ON FUNCTORIAL DECOMPOSITIONS OF SELF-SMASH PRODUCTS

PAUL SELICK AND JIE WU

Abstract. We give a decomposition formula for n-fold self smash of a two-cell suspension X localized at 2, in which the mod 2 homology of each factor in the decomposition is explicitly given and is indecomposable over the Steenrod algebra if X is a suspension of ℝP^{2}, ℂP^{2}, ℍP^{2} or 𝕂P^{2}. Furthermore, we give an explicit formula for the multiplicity of indecomposable projective symmetric group modules over ℤ/2 which correspond to two-row Young diagrams. Moreover we give an explicit formula to compute the submatrix of the decomposition matrix of symmetric group algebra over ℤ/2 corresponding to two-row Young diagrams.

1. Introduction

Let X = ΣY be the suspension of a p-local CW-complex Y and let X^{(n)} be the n-fold self smash product of X. In this paper, we study functorial decompositions of X^{(n)} using canonical representation of symmetric group S_{n}. We start with some observations. Let S_{n} acts on X^{(n)} by permuting positions. Then we obtain a representation θ: S_{n} → [X^{(n)}, X^{(n)}]. For n ≥ 2, the function θ extends to a ℤ-linear map ℤ(θ): ℤ(S_{n}) → [X^{(n)}, X^{(n)}] because the co-H-space X^{(n)} is homotopy co-associative and co-commutative and each map σ: X^{(n)} → X^{(n)} is an co-H-map for σ ∈ S_{n}. Since X is a p-local suspension, the ℤ-linear map ℤ(θ) extends to a ℤ_{(p)}-linear map ℤ_{(p)}(θ): ℤ_{(p)}(S_{n}) → [X^{(n)}, X^{(n)}], where ℤ_{(p)} is the ring of p-local integers. This gives a homotopy representation of ℤ_{(p)}(S_{n}) on X^{(n)}. Now let

1 = ∑_{α} e_{α} (1)

be an orthogonal decomposition of the identity in ℤ_{(p)}(S_{n}) in terms of primitive idempotents. Let f_{α}: X^{(n)} → X^{(n)} be a map such that the homotopy class [f_{α}] = ℤ_{(p)}(e_{α}). Then we obtain a homotopy orthogonal decomposition of the identity map id_{X^{(n)}} : X^{(n)} → X^{(n)}

id_{X^{(n)}} ≃ ∑_{α} f_{α}.

Let e_{α}(Y) be the homotopy colimit

e_{α}(Y) = hocolim_{f_{α}} X^{(n)}

for each α. Then e_{α}(Y) is a homotopy retract of X^{(n)} because f_{α} is a homotopy idempotent. Since f_{α} ∘ f_{β} is null homotopic for α β̸, one gets a homotopy decomposition

X^{(n)} = (ΣY)^{(n)} ≃ ⋁_{α} e_{α}(Y). (2)

Furthermore, this decomposition is functorial with respect to Y. In other words, each e_{α} is a functor from spaces to co-H-spaces. (Note: for each α, we can fix a choice of the representative f_{α}. This gives a particular choice of space e_{α}(Y) such that e_{α}(Y) is strictly functorial with respect to Y. Furthermore, there is a choice of map

ϕ: X^{(n)} = (ΣY)^{(n)} → ⋁_{α} e_{α}(Y)

such that ϕ is strictly functorial on Y and is a homotopy equivalence for each Y.)

Recall that primitive idempotents, up to conjugates, in ℤ_{(p)}(S_{n}) are one-to-one corresponds to p-regular partitions of n. A (proper) partition of n is a sequence of positive integers λ = (λ_{1}, ⋯, λ_{s}) such that

λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{s} and ∑_{j=1}^{s} = n.

s is called the length of λ, which is denoted by Len(λ). A partition λ = (λ_{1}, ⋯, λ_{s}) is called p-regular if there is NO subscript i with 1 ≤ i ≤ s such that λ_{i} = λ_{i+1} = ⋯ = λ_{i+p−1}. For each p-regular partition λ, there is an idempotent Q_{λ} ∈ ℤ_{(p)}(S_{n}) such that the (left) ideal generated by Q_{λ} is a projective cover of the Specht module corresponding to λ (See [7] for details). Any primitive idempotent in ℤ_{(p)}(S_{n}) is conjugate to one and only one Q_{λ}. Let d_{λ} be the number of idempotents in decomposition 8 which are conjugate to Q_{λ}. We called d_{λ} the multiplicity of primitive idempotent Q_{λ} in ℤ_{(p)}(S_{n}). Let Q_{λ}(Y) be the homotopy colimit

Q_{λ}(Y) = hocolim_{e^{λ}} X^{(n)}.

Then decomposition 7 can be rewritten as

X^{(n)} = (ΣY)^{(n)} ≃ ⋁_{λ} ⋁^{d_{λ}} Q_{λ}(Y), (3)

where λ runs over all p-regular partitions of n. Let H̄_{∗}(X) denote the reduced mod p homology.

Theorem 1.1. Let X = ΣY be a p-local suspension. Suppose that p = 2 or H̄_{∗}(X)^{odd} = 0. Then Q_{λ}(Y) is contractible if and only if Len(λ) > dim H̄_{∗}(X).

Note: If p > 2 and H̄_{∗}(X)^{odd} ≠ 0, then Q_{λ}(Y) may not be contractible even if Len(λ) > dim H̄_{∗}(X). For instance, let Y be an even sphere and let X = ΣY, then Q_{(1,1)}(Y) ≃ X^{(2)}.

