{2m−1} = Q_{Γ},

where

F_{r} = U(n − 1)⟨V_{1}, V_{2}, …, V_{r}⟩, r = 1, 2, …, 2m − 1.

The filtration (Φ) is a refinement of (F) such that every consecutive quotient will turn out to be a module generated by a high weight vector. First we need the following total order on pairs {(j, k) | 1 ≤ j ≤ k ≤ m}: set (j, k) ≺ (j′, k′) ⇔ 2k − j < 2k′ − j′ or 2k − j = 2k′ − j′ and j < j′. (10) Put

Φ_{jk} = U(n)⟨v(j′, k′) | (j′, k′) ⪯ (j, k)⟩, Φ′_{jk} = U(n)⟨v(j′, k′) | (j′, k′) ≺ (j, k)⟩. Then the submodules {Φ_{jk} | 1 ≤ j ≤ k ≤ m} taken in our total order form a filtration of Q_{Γ} with consecutive quotients Φ_{jk}/Φ′_{jk}. We call this filtration (Φ).

We concentrate on (F) first. To prove some of its properties we need the following technical result which will be used to show that certain expressions are equal to zero.

Lemma 2.3. Let 1 ≤ l ≤ m, c_{u} ∈ C_{l} = {c_{1} < c_{2} < ⋯ < c_{d}}. Then for any D ⊆ [i_{l+1}, n) we have T_{c_{u},n}({c_{u+1}, …, c_{d}} ∪ D)v_{+} = 0.

Proof. Let B_{l} = {b_{1} < b_{2} < ⋯ < b_{d}}, see Lemma 2.1(i). Then, by Lemma 2.1(ii), c_{j} ≤ b_{j}, and B(c_{j}, b_{j}) = B(i_{j}, b_{j}) − C(i_{j}, c_{j}) ≡ 0 for j = u, u + 1, …, d. Moreover, b_{j} ≤ i_{l+1} −2 (interpret as b_{j} ≤ n−1 if l = m), in view of (8), hence also c_{j} ≤ i_{l+1} −2 (j = u, u + 1, …, d). Now apply Lemma 1.5 with

i = c_{u}, j = n, M = {c_{u+1}, …, c_{d}} ∪ D, k = i_{l+1} − 1 (interpret as k = n if l = m), M ∩ (i, k) = {c_{u+1}, …, c_{d}}, and φ(c_{v}) = b_{v}, v = u, u + 1, …, d.

Corollary 2.4. Let i_{k} < t < n. Then L_{t}v(j, k) = 0.

Proof. Set

M = C_{j} ∪ C̄_{j+1,k} ∪ C_{k+1,m}.

Then L_{t}v(j, k) = L_{t}T_{i_{j},n} (M) v_{+}. So, in view of Lemma 1.11, we may assume that t ∈ C_{km}. In this case, again by Lemma 1.11,

L_{t}v(j, k) = −T_{i_{j},t}(M ∩ (i_{j}, t))T_{tn}(M ∩ (t, n))v_{+},

which is 0 by Lemma 2.3.

Proposition 2.5. Let v(j, k) ∈ V_{r}. Then v(j, k) + F_{r−1} is a U(n − 1)-primitive vector in F_{r}/F_{r−1}.

Proof. Note that the weight of v(j, k) is λ − α(i_{j}, n), so we only have to check that E_{l,l+1}v(j, k) ∈ F_{r−1} for l ∈ [i_{j}, n − 2]. Recall that v(j, k) = T_{i_{j},n}(M)v_{+}, where

M = C_{j} ∪ {i_{j+1}} ∪ C_{j+1} ∪ ⋯ ∪ {i_{k}} ∪ C_{k} ∪ C_{k+1} ∪ ⋯ ∪ C_{m}.

