FIXED-POINT FUNCTORS, ROW AND COLUMN REMOVAL FOR SYMMETRIC GROUPS AND SCHUR ALGEBRAS

DAVID J. HEMMER

Abstract. Let Σ_{d} be the symmetric group. For 1 < m < d let F_{m} be the functor which takes a Σ_{d} module U to the fixed points U^{Σ_{m}}, which is naturally a module for Σ_{d−m}. This functor was used in [7] to study cohomology of the symmetric group, but little is known about it. This paper initiates a study of F_{m}. First we relate it to James’ work on row and column removal and decomposition numbers for the Schur algebra in [11]. Next we determine the image of F_{m} on dual Specht modules, permutation and twisted permuation modules, and some Young and twisted Young modules. In particular F_{m} acts as first row removal on dual Specht modules S_{λ} with λ_{1} = m and as first column removal on twisted Young and twisted permutation modules corresponding to partitions with m parts. Finally we prove that determining F_{m} on the Young modules is equivalent to determining the decomposition numbers for the Schur algebra. To prove this we first determine exactly which Young modules have a trivial submodule.

1. Notation and Preliminaries

We will assume familiarity with the representation theory of the symmetric group Σ_{d} and of the Schur algebra S(n, d) as found in [6], [9] or [15]. Let k be an algebraically closed field of characteristic p > 2. We write λ ⊢ d for λ = (λ_{1}, …, λ_{k}) a partition of d and λ ⊧ d for a composition of d. Let Λ^{+}(n, d) denote the set of partitions of d with at most n parts and let Λ(n, d) denote the set of compositions of d with at most n parts. 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}. It is p-restricted if its conjugate partition, denoted λ′, is p-regular. We write ⊵ 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}) ⊢ d − λ_{1}.

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

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

Date: March 2004. 1991 Mathematics Subject Classification. Primary 20C30. Research of the author was supported in part by NSF grant DMS-0102019.

The complex simple Σ_{d} modules are the Specht modules {S^{λ} | λ ⊢ d}. Simple kΣ_{d} modules can be indexed by p-restricted partitions or by p-regular partitions. 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 kΣ_{d} modules. The two indexings are related by D^{λ} ≅ D_{λ′} ⊗ sgn, where sgn is the one-dimensional signature representation. We also recall that:

S^{λ} ⊗ sgn ≅ S_{λ′}  (1.1)

where S_{μ} denotes the dual of the Specht module S^{μ}.

We will also consider the Young modules {Y^{λ} | λ ⊢ d}, the permutation modules {M^{λ} | λ ∈ Λ(n, d)}, and their twisted versions obtained by tensoring with sgn. If λ is p-restricted, then Y^{λ} is the projective cover of D_{λ}. All these modules, and the S(n, d) modules below, are described in [15].

Let V = k^{n} be the natural module for the general linear group GL_{n}(k). Then V^{⊗d} is a GL_{n}(k) module with the diagonal action and a kΣ_{d} module by place permutation. The Schur algebra S(n, d) is defined by:

S(n, d) := End_{kΣ_{d}}(V^{⊗d}).

The actions of GL_{n}(k) and Σ_{d} commute, so we get a map GL_{n}(k) → S(n, d). This map identifies the category mod-S(n, d) with homogeneous polynomial representations of GL_{n}(k) of degree d.

For λ ∈ Λ^{+}(n, d) we denote the irreducible S(n, d) module with highest weight λ by L(λ). We also write Δ(λ) and ∇(λ) (sometimes written V (λ) and H^{0}(λ)) for the standard and costandard modules with highest weight λ, respectively. P (λ) will be the projective cover of L(λ), I(λ) the injective hull, and T (λ) the corresponding tilting module. Finally S^{λ}(V) and Λ^{λ}(V) will be the symmetric and exterior powers.

Let I(n, d) denote the set of all d-tuples i = (i_{1}, i_{2}, …, i_{d}) with i_{s} ∈ {1, 2, …, n}. The symmetric group Σ_{d} acts on the right on I(n, d) by:

iπ = (i_{π(1)}, i_{π(2)} …, i_{π(d)}).

This action extends to an action on I(n, d) × I(n, d), and we write (i, j) ∼ (k, l) if k = iπ and l = jπ for some π ∈ Σ_{d}. Let Ω be a set of equivalence class representatives under ∼ . Recall [6] that S(n, d) has a basis {ξ_{i,j}} indexed by Ω. For i ∈ I(n, d) define the weight of i by wt(i) = (λ_{1}, λ_{2}, …, λ_{n}) ∈ Λ(n, d), where λ_{k} is the number of times k appears in i. Then ξ_{i,i} is an idempotent, usually denoted ξ_{λ}.

1.1. The Schur and adjoint Schur functor. Henceforth assume n ≥ d. Let ω = (1^{d}) ∈ Λ(n, d) and let e denote the idempotent ξ_{ω} ∈ S(n, d). Then eS(n, d)e ≅ kΣ_{d}, and the Schur functor F : mod-S(n, d) → mod-kΣ_{d} is defined by F(U) := eU. This is an exact, covariant functor with:

\begin{matrix}F(∇(λ)) = S^{λ} & F(Δ(λ)) = S_{λ} & F(L(λ)) = D_{λ} or 0 \\ F(P (λ)) = Y^{λ} & F(I(λ)) = Y^{λ} & F(T (λ)) = Y^{λ′} ⊗ sgn \end{matrix} .  (1.2)

The Schur functor admits a right adjoint functor G : mod-kΣ_{d} → mod-S(n, d) defined by:

G(N) := Hom_{kΣ_{d}}(V^{⊗d}, N)
≅ Hom_{eS(n,d)e}(eS(n, d), N).

The functor G is only left exact, and so has higher right dervied functors:

R^{i}G(N) = Ext_{kΣ_{d}}^{i}(V^{⊗d}, N).

We now collect some known results about G and R^{1}G. Recall that we are assuming p > 2 throughout:

Proposition 1.1.

(i) [14, 3.2]. G(S_{λ}) = Δ(λ).

(ii) [2, 5.2.4] G(Y^{λ} ⊗ sgn) = T (λ′)

(iii) [8, 3.8.2] For p > 3 G(Y^{λ}) = P (λ).

(iv) [5, 3.3] G(D_{λ}) has simple socle L(λ).