Let X = Σ(Y) be a suspension of a two-cell complex localized at 2. By Theorem 1.1, we have

X^{(n)} ≃ ⋁_{n/2<a≤n} ⋁^{c_{(a,n−a)}} Q_{(a,n−a)}(Y). (4)

We can determine the homology of Q_{(a,n−a)}(Y) for each n/2 < a ≤ n and the multiplicity d_{(a,n−a)}.

Let u, v be a basis for V = H̄_{∗}(X) with |u| ≤ |v|. Let S(V) be the free commutative algebra generated by V and let P (V) be the set of primitive elements in S(V). Then

P (V) = ⨁_{k=0}^{∞} P_{2^{k}}(V)

and P_{2^{k}}(V) has a basis u^{2^{k}}, v^{2^{k}}. Since V is a module over the Steenrod algebra, P_{2^{k}}(V) is a module over the Steenrod algebra in the sense that P_{2^{k}}(V) is a submodule of S_{2^{k}}(V) over the Steenrod algebra. Let s(n) be the integer such that 2^{s(n)} − 1 ≤ n ≤ 2^{s(n)+1} − 2 and let

n − 2^{s(n)} + 1 = 2^{a(1;n)} + 2^{a(2;n)} + ⋯ + 2^{a(l(n);n)}

with 0 ≤ a(1; n) < a(2; n) < ⋯ < a(l(n); n).

Theorem 1.2. Let X = ΣY be a 2-local suspension and let V = H̄_{∗}(X) with a basis |u|, |v|. Then

1) there is an isomorphism of modules over the Steenrod algebra

H̄_{∗}(Q_{(n)}(Y)) ≅ ⨂_{j=0}^{s(n)−1} P_{2^{j}}(V) ⊗ ⨂_{j=1}^{l(n)} P_{2^{a(j;n)}}(V);

2) there is a homotopy equivalence

Q_{(a,n−a)}(Y) ≃ Σ^{(|u|+|v|)(n−a)}Q_{(n−2(n−a))}(Y)

for n/2 < a ≤ n.

3) the (mod 2) homology of the decomposition 5 is a complete decomposition over the Steenrod algebra if H̄_{∗}(X) has a non-trivial Steenrod operation.

This theorem follows from Proposition 4.3, Theorems 4.5, 5.7 and 5.8.

Assertion 3 says that any idempotent ϕ: H̄_{∗}(X^{(n)}) → H̄_{∗}(X^{(n)}) over the Steenrod algebra can be geometrically realizable by an idempotent f: X^{(n)} → X^{(n)}, that is f_{∗} = ϕ. Furthermore the map f can be chosen to be a functorial map

f = ∑_{σ∈S_{n}} k_{σ}σ: X^{(n)} → X^{(n)}

for some k_{σ} ∈ ℤ_{(2)}.

For each a with n/2 < a ≤ n, we have

2^{s(2a−n)} − 1 ≤ 2a − n ≤ 2^{s(2a−n)+1} − 2.

Let

2a − n − 2^{s(2a−n)} + 1 = η_{0} + η_{1}2 + ⋯ + η_{s(2a−n)−1}2^{s(2a−n)−1)}

be the 2-adic resolution, where η_{i} = 0, 1. Let

ω(a) = (η_{0} + 1, η_{1} + 1, ⋯, η_{s(2a−n)−1} + 1).

Let m be a non-negative integer. Define

(5) ϵ(m) = \begin{cases}2 & if & m > 0 and m is even, \\ 1 & if & m is odd, \\ 0 & if & m = 0. \end{cases}

Let I = (i_{1}, ⋯, i_{s}) be a sequence of non-negative integers. Let

ϵ(I) = (ϵ(i_{1}), ⋯, ϵ(i_{s})).

Theorem 1.3. For each n/2 < a ≤ n, the multiplicity d_{(a,n−a)} of Q_{(a,n−a)} is given by the formula

d_{(a,n−a)} = ∑ ; \begin{matrix}I = (i_{0}, ⋯, i_{s}), i_{0} = n \\ ϵ(I) = ω(a) \\ 0 ≤ i_{l} ≤ \frac{i_{l−1}−ϵ(i_{l−1})}{2} \\ i_{l−1} > ϵ(i_{l−1}) for 0 ≤ l ≤ s \\ 0 ≤ i_{s} ≤ ϵ(i_{s}) \end{matrix} k_{I}, (6)

where

k_{I} = ( i_{1}^{\frac{i_{0}−ϵ(i_{0})}{2}} )·( i_{2}^{\frac{i_{1}−ϵ(i_{1})}{2}} ) ⋯ ( i_{s}^{\frac{i_{s−1}−ϵ(i_{s−1})}{2}} ) 2^{(n−(ϵ(i_{0})+i_{1}+ϵ(i_{1})+···+i_{s−1}+ϵ(i_{s−1})+i_{s}))/2}.

This Theorem follows from Proposition 4.3, Theorems 5.7 and 5.8.

The representation of general linear groups and symmetric groups on tensor products has been studied by various people (for example [1, 5, 6, 7, 12]), where the hyper algebras play an important role. There are certain connections between the hyper algebras and the Steenrod algebra (See Section 3). Basic ideas in this paper are to study the representation of symmetric group on self tensor products of twodimensional modules in mod 2 case. We find that there is a special property on the 3-fold self tensor (Lemma 4.2.) From this, we find a recursive algorithm to produce an explicit decomposition of n-fold self tensor (Proposition 4.3. Theorem 4.5 says that this decomposition is in fact a complete decomposition over the Steenrod algebra. By using general results in representation theory, Theorem 5.8 concludes that this decomposition is a complete functorial decomposition (See section 2 for the definition of functorial decompositions). In particular, we have Theorems 1.2 and 1.3 above.