By Lemma 1.4, we may assume that l ∉ M and l + 1 ∈ M. Suppose first that l + 1 ∈ {i_{j+1}, …, i_{k}}, i.e. l = i_{t} − 1 for j + 1 ≤ t ≤ k. Then

M ∩ (i_{t}, l] = C_{t} ∪ C̄_{t+1,k} ∪ C_{k+1,m}.

So Lemma 1.4 yields

E_{l,l+1}v(j, k) = E_{i_{t}−1,i_{t}}T_{i_{j},n}(M)v_{+} = ST_{i_{t},n}(M ∩ (i_{t}, n))v_{+} = Sv(t, k),

for some S ∈ U(n−1). But t > j, hence 2k −t < 2k −j = r, and so Sv(t, k) belongs to F_{r−1}.

Now, let l + 1 ∈ C_{j} ∪ C_{j+1} ∪ ⋯ ∪ C_{m}. Then l + 1 = c_{u} ∈ C_{t} = {c_{1} < ⋯ < c_{d}}, for some j ≤ t ≤ m.

By Lemma 1.4, we have

E_{l,l+1}v(j, k) = E_{c_{u}−1,c_{u}}T_{i_{j},n}(M)v_{+} = ST_{c_{u},n}({c_{u+1}, …, c_{d}} ∪ D)v_{+},

14 ALEXANDER KLESHCHEV

where S ∈ U(n − 1) and D = M ∩ [i_{t+1}, n) ⊆ [i_{t+1}, n). Now,

T_{c_{u},n}({c_{u+1}, …, c_{d}} ∪ D)v_{+} = 0

by Lemma 2.3 Corollary 2.6. Q_{Γ} = U(n − 1)^{−}⟨V ⟩.

The next lemma provides us with a central element z_{n−1} + x which will be used to construct endomorphisms of the first level. Lemma 2.7. There exists a constant x such that

(z_{n−1} + x)v(j, k) = v(j, k − 1) if j < k, and (z_{n−1} + x)v(j, j) = 0 for all 1 ≤ j ≤ k ≤ m. Proof. Recall that z_{n−1} = θ_{n−1} − L_{2} − L_{3} − … L_{n−1}, where θ_{n−1} ∈ U(n − 1)^{0}. Therefore

θ_{n−1}v(j, k) = (λ − α(i_{j}, n))(θ_{n−1}).

By Lemma 1.12, for any 1 ≤ j < j′ ≤ m, we have

(λ − α(i_{j}, n))(θ_{n−1}) − (λ − α(i_{j′}, n))(θ_{n−1}) = C(i_{j}, i_{j′}) ≡ 0,

so we can choose a scalar x (one for all j) such that

(θ_{n−1} + x)v(j, k) = 0, 1 ≤ j ≤ k ≤ m.

So

(z_{n−1} + x)v(j, k) = (θ_{n−1} + x − L_{2} − L_{3} − … L_{n−1})v(j, k)
= (−L_{2} − L_{3} − … L_{n−1})v(j, k)
= (C(i_{j}, i_{k}) − L_{2} − L_{3} − … L_{n−1})v(j, k)

since C(i_{j}, i_{k}) ≡ 0. By Lemma 1.11, the last expression equals

(C(i_{j}, i_{k}) − L_{i_{j}+1} − L_{i_{j}+2} − ⋯ − L_{i_{k}})v(j, k) − ∑_{t=i_{k}+1}^{n−1} L_{t}v(j, k). (11)

By Lemma 1.8, we have

(C(i_{j}, i_{k}) − L_{i_{j}+1} − L_{i_{j}+2} − ⋯ − L_{i_{k}})v(j, k) = v(j, k − 1)

(or is absent if j = k). So it suffices to apply Corollary 2.4 to show that the second term in (11) equals zero.

The following result shows that the filtration (F) is as strict as it can be. Proposition 2.8. Let v(j, k) ∈ V_{r}. Then v(j, k) ∉ F_{r−1}.

