PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 133, Number 2, Pages 403–414 S 0002-9939(04)07575-6 Article electronically published on August 4, 2004

A ROW REMOVAL THEOREM FOR THE EXT^{1} QUIVER OF SYMMETRIC GROUPS AND SCHUR ALGEBRAS

DAVID J. HEMMER

(Communicated by Jonathan I. Hall)

Abstract. In 1981, G. D. James proved two theorems about the decomposition matrices of Schur algebras involving the removal of the first row or column from a Young diagram. He established corresponding results for the symmetric group using the Schur functor. We apply James’ techniques to prove that row removal induces an injection on the corresponding Ext^{1} between simple modules for the Schur algebra.

We then give a new proof of James’ symmetric group result for partitions with the first part less than p. This proof lets us demonstrate that first-row removal induces an injection on Ext^{1} spaces between these simple modules for the symmetric group. We conjecture that our theorem holds for arbitrary partitions. This conjecture implies the Kleshchev-Martin conjecture that Ext_{Σ_{r}}^{1} (D_{λ}, D_{λ}) = 0 for any simple module D_{λ} in characteristic p ≠ 2. The proof makes use of an interesting fixed-point functor from Σ_{r}-modules to Σ_{r−m}-modules about which little seems to be known.

1. Introduction

We will assume familiarity with representation theory of the symmetric group Σ_{r} as found in [7] and of the Schur algebra S(n, r) as found in [4]. We write λ ⊢ r for λ = (λ_{1}, λ_{2}, …, λ_{k}) a partition of r. Let N = {0, 1, 2, …}. We do not distinguish between λ and its Young diagram:

λ = {(i, j) ∈ N × N | j ≤ λ_{i}}.

A partition λ is p-regular if there is no i such that λ_{i} = λ_{i+1} = ⋯ = λ_{i+p−1}. A partition is p-restricted if its conjugate partition, denoted λ′, is p-regular.

We write λ⊵μ for the usual dominance order on partitions. For λ = (λ_{1}, λ_{2}, …, λ_{k}) we write \overline{λ} for λ with its first row removed, i.e.

\overline{λ} = (λ_{2}, …, λ_{k}) ⊢ r − λ_{1}.

We write λ̂ for λ with its first column removed, i.e.

λ̂ = (λ_{1} − 1, λ_{2}, −1, …, λ_{k} − 1) ⊢ r − k.

The complex simple Σ_{r}-modules are the Specht modules {S^{λ} | λ ⊢ r}. The simple modules in characteristic p can be indexed by p-restricted partitions or by

Received by the editors May 23, 2003 and, in revised form, October 15, 2003. 2000 Mathematics Subject Classification. Primary 20C30. The author’s research was supported in part by NSF grant DMS-0102019.

\begin{matrix}c \\ ◯ \end{matrix}2004 American Mathematical Society

p-regular partitions. If we let S_{λ} denote (S^{λ})^{∗}, then both

{D^{λ} := S^{λ}/rad(S^{λ}) | λ is p-regular}

and

{D_{λ} = soc(S^{λ}) | λ is p-restricted}

are complete sets of nonisomorphic simple Σ_{r}-modules in characteristic p. The two indexings are related by D^{λ} ≅ D_{λ′} ⊗ sgn, where sgn denotes the one-dimensional signature representation.

For a module M and a simple module S we let [M : S] denote the composition multiplicity of S in M. James proved the following row removal theorems in [6].

Theorem 1.1 (James). Let λ and μ be partitions of r with λ_{1} = μ_{1} = m, and let λ be p-restricted. Then [S_{μ} : D_{λ}] = [S\overline{μ} : D\overline{λ}].

Theorem 1.2 (James). Let λ and μ be partitions of r with λ_{1} = μ_{1} = m, and let λ be p-regular. Then [S^{μ} : D^{λ}] = [S\overline{μ} : D\overline{λ}].

From these results, James deduced corresponding results for first-column removal by tensoring with the sign representation.

We apply James’ technique to prove that row removal gives an injection on the corresponding Ext^{1} space between simple modules for the Schur algebra. Then we present a new proof of James’ theorem for symmetric groups in the case when λ_{1} < p. We apply this proof, together with a theorem of Kleshchev and Sheth, to prove the corresponding Ext^{1} result for symmetric groups.

We remark that Theorems 1.1 and 1.2 have been generalized to removing multiple rows and columns by Donkin [1, 2]; however, we will not use these generalizations. We would like to thank Gordon James and Dan Nakano for useful discussions about this paper.