We should point out that Theorems 5.4 and 5.6 suggest that we can filter the symmetric group algebra and its decomposition matrix according to the length of partitions. Theorems 6.1 and 6.4 give an explicit formula to compute entries in the submatrix of the decomposition matrix of symmetric groups corresponding to tworow Young diagrams. This gives a different approach from the classical methods in the representation theory of symmetric groups.

The article is organized as follows. In section 2, we establish a relation between functorial decompositions of tensor products and the representation of symmetric groups on the tensor products. We give some relations between the formal Steenrod algebras and the hyper algebras in section 3. In section 4, we give an explicit decomposition of self tensor products of two-dimensional modules and show that this decomposition is complete over the Steenrod algebra. We review and give some general results on representations of symmetric groups in section 5. The proof of Theorem 1.1 is given inn this section. In section 6, we give some applications. In particular, we have Theorems 6.1 and 6.4 in this section.

2. Functorial Decompositions of Tensor Products

In this section, the ground ring k is a field. Let V be a graded (ungraded) module and let S_{n} acts on V^{⊗n} by permuting positions in graded (ungraded) sense. Let

1 = ∑_{i=1}^{t} e_{i}

be an orthogonal decomposition of the identity in terms of idempotents in k(S_{n}). Let

e_{i}(V) = Im(e_{i}: V^{⊗n} → V^{⊗n}).

Then there is a decomposition

V^{⊗n} ≅ ⨁_{i=1}^{t} e_{i}(V).

Definition 2.1. Let V be a fixed module. A decomposition

V^{⊗n} ≅ ⨁_{i=1}^{t} W_{i}

is called a functorial decomposition if there exist idempotents e_{i} ∈ k(S_{n}) for 1 ≤ i ≤ t such that

1) there is an orthogonal decomposition of the identity

1 = ∑_{i=1}^{t} e_{i}

in k(S_{n});

2) there is an isomorphism of modules

e_{i}(V) ≅ W_{i}

for each 1 ≤ i ≤ t.

Let T_{n}(V) = V^{⊗n}. For an ungraded module V, functorial decompositions of V^{⊗n} is equivalent to decompositions of the functor T_{n} by the following theorem.

Theorem 2.2 ( [11]). Let V be a fixed ungraded module. Then

V^{⊗n} ≅ ⨁_{i=1}^{t} W_{i}

is a functorial decomposition if and only if there is a decomposition of the functor T_{n}, from ungraded modules to ungraded modules,

T_{n} ≅ ⨁_{i=1}^{t} A_{i}

such that

A_{i}(V) ≅ W_{i}

for 1 ≤ i ≤ t.

Let k be of characteristic p and let Q_{λ} be the primitive idempotent in k(S_{n}) corresponding to a p-regular partition λ. Then there is a complete functorial decomposition

V^{⊗n} ≅ ⨁_{λ} Q_{λ}(V)^{⊕d_{λ}}.

Let X = ΣY be a suspension and let V = H̄_{∗}(X; k). Then

H̄_{∗}(Q_{λ}(Y)) ≅ Q_{λ}(V)

and so the homology of Q_{λ}(Y) can be studied by looking at functorial decompositions of V^{⊗n}.

One of natural tools to study functorial decomposition of V^{⊗n} is to Hopf algebra actions on V^{⊗n} which is described as follows. Let A be a graded Hopf algebra and let V be a graded module over A. Then V^{⊗n} is a module over A via comultiplication. More precisely, the map A ⊗ V^{⊗n} → V^{⊗n} is given by the composite

A ⊗ V^{⊗n} ^{ψ^{n}⊗V^{⊗n}}^{}A^{⊗n} ⊗ V^{⊗n} _{≅}^{}(A ⊗ V)^{⊗n} ^{}V^{⊗n}.

Proposition 2.3. Let A be a graded cocommutative Hopf algebra and let V be a graded module over A. Then any functorial decomposition of V^{⊗n} is a decomposition of V^{⊗n} over A.

The proof is straight forward.

In particular, if V is a module over the Steenrod algebra (whatever it is geometrically realizable or not), then any functorial decomposition of V^{⊗n} is a decomposition over the Steenrod algebra. We will show that functorial decompositions of V^{⊗n} is the same as decompositions of V^{⊗n} over the Steenrod algebra when p = 2, dim V = 2 and V has a non-trivial Steenrod operation.

3. Hyper algebras and formal Steenrod algebras

In this section, the ground ring k is any field.

3.1. Cofree Bi-algebra Cover of an Algebra. Let M be a graded module of finite type. Let Γ(M) = Λ(M^{∗})^{∗} be the (graded) dual of the free commutative (graded) algebra generated by the (graded) dual of M. Then Γ(M) is a (co-free) coalgebra. Let q_{M}: Γ(M) → M be the dual of the inclusion M^{∗} → Λ(M^{∗}). By the universal property of the functor Λ, we have that Γ(M) satisfies the following universal property:

Let C be any cocommutative algebra and let f: C → M be any k-linear map. Then there exists a unique morphism of coalgebras f̃: C → Γ(M) such that q_{M} ∘ f̃ = f.

Γ defines a functor from modules of finite type to cocommutative coalgebras of finite type. Now let M be a general free graded module. Then M is a colimit of submodules of finite type, that is, M = colim_{α} M_{α}, where M_{α} is a submodule of M of finite type. We define

Γ(M) = colim_{α} Γ(M_{α}).

The quotient map q_{M_{α}}: Γ(M_{α}) → M_{α} induces the quotient map q_{M}: Γ(M) → M. The functor Γ is just the adjoint functor of the forgetful functor from cocommutative coalgebras to modules. The following proposition is well-known.

Proposition 3.1. The coalgebra Γ(M) with the quotient map q_{M}: Γ(M) → M satisfies the following universal property:

Let C be any cocommutative coalgebra and let f: C → M be any k-linear map. Then there is a unique morphism of coalgebras f̃: C → Γ(M) such that q_{M} ∘ f̃ = f.