Proof. If v(j, k) ∈ F_{r−1} then, considering weights and using Corollary 2.6, we can write

v(j, k) = ∑ y_{t,u}v(t, u), (12)

where the sum is over all t, u such that j ≤ t ≤ u ≤ m, 2u − t < r, and y_{t,u} ∈ U(n)_{−α(i_{j},i_{t})}^{−}.

By Lemma 2.7, there exists a scalar x such that (z_{n−1}+x)v(j, k) = v(j, k−1) if j < k and (z_{n−1}+x)v(j, j) = 0. Since z_{n−1} is central in U(n−1), so is (z_{n−1}+x). Multiply both sides of (12) by (z_{n−1} + x)^{k−j}. The LHS becomes v(j, j) = T_{i_{j},k}(C_{jm})v_{+} ≠ 0, by Lemma 1.5. On the other hand, since all y_{t,u} ∈ U(n − 1), the RHS equals

∑ y_{t,u}(z_{n−1} + x)^{k−j}v(t, u),

and we claim that this is zero. Well, in view of Lemma 2.7, we only have to show that u − t < k − j. But this is true since 2u − t < r implies u − t < \frac{r−t}{2}, 2k − j = r implies k − j = \frac{r−j}{2} , and t ≥ j. (This argument also shows that if j = k in (12) then there are no t, u such that j ≤ t ≤ u ≤ m, 2u − t < r, and so (12) should be interpreted as v(j, j) = 0, which is not true).

Let us choose x as in Lemma 2.7. Since z_{n−1} + x is a central element of U(n − 1) (see (5)), the multiplication by this element defines a U(n − 1)-endomorphism of the first level L^{(1)}. Moreover, in view of Lemma 2.7, the submodule Q_{Γ}, defined in (9), is invariant with respect to this endomorphism. So we can give the following

Definition 2.9. We define an endomorphism φ_{Γ} ∈ End_{U(n−1)}(Q_{Γ}, Q_{Γ}) by setting

φ_{Γ}(v) = (z_{n−1} + x)v, v ∈ Q_{Γ}.

The next result shows that Q_{Γ} is a cyclic module generated by the vector v(1, m).

Lemma 2.10. Q_{Γ} = U(n − 1)⟨v(1, m)⟩.

Proof. Let 2 ≤ j ≤ m. By Lemma 2.1(ii) and (8), we have i_{j} − 1 ∉ C_{1} ∪ C̄_{2,m}. Since v(1, m) = T_{i_{1},n}(C_{1} ∪ C̄_{2,m})v_{+}, Lemma 1.4 yields

E_{i_{j}−1,i_{j}}v(1, m) = −T_{i_{1},i_{j}−1}(C_{1} ∪ C̄_{2,j−1})T_{i_{j},n}(C_{j} ∪ C̄_{j+1,m})v_{+}. (13)

In view of Lemma 1.5, we have T_{i_{1},i_{j}−1}(C_{1} ∪ C̄_{2,j−1})v_{+} ≠ 0, hence there exists an element u ∈ U(n − 1)_{α(i_{1},i_{j}−1)}^{+} such that

uT_{i_{1},i_{j}−1}(C_{1} ∪ C̄_{2,j−1})v_{+} = cv_{+}

for a non-zero constant c. Since both u and T_{i_{1},i_{j}−1}(C_{1} ∪ C̄_{2,j−1}) commute with T_{i_{j},n}(C_{j} ∪ C̄_{j+1,m}), we conclude from (13) that

uE_{i_{j}−1,i_{j}}v(1, m) = −cT_{i_{j},n}(C_{j} ∪ C̄_{j+1,m})v_{+} = −cv(j, m).

Now, using Lemma 2.7, we get any v(j, k) in U(n−1)⟨v(1, m)⟩, because (z_{n−1} +x) ∈ U(n − 1).