2. An Ext^{1}-theorem for Schur algebras

Let k be an algebraically closed field of characteristic p > 0, and let n ≥ r. The simple, complex, polynomial representations of GL_{n} of homogeneous degree r are the Weyl modules {V (λ) | λ ⊢ r}. Over k the Weyl modules may no longer be simple, but each Weyl module has a simple head denoted by L(λ).

The category of polynomial representations of GL_{n}(k) of homogeneous degree r is equivalent to the category mod-S(n, r). The Schur functor is an exact functor from mod-S(n, r) to mod-kΣ_{r} that maps V (μ) to S_{μ} and, for μ p-restricted, maps L(μ) to D_{μ}. Using this functor (specifically Theorem 6.6g in [4]), it sufficed for James to prove Theorem 1.1 for S(n, r), namely that [V (μ) : L(λ)] = [V (\overline{μ}) : L(\overline{λ})].

To do this, James defined [6, p. 117] an idempotent η ∈ S(n, r) such that ηS(n, r)η contains a subalgebra isomorphic to S(n − 1, r − m). Let

F_{m} : mod-S(n, r) → mod-S(n − 1, r − m)

be defined by F_{m}(U) = Res_{S(n−1,r−m)}^{ηS(n,r)η}(ηU). James showed:

Theorem 2.1 ([6, pp. 117-120]). Let μ ⊢ r with μ_{1} = m. Then:

(i) F_{m}(V (μ)) ≅ V (\overline{μ}). (ii) F_{m}(L(μ)) ≅ L(\overline{μ}). (iii) F_{m}(rad(V (μ)) ≅ rad(V (\overline{μ})).

The only other tool we need for our first theorem is the following result.

Lemma 2.2 ([8, II 2.14]). Suppose λ and μ are partitions of r and μ ▷̸λ. Then

Ext_{S(n,r)}^{1}(L(λ), L(μ)) ≅ Hom_{S(n,r)}(rad(V (λ), L(μ))).

Although Lemma 2.2 is actually stated in Jantzen’s book for Ext^{1} in the category of rational GL_{n}(k)-modules, this is known to agree with Ext^{1} in mod-S(n, r) by [3, 2.2d]. We can now prove:

Theorem 2.3. Let λ and μ be partitions of r with λ_{1} = μ_{1} = m. Then there is an injection

0 → Ext_{S(n,r)}^{1}(L(λ), L(μ)) → Ext_{S(n−1,r−m)}^{1}(L(\overline{λ}), L(\overline{μ})).

Proof. Since the modules L(τ) are self-dual we can assume μ ▷̸λ. The functor F_{m} is exact, so any copies of L(μ) in the second radical layer of V (λ) will map to copies of L(\overline{μ}) in the second radical layer of V (\overline{λ}), by Theorem 2.1. The injection then follows from Lemma 2.2. It is an injection rather than an isomorphism because other copies of L(\overline{μ}) may “float up” to the second radical layer of V (\overline{λ}). There is no assurance that F_{m} preserves the radical layers of V (λ). We know only that it preserves the radical. □

We remark that the situation for column removal is much simpler. Namely, if λ and μ have m parts, then

Ext_{S(n,r)}^{1}(L(λ), L(μ)) ≅ Ext_{S(n−m,r−m)}^{1}(L(λ̂), L(μ̂))

is clear by tensoring with the determinant representation.

3. Symmetric group preliminary results

We desire a result like Theorem 2.3 for the symmetric group. To do so it is necessary to first reprove James’ results without using the Schur functor. Then we can use a theorem of Kleshchev and Sheth to play the role of Lemma 2.2.

We begin by gathering information on the modules S^{λ}, D^{λ}, S_{λ} and D_{λ}, and establish our notation. For more details on results presented in this section see [7]. For λ ⊢ r, a λ-tableau is one of the r! arrays of integers obtained by replacing each node of λ bijectively with the integers 1, 2, …, r. There is a natural action of Σ_{r} on the set of tableaux. For a tableau t, let R(t) denote the set of permutations in Σ_{r} keeping the rows of t fixed setwise; and similarly let C(t) denote the column stabilizer. A tableau is standard if its rows and columns are increasing.

For a λ-tableau t, define the signed column sum

κ_{t} = ∑_{σ∈C(t)} sgn(σ)σ

and the row sum

ρ_{t} = ∑_{σ∈R(t)} σ.

There is an equivalence relation on λ-tableaux given by t_{1} ∼ t_{2} if t_{2} = πt_{1} for some π ∈ R(t_{1}). The equivalence classes are called λ-tabloids and are denoted by {t}. There is a natural action of Σ_{r} on the set of λ-tabloids, and this permutation module is denoted by M^{λ}. For a tableau t, the corresponding polytabloid is defined as e_{t} := κ_{t}{t} ∈ M^{λ}. The following theorem is fundamental.