Proof. Since any cocommutative coalgebra is a colimit of subcoalgebras of finite type (See [8]), C = colim_{α} C_{α} with C_{α} subcoalgebras of C of finite type. For each C_{α}, f(C_{α}) is contained in a submodule of M of finite type. Thus there is a unique morphism of coalgebras f̃_{α}: C_{α} → Γ(M) such that q_{M} ∘ f̃_{α} = f|_{C_{α}} for each α. Now the morphism {f̃_{α}} defines a unique morphism of coalgebras f̃: C → Γ(M) and hence the result.

Let x be a graded element. Then the cofree cocommutative coalgebra Γ(x) is given as follows.

1) If k is of characteristic 2 or |x| is even, then Γ(x) has a basis {γ_{j}(x)|j ≥ 0} with γ_{0}(x) = 1, the co-identity, and

ψ(γ_{n}(x)) = ∑_{i+j=n} γ_{i}(x) ⊗ γ_{j}(x). (7)

2) If k is of characteristic ≠ 2 and |x| is odd, then Γ(x) has a basis {γ_{j}(x)|j = 0, 1} with γ_{0}(x) = 1, the co-identity, and ψ(γ_{1}(x)) = 1 ⊗ γ_{1}(x) + γ_{1}(x) ⊗ 1.

Let {x_{α}} be a (graded) basis for M. Then there is an isomorphism of coalgebras

Γ(M) ≅ ⨂_{α} Γ(x_{α}).

Now suppose that A is an algebra. Let μ: A ⊗ A → A be the multiplication and let η: k → A be the unit map. By the universal property of the functor Γ, we have a unique morphism of coalgebras μ: Γ(A) ⊗ Γ(A) → Γ(A ⊗ A) such that the following diagram commutes

[figure: Γ(A) ⊗ Γ(A); μ; Γ(A); A ⊗ A; q_{A} ⊗ q_{A}; μ; A; q_{A}]

and a unique morphism of coalgebras η: k → Γ(A) such that q_{A} ∘ η = η: k → A. Thus Γ(A) is a bialgebra which satisfies the following universal property.

Proposition 3.2. Let A be an algebra. Then Γ(A) satisfies the following universal property:

Let B be any cocommutative bialgebra and let ϕ: B → A be any morphism of algebras. Then there is unique morphism of bialgebras ϕ̃: B → Γ(A) such that q_{A} ∘ ϕ̃ = ϕ.

Proof. Let ϕ̃: B → Γ(A) be the (unique) morphism of coalgebras such that q_{A}∘ϕ̃ = ϕ. Note that both maps

ϕ̃ ∘ μ_{B}, μ_{Γ(A)} ∘ (ϕ̃ ⊗ ϕ̃): B ⊗ B → Γ(A)

are morphisms of coalgebras such that

q_{A} ∘ ϕ̃ ∘ μ_{B} = q_{A} ∘ μ_{Γ(A)} ∘ (ϕ̃ ⊗ ϕ̃): B ⊗ B → A.

Thus

ϕ̃ ∘ μ_{B} = μ_{Γ(A)} ∘ (ϕ̃ ⊗ ϕ̃)

and so ϕ is a morphism of bialgebras and hence the result.

3.2. The Graded Hyper algebra And The Formal Steenrod Algebra. Let V be a graded k-module. Let End(V) be the set of graded endomorphisms of V, that is,

End(V)_{k} = {f: V → V |f(V_{i}) ⊆ V_{i+k} foreach i}.

Then End(V) is a graded (finite) algebra under the multiplication

fg = g ∘ f: V → V

and V is a left End(V)-module. Since q: Γ(End(V)) → End(V) is a morphism of algebras, V is a module over Γ(End(V)). It follows that V^{⊗n} is module over Γ(End(V)) via comultiplication. We write U^{V} for Γ(End(V)). By Proposition 3.2, the bialgebra U^{V} satisfies the following universal property:

Let B be a cocommutative graded bialgebra. If V is a module over B, then there is a unique morphism of bi-algebras ϕ: B → U^{V} such that

μ_{B} = μ_{A^{V}} ∘ (ϕ ⊗ V): B ⊗ V → V.

Equation 7 gives the basic formula for calculations of U^{V} -action on V^{⊗n}. One can easily show that if V is an ungraded k-module, then U^{V} is the Hyper algebra defined in [1, 7]. For this reason, we call U^{V} is the Hyper algebra induced by V whatever V is a graded or ungraded k-module.

Let End(V)^{+} = ⨁_{k>0} End(V)_{k} and let End(V)_{k}^{−} = End(V)_{−k}^{+}. Then

End(V) = End(V)^{+} ⊕ End(V)_{0} ⊕ End(V)^{−}

as graded modules and so there is an isomorphism of coalgebras

(8) U^{V} = Γ(End(V)) ≅ Γ(End(V)^{+}) ⊗ Γ(End(V)_{0}) ⊗ Γ(End(V)^{−}).

Since each of End(V)^{+}, End(V)_{0} and End(V)^{−} is closed under the multiplication of End(V), each of Γ(End(V)^{+}), End(V)_{0} and End(V)^{−} is a sub bialgebra of U^{V}. Let A^{V} denote for Γ(End(V)^{+}). Then A^{V} is a connected graded bialgebra and so it is a connected Hopf algebra. A^{V} is called the formal Steenrod algebra induced by V since it canonically links to the Steenrod algebra. By Formula 8, the action of U^{V} on V^{⊗n} is generated by

1) the action of A^{V};

2) the action of Γ(End(V)_{0});

3) the dual action of A^{V}.