We will also need some information about R^{1}G:

Proposition 1.2. Let G : mod-kΣ_{d} → mod-S(n, d) be the adjoint Schur functor and let U ∈ mod-kΣ_{d}. Then:

(i) [14, 6.4]For p > 3, R^{1}G(S_{λ}) = 0.

(ii) [5, 5.1] As an S(n, d) module, R^{1}G(U) has no composition factors L(τ) where τ

is p-restricted. (iii) [10, 2.8] If L(μ) lies in soc(Δ(λ)) then μ must be p-restricted.

1.2. Decomposition numbers, Specht and dual Specht filtrations. We collect here material that will be used repeatedly in later sections. For a thorough treatment see Martin’s book [15].

We say a kΣ_{d} module U has a Specht filtration, or a Specht series, if it has a filtration with successive quotients isomorphic to Specht modules. Similarly, we will say U has a a dual Specht filtration. For S(n, d) modules we refer to a Weyl filtration (by Δ(μ)’s) or a good filtration (by ∇(μ)’s). The multiplicities in a good or Weyl filtration are independent of the choice of filtration. The same holds for Specht and dual Specht filtrations in mod-kΣ_{d} when p > 3 (but not for p = 2 or 3) by work in [8].

The Young modules Y^{λ} are self-dual and are known to have both Specht and dual Specht filtrations. If [Y^{μ} : S_{λ}] denotes the multiplicity of S_{λ} in a dual Specht filtration of Y^{μ} and [Δ(λ) : L(μ)] denotes a decomposition number for S(n, d) then a well-known reciprocity theorem [15, p. 118] gives:

[Δ(λ) : L(μ)] = [I(μ) : ∇(λ)]
= [Y^{μ} : S^{λ}]  (1.3)
= [Y^{μ} : S_{λ}].

So knowledge of the decomposition numbers for S(n, d) is equivalent to knowledge of multiplicities in dual Specht filtrations of Young modules, a fact we will use repeatedly later.

The category mod-S(n, d) is a highest weight category. In particular [Δ(λ) : L(λ)] = 1 and [Δ(λ) : L(μ)] = 0 unless λ ⊵ μ. This triangular structure in the decomposition matrix, together with reciprocity (1.3, will be very useful to us, since it gives a triangular structure to the matrix of filtration multiplicities [Y^{μ} : S_{λ}]. In particular suppose we know a kΣ_{d} module is a direct sum of Young modules. If we know the multiplicities in a dual Specht filtration of the module, then we can determine the multiplicities of the Young module summands.

A permutation module M^{λ} is a direct sum of Young modules:

M^{λ} ≅ Y^{λ} ⨁_{μ▷λ}K_{λμ}Y^{μ}.

The p-Kostka numbers K_{λμ} are not known. However Young’s rule (see [9, Chpt. 14]) gives a nice formula for the multiplicities in a Specht or dual Specht filtration of M^{λ}. Thus if we know the decomposition numbers for S(n, d), then Young’s rule lets us calculate the p-Kostka numbers.

2. Relating the fixed-point functor to James’ idempotent

In [7] we proved some theorems on extensions between simple modules for the symmetric group when the first row or column of the corresponding partitions is removed. The proof used the fixed-point functor

F_{m} : mod-kΣ_{d} → mod-kΣ_{d−m}

defined by:

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

The proofs were motivated by, but did not really use, James’ work in [11]. In this paper we will first relate James’ paper to the functor F_{m}. Throughout the paper we will consider Σ_{d−m} as the subgroup of Σ_{d} fixing 1, 2, … m. It is sometimes convenient to think of F_{m} as first restricting to Σ_{m} × Σ_{d−m} ≤ Σ_{d} and then taking the largest subspace on which Σ_{m} acts trivially.

If we define the Hecke algebra

S(λ) := End_{kΣ_{d}}(M^{λ})

then S((m, 1^{d−m})) acts on M^{(m,1^{d−m})} on the right, and hence acts on the left on F_{m}(U). So we could consider F_{m} as a functor to mod-S((m, 1^{d−m})) instead of to mod-kΣ_{d−m}. Of course kΣ_{d−m} sits naturally inside S((m, 1^{d−m})). The algebras S(λ) are of interest in their own right, see for example [4].

In order to describe James’ work we need some more notation. Fix 1 < m < d and let I^{∗}(n, d) be the subset of i ∈ I(n, d) consisting of those d-tuples with i_{d−m+1} = i_{d−m+2} = ⋯ = i_{d} = 1 and i_{k} ≠ 1 for 1 ≤ k ≤ d − m. James defined an idempotent η = Σξ_{i,i}, the sum being over distinct elements ξ_{i,i} with i ∈ I^{∗}(n, d). James also defined:

S_{1} := span{ξ_{i,j} | i, j ∈ I^{∗}(n, d)} ⊂ S(n, d).

Then S_{1} is a subalgebra of ηS(n, d)η, and James proved [11] that S_{1} ≅ S(n − 1, d − m). Thus we have the following (only partly commutative) diagram of functors, where F and G are as above while F̃ and G̃ are the corresponding Schur and adjoint Schur functors for the smaller symmetric group Σ_{d−m}. Also we use J to denote multiplication by η followed by restriction to S_{1}.

[figure: J; mod-S_{1}; mod-S(n, d); mod-ηS(n, d)η; ·η; res; F̃; G̃; F; G; mod-kΣ_{d−m}; mod-kΣ_{d}; F_{m}]

Our first theorem is that James’ functor J is closely related to the fixed-point functor F_{m}:

Theorem 2.1. Let U ∈ mod-kΣ_{d} and let the functors be as in the diagram above. Then:

F_{m}(U) ≅ F̃(J (G(U))).

Proof. We begin the proof with a well-known lemma:

Lemma 2.2. Recall that S(n, d) ≅ End_{kΣ_{d}}(V^{⊗d}). Then:

(i) V^{⊗d} ≅ ⨁_{λ∈Λ(n,d)} M^{λ} as kΣ_{d}-modules.

(ii) ξ_{i,j} ∈ S(n, d) corresponds to an element in Hom_{kΣ_{d}}(M^{λ}, M^{μ}) where λ = wt(i)

and μ = wt(j). (iii) ξ_{λ} corresponds to projection onto M^{λ}. (iv) η corresponds to projection onto

⨁_{λ_{1}=m^{λ∈Λ(n,d)}} M^{λ}.