We are going to use some special U(n − 1)-weights which are the high weights of certain composition factors of L^{(1)}. Set

μ^{j} = (λ − α(i_{j}, n)) ↓ ∈ X(n − 1), j = 1, 2, …, m. (14)

Proposition 2.11.

(i) Let 1 ≤ t < u ≤ n and μ = (λ − α(t, u)) ↓∈ X_{+}(n − 1). Then L_{n−1}(μ) is a composition factor of L^{(1)} if and only if u = n and t is normal;

(ii) Let 1 ≤ j ≤ m. Then L_{n−1}(μ^{j}) is a composition factor of F_{r}/F_{r−1} if and only if r = 2k −j for some k ∈ {j, j +1, …, m}. In this case the multiplicity [F_{r}/F_{r−1} : L_{n−1}(μ^{j})] is equal to 1.

Proof. (i) is proved in [16, 9.4].

(ii) Let j ≤ k ≤ m. Put 2k − j = r. Then v(j, k) ∈ F_{r}. By virtue of Propositions 2.8 and 2.5, v(j, k) + F_{r−1} is a non-zero U(n − 1)^{+}-primitive vector of F_{r}/F_{r−1}. So [F_{r}/F_{r−1} : L_{n−1}(μ^{j})] ≥ 1. To prove that the multiplicity is actually equal to 1, and that L_{n−1}(μ^{j}) does not appear in the layer F_{r}/F_{r−1} for r not of the form 2k − j with k ≥ j, it suffices to know that

[Q_{Γ} : L_{n−1}(μ^{j})] ≤ m − (j − 1).

But this follows from the fact that

[L^{(1)} : L_{n−1}(μ^{j})] = m − (j − 1),

proved in [16, 9.4(ii)].

Now we return to the filtration (Φ).

Lemma 2.12. Let 1 ≤ j ≤ k ≤ m. Then

(i) Φ_{jk}/Φ′_{jk} is a non-zero module generated by a high weight vector v(j, k)+Φ′_{jk} of weight μ^{j};

(ii) Φ_{jk}/Φ′_{jk} does not have composition factors of the form L_{n−1}((λ−α(t, u)) ↓),

1 ≤ t < u ≤ n, except L_{n−1}(μ^{j}); (iii) Φ_{j,k+1}/Φ′_{j,k+1} is a quotient of Φ_{jk}/Φ′_{jk}. Moreover,

φ_{Γ}(Φ_{j,k+1}) ⊆ Φ_{jk}, φ_{Γ}(Φ′_{j,k+1}) ⊆ Φ′_{jk},

and in this way φ_{Γ} induces a surjective homomorphism from Φ_{j,k+1}/Φ′_{j,k+1} to Φ_{jk}/Φ′_{jk}.

Proof. (i) In view of Proposition 2.5, to prove (i), we only have to show that v(j, k) ∉ Φ′_{jk}. Let 2k − j = r. By Corollary 2.6, we have

Φ′_{jk} = U(n)^{−}⟨F_{r−1}, v(j′, k′) | 2k′ − j′ = r, j′ < j⟩

However, the weight of v(j′, k′) is λ − α(i_{j′}, n) < λ − α(i_{j}, n), for any j′ < j. Therefore

(Φ′_{jk})_{λ−α(i_{j},n)} = (F_{r−1})_{λ−α(i_{j},n)}.

So v(j, k) ∈ Φ′_{jk} implies v(j, k) ∈ F_{r−1}, which contradicts Proposition 2.8. (ii) follows from Proposition 2.11 and (i).

(iii) In view of Lemma 2.7, φ_{Γ}(Φ_{j,k+1}) ⊆ Φ_{jk} and φ_{Γ}(Φ′_{j,k+1}) ⊆ Φ′_{jk}. So φ_{Γ} induces a homomorphism from Φ_{j,k+1}/Φ′_{j,k+1} to Φ_{jk}/Φ′_{jk}. It is surjective by (i), since φ_{Γ}(v(j, k + 1)) = v(j, k) in view of Lemma 2.7.