Let V be a graded module such that dim V_{k} ≤ 1 for each k and let {x_{i}}_{i∈I} be a graded basis for V, where I is an ordered subset of ℤ. For each pairing (i, j) with i, j ∈ I, let e_{ij} ∈ End(V) such that

e_{ij}(x_{k}) = δ_{jk}x_{i}

for k ∈ I, where δ_{st} = 1 if s = t and δ_{st} = 0 if s ≠ t. The degree of e_{ij} is given by

|e_{ij}| = |x_{i}| − |x_{j}|.

Let w = x_{i_{1}} ⋯ x_{i_{n}} be a monomial in V^{⊗n} and let

c_{i}(w) = |{i_{k} = i|1 ≤ k ≤ n}|

the occurrence of x_{i} in w. Then

γ_{k}(e_{ii})(w) = ( \begin{matrix}c_{i} \\ k \end{matrix} ) w (9)

for each k ≥ 0. Let k be of characteristic p with p finite or infinite and let c_{I} = {c_{i}}_{i∈I} be an ordered set of non-negative integers and let V (c_{I}; n) be the submodule of V^{⊗n} spanned by the monomial w with c_{i}(w) = c_{i} for each i ∈ I. By Formula 9, we have Proposition 3.3. Let V be a graded module. Suppose that V has at most one cell in each dimension. Then ϕ: V^{⊗n} → V^{⊗n} is a morphism over Γ(End(V)_{0}) if and only if ϕ maps V (c_{I}; n)) into V (c_{I}; n) for each ordered set of non-negative integers c_{I}.

A k-linear endomorphism of V^{⊗n} which sends each V (c_{I}; n)) to itself is called a weighted map according to [5]. By Formula 8, we have the following connection between the Hyper algebra and the formal Steenrod algebra. Note that V^{⊗n} is a weighted bi-module over A^{V}, that is, V^{⊗n} is a left and right A^{V} -module, where the right A^{V} action on V^{⊗n} is given by the dual action of A^{V}.

Proposition 3.4. Let V be graded module with a graded basis {x_{i}|i ∈ I}. Suppose that V has at most one cell in each dimension. Let ϕ: V^{⊗n} → V^{⊗n}. Then ϕ is a morphism of modules over U^{V} if and only if ϕ is a morphism of weighted bi-modules over A^{V}.

Note:

1) If dim(V) = 2 with a graded basis u, v and |u| < |v|, then a map ϕ: V^{⊗n} → V^{⊗n} is a weighted map if and only if ϕ is a graded map. In this case, ϕ: V^{⊗n} → V^{⊗n} is a morphism over U^{V} if and only if ϕ is a morphism of graded bi-modules over A^{V};

2) Let V be an ungraded module with dim(V) = k. Let {x_{1}, ⋯, x_{k}} be a basis for V. Given an positive integer n, we can choose a grading on V such that the following property holds:

A map ϕ: V^{⊗n} → V^{⊗n} is a weighted map if and only if ϕ is a graded map. For instance, we can choose a grading on V such that |x_{1}| < |x_{2}| and n|x_{j}| <

|x_{j+1}| for 2 ≤ j ≤ k − 1.

Let V be an ungraded module with dim(V) = k. Then V is a module over the general linear group GL(k; k) and so is V^{⊗n} for any n. Let |k| denote the cardinality of the set k.

Theorem 3.5. [7, Theorems 8.2.13 and 8.2.14] Let V be an ungraded module with dim(V) = k. Then

1) Every U^{V} -submodule of V^{⊗n} is a GL(k, k)-submodule.

2) If |k| ≥ n + 2, every GL(k, k)-submodule of V^{⊗n} is a U^{V} -submodule.

3) If W is a finitely generated U^{V} -submodule of V^{⊗n}, then every U^{V} -homomorphism from W into T (V) is a GL(k; k)-homomorphism;

4) If |k| ≥ max m, n + 2 and W is a GL(k; k)-submodule of V^{⊗n}, then every GL(k, k)-homomorphism from W into V^{⊗m} is a U^{V} -homomorphism.

4. Decompositions of V^{⊗n} over Hopf algebras

In this section, the ground field k is a field of characteristic two and V is a twodimensional module over the formal Steenrod algebra A^{V}. Let {u, v} be a basis for V with |u| = 0 and |v| = 1. Let {e_{12}} and {e_{21}} be the basis for End(V)^{+} and End(V)^{−}, respectively. We write Sq^{k} for the element γ_{k}(e_{12}) ∈ U_{k}^{V} and Sq_{∗}^{k} for γ_{k}(e_{21}) ∈ U_{−k}^{V}. In particular, Sq^{1}(u) = v and Sq_{∗}^{1}(v) = u. Let W be any module. Let L_{n}(W) denote the set of Lie elements of degree n in the free Lie algebra L(W). Let a_{1}, ⋯, a_{n} be elements in W. We write β_{n}(a_{1}a_{2} ⋯ a_{n}) = [[a_{1}, a_{2}, ⋯, a_{n}] for the n-fold commutator from left to right. Let P_{2^{k}}(W) be the set of primitive elements of degree 2^{k} in the symmetric algebra S(W). Note L_{1}(V) = P_{1}(V) = V.

Proposition 4.1. Let V be a two-dimensional module. Then there are isomorphisms of modules over U^{V}.

1) L_{2}(P_{2^{k}}(V)) ≅ P_{2^{k}}(L_{2}(V)) ≅ L_{2}(V)^{⊗k+1} is a one-dimensional module for any k ≥ 0;

2) P_{2^{s}}(P_{2^{k}}(V)) ≅ P_{2^{s+k}}(V) is a two-dimensional module for any k, s ≥ 0;

3) L_{3}(V) ≅ L_{2}(V) ⊗ V;

4) S_{3}(V) ≅ P_{2}(V) ⊗ V.