Proof. To see (i), recall that V^{⊗d} has standard basis {e_{i} := e_{i_{1}}⊗e_{i_{2}}⊗⋯⊗e_{i_{d}} | i ∈ I(n, d)}. Under this identification M^{λ} is spanned by the basis vectors {e_{i} | wt(i) = λ}. Parts (ii), and (iii) are immediate from the description in [6, 2.6a] of the action of ξ_{i,j} on V^{⊗d}. Part (iv) follows from (iii). □

To prove the theorem, let U ∈ mod-kΣ_{d}. Let i = (m + 1, m + 2, …, d, 1, 1, …, 1) ∈ I^{∗}(n, d) and let ẽ = ξ_{i,i}. The ẽ is an idempotent, ẽS_{1}ẽ ≅ kΣ_{d−m} and the Schur functor F̃ is multiplication by ẽ.

Now G(U) ≅ Hom_{kΣ_{d}}(V^{⊗d}, U) is a left S(n, d) module with action obtained from the right action of S(n, d) on V^{⊗d}. Thus the action of S(n, d) (and hence of ηS(n, d)η and of S_{1}) is given by precomposing functions. That is if f : V^{⊗d} → V^{⊗d} is in S(n, d) and g : V^{⊗d} → U is in G(U) then fg = g ∘ f : V^{⊗d} → U. So Lemma 2.2(iv) gives:

ηG(U) ≅ Hom_{kΣ_{d}}( ⨁_{λ_{1}=m^{λ∈Λ(n,d)}} M^{λ}, U).

But ẽ is projection onto M^{(m,1^{d−m})} so we get:

ẽηG(U) ≅ Hom_{kΣ_{d}}(M^{(m,1^{d−m})}, U)
≅ F_{m}(U).
□

3. Some general properties of F_{m}

In this brief section we collect a few general properties of F_{m} which will be useful in determining how F_{m} acts on specific modules. We first remark that F_{m} has a left adjoint functor G_{m} : mod-kΣ_{d−m} → mod-kΣ_{d} given by:

G_{m}(U) := Ind_{Σ_{m}×Σ_{d−m}}^{Σ_{d}}(k ⊗ U).

The functor G is exact, but F_{m} is exact only when m < p, and is left exact in general. Thus F_{m} has higher right derived functors

R^{i}F_{m}(U) = Ext_{kΣ_{d}}^{i}(M^{(m,1^{d−m})}, U).

A key fact in understanding F_{m} is that R^{1}F_{m} vanishes on dual Specht modules. A closely related fact (That the first higher right derived functor of the adjoint Schur functor G vanishes on dual Specht modules) played a key role in [8]. Some of the results below are false unless we assume p > 3. This seems to be similar to what was observed in [8]. Specifically when p = 3 it is possible for H^{1}(Σ_{d}, S_{λ}) to be nonzero, and this has a dramatic effect on the results.

Proposition 3.1. Let p > 3 and λ ⊢ d. Then R^{1}F_{m}(S_{λ}) = 0.

Proof. We have

R^{1}F_{m}(S_{λ}) ≅ Ext_{kΣ_{d}}^{1}(M^{(m,1^{d−m})}, S_{λ})
≅ Ext_{kΣ_{d}}^{1}(S^{λ}, M^{(m,1^{d−m})}).

But the permutation modules M^{μ} are direct sums of Young modules, and Ext_{kΣ_{d}}^{1}(S^{λ}, Y^{μ}) is always zero when p > 3 [14, 6.4b], so the result follows. □

Proposition 3.1 is false when p = 3. For example if d = 5 and m = 3 then:

R^{1}F_{3}(S_{(1^{5})}) ≅ Ext_{kΣ_{3}}^{1}(k, Res_{Σ_{d}}(S^{(1^{5})}))
≅ Ext_{kΣ_{3}}^{1}(k, sgn) ≠ 0.

Finally we recall that F_{m} was used in [7], where its image was determined on the modules S^{λ}, S_{λ} and D_{λ} if λ_{1} ≤ m < p. (In particular where F_{m} is exact.) In that case when m = λ_{1}, the functor F_{m} “removes the first row” from S^{λ}, S_{λ} and D_{λ} (i.e. maps S_{λ} to S\overline{λ}, etc...). In the general case considered here (with m arbitrary), we will see that F_{m} acts as first row or first column removal only on the twisted modules, namely S^{λ} ⊗ sgn, Y^{λ} ⊗ sgn and M^{λ} ⊗ sgn.

4. F_{m} on dual Specht modules.

In this section we show that F_{m} behaves very nicely on dual Specht modules S_{λ}. We will need the notions of semistandard λ tableaux of type μ and the basis of M^{λ} given by λ-tabloids, which are described in the book [9]. First we recall:

Lemma 4.1. The dimension of F_{m}(S_{λ}) is the number of semistandard λ-tableaux of type (m, 1^{d−m}). In particular, if λ_{1} < m then F_{m}(S_{λ}) = 0.

Proof. Since

F_{m}(S_{μ}) ≅ Hom_{kΣ_{d}}(M^{(m,1^{d−m})}, S_{λ})
≅ Hom_{kΣ_{d}}(S^{λ}, M^{(m,1^{d−m})}),

the result follow from Theorem 13.13 in [9], where a basis for Hom_{kΣ_{d}}(S^{λ}, M^{μ}) is described for arbitrary λ and μ. In particular, if λ_{1} < m then there is no such tableau, so F_{m}(S_{λ}) = 0. □

Thus we must consider F_{m}(S_{λ}) for λ_{1} ≥ m. The case λ_{1} = m follows from our work in Section 2 together with work of James:

Theorem 4.2. Let λ_{1} = m. Then F_{m}(S_{λ}) ≅ S\overline{λ}.