Theorem 3.1 ([7, Thm. 8.4]). {e_{t} | t is a standard λ-tableau} is a basis for S^{λ}.

In particular, S^{λ} is a submodule of M^{λ}.

We will find it useful to identify the various kΣ_{r}-modules as left ideals in the group algebra kΣ_{r}. In particular (see for example [10, Thm. 4.2.2]):

Lemma 3.2. Choose any λ-tableau T. Then

(i) M^{λ} ≅ kΣ_{r}ρ_{T};
(ii) S^{λ} ≅ kΣ_{r}κ_{T} ρ_{T};
(iii) S_{λ} ≅ kΣ_{r}ρ_{T} κ_{T};
(iv) D^{λ} ≅ kΣ_{r}κ_{T} ρ_{T} κ_{T};
(v) D_{λ} ≅ kΣ_{r}ρ_{T} κ_{T} ρ_{T};
(vi) a basis for S_{λ} is given by {πρ_{T} κ_{T} | πT is standard}.

If λ′ denotes the conjugate partition to λ, the following is well known.

Lemma 3.3.

(i) For λ p-restricted, D_{λ} ≅ head(S_{λ}) ≅ soc(S^{λ}). (ii) For λ p-regular, D^{λ} ≅ head(S^{λ}) ≅ soc(S_{λ}).

To obtain his results on decomposition numbers from Theorem 2.1, James applied the following lemma.

Lemma 3.4 ([4, Lemma 6.6b]). Let S be an algebra and η ∈ S an idempotent. Suppose V is an S-module and F is a simple S-module such that ηF ≠ 0. Then ηF is a simple ηSη-module and [V : F] = [ηV : ηF].

As in James’ proof for S(n, r), we will find an idempotent η in kΣ_{r} such that ηkΣ_{r}η has a subalgebra isomorphic to kΣ_{r−m}, and such that λ_{1} = m < p implies ηS^{λ} ≅ S\overline{λ} and ηD_{λ} ≅ D\overline{λ} as kΣ_{r−m}-modules.

Our idempotent exists only when m < p, which (coincidentally?) is the only case where the symmetric group result corresponding to Lemma 2.2 is known. Thus we can obtain a result on Ext^{1} for symmetric groups in this case.

4. Determining the row removal functor on S^{λ}

Our main result in the next two sections is a new proof of a weaker version (Theorems 4.1, 4.2) of James’ theorems, which is entirely contained in symmetric group theory. This proof will lead to new results in Section 6. Henceforth we assume p > 2. This eliminates problems with semistandard homomorphisms (see [7, 13.14]) and is not relevant to our results for kΣ_{r}, because for p = 2 the only λ ⊢ r with λ_{1} < 2 is (1^{r}).

Theorem 4.1. Let λ, μ ⊢ r with λ_{1} = μ_{1} = m < p. Then [S_{μ} : D_{λ}] = [S\overline{μ} : D\overline{λ}].

Tensoring with the sign representation gives:

Theorem 4.2. Let λ, μ ⊢ r have m parts, for m < p. Then [S^{μ} : D^{λ}] = [Sμ̂ : Dλ̂].

Henceforth when we write Σ_{m} it will be acting on {1, 2, …, m}. When we write Σ_{r−m} it will be acting on {m + 1, m + 2, …, r} and will be embedded in Σ_{r} in the natural way. Similarly, when \overline{λ} ⊢ r − m, a \overline{λ}-tableau will be labelled with the numbers {m + 1, m + 2, …, r} rather than by {1, 2, …, r − m}.

We begin by defining

η = \frac{1}{m!} ∑_{σ∈Σ_{m}} σ.

Notice that η is a nonzero idempotent (and η exists only because m < p). So for U a kΣ_{r}-module, ηU is a ηkΣ_{r}η-module. The group algebra kΣ_{r−m} sits naturally inside ηkΣ_{r}η (namely as ηkΣ_{r−m}η) and η commutes with Σ_{r−m}. So we can regard ηU as a kΣ_{r−m}-module by restriction. Left multiplication by η then restriction to kΣ_{r−m} gives an exact functor

F_{m} : mod-kΣ_{r} → mod-kΣ_{r−m}

defined by F_{m} := Res_{kΣ_{r−m}}^{ηkΣ_{r}η}(ηU).

We would like to see how this functor behaves on the Specht, dual Specht, and simple modules. In this section we determine how it acts on Specht modules and use the information to determine which simple modules it annihilates. In the next section we consider dual Specht modules and use the information to determine F_{m} on simple modules. We begin with an easy lemma.