The proof is immediate.

Lemma 4.2. Let V be a two-dimensional module. Then there is a complete decomposition over A^{V}

P_{2^{k}}(V)^{⊗3} ≅ ((L_{2}(V)^{⊗k+1})^{⊕2} ⊕ P_{2^{k+1}}(V)) ⊗ P_{2^{k}}(V)

for any k ≥ 0.

The proof is immediate.

Let m be a non-negative integer. Let ϵ(m) be defined by equation 5. Let B be a functor. We write B^{i} for B^{⊗i} when i > 0 and B^{0} for k. Let B and C be functors. We write BC for B ⊗ C. We write kB for B^{⊕k} when k > 0 and 0B = 0. In the following proposition, P_{2^{s}} = P_{2^{s}}(V), L_{i} = L_{i}(V) and L_{i} ∘ P_{j} = L_{i}(P_{j}(V)).

Proposition 4.3. There is a decomposition over U^{V}

V^{⊗n} ≅ ⨁ ; \begin{matrix}0 ≤ i_{l} ≤ \frac{i_{l−1}−ϵ(i_{l−1})}{2} \\ i_{l−1} > ϵ(i_{l−1}) \\ 1 ≤ l ≤ s \\ 0 ≤ i_{s} ≤ ϵ(i_{s}) \end{matrix} k_{I}B_{I} · P_{2^{s}}^{ϵ(i_{s})} · P_{2^{s−1}}^{ϵ(i_{s−1})} ⋯ P_{2}^{ϵ(i_{1})}P_{1}^{ϵ(n)},

where

B_{I} = L_{2}^{\frac{n−ϵ(n)}{2} −i_{1}} · (L_{2} ∘ P_{2})^{\frac{i_{1}−ϵ(i_{1})}{2} −i_{2}} ⋯ (L_{2} ∘ P_{2^{s−1}})^{\frac{i_{s−1}−ϵ(i_{s−1})}{2} −i_{s}}

and

k_{I} = ( i_{1}^{\frac{n−ϵ(n)}{2}} ) · ( i_{2}^{\frac{i_{1}−ϵ(i_{1})}{2}} ) ⋯ ( i_{s}^{\frac{i_{s−1}−ϵ(i_{s−1})}{2}} ) 2^{(n−(ϵ(n)+i_{1}+ϵ(i_{1})+···+i_{s−1}+ϵ(i_{s−1})+i_{s}))/2}.

Note: dim B_{I} = 1 and so B_{I} ⊗ M is just a suspension of M.

Proof. Notice that V^{⊗3} ≅ (2L_{2} ⊕ P_{2}) ⊗ P_{1}. It follows that

V^{⊗2k+1} ≅ (2L_{2} ⊕ P_{2})^{k} ⊗ P_{1} and
V^{⊗2k+2} ≅ (2L_{2} ⊕ P_{2})^{k} ⊗ P_{1}^{2}.

Thus

V^{n} ≅ ⨁_{i=0}^{\frac{n−ϵ(n)}{2}} 2^{\frac{n−ϵ(n)}{2} −i} ( i^{\frac{n−ϵ(n)}{2}} ) L_{2}^{\frac{n−ϵ(n)}{2} −i}P_{2}^{i}P_{1}^{ϵ(n)}.

Notice that dim P_{2^{l}} = 2 for any l ≥ 0 with a basis u^{2^{l}} and v^{2^{l}} and the formal Steenrod operation given by Sq_{∗}^{2^{l}}(v^{2^{l}}) = u^{2^{l}}. It follows that

P_{2^{l}}^{k} ∼ ⨁_{i=0}^{\frac{k−ϵ(k)}{2}} 2^{\frac{k−ϵ(k)}{2} −i} ( i^{\frac{k−ϵ(k)}{2}} ) (L_{2} ∘ P_{2^{l}})^{\frac{k−ϵ(k)}{2} −i}(P_{2} ∘ P_{2^{l}})^{i}P_{2^{l}}^{ϵ(k)}

for any l, k. Observe that P_{2} ∘ P_{2^{l}} ≅ P_{2^{l+1}}. The assertion follows.

Now we are going to show that this is a complete decomposition over the formal Steenrod algebra A^{V}. Notice each B_{I} is just a suspension. It suffices to show that each P_{2^{s}}^{ϵ(i_{s})}(V) ⊗ ⋯ ⊗ P_{2}^{ϵ(i_{1})}(V) ⊗ V^{ϵ(n)} is indecomposable over A^{V}, where ϵ(i_{j}) = 1 or 2.

Lemma 4.4. Let 0 ≤ c_{1} < c_{2} < ⋯ < c_{k} and let ϵ_{j} = 1 or 2 for 1 ≤ j ≤ k. Then

M = P_{2^{c_{1}}}(V)^{⊗ϵ_{1}} ⊗ P_{2^{c_{2}}}(V)^{⊗ϵ_{2}} ⊗ ⋯ ⊗ P_{2^{c_{k}}} (V)^{⊗ϵ_{k}}

is an indecomposable module over the Steenrod algebra.

Proof. The proof is given by induction on k. The assertion holds when k = 1. Suppose that the assertion holds for all sequences 0 ≤ c′_{1} < ⋯ < c′_{k′} with k′ < k and k ≥ 2. Let 0 ≤ c_{1} < ⋯ < c_{k}. Let f: M → M be any idempotent over the Steenrod algebra such that f sends the top cell v^{2^{c_{1}+ϵ_{1}}+···+2^{c_{k}+ϵ_{k}}} to itself. We show that f is an isomorphism from which the assertion will follow. Let N = P_{2^{c_{2}}}(V)^{⊗ϵ_{2}} ⊗ ⋯ ⊗ P_{2^{c_{k}}} (V)^{⊗ϵ_{k}}. Let

C = Im(Sq_{∗}^{2^{c_{1}}} : P_{2^{c_{1}}}(V) → P_{2^{c_{1}}}(V)) = Ker(Sq_{∗}^{2^{c_{1}}} : P_{2^{c_{1}}}(V) → P_{2^{c_{1}}}(V)).