Proof. James showed in [11] that J (Δ(λ)) ≅ Δ(\overline{λ}). So:

F_{m}(S_{λ}) ≅ F̃(J (G(S_{λ}))) by Thm. 2.1.
≅ F̃(J (Δ(λ)) by Prop. 1.1(i)
≅ F̃(Δ(\overline{λ}))
≅ S\overline{λ} by (1.2).
□

In order to describe F_{m}(S_{λ}) in general we must discuss skew diagrams and the corresponding skew Specht modules. Suppose λ = (λ_{1}, λ_{2} … λ_{r}) ⊢ d and μ = (μ_{1}, μ_{2} … μ_{s}) ⊢ t for t < d. Suppose μ_{i} ≤ λ_{i} for all i (where μ_{i} is interpreted as 0 for i > r). Then the skew diagram λ\μ is defined as:

λ\μ := {(i, j) ∈ N × N | 1 ≤ i ≤ r, μ_{i} ≤ j ≤ λ_{i}.}

To a skew diagram (or even some more general diagrams) there is a associated a skew Specht module S^{λ\μ} and its dual, which we denote S_{λ\μ}. Their construction can be found in [12], where it is shown that S^{λ\μ} has a Specht filtration, and the filtration multiplicities are determined combinatorially. We will not describe the details of the construction here, but we will need the following from the paper of James and Peel:

Proposition 4.3. [12, Thm. 3.1] Let λ ⊢ d. When restricted to Σ_{m} × Σ_{d−m}, the module S^{λ} has a chain of submodules with factors isomorphic to S^{β} ⊗S^{λ\β}, where each partition β of m such that λ\β exists occurs exactly once.

Note that James and Peel constructed a filtration of S^{λ}, but of course taking duals gives a filtration of S_{λ} by modules S_{β} ⊗ S_{λ\β}. We will also need the fact that James and Peel constructed the filtration explicitly. In particular the filtration of S_{λ} can be chosen so that the term S_{m} ⊗ S_{λ\(m)} is a submodule. We also need the following well-known lemma:

Lemma 4.4. [9, 13.17]

Hom_{Σ_{d}}(k, S_{λ}) ≅ \{ \begin{matrix}k if λ = (d) \\ 0 else \end{matrix} .

We can now determine F_{m}(S_{λ}) in general:

Theorem 4.5. Let λ ⊢ d. Then:

F_{m}(S_{λ}) ≅ \{ \begin{matrix}S_{λ\(m)} & if λ_{1} > m \\ S\overline{λ} & if λ_{1} = m \\ 0 & if λ_{1} < m. \end{matrix}

Furthermore F_{m} takes modules with dual Specht filtrations to modules with dual Specht filtrations.

Proof. Only the case λ_{1} > m is not handled by Lemma 4.1 or Theorem 4.2. It’s clear from the description of F_{m} that one way to get F_{m}(U) is to first restrict U to Σ_{m}×Σ_{d−m}, and then take the subspace on which Σ_{m} acts trivially, which is a Σ_{d−m} submodule. But we know the filtration in Proposition 4.3 can be chosen so S_{(m)} ⊗ S_{λ\(m)} occurs as a submodule. Lemma 4.4 guarantees that no other S_{β}’s have any Σ_{m} fixed-points, so the first part of result is immediate.

The second part follows by induction on the number of dual Specht modules in the filtration. We know from the filtration of James and Peel that F_{m}(S_{λ}) ≅ S_{λ\(m)} has a dual Specht module filtration. But now suppose U has a dual Specht filtration. Then we have:

0 → S_{μ} → U → N

where N has a dual Specht filtration. Applying F_{m} gives:

0 → F_{m}(S_{μ}) → F_{m}(U) → F_{m}(N) → R^{1}F_{m}(S_{μ}).

However R^{1}F_{m}(S_{μ}) = 0 Lemma 1.2, and F_{m}(N) has a dual Specht filtration by induction, so F_{m}(U) has a dual Specht filtration. □

We close this section with a few observations. First, the statement corresponding to Lemma 4.4 is definitely false for S^{λ}. This will explain the much greater difficulty in determining F_{m}(S^{λ}). Also F_{m}(S_{λ}) ≅ Hom_{Σ_{d}}(M^{(m,1^{d−m})}, S_{λ}) has a basis of semistandard homomorphisms indexed by semistandard λ-tableau of type (m, 1^{d−m}). (See [9, Chpt. 13] for details.) These tableaux are in obvious bijection with the set of standard λ\(m) tableaux, which index a basis for S_{λ\(m)}. In this case the obvious bijection between the two bases does not extend to a kΣ_{d−m} homomorphism.

5. F_{m} on permutation and twisted permutation modules.

In [9, 13.19], James gives a basis for Hom_{Σ_{d}}(M^{λ}, M^{μ}) indexed by row-standard λtableau of type μ. This basis can be used to prove:

Theorem 5.1.

F_{m}(M^{μ}) ≅ ⨁_{τ_{i}≤μ_{i}∀i^{τ⊧d−m}} M^{τ} .

Proof. Since F_{m}(M^{μ}) ≅ Hom_{Σ_{d}}(M^{(m,1^{d−m})}, M^{μ}), we have a basis for F_{m}(M^{μ}) indexed by (m, 1^{d−m})-tableaux of type μ. Now let

T = t_{1} t_{2} ⋯ t_{m} ; t_{m+1} ; t_{m+2} ; ⋮ ; t_{d}

be a row standard (m, 1^{d−m})-tableau of type μ and let Θ_{T} be the corresponding homomorphism from M^{(m,1^{d−m})} to M^{μ}. Then it is an easy calculation (using the description of Θ_{T} in [9] and the right action of Σ_{d−m} on M^{(m,1^{d−m})}) that for σ ∈ Σ_{d−m} we have σ(Θ_{T}) = Θ_{σT} where:

σT = t_{1} t_{2} ⋯ t_{m} ; t_{σ(m+1)} ; t_{σ(m+2)} ; ⋮ ; t_{σ(d)} .

Now it is just a question of studying the permutation action of Σ_{d−m} on these tableaux, and the result follows. In particular, for τ = (τ_{1}, τ_{2}, …, τ_{k}) ⊧ d − m, the summand M^{τ} in the theorem corresponds to the set of tableaux T with τ_{i} i’s in the “leg” of T. □

As an example of Theorem 5.1:

F_{3}(M^{321}) = M^{3} ⊕ M^{210} ⊕ M^{201} ⊕ M^{120} ⊕ M^{021} ⊕ M^{111}
≅ M^{3} ⊕ (M^{21})^{⊕4} ⊕ M^{111}.

The analysis of F_{m} on twisted permutation modules is different. We begin with a lemma:

Lemma 5.2.

(i) sgn ≅ S^{(1^{d})}.