Lemma 4.3. If λ_{1} < m, then F_{m}(S^{λ}) = 0.

Proof. Since λ_{1} < m, any standard tableau t must have a column with more than one entry from {1, 2, …, m}. This implies ηκ_{t} = 0; so ηe_{t} = 0. Thus, by Theorem 3.1, η annihilates a basis for S^{λ}; hence ηS^{λ} = 0. □

Next we determine which simple modules are annihilated by F_{m}.

Lemma 4.4. F_{m}(D_{λ}) = 0 if and only if λ_{1} < m.

Proof. Suppose λ_{1} < m. Then by Lemma 4.3, ηS^{λ} = 0. Since D_{λ} = soc(S^{λ}), we know ηD_{λ} = 0. Conversely suppose λ_{1} ≥ m. Choose a λ-tableau T with {1, 2, …, m} in its first row, so that ηρ_{T} = ρ_{T}. We have

D_{λ} = kΣ_{r}ρ_{T} κ_{T} ρ_{T}

by Lemma 3.2(v), so

0 ≠ ρ_{T} κ_{T} ρ_{T} = ηρ_{T} κ_{T} ρ_{T} ∈ ηD_{λ}.

Thus ηD_{λ} ≠ 0. □

We will now determine how F_{m} acts on the Specht modules S^{λ} when λ_{1} = m.

Theorem 4.5. Let λ_{1} = m, and let t_{1}, …, t_{s} be the standard λ -tableaux with first row 1 2 ⋯ m. Then for any standard λ-tableau t,

ηe_{t} ≠ 0 iff t = t_{i} for some i.

The set \{ηe_{t_{i}}\}_{i=1}^{s} is linearly independent. Furthermore, ηS^{λ} ≅ S\overline{λ} as Σ_{r−m}- modules, i.e. F_{m}(S^{λ}) ≅ S\overline{λ}.

Proof. Suppose t is a standard λ-tableau but t ∉ {t_{i}}_{i=1}^{s}. Then the first column of t must contain one, plus at least one other number ≤ m. Thus ηκ_{t} = 0, so ηe_{t} = 0 as desired. To prove the linear independence of the set {ηe_{t_{i}}}_{i=1}^{s} we first recall the total order on λ-tabloids from [7, p. 10]:

Definition: {t_{1}} < {t_{2}} if ∃i such that:

(i) {i + 1, …, r} are in the same row of {t_{1}} and {t_{2}}.

(ii) i is higher in {t_{1}} than in {t_{2}}.

By [7, Lemma 8.2], to prove the linear independence of the set {ηe_{t_{i}}}_{i=1}^{s}, it is sufficient to prove the following.

Lemma 4.6. The tabloid {t_{i}} is the greatest (in the total order) tabloid that occurs in ηe_{t_{i}}, and it occurs with nonzero coefficient.

Proof. It is easy to check that for any tableau T,

(4.1)  κ_{πT} = πκ_{T} π^{−1}.

Set {t} = {t_{i}}. Since 1 2 ⋯ m is the first row of {t}, we know Σ_{m} ≤ R(t), and hence

(4.2)  σ{t} = {t} = {σt} ∀σ ∈ Σ_{m};

so η{t} = {t}. Using this plus Equation (4.1) we determine

ηe_{t} = ηκ_{t}{t}
= (\frac{1}{m!} ∑_{σ∈Σ_{m}} σ) κ_{t}\{t\}

= \frac{1}{m!} (κ_{t}\{t\} + ∑_{1≠σ∈Σ_{m}} κ_{σt}\{σt\})

= \frac{1}{m!} ( ∑_{π∈C(t)} sgn(π)π\{t\} + ∑_{1≠σ∈Σ_{m}} κ_{σt}\{σt\})

(4.3)  = \frac{1}{m!}\{t\} + \frac{1}{m!} ∑_{1≠π∈C(t)} sgn(π)π\{t\} + \frac{1}{m!} ∑_{1≠σ∈Σ_{m}} κ_{σt}\{σt\}.

Now we will show that all the tabloids that occur in Equation (4.3) except {t} are tabloids smaller than {t}, and that {t} occurs with coefficient one. First we recall that by [7, 3.15]:

t standard and 1 ≠ π ∈ C(t) ⟹ {t} > {πt}.

This implies that the tabloids in the second summand of (4.3) are all < {t}.

Next we consider the third summand in (4.3). These are tabloids of the form {s} := π{σt}, for π ∈ C(σt). For π = 1 we get {s} = {σt} = {t}. This yields another (m! − 1)/m copies of {t}, bringing the coefficient of {t} in (4.3) to one. Now suppose π ≠ 1, and π ∈ C(σt). Then

{t} = {σt}
> π{σt} since 1 ≠ π ∈ C(σt).