Notice that Sq_{∗}^{2^{c_{1}}} : N → N is zero. Thus

Im(Sq_{∗}^{2^{c_{1}}} : M → M) = Ker(Sq_{∗}^{2^{c_{1}}} : M → M) = C ⊗ N.

Notice that f: M → M is morphism over the Steenrod algebra. Thus f sends M_{1} ⊗ N into itself. Let f′: C ⊗ N → C ⊗ N be the resulting map. Notice that C is a submodule of P_{2^{c_{1}}}(V) over the Steenrod algebra. Thus C ⊗ N is a submodule of M over the Steenrod algebra and f′: C ⊗ N → M_{2} ⊗ N is a morphism over the Steenrod algebra. We consider two cases: Case I. ϵ_{1} = 1.

Then C is a one-dimensional k-module generated by u^{2^{c_{1}}}. Notice that

Sq_{∗}^{2^{c_{1}+ϵ_{1}}+···+2^{c_{k}+ϵ_{k}}} (v^{2^{c_{1}+ϵ_{1}}+···+2^{c_{k}+ϵ_{k}}}) = u^{2^{c_{1}+ϵ_{1}}+···+2^{c_{k}+ϵ_{k}}} .

by the Cartan formula. Thus f′: C⊗N → C⊗N sends the bottom cell u^{2^{c_{1}+ϵ_{1}}+···+2^{c_{k}+ϵ_{k}}} to itself. Notice that C ⊗ N ≅ N as modules over the Steenrod algebra. By induction, f′ is an isomorphism. Let D = P_{2^{c_{1}}}(V)/C. Then there is a commutative diagram of short exact sequences over the Steenrod algebra

[figure: C ⊗ N; ⊂; M; D ⊗ N; C ⊗ N; f′; ⊂; M; f; D ⊗ N.; f′′]

Notice that f′′: D ⊗ N → D ⊗ N sends the top cell u^{2^{c_{1}+ϵ_{1}}+···+2^{c_{k}+ϵ_{k}}} to itself. Thus f′′: D ⊗ N → D ⊗ N is an isomorphism by induction and so f is an isomorphism by the five-lemma. Case II. ϵ_{1} = 2.

Then C is generated by u^{2^{c_{1}+1}} and [u^{2^{c_{1}}}, v^{2^{c_{1}}}] = u^{2^{c_{1}}}v^{2^{c_{1}}} − v^{2^{c_{1}}}u^{2^{c_{1}}}. Let C^{(1)} and C^{(2)} be the submodule of C generated by u^{2^{c_{1}+1}} and u^{2^{c_{1}}}v^{2^{c_{1}}} + v^{2^{c_{1}}}u^{2^{c_{1}}}, respectively. Then C ⊗ N = C^{(1)} ⊗ N ⊕ C^{(2)} ⊗ N as modules over the Steenrod algebra. Let n be integer such that C^{(1)} ⊗ N is not zero in dimension n. Then

n = 2(2^{c_{i_{1}}} + ⋯ + 2^{c_{i_{t}}}) + 2^{c_{j_{1}}} + ⋯ + 2^{c_{js}}

for some sequence 2 ≤ i_{1} < i_{2} < ⋯ < i_{t} ≤ c_{k}, 2 ≤ j_{1} < i_{2} < ⋯ < j_{s} ≤ c_{k} and {i_{1}, ⋯, i_{t}} ∩ {j_{1}, ⋯, j_{s}} = ∅ according to occurrence of v’s. In particular, n is divisible by 2^{c_{2}+1}. Let m be integer such that C^{(2)} ⊗ N is not zero in dimension m. Then m = 2(2^{c_{1}} +m′), where m′ is divisible by 2^{c_{2}}. In particular, m is not divisible by 2^{c_{2}+1} because c_{1} < c_{2}. It follows that f sends C^{(1)} ⊗ N and C^{(2)} ⊗ N into themselves, respectively, by dimensional reason. Let f^{(i)} : C^{(i)} ⊗ N → C^{(i)} ⊗ N be the resulting map for i = 1, 2. Now we need to show that f^{(1)} and f^{(2)} are nontrivial maps. Notice that f sends the bottom cell to the bottom cell. Thus f^{(1)} sends the bottom cell to the bottom cell and so f^{(1)} : C^{(1)} ⊗ N ≅ N → C^{(1)} ⊗ N ≅ N is a nontrivial map and so an isomorphism. Notice that

Sq_{∗}^{2^{c_{1}}+m}(v^{2^{c_{1}+ϵ_{1}+m}}) = (u^{2^{c_{1}}}v^{2^{c_{1}}} + v^{2^{c_{1}}}u^{2^{c_{1}}})u^{m},