(ii) Hom_{kΣ_{d}}(S^{(1^{d})}, M^{λ}) ≅ \{ \begin{matrix}k if λ ∈ Λ(n, d) has d ones and n − d zeroes. \\ 0 else. \end{matrix} (iii)

Res_{Σ_{m}}^{Σ_{d}}(M^{λ}) ≅ ⨁_{τ_{i}≤λ_{i}^{τ∈Λ(d,m)}}(M^{τ})^{⊕dimM^{λ\τ}} .

where dimM^{λ\τ} is \frac{d!}{Π(λ_{i}−τ_{i})!}.

Proof. Part (i) is well-known and (ii) is immediate from [9, 13.13]. Part (iii) can be seen either from Mackey’s theorem or directly by considering the action of Σ_{m} on λtabloids. □

We can now determine how F_{m} behaves on twisted permutation modules. In general the image is much smaller than on permutation modules. In particular, if λ has m parts then F_{m} just sends M^{λ} ⊗ sgn to M λ̂ ⊗ sgn. In general:

Theorem 5.3. For λ ⊢ d we have:

F_{m}(M^{λ} ⊗ sgn) ≅ ⨁_{λ_{i}−1≤ρ_{i}≤λ_{i}^{ρ⊧d−m}} M^{ρ} ⊗ sgn.

In particular if λ has fewer than m parts than F_{m}(M^{λ} ⊗ sgn) = 0 and if λ has exactly m parts, then F_{m}(M^{λ} ⊗ sgn) ≅ M λ̂ ⊗ sgn.

Proof.

F_{m}(M^{λ} ⊗ sgn) ≅ Hom_{kΣ_{m}}(k, M^{λ} ⊗ sgn)
≅ Hom_{kΣ_{m}}(S^{1^{m}}, M^{λ}) since S^{1^{m}} ≅ sgn  (5.1)
≅ Hom_{kΣ_{m}}(S^{1^{m}}, ⨁_{τ_{i}≤λ_{i}^{τ⊧m}}(M^{τ})^{⊕dim M^{λ\τ}}) by Lemma 5.2(iii).

By Lemma 5.2(ii), the only τ which contribute to the direct sum in (5.1) are those with m ones. For each such τ, Hom_{kΣ_{d}}(S^{1^{d}}, M^{τ}) is one-dimensional. So we get a space of dimension equal to that of M^{ρ}, where ρ = λ\τ has λ_{i} − 1 ≤ ρ_{i} ≤ λ_{i}. Then it is just a matter of checking that the action of Σ_{d−m} on these maps is the same as the action on M^{ρ}. We leave this to the reader.

When λ has < m parts, there are no such ρ, so F_{m}(M^{λ} ⊗ sgn) = 0. When λ has exactly m parts, then λ̂ is the only such ρ and F_{m}(M^{λ} ⊗ sgn) ≅ M λ̂ ⊗ sgn.

□

It is known that G(M^{λ}) ≅ S^{λ}(V) and G(M^{λ} ⊗ sgn) ≅ Λ^{λ}(V). Thus an alternate way to prove Theorems 5.1 and 5.3 would be to analyze how James’ idempotent acts on the standard bases of S^{λ}(V) and Λ^{λ}(V), then apply Theorem 2.1. Combinatorially the analysis would be of similar difficulty to our proof..

Since we are discussing semistandard homomorphisms, we will take the opportunity to observe that the tools being used in this paper gives a very short proof of a theorem originally proved by James via a long combinatorial argument. In [9, Chpt. 13], James constructed a basis for Hom_{Σ_{d}}(M^{λ}, M^{μ}) indexed by row-standard λ -tableaux of type μ. He then showed that those maps corresponding to semistandard tableaux, when restricted to S^{λ} ⊆ M^{λ}, give a basis for Hom_{Σ_{d}}(S^{λ}, M^{μ}). This indirectly shows that every element of Hom_{kΣ_{d}}(S^{λ}, M^{μ}) extends to a map on M^{λ}. James remarked that he knew no direct proof of this fact, so we give one below when p > 3:

Proposition 5.4. [9, 13.15] Suppose p > 3. Then every element of Hom_{kΣ_{d}}(S^{λ}, M^{μ}) can be extended to an element of Hom_{kΣ_{d}}(M^{λ}, M^{μ}).

Proof. We have:

0 → S^{λ} → M^{λ} → Q → 0

where Q has a Specht filtration. Apply Hom_{kΣ_{d}}(−, M^{μ}) to the sequence to get:

⋯ → Hom_{kΣ_{d}}(M^{λ}, M^{μ}) →^{π} Hom_{kΣ_{d}}(S^{λ}, M^{μ}) → Ext_{kΣ_{d}}^{1}(Q, M^{μ}) → · · · .  (5.2)

But

dim_{k}Ext_{kΣ_{d}}^{1}(Q, M^{μ}) = dim_{k}Ext_{kΣ_{d}}^{1}(M^{μ}, Q^{∗})
≤ dim_{k}Ext_{kΣ_{d}}^{1}(V^{⊗d}, Q^{∗}) by Lemma 2.2(i)
= dim_{k}R^{1}G(Q^{∗})
= 0 by Prop.1.2(i)

since Q^{∗} has a dual Specht filtration. Thus the map π in (5.2) must be a surjection, and the proposition follows. □

6. F_{m} on Young and twisted Young modules

Having determined how the fixed point functor acts on permutation modules and twisted permutation modules, we now turn to their direct summands, namely the Young modules Y^{λ} and twisted Young modules Y^{λ} ⊗sgn. We will see, as in the previous section, that the twisted case is better, in the sense that when λ has m parts, F_{m} acts as first column removal on Y^{λ} ⊗sgn.. No such result holds for F_{m}(Y^{λ}). Indeed we will show that determining all the F_{m}(Y^{λ}) is essentially equivalent to determining the decomposition matrix for the Schur algebra, a very difficult problem indeed!

We begin with a lemma made easy by the calculations in the previous section:

Lemma 6.1. F_{m}(Y^{λ}) is a direct sum of Young modules for kΣ_{d−m} and F_{m}(Y^{λ} ⊗ sgn) is a direct sum of twisted Young modules for kΣ_{d−m}.