Hence the remaining tabloids in (4.3) are also smaller than {t}, completing the proof of Lemma 4.6. □

To complete the proof of Theorem 4.5, it remains to show that ηS^{λ} ≅ S\overline{λ} as kΣ_{r−m}-modules. The linear independence of the set {ηe_{t_{i}}} proves that both modules have dimension equal to the number of standard \overline{λ}-tableaux. For a λ-tableau T, we let \overline{T} denote T with its first row removed. Notice that the \overline{λ}-tableaux are all of the form \overline{T} where T is a λ-tableau with first row 1 2 ⋯ m. We use Lemma 3.2 to deduce:

Lemma 4.7.

(i) {e_{T} | T is a λ-tableau}is a spanning set for S^{λ}.

(ii) \{e\overline{T} | T ∈ \{t_{i}\}_{i=1}^{s}\}is a spanning set for S\overline{λ}.

We remark that \{\overline{t}_{i}\}_{i=1}^{s} is a complete set of standard \overline{λ}-tableaux.

Now, following James [6], we define a kΣ_{r−m}-homomorphism Θ : M^{λ} → M\overline{λ} by

(4.4)  Θ(\{t\}) = \{ \begin{matrix}0 & if the first row of \{t\} is not 1 2 ⋯ m, \\ \{\overline{t}\} & otherwise. \end{matrix}

Notice that by Equation (4.2),

(4.5)  Θ({ηt}) = Θ({t}).

It is a simple computation that for a λ-tableau t,

(4.6)  Θ(e_{t}) = \{ \begin{matrix}sgn(π)e\overline{πt} & if \{1, 2, …, m\} are in distinct columns of t \\ 0 & otherwise, \end{matrix}

where π is any element of C(t) such that πt has {1, 2, …, m} in the first row.

By Lemma 4.7, the set \{ηe_{T} | T a λ-tableau\} spans ηS^{λ} and \{e\overline{T} | T ∈ \{t_{i}\}_{i=1}^{s}\} spans S\overline{λ}. Thus, Equations (4.5) and (4.6), plus the known equality of the dimensions, prove that restricting Θ to ηS^{λ} gives an isomorphism onto S\overline{λ}. This completes the proof of Theorem 4.5. □

5. Determining F_{m}(S_{λ}) and F_{m}(D_{λ})

In order to determine F_{m}(D_{λ}) we will need to understand F_{m}(S_{λ}). The analysis will be similar to the last section, but the proof is subtly different. Namely, we will use other means to determine the dimension of ηS_{λ} before we determine its module structure. In Section 7 we will say more about why we believe the two cases are fundamentally different.

To begin, observe that for a kΣ_{r}-module U, the subspace ηU is exactly the space of fixed points U^{Σ_{m}} under Σ_{m}. Since Σ_{m} commutes with Σ_{r−m}, this space carries the structure of a kΣ_{r−m}-module. We will say more about this in Section 7, but for now we use it to prove:

Lemma 5.1. dim_{k}(ηS_{λ}) = dim_{k}S\overline{λ}.

Proof. As we remarked above,

dim_{k}(ηS_{λ}) = dim_{k}(S_{λ})^{Σ_{m}}
= dim_{k}Hom_{kΣ_{m}}(k, S_{λ})
= dim_{k}Hom_{kΣ_{r}}(S^{λ}, M^{(m,1^{r−m})}).

But since p > 2, this is just the number of semistandard λ-tableaux of type (m, 1^{r−m}) by [7, 13.14], which (since λ_{1} = m) is the number of standard \overline{λ}-tableaux, i.e. the dimension of S\overline{λ}. □

Recall that since S_{λ} ≅ S^{λ′} ⊗ sgn, we can consider S_{λ} as sitting inside M^{λ′} ⊗ sgn. So S_{λ} has a basis of the form {e_{t} ⊗ ϵ} where t is a standard λ′-tableaux and ϵ is such that σϵ = sgn(σ)ϵ. As in Theorem 4.5 we have:

Theorem 5.2. Let t_{1}, t_{2}, …, t_{s} be the standard λ′-tableaux that have 1, 2, …, m as their first column. Then:

(i) η(e_{t_{i}} ⊗ ϵ) = e_{t_{i}} ⊗ ϵ.

(ii) {e_{t_{i}} ⊗ ϵ}_{i=1}^{s} is a basis for ηS_{λ}.

(iii) ηS_{λ} ≅ S\overline{λ} as kΣ_{r−m}-modules.