Next we want to show that Q_{Γ}’s are actually direct summands of L^{(1)}. We now return to the more detailed notation with α’s in the indices: Γ_{α}, Q_{α}, i_{j}^{(α)}, etc. Recall that, by definition, Q_{α} ≠ 0 if and only if there is a normal i with res(i, λ_{i}) = α.

Theorem 2.13.

(i) L^{(1)} = ⊕_{α∈ℤ/pℤ}Q_{α}; (ii) If Q_{α} ≠ 0 then Q_{α} has a simple socle and a simple head both isomorphic to

L_{n−1}((λ − α(i_{1}^{α}, n)) ↓); (iii) Q_{α} is indecomposable (for any α). (iv) Q_{α} and Q_{β} are in distinct blocks (for any pair α ≠ β);

(v) Q_{α} is (contravariantly) selfdual (for any α).

Proof. Put M = ∑_{α} Q_{α}. We first show that M = L_{1}. By (1), it suffices to show that every F_{jn}v_{+}, j = 1, 2, …, n − 1, is contained in M. We apply downward induction on j. Let j = n − 1. If λ_{n−1} − λ_{n} = λ_{n−1} ≡ 0 then F_{n−1,n}v_{+} = 0. Otherwise n − 1 is normal, so n − 1 is equal to some i_{m_{α}}^{α}, i.e. belongs to the corresponding Γ_{α}. Now F_{n−1,n}v_{+} = T_{n−1,n}(∅)v_{+} = v(m_{α}, m_{α}) ∈ Q_{α} ⊆ M. Let j < n − 1. If j is not normal then F_{j,n}v_{+} can be expressed through non-trivial subdivisions, see Lemma 1.7, and the inductive hypothesis applies. So we may assume that j is normal. Then j belongs to some Γ_{α} of the form (7). Let j = i_{t}^{α}. By Lemmas 2.1(iii) and 1.2, we have for the corresponding vector v(t, m_{α}):

v(t, m_{α}) = aF_{jn}v_{+} + [other terms]

for some non-zero constant a. Applying inductive hypothesis to [other terms] and using the fact that v(t, m_{α}) ∈ Q_{α}, we conclude that F_{jn}v_{+} ∈ Q_{α} ⊆ M.

Now, let α(1), …, α(q) be all the residues α such that Q_{α} ≠ 0 (equivalently, Γ_{α} ≠ ∅). It is proved in [14, 4.11] that the socle of L^{(1)} is

⊕_{j=1}^{q}L_{n−1}((λ − α(i_{1}^{α(j)}, n)) ↓).

Note that the Young diagrams of the weights (λ − α(i_{1}^{α(j)}, n)) ↓ all have different residue contents, just because all α(j) are different. By the linkage principle for GL(n) (see the explanation in the proof of [5, 4.3]), all composition factors of the socle belong to different blocks. This immediately implies that all Q_{α(j)} are indecomposable, belong to different blocks, and have the simple socles as claimed in (ii).

Moreover, it is proved in [11] (see also [12, 2.9]) that L^{(1)} is (contravariantly) selfdual. (We refer the reader to [10, p.205] for the definition of the contravariant dual ^{τ}V of a module V.) Since τ preserves the blocks, all ^{τ}Q_{α} are also in different blocks, and we may conclude that ^{τ}Q_{α} ≅ Q_{α}. This completes the proof of the theorem.

Theorem 2.14. Let

Γ = Γ_{α} = {i_{1} < i_{2} < ⋯ < i_{m}}.

Then

{id_{Q_{Γ}}, φ_{Γ}, …, φ_{Γ}^{m−1}}

is a basis of End_{U(n−1)}(Q_{Γ}, Q_{Γ}).