Proof. Since Y^{λ} is a direct summand of M^{λ} this is immediate from Theorems 5.1 and 5.3. □

We can determine F_{m}(Y^{λ} ⊗ sgn) precisely in some cases:

Theorem 6.2. Let p > 3. If λ has fewer than m parts than F_{m}(Y^{λ} ⊗ sgn) = 0. If λ has exactly m parts then

F_{m}(Y^{λ} ⊗ sgn) ≅ Y λ̂ ⊗ sgn.

Proof. From Lemma 6.1 we know F_{m}(Y^{λ}⊗sgn) is a direct sum of twisted Young modules. Twisted Young modules have filtrations by both Specht and dual Specht modules, and the matrix giving the multiplicities [Y^{λ} : S_{μ}] is triangular. Thus if we can prove F_{m}(Y^{λ}⊗sgn) has a filtration by dual Specht modules with the same multiplicities as in a dual Specht filtration of Y λ̂ ⊗ sgn, we can conclude that it is indeed Y λ̂ ⊗ sgn. So let

Y^{λ} ⊗ sgn = S^{λ} ⊗ sgn + ∑_{μ▷λ} a_{λμ}(S^{μ} ⊗ sgn)
= S_{λ′} + ∑_{μ▷λ} a_{λμ}S_{μ′},  (6.1)

where by the “sum” in (6.1), we mean the modules have filtrations with the factors listed in the sum. Now, appealing to the second part of Theorem 4.5, it makes sense to apply F_{m} to both sides of (6.1). Notice first that if λ has fewer than m parts, then λ′ and all the μ′’s in the sum have first row less than m, so the corresponding S_{μ′}’s are annihilated by F_{m}. So Y^{λ} ⊗ sgn has a filtration by dual Specht modules, all of which are annihilated by F_{m}. Any such module must be annihilated by F_{m}. So now assume λ has m parts. Then applying F_{m} to (6.1) gives:

F_{m}(Y^{λ} ⊗ sgn) = S\overline{λ′} + ∑_{μ_{1}=m} a_{λμ}S\overline{μ′}  (6.2)

Now suppose

Y λ̂ ⊗ sgn = Sλ̂ ⊗ sgn + ∑_{τ▷λ̂}b_{λ̂τ} S^{τ} ⊗ sgn

= Sλ̂′ + ∑_{τ▷λ̂}b_{λ̂τ} S_{τ′}.  (6.3)

Since τ ▷ λ̂ and λ̂ has ≤ m parts, then τ has ≤ m parts. Thus all the τ appearing in (6.3) are of the form μ̂ for some μ ⊢ d with μ_{1} = m. This fact plus the observation that λ̂′ = \overline{λ′} for any λ lets us rewrite (6.3) as:

Y λ̂ ⊗ sgn = S\overline{λ′} + ∑_{μ_{1}=m} b_{λ̂μ̂}S\overline{μ′}.  (6.4)

However by (1.3) the a_{λμ} are decomposition numbers for S(n, d), specifically:

a_{λμ} = [Δ(μ) : L(λ)].

But James proved in [11] that first column removal preserves decomposition numbers (essentially just by tensoring with the determinant.) Thus a_{λμ} = b_{λ̂μ̂}, so the sums in (6.2) and (6.4) coincide, as desired.

□

We remark that if λ is p-restricted then a twisted Young module is also a Young module. In particular Y^{λ} ⊗ sgn ≅ Y^{m(λ)} where m(λ) is the Mullineux map given by D_{λ}⊗sgn ≅ D_{m(λ)}. Thus we have determined how F_{m} behaves on certain Young modules. We are not sure if the theorem holds for p = 3.

We also remark that Theorems 5.3 and 6.2 give a column removal theorem for p-Kostka numbers, which is already known. In particular:

Theorem 6.3. Let λ, μ ⊢ d both have m parts. Then:

[M^{μ} : Y^{λ}] = [M μ̂ : Y λ̂].

Proof. As we mentioned the result would follow from Theorems 5.3 and 6.2 since for Y^{λ} a summand of M^{μ}, λ cannot have more parts than μ. However it is easier to recall the well known fact that the p-Kostka number [M^{μ} : Y^{λ}] is the same as the dimension of the μ-weight space in L(λ), so tensoring with the determinant gives the result. □

Unlike the case for decomposition numbers, first row removal definitely does not preserve p-Kostka numbers!

We wish now to convince the reader that determining the multiplicities of Young modules for Σ_{d−m} in F_{m}(Y^{λ}) is a very difficult problem by showing it is equivalent to knowing the decomposition numbers for the Schur algebra.

For the base case in the inductive proof of the main theorem of this section, we will need to know Hom_{kΣ_{d}}(k, Y^{λ}). Now S^{λ} ⊆ Y^{λ} ⊆ M^{λ}, and Frobenius reciprocity immediately gives Hom_{kΣ_{d}}(k, M^{λ}) ≅ k, so S^{λ} and Y^{λ} each have at most one dimension of fixed points. The partitions λ for which Hom_{kΣ_{d}}(k, S^{λ}) is nonzero are determined in [9, 24.4], and certainly if S^{λ} has a fixed point then so does Y^{λ}. However there are other Young modules with fixed points. First we need:

Lemma 6.4. Let p > 3. Then

[Y^{μ} : S_{(d)}] = dim_{k}Hom_{kΣ_{d}}(k, Y^{μ}).

Proof. The construction of Y^{μ} given [8, Section 4] shows that a dual Specht filtration of Y^{μ} can be built with any copies of S_{(d)} ≅ k on the bottom. (Of course we know, from Proposition 6.7, that there will be at most one copy.) □

Recall from (1.3) that

[Y^{μ} : S_{(d)}] = [∇(d) : L(μ).

But these numbers are known. Let μ = (μ_{1}, μ_{2}, …, μ_{k} ⊢ d. Define a sequence of nonnegative integers α(μ) as follows. First write the μ_{i} out base p. Then add them all together. For i ≥ 1 let α_{i} be the number that is “carried” to the top of the p^{i} column during the addition. For example let p = 3 and μ = (552). Then adding 5 + 5 + 2 base three gives:

1 2 1 2 1 2 + \underline{0} \underline{2} 1 1 0

Lemma 6.5. [3, Sec. 2.4]

Lemma 6.6. Let λ ⊢ d and p > 3. Then:

Hom_{kΣ_{d}}(k, Y^{λ}) = [∇(d) : L(λ)].

Proof.