Proof. Part (i) follows from the fact that for σ ∈ Σ_{m} we have σ(e_{t_{i}}) = sgn(σ)e_{t_{i}}. Part (ii) follows from the fact that the e_{t_{i}} are linearly independent and the number of them is the same as the dimension of ηS_{λ} determined in Lemma 5.1. Part (iii) is then clear from looking at the action of Σ_{r−m} on the {e_{t_{i}}}. (See also Lemma 5.4(iii).) □

We remark that this situation is different from Theorem 4.5, because for t not in the set {t_{1}, t_{2}, …, t_{s}}, we may still have ηe_{t} ≠ 0.

Let ⟨ , ⟩ denote the bilinear form on M^{λ} defined by setting the basis of tabloids to be orthonormal. This immediately gives a form on M^{λ} ⊗ sgn as well. Let ⟨⟨ , ⟩⟩ denote the form similarly defined on M^{\overline{λ}′} ⊗ sgn. Then S_{λ} ⊆ M^{λ′} ⊗ sgn and from [7] we have:

Lemma 5.3. Let λ be p-restricted.

(i) For S_{λ} ⊆ M^{λ′} ⊗ sgn we have rad(S_{λ}) = S_{λ} ∩ (S_{λ})^{⊥}. (ii) For S\overline{λ} ⊆ M^{\overline{λ}′} ⊗ sgn we have rad(S\overline{λ}) = S\overline{λ} ∩ (S\overline{λ})^{⊥}.

As in the proof of Theorem 4.5 we define Ψ : M^{λ′} ⊗ sgn → M^{\overline{λ}′} ⊗ sgn by

Ψ(\{t\} ⊗ ϵ) = \begin{cases}ˆ\{t\} ⊗ ϵ & if 1, 2, …, m are in distinct rows of t, \\ 0 & otherwise, \end{cases}

where ˆ\{t\} is the \overline{λ}′-tableau obtained by removing the numbers 1, 2, …, m from \{t\}. We let \overline{Ψ} denote the restriction of Ψ to ηS_{λ}. The following lemma is a straightforward calculation. For part (iv) it suffices to check on the basis of ηS_{λ} given in Theorem 5.2(ii).

Lemma 5.4. Let t_{1}, t_{2}, …, t_{s} and Ψ be as above. Let t̂_{i} be the standard \overline{λ}′-tableau given by removing the first column of t_{i}. Then:

(i) Ψ is a kΣ_{r−m}-homomorphism. (ii) Ψ(ηx) = Ψ(x) ∀x ∈ M^{λ′}. (iii) Ψ(e_{t_{i}} ⊗ ϵ) = m!(et̂_{i} ⊗ ϵ). In particular \overline{Ψ} is an isomorphism from ηS_{λ} to

S\overline{λ}. (iv) For any x, y ∈ ηS_{λ} we have ⟨x, y⟩ = m!(⟨⟨\overline{Ψ}(x), \overline{Ψ}(y)⟩⟩).

(v) For any u, v ∈ S_{λ} we have ⟨u, v⟩ = ⟨ηu, ηv⟩.

Finally we can determine F_{m}(D_{λ}) as a kΣ_{r−m}-module:

Theorem 5.5. ηD_{λ} ≅ D\overline{λ} as Σ_{r−m}-modules, i.e. F_{m}(D_{λ}) ≅ D\overline{λ}.

Proof. We know ηD_{λ} = η(S_{λ}/rad(S_{λ})). Since D\overline{λ} = S\overline{λ}/rad(S\overline{λ}), it is enough to show that \overline{Ψ} maps η(rad(S_{λ})) onto rad(S\overline{λ}).

So choose an arbitrary x ∈ rad(S\overline{λ}). Then x = \overline{Ψ}(ηu) for some u in S_{λ}. We must show that ηu is in η(rad(S_{λ})), so we prove u ∈ rad(S_{λ}). To do this, choose any v ∈ S_{λ}. Then Lemma 5.4 gives

⟨u, v⟩ = ⟨ηu, ηv⟩
= m!(⟨⟨\overline{Ψ}(ηu), \overline{Ψ}(ηv)⟩⟩)
= m!(⟨⟨x, \overline{Ψ}(ηv)⟩⟩)
= 0 since x ∈ rad(S\overline{λ}).

Thus u ∈ rad(S_{λ}) and so ηu ∈ η(rad(S_{λ})) as desired. □

All the pieces are now in place to prove Theorem 4.1. It is well known that any composition factor D_{μ} of S^{λ} has λ⊵μ. In particular, S^{λ} has no composition factors D_{μ} with μ_{1} > λ_{1} = m. Thus η annihilates all the composition factors of S^{λ} except those D_{μ} with μ_{1} = λ_{1} = m. We have proved that ηS^{λ} ≅ S\overline{λ} and that ηD_{μ} ≅ D\overline{μ}. In particular ηD_{μ}, which is guaranteed by Lemma 3.4 to be a simple ηΣ_{r}η-module, remains simple as a ηkΣ_{r−m}η ≅ kΣ_{r−m}-module.