Proof. Let us write φ and Q for φ_{Γ} and Q_{Γ}. We first prove that the homomorphisms φ^{0} = id_{Q}, φ, …, φ^{m−1} are linearly independent. Assume a linear combination

c_{0}φ^{0} + c_{1}φ + ⋯ + c_{m−1}φ^{m−1}

is zero. If not all c_{t} are equal to zero, choose the smallest number s with c_{s} ≠ 0. By Lemma 2.7, we have

φ^{s}(v(1, m)) = v(1, m − s),

and

φ^{t}(v(1, m)) = v(1, m − t) ∈ F_{2(m−t)−1}.

In view of Proposition 2.8, we have v(1, m − s) ∉ F_{2(m−t)−1} for any t > s. So

(c_{0}φ^{0} + c_{1}φ + ⋯ + c_{m−1}φ^{m−1})(v(1, m)) ≠ 0,

giving a contradiction.

Now, we show that {φ^{0}, φ, …, φ^{m−1}} span the space End_{U(n−1)}(Q). Pick an element f ∈ End_{U(n−1)}(Q). There exists a pair (j, k) with 1 ≤ j ≤ k ≤ m such that f(v(1, m)) ∈ Φ_{jk} \ Φ′_{jk}. We prove the result by induction on the (j, k), taken in the total order (10), using the case f(v(1, m)) = 0 as the induction base. Indeed, if f(v(1, m)) = 0 then f = 0 by Lemma 2.10.

By Theorem 2.13, Q has a simple head isomorphic to L_{n−1}(μ) where μ = (λ − α(i_{1}, n)) ↓. Let J be the submodule of Q such that Q/J ≅ L_{n−1}(μ). Then v(1, m)+J is a (non-zero) high weight vector of Q/J.

We claim that L_{n−1}(μ) is a composition factor of Φ_{jk}/Φ′_{jk}. Indeed, f(Q), being a non-zero quotient of Q, has the same simple head, and f(v(1, m)) + f(J) is a nonzero high weight vector of f(Q)/f(J). Since f(J) is the unique maximal proper submodule of f(Q), we have f(J) ⊇ f(Q) ∩ Φ′_{jk}. We conclude that the module

(f(Q) + Φ′_{jk})/Φ′_{jk} ≅ f(Q)/(f(Q) ∩ Φ′_{jk})

contains a composition factor L_{n−1}(μ). But this module is a submodule of Φ_{jk}/Φ′_{jk} because f(v(1, m)) ∈ Φ_{jk} and v(1, m) generates Q in view of Lemma 2.10.

Now, by Lemma 2.12, we have j = 1 and

f(v(1, m)) ≅ cv(1, k) (mod Φ′_{1k})

for some non-zero constant c. However φ^{m−k}(v(1, m)) = v(1, k), so the inductive hypothesis applies to f − cφ^{m−k}.

3. From GL(n) to Σ_{n}.

The information about Schur functors used in this section can be found in [8], [11] and [13]. Let M_{F} (n, n) be the category of the (left) finite dimensional polynomial representations of GL(n) in which all coefficients are homogeneous polynomials of degree n in the matrix coefficients of GL(n). We also denote by mod _{F} Σ_{n} the category of the (left) finite dimensional F Σ_{n}-modules. If V ∈ M_{F} (n, n) and ω = (1, 1, …, 1) ∈ Λ(n) then the weight space V_{ω} is invariant with respect to the subgroup of monomial matrices in GL(n). This subgroup is isomorphic to Σ_{n}, and so V → V_{ω} defines an exact functor

f_{n} : M_{F} (n, n) → mod _{F} Σ_{n}.

Under this functor, a homomorphism φ : V → W goes to f_{n}(φ) : V_{ω} → W_{ω} which is just the restriction of φ to V_{ω}.