Hom_{kΣ_{d}}(k, Y^{λ}) ≅ Hom_{kΣ_{d}}(S^{(d)}, Y^{λ})
≅ Hom_{S(n,d)}(∇(d),

=

□

Proposition 6.7. Let p > 3. Then

Hom_{kΣ_{d}}(k, S^{λ}) ≅ Hom_{kΣ_{d}}(k, Y^{λ}).

Proof. Of course the proposition is obvious when λ is p-restricted in which case S^{λ} and Y^{λ} both have simple socle D_{λ}. Since S^{λ} ⊂ Y^{λ} the proposition can only fail if Y^{λ} has a fixed point but S^{λ} does not. But, since p > 3, Ext_{Σ_{d}}^{1}(S^{λ}, k) = 0 by the main result of [1]. So if there is a trivial submodule of Y^{λ} not lying in S^{λ} then there must be a short exact sequence:

0 → S^{λ} ⊕ k → Y^{λ} → Q → 0

which gives:

0 → Q^{∗} → Y^{λ} → S_{λ} ⊕ k → 0.

Young modules have dual Specht filtrations so Lemma 1.2(i) implies R^{1}G(Y^{λ}) = 0. So applying G and using Proposition 1.1 yields:

0 → G(Q^{∗}) → P (λ) →^{π} Δ(λ) ⊕ G(k) → R^{1}G(Q^{∗}) → 0.  (6.5)

The adjointness of F and G shows:

Hom_{S(n,d)}(Δ(λ), G(k)) ≅ Hom_{kΣ_{d}}(S_{λ}, k)
≅ Hom_{kΣ_{d}}(k, S^{λ})
= 0 by assumption.

Since P (λ) is the projective cover of Δ(λ), this forces Hom_{S(n,d)}(P (λ), G(k)) = 0 as well. But the map π in (6.5) cannot be zero since that would force Δ(λ) to inject into R^{1}G(Q^{∗}), contradicting Proposition 1.2 (ii), (iii). So (6.5) is actually a direct sum of two exact sequences, namely:

0 → G(Q^{∗}) → P (λ) → Δ(λ) → 0 → 0  (6.6)

and

0 → 0 → 0 → G(k) → R^{1}G(Q^{∗}) → 0.  (6.7)

Since P (λ) and Δ(λ) both have Weyl filtrations, (6.6) implies that G(Q^{∗}) also has Weyl filtration by [13, 4.17]. But this implies Q^{∗} has a dual Specht filtration by [8, 3.5.1]. Hence R^{1}G(Q^{∗}) = 0, so G(k) = 0. But this is absurd. Since F ∘ G is the identity functor, G does not annihilate any module. □

We are not aware of any counterexamples to the proposition when p = 3.

Now we will show that determining F_{m}(Y^{λ}) is essentially equivalent to determining the decomposition numbers for S(n, d). Specifically:

Theorem 6.8. Let p > 3. Suppose we know the decomposition numbers for S(n, r) for r ≤ d. Then we can figure out F_{m}(Y^{λ}) for all m and all λ ⊢ d. Conversely suppose we know F_{m}(Y^{λ}) for all m and all partitions λ ⊢ r, for r ≤ d. Then we can determine all the decomposition numbers for S(n, r) for r ≤ d.

Proof. First suppose we know all [Δ(λ) : L(μ)], or equivalently by (1.3), suppose we know all the multiplicities [Y^{μ} : S_{λ}] in dual Specht filtrations of the Young modules. We will calculate F_{m}(Y^{λ}) inductively. Y^{(d)} ≅ k so F_{m}(Y^{(d)}) = Y^{(d−m)}. Now assume we know F_{m}(Y^{τ}) for all τ ▷ μ, we must determine F_{m}(Y^{μ}).

We know how to write F_{m}(M^{μ}) as a direct sum of permutation modules. However Young’s rule [9, Chpt. 14] tells us the filtration multiplicities in a Specht filtration of any permutation module, so we can write:

F_{m}(M^{μ}) = ∑_{λ⊢d−m} c_{μλ}S_{λ}.  (6.8)

where by the summation we again mean the module has a dual Specht filtration with the factors in the sum. We also know

M^{μ} ≅ Y^{μ} ⨁_{τ▷μ}K_{μτ} (Y^{τ})  (6.9)

where the p-Kostka numbers K_{μτ} can be determined from the decomposition numbers. So we have a Specht series of F_{m}(M^{μ}). We also know F_{m}(Y^{τ}) for all τ ▷μ. So Equations (6.8) and (6.9) together with knowledge of the F_{m}(Y^{τ})’s let us calculate the Specht filtration multiplicities in F_{m}(Y^{μ}). But we already know that F_{m}(Y^{μ}) is a direct sum of Young modules for kΣ_{d−m}, so the decomposition numbers of S(n, d−m) let us determine precisely which Young modules.

Conversely suppose we know F_{m}(Y^{μ}) for all m, and for all μ ⊢ r with r ≤ d. Inductively we can assume we know the decomposition numbers for S(n, r) with r < d, and we will obtain the decomposition numbers for S(n, d). That is, we need to get all the [Y^{μ} : S_{λ}]. We will proceed inductively on λ. Theorem 6.7 together with the following lemma provides the base case:

So James’ determination of which S^{λ} have a trivial submodule gives us the base case of our induction, i.e. gives us [Y^{μ} : S_{(d)}]. Now assume we have calculated [Y^{μ} : S_{λ}] for all μ and for all λ ▷ τ. We must determine [Y^{μ} : S_{τ} .] We will actually simultaneously get [Y^{μ} : S_{σ}] for all σ with σ_{1} = τ_{1}. Write:

Y^{μ} = ∑_{ρ_{1}<τ_{1}} a_{μρ}S_{ρ} + ∑_{ρ_{1}=τ_{1}} b_{μρ}S_{ρ} + ∑_{ρ_{1}>τ_{1}} c_{μρ}S_{ρ}  (6.10)

where we know the {c_{μρ}} by inductive hypothesis. Now apply F_{τ_{1}} to (6.10) and use Theorem 4.5 to get:

F_{m}(Y^{μ}) = ∑_{ρ_{1}=τ_{1}} b_{μρ}S\overline{ρ} + ∑_{ρ_{1}>τ_{1}} c_{μρ}(S_{ρ\(τ_{1})}).  (6.11)

The dual Specht filtration multiplicities in the right hand sum in (6.11) are known since James and Peel [12] give a dual Specht filtration of the S_{ρ\(τ_{1})}. Also by assumption we can write F_{m}(Y^{μ}) as a direct sum of Young modules, and we can then write out a dual Specht filtration for it, since we are assuming we know the decomposition numbers for S(n, d − τ_{1}). Equating the multiplicities on both sides of (6.11) lets us solve for all the {b_{μρ}} with ρ_{1} = τ_{1} as desired. □

It is elementary to carry out either of the computations described in the proof of the theorem. In Appendix A we present the images of Y^{λ} for d = 10, p = 5 (where all the decomposition numbers are known) under F_{5}. It would be interesting to get a more explicit relation between the data in Appendix A and decomposition numbers.