Thus Lemma 3.4 implies [S^{λ} : D_{μ}] = [S\overline{λ} : D\overline{μ}]. Theorem 4.1 then follows because S_{λ} = (S^{λ})^{∗} and all the D_{μ} are self-dual; so [S^{λ} : D_{μ}] = [S_{λ} : D_{μ}]. Theorem 4.2 follows from Theorem 4.1 by tensoring with sgn and recalling that S^{λ} ⊗ sgn ≅ S_{λ′}.

6. A row removal theorem for Ext_{kΣ_{r}}^{1}(D_{λ}, D_{μ})

In this section we combine information from our partial proof of James’ result with a theorem of Kleshchev and Sheth to derive a new result about the Ext^{1}-quiver of the symmetric group.

Given any finite-dimensional algebra S and an idempotent e ∈ S there is an exact functor F : mod-S → mod-eSe given by multiplication by e. If eL(λ) and eL(μ) are nonzero, then they are irreducible and there is an injection

0 → Ext_{S}^{1}(L(λ), L(μ)) → Ext_{eSe}^{1}(eL(λ), eL(μ)).

However both James’ proof and our proof involve a restriction functor after multiplication by the idempotent, in our case restricting from ηkΣ_{r}η to kΣ_{r−m}. But restriction does not in general induce an injection on Ext^{1}. The following result of Kleshchev and Sheth lets us use our row removal functor to obtain a result on extensions between simple modules. We have translated the theorem to index irreducibles with p-restricted partitions rather than p-regular.

Theorem 6.1 ([9, Theorem 2.10]). Let λ, μ be partitions of r with λ_{1}, μ_{1} < p and assume μ ▷̸λ. Then

Ext_{kΣ_{r}}^{1}(D_{λ}, D_{μ}) ≅ Hom_{kΣ_{r}}(D_{μ}, S^{λ}/D_{λ}).

This is all we need to prove:

Theorem 6.2. Let λ_{1}, μ_{1} = m < p. Then there is an injection

0 → Ext_{kΣ_{r}}^{1}(D_{λ}, D_{μ}) → Ext_{kΣ_{r−m}}^{1}(D\overline{λ}, D\overline{μ}).

Equivalently, if λ and μ both have m < p parts, then there is an injection

0 → Ext_{kΣ_{r}}^{1}(D^{λ}, D^{μ}) → Ext_{kΣ_{r−m}}^{1}(Dλ̂, Dμ̂).

Proof. Since the irreducible modules are self-dual we can assume μ ▷̸ λ without loss of generality, so of course \overline{μ} ▷̸\overline{λ} as well. We have

0 → D_{λ} → S^{λ} → S^{λ}/D_{λ} → 0.

Multiplying by η gives

0 → D\overline{λ} → S\overline{λ} → η(S^{λ}/D_{λ}) → 0.

Thus,

η(S^{λ}/D_{λ}) ≅ S\overline{λ}/D\overline{λ}.

So each D_{μ} in the socle of S^{λ}/D_{λ} maps to a D\overline{μ} in the socle of S\overline{λ}/D\overline{λ}. We get

0 → Hom_{kΣ_{d}}(D_{μ}, S^{λ}/D_{λ}) → Hom_{kΣ_{d−m}}(D\overline{μ}, S\overline{λ}/D\overline{λ}),

which, together with Theorem 6.1, completes the proof. □

We have verified that Theorem 6.2 holds for all known Ext^{1}-quivers for the symmetric group, including blocks of small defect and for various small r. This data together with Theorem 6.2 leads us to the following conjecture.

Conjecture 6.3. Let p ≥ 3, and let λ and μ be p-restricted partitions of r with λ_{1} = μ_{1} = m. Then there is an injection

0 → Ext_{kΣ_{r}}^{1}(D_{λ}, D_{μ}) → Ext_{kΣ_{r−m}}^{1}(D\overline{λ}, D\overline{μ}).

Equivalently, for τ and ρ p-regular partitions of r with m parts, there is an injection

0 → Ext_{kΣ_{r}}^{1}(D^{τ} , D^{ρ}) → Ext_{kΣ_{r−m}}^{1}(Dτ̂, Dρ̂).

Conjecture 6.3 immediately implies the following conjecture.