A dominant weight λ ∈ X_{+}(n) is called p-restricted if λ_{i}−λ_{i+1} < p for 1 ≤ i < n. It is known that

f_{n}(L_{n}(λ)) = \begin{cases}D^{λ′} ⊗ sgn_{n} & if λ is p-restricted, \\ 0 & otherwise, \end{cases}

and

f_{n}(Δ_{n}(λ)) = S^{λ′} ⊗ sgn_{n},

where λ′ denotes the partition conjugate to λ, and sgn_{n} is the alternating representation of Σ_{n}.

It was first noticed in [11] (see also [12, 4.7(i),4.8]) that the first level L^{(1)} is the ‘right object’ to look at if we are interested in restriction from Σ_{n} to Σ_{n−1}:

L^{(1)} ∈ M_{F} (n − 1, n − 1), and f_{n−1}(L^{(1)}) = (D^{λ′} ⊗ sgn_{n}) ↓_{Σ_{n−1}} .

In [11], a certain subcategory M_{F} (n, n)^{c} of M_{F} (n, n) was shown to be equivalent to mod _{F} Σ_{n} under the Scur functor and certain inverse functor

h_{n} : mod _{F} Σ_{n} → M_{F} (n, n)^{c}.

This subcategory can be defined as a full subcategory of M_{F} (n, n) having p-restricted socles and heads. It is shown in [12, Theorem B] that L^{(1)} is such a module. This explains how to deduce Theorems A(i), B, E in the Introduction from Theorem 2.14 and Lemma 2.12.

4. On a Conjecture of Benson

In this section we show how to deduce Theorems A(ii), C and Corollary D from Theorem A(i).

Lemma 4.1. Let V and W be F Σ_{n}-modules. Then

dim Hom_{Σ_{n+1}}(V ↑^{Σ_{n+1}}, W ↑^{Σ_{n+1}})
= dim Hom_{Σ_{n}}(V, W) + dim Hom_{Σ_{n−1}}(V ↓_{Σ_{n−1}}, W ↓_{Σ_{n−1}}).

Proof. This is the Intertwining Number Theorem from [7, 44.5] with G = Σ_{n}, H_{1} = H_{2} = Σ_{n−1}.

Corollary 4.2. Let D^{λ} be an irreducible F Σ_{n}-module. Then

dim End_{Σ_{n+1}}(D^{λ} ↑^{Σ_{n+1}}) = 1 + dim End_{Σ_{n−1}}(D^{λ} ↓_{Σ_{n−1}}).

This corollary together with Theorem A(i) yields Theorem A(ii). To deduce Theorem C it now suffices to prove the following

Lemma 4.3. Let D^{λ} be an irreducible F Σ_{n}-module. Suppose λ has r good addable nodes, say, B_{1}, B_{2}, …, B_{r}. Then D^{λ} ↑^{Σ_{n+1}} is semisimple if and only if

dim End_{Σ_{n+1}}(D^{λ} ↑^{Σ_{n+1}}) = r.

Moreover, in this case we have

D^{λ} ↑^{Σ_{n+1}} ≅ D^{λ^{B_{1}}} ⊕ D^{λ^{B_{2}}} ⊕ ⋯ ⊕ D^{λ^{Br}} .

Proof. By [15, 3.2], we have

D^{λ} ↑^{Σ_{n+1}} ≅ J_{1} ⊕ J_{2} ⊕ ⋯ ⊕ J_{r}

where every J_{i} is indecomposable and has a simple socle and a simple head both isomorphic to D^{λ^{B_{i}}}. So dim End_{Σ_{n+1}}(J_{i}) = 1 if J_{i} is irreducible, and dim End_{Σ_{n+1}}(J_{i}) > 1 otherwise. The lemma follows.

Finally we prove Corollary D.

Corollary 4.4. Let p = 2, λ = (λ_{1} > λ_{2} > ⋯ > λ_{t} > 0) be a 2-regular partition, and D^{λ} be the corresponding irreducible. Then D^{λ} ↑^{Σ_{n+1}} is semisimple if and only if all λ_{i} are even. In this case,

D^{λ} ↑^{Σ_{n+1}} ≅ D^{(λ_{1},...,λ_{t−1},λ_{t}+1)} ⊕ D^{(λ_{1},...,λ_{t},1)}.