7. F_{m} on simple modules and Specht modules.

Determining F_{m}(D_{λ}) and F_{m}(S^{λ}) seems to be very difficult. Indeed we suspect, although cannot prove, that either or both problem may be as difficult as determining decomposition numbers, similar to Theorem 6.8. The only thing we can say comes from the following lemma:

Lemma 7.1. [5] G(D_{λ}) has simple socle isomorphic to L(λ).

Thus we have:

Proposition 7.2. Let λ_{1} = m. Then:

(i) D\overline{λ} ∈ soc(F_{m}(D_{λ})).

(ii) If G(D_{λ}) is simple then F_{m}(D_{λ}) ≅ D\overline{λ}.

(iii) [7, 5.5] If m < p then F_{m}(D_{λ}) ≅ D\overline{λ}

Proof. Parts (i) and (ii) are both immediate from Theorem 2.1 since James proved in [11] that J (L(λ)) ≅ L(\overline{λ}). □

We have some evidence that G(D_{λ}) is simple about half the time, usually either G(D_{λ}) or G(D_{λ} ⊗ sgn) is simple. Another case where G(D_{λ}) is simple is if D_{λ} is a completely splittable module [14].

We cannot say much about F_{m}(S^{λ}) either. In [7] it was shown that if λ_{1} = m < p then F_{m}(S^{λ}) ≅ S\overline{λ}. The corresponding statement to Theorem 4.5 is definitely false. That is, even if λ_{1} < m, it is possible for F_{m}(S^{λ}) to be nonzero. The filtration of S^{λ} as a Σ_{m} × Σ_{d−m} module given in Theorem 4.3 would not have any terms of the form S^{(m)}⊗S^{λ\(m)}. But this does not rule out Σ_{m} fixed points since nontrivial Specht modules can still have fixed points. We renew our conjecture from [7]:

Conjecture 7.3. F_{m}(S^{λ}) has a Specht filtration as a kΣ_{m} module.

Perhaps a better understanding of the filtration given in Theorem 4.3 would be useful to attack the conjecture. If S^{β} ⊗S^{λ\β} occurs in the filtration, it is known whether S^{β} has a fixed point under Σ_{m}. The question is whether this fixed point ”drops” to the bottom of the filtration, and whether it takes all of S^{λ\β} with it, so to speak. This seems to be difficult to determine.

Appendix A. F_{5} on Young modules for Σ_{10}

Below are the images of the Young modules for Σ_{10} under the functor F_{5} in characteristic p = 5. Young modules not listed are annihilated by the functor.

\begin{aligned}F_{5}(Y^{10}) & ≅ & Y^{5} \\ F_{5}(Y^{91}) & ≅ & Y^{5} ⊕ Y^{41} \\ F_{5}(Y^{82}) & ≅ & Y^{41} ⊕ Y^{32} \\ F_{5}(Y^{81^{2}}) & ≅ & Y^{41} ⊕ Y^{31^{2}} \\ F_{5}(Y^{73}) & ≅ & Y^{41} ⊕ Y^{32} \\ F_{5}(Y^{721}) & ≅ & Y^{32} ⊕ Y^{31^{2}} ⊕ Y^{2^{2}1} \\ F_{5}(Y^{71^{3}}) & ≅ & Y^{31^{2}} ⊕ Y^{21^{3}} \\ F_{5}(Y^{64}) & ≅ & (Y^{41})^{2} ⊕ Y^{32} \\ F_{5}(Y^{631}) & ≅ & (Y^{32})^{2} ⊕ (Y^{31^{2}})^{2} ⊕ Y^{2^{2}1} \\ F_{5}(Y^{62^{2}}) & ≅ & Y^{32} ⊕ Y^{2^{2}1} \\ F_{5}(Y^{621^{2}}) & ≅ & Y^{2^{2}1} ⊕ Y^{21^{3}} \\ F_{5}(Y^{61^{4}}) & ≅ & Y^{21^{3}} ⊕ Y^{1^{5}} \\ F_{5}(Y^{5^{2}}) & ≅ & Y^{5} ⊕ Y^{41} \end{aligned}  \begin{aligned}F_{5}(Y^{541}) & ≅ & (Y^{41})^{2} ⊕ Y^{31^{2}} \\ F_{5}(Y^{532}) & ≅ & Y^{32} \\ F_{5}(Y^{531^{2}}) & ≅ & (Y^{31^{2}})^{2} ⊕ Y^{21^{3}} \\ F_{5}(Y^{52^{2}1}) & ≅ & Y^{2^{2}1} \\ F_{5}(Y^{521^{3}}) & ≅ & (Y^{21^{3}})^{2} ⊕ Y^{1^{5}} \\ F_{5}(Y^{51^{5}}) & ≅ & (Y^{1^{5}})^{2} \\ F_{5}(Y^{4^{2}2}) & ≅ & (Y^{41})^{2} \\ F_{5}(Y^{43^{2}}) & ≅ & Y^{32} ⊕ Y^{31^{2}} \\ F_{5}(Y^{4321}) & ≅ & Y^{31^{2}} \\ F_{5}(Y^{42^{2}1^{2}}) & ≅ & Y^{21^{3}} \\ F_{5}(Y^{41^{6}}) & ≅ & Y^{1^{5}} \\ F_{5}(Y^{3^{3}1}) & ≅ & Y^{41} \\ F_{5}(Y^{3^{2}2^{2}}) & ≅ & Y^{41} \end{aligned}

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University of Toledo, Department of Mathematics, 2801 W. Bancroft, Toledo, OH 43606, USA

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