Conjecture 6.4 (Kleshchev, Martin). For p ≥ 3, Ext_{kΣ_{r}}^{1}(D_{λ}, D_{λ}) = 0.

The reason for this is that if Conjecture 6.3 holds and if λ = (λ_{1}, λ_{2}, …, λ_{k}), then we can remove rows one at a time. This eventually gives an injection from Ext_{kΣ_{r}}^{1}(D_{λ}, D_{λ}) into Ext_{kΣ_{λ_{k}}}^{1} (D_{λ_{k}}, D_{λ_{k}}) ≅ Ext_{kΣ_{λ_{k}}}^{1} (k, k), which is known to be zero.

We present an example where the injection in Theorem 6.2 is proper. Let p = 3, m = 2 and choose λ = (2^{3}, 1^{6}) and μ = (2, 1^{10}). Then

S^{(2^{3},1^{6})} ≅ \begin{matrix}D_{(2,1^{10})} \\ D_{(1^{12})} \\ D_{(2^{3},1^{6})}, \end{matrix}

so Ext_{kΣ_{12}}^{1}(D_{(2,1^{10})}, D_{(2^{3},1^{6})}) = 0. But

ηS^{(2^{3},1^{6})} ≅ S^{(2^{2},1^{6})} ≅ \begin{matrix}D_{(1^{10})} \\ D_{(2^{2},1^{6})}. \end{matrix}

Notice that η annihilated the D_{(1^{12})} term. So the D_{(2,1^{10})} term dropped down, and

Ext_{kΣ_{10}}^{1}(D_{(2^{2},1^{6})}, D_{(1^{10})}) ≅ k.

James also proved a result corresponding to removing the first column from D_{λ}. Using the idempotent

η′ := \frac{1}{m!} ∑_{σ∈Σ_{m}} sgn(σ)σ

and proceeding with a similar analysis we could obtain Theorem 6.2 for first-column removal from D_{λ}. However, for fixed p there are only finitely many p-restricted partitions with less than p parts, so the corresponding theorem is weaker. We are not aware of any counterexamples to the column removal statement for D_{λ} corresponding to Conjecture 6.3, so perhaps this holds as well.

7. Remarks on fixed point functors

As above we consider Σ_{m} and Σ_{r−m} as subgroups of Σ_{r}, but we drop the assumption that m < p. For U ∈ mod-kΣ_{r}, the fixed points of U under Σ_{m} are clearly invariant under the action of Σ_{r−m}. So we can define

F_{m} : mod-kΣ_{r} → mod-kΣ_{r−m}

by

F_{m}(U) = U^{Σ_{m}} ≅ Hom_{kΣ_{m}}(k, U) ≅ Hom_{kΣ_{r}}(M^{(m,1^{r−m})}, U).

When m < p this functor agrees with the functor F_{m} defined previously, namely it is multiplication by an idempotent and then restriction (and hence is exact). When m ≥ p, the module k is not projective as a kΣ_{m}-module, so the functor F_{m} is only left exact. Very little seems to be known about this functor. For example what is F_{m}(S^{λ})? In Section 4 we determined this in the special case λ_{1} = m < p.

We also determined F_{m}(S_{λ}) in this case by a similar but not identical proof. It is clear the two situations are very different. In particular the dimension F_{m}(S_{λ}) is independent of the characteristic and is the number of semistandard λ-tableaux of type (m, 1^{r−m}). We will study this functor in more detail in [5]. In particular we can show:

Theorem 7.1. Let λ = (λ_{1}, λ_{2}, …, λ_{k}). Then:

(i) If m > λ_{1}, then F_{m}(S_{λ}) = 0. (ii) If m = λ_{1}, then F_{m}(S_{λ}) ≅ S\overline{λ}. (iii) If m < λ_{1}, then F_{m}(S_{λ}) ≅ S_{λ\(m)}, the dual of a skew Specht module.

The situation for Specht modules in characteristic p is much more difficult; not even the dimension of F_{m}(S^{λ}) is known. Of course F_{m}(S^{λ}) is not just a vector space, but has the structure of a kΣ_{r−m}-module. The author is not aware of any investigation of this module structure.

We remark that part (i) of Theorem 7.1 is not true for Specht modules. For example, when p = 3, λ = (7, 2, 2) and m = 8 > λ_{1},

dimF_{8}(S^{λ}) = 3 > 0.

We make the following conjecture.

Conjecture 7.2. F_{m}(S^{λ}) has a filtration by Specht modules for kΣ_{r−m}.

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Department of Mathematics, University of Toledo, 2801 W. Bancroft, Toledo, Ohio 43606

E-mail address: david.hemmer@utoledo.edu