Proof. If all λ_{i} are even then the top removable node (1, λ_{1}) is the only normal node for λ, and the two bottom addable nodes, (t+1, 1) and (t, λ_{t} +1), are the only good addable nodes for λ. In view of Theorem C, this proves the result in one direction.

Conversely, let D^{λ} ↑^{Σ_{n+1}} be semisimple. In characteristic 2 a partition has at most two good addable nodes, see [15, 1.4]. In our case there should be exactly two, since otherwise, in view of Theorem C, D^{λ} ↑^{Σ_{n+1}} is irreducible, which implies dim End_{Σ_{n+1}}(D^{λ} ↑^{Σ_{n+1}}) = 1, contradicting Corollary 4.2.

So we have to show that if λ has two good addables and one normal removable then all λ_{i} are even. But a 2-regular partition has only one removable if and only if λ_{1} ≡ λ_{2} ≡ ⋯ ≡ λ_{t} (mod 2). If all λ_{i} are odd then the only good addable node is the next to the bottom one. This proves the corollary.

References

1. D. Benson. Some remarks on the decomposition numbers for the symmetric groups, p. 381-394 in Proceedings Arcata Conference on Representations of Finite Groups, Proc. Symp. Pure Math. 47, Part 1, Amer. Math. Soc., Providence, 1987.

2. J. Brundan. Lowering operators for GL(n) and quantum GL(n), to appear in Proceedings Seattle Conference on Representation Theory, Proc. Symp. Pure Math.

3. J. Brundan. Double Cosets in Algebraic Groups, Ph. D. Thesis, Imperial College, London, 1996.

4. J. Brundan. Modular branching rules and the Mullineux map for Hecke algebras of type A, Proc. London Math. Soc. (3), to appear.

5. J. Brundan, A. Kleshchev and I. Suprunenko. Semisimple restrictions from GL(n) to GL(n − 1), preprint, 1997.

6. R. W. Carter and G. Lusztig, On the modular representations of the general linear and symmetric groups, Math. Z. 136(1974), 193–242.

7. C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Interscience Publ., New York/London, 1962.

8. J. A. Green. Polynomial Representations of GL_{n}, volume 830 of Lecture Notes in Mathematics. Springer-Verlag, Berlin/Heidelberg/New York, 1980.

9. G. D. James. The Representation Theory of the Symmetric Groups, volume 682 of Lecture Notes in Mathematics. Springer, New York/Heidelberg/Berlin, 1978.

10. J. C. Jantzen. Representations of Algebraic Groups, Academic Press, Orlando, 1987.

11. J. C. Jantzen and G. M. Seitz. On the representation theory of the symmetric groups. Proc. London Math. Soc. (3), 65(1992), 475–504.

12. A. S. Kleshchev. On restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups, I. Proc. London Math. Soc. (3), 69(1994), 515–540.

13. A. S. Kleshchev. Branching rules for modular representations of symmetric groups, I. J. Algebra 178(1995), 493-511.

14. A. S. Kleshchev. Branching rules for modular representations of symmetric groups, II. J. Reine Angew. Math., 459(1995), 163–212.

15. A. S. Kleshchev. Branching rules for modular representations of symmetric groups, III. J. London Math. Soc. (2) 54(1996), 25-38.

16. A. S. Kleshchev. On decomposition numbers and branching coefficients for symmetric and special linear groups, Proc. Lond. Math. Soc. (3) 75(1997), to appear.

17. R. Steinberg. Lectures on Chevalley Groups. Yale University, 1967.

Department of Mathematics, University of Oregon, Eugene OR 97403, USA E-mail address: klesh@math.uoregon.edu