where m = 2^{c_{2}+ϵ_{2}}+⋯+2^{c_{k}+ϵ_{k}}. Thus f^{(2)} : C^{(2)}⊗N → C^{(2)}⊗N is a nontrivial map and so an isomorphism. Let E = P_{2^{c_{1}}}(V)^{⊗2}/C. Then there is a short exact sequence of modules over the Steenrod algebra C⊗N → M → E⊗N. Let f′′: E⊗N → E⊗N be the morphism over the Steenrod algebra induced by f: M → M. Now E has a basis represented by v^{2^{c_{1}+1}} and u^{2^{c_{1}}}v^{2^{c_{1}}}. Let E^{(1)} and E^{(2)} be the submodule of E generated by v^{2^{c_{1}+1}} and u^{2^{c_{1}}}v^{2^{c_{1}}}, respectively. Then E ⊗ N = E^{(1)} ⊗ N ⊕ E^{(2)} ⊗ N as modules over the Steenrod algebra. By the arguments similar to above, E^{(1)} ⊗N and E^{(2)} ⊗N do not have nontrivial elements in the same dimension and so f′′: E ⊗ N → E ⊗ N is decomposed as morphisms f̃^{(1)} : E^{(1)} ⊗N → E^{(1)} ⊗N and f̃^{(2)} : E^{(2)} ⊗N → E^{(2)} ⊗N. Notice that f sends the top cell to the top cell. Thus f̃^{(1)} sends the top cell to the top cell and so f̃^{(1)} : E^{(1)} ⊗ N → E^{(1)} ⊗ N is an isomorphism. Now we need show that f̃^{(2)} : E^{(2)} ⊗ N → E^{(2)} ⊗ N is a nontrivial map. Notice that a basis for M_{t} with t ≤ 2^{c_{1}} is given as follows:

u^{2^{c_{1}+1}}u^{m}, v^{2^{c_{1}}}u^{2^{c_{1}}}u^{m}, u^{2^{c_{1}}}v^{2^{c_{1}}}u^{m},

where

f(u^{2^{c_{1}+1}}u^{m}) = u^{2^{c_{1}+1}}u^{m} f((u^{2^{c_{1}}}v^{2^{c_{1}}} + v^{2^{c_{1}}}u^{2^{c_{1}}}u^{m}) = (u^{2^{c_{1}}}v^{2^{c_{1}}} + v^{2^{c_{1}}}u^{2^{c_{1}}})u^{m}
Sq_{∗}^{2^{c_{1}}}(v^{2^{c_{1}}}u^{2^{c_{1}}}u^{m}) = Sq_{∗}^{2^{c_{1}}}(u^{2^{c_{1}}}v^{2^{c_{1}}}u^{m}) = u^{2^{c_{1}+1}}u^{m}.

Let

f(v^{2^{c_{1}}}u^{2^{c_{1}}}u^{m}) = k_{11}v^{2^{c_{1}}}u^{2^{c_{1}}}u^{m} + k_{12}u^{2^{c_{1}}}v^{2^{c_{1}}}u^{m}
f(u^{2^{c_{1}}}v^{2^{c_{1}}}u^{m}) = k_{21}v^{2^{c_{1}}}u^{2^{c_{1}}}u^{m} + k_{22}u^{2^{c_{1}}}v^{2^{c_{1}}}u^{m}.

Then we have

k_{11} + k_{12} = 1, k_{21} + k_{22} = 1, k_{21} + k_{11} = 1, k_{22} + k_{12} = 1.

Thus the determinant of f: M_{2^{c_{1}}} → M_{2^{c_{1}}} is 2k_{11} − 1 and so f: M_{2^{c_{1}}} → M_{2^{c_{1}}} is an isomorphism. It follows that f̃^{(2)} : E^{(2)} ⊗ N → E^{(2)} ⊗ N is nontrivial and so an isomorphism. Now f′′: E ⊗ N → E ⊗ N is an isomorphism and so f: M → M is an isomorphism. The induction is finished and the assertion follows.

Thus we have

Theorem 4.5. Let V be a two-dimensional module. Then the decomposition given in Proposition 4.3 is a complete decomposition over A^{V}. In particular, it is a complete decomposition over U^{V}.

5. Relationship with Representations of Σ_{n}

For a vector space V, write |V | for the dimension of V. For a partition λ = (λ_{1}, λ_{2}, …, λ_{r}), write Len(λ) = r. We write Σ_{λ} for the subgroup Σ_{λ_{1}} ×Σ_{λ_{2}} ×⋯×Σ_{λ_{r}} of Σ_{n} and let ξ_{λ} denote the representation of n induced from the trivial representation on Σ_{λ}. We write α_{λ} for the characteristic 0 irreducible corresponding to λ. Thus α_{λ} = Im e_{λ}.

In this section it suffices to consider the case where k = ℤ/pℤ. The projective indecomposable k(Σ_{n}) modules are in 1 − 1 correspondence with the p-regular partitions of n. Given a p-regular partition λ, let Q_{λ} denote the corresponding projective and let f_{λ} denote the idempotent such that Q_{λ} = k(Σ_{n})f_{λ}. For a vector space V, write Q_{λ}(V) = f_{λ}(V^{⊗n}).

Remark 5.1. f_{λ} is not the mod p reduction of e_{λ}.

For any projective Q in ℤ/pℤ(G) there is a unique lift to a projective in Q̂ over ℤ_{(p)}. We write Q̃ for Q̂ ⊗ ℚ.

For partitions λ, μ of n, as in [7, p. 23] we define a partial order by λ ≤ μ if and only if ∑_{j=1}^{t} λ_{j} ≤ ∑_{j=1}^{t} μ_{j} for all t = 1, … Len(λ).

Lemma 5.2. If λ, μ are partitions of n with λ ≤ μ then Len(λ) ≥ Len(μ).

Proof. If Len(λ) < Len(μ) then for t = Len(λ) we have

∑_{j=1}^{t} λ_{j} = n = ∑_{j=1}^{Len(μ)} λ_{j} > ∑_{j=1}^{t} λ_{j},

contradicting λ ≤ μ.

For a partition μ we define μ^{R} as in [7, p. 282] and observe that μ ≤ μ^{R} for any μ. According to [7, Theorem 6.3.51, p. 283], if the intertwining number (Q̃_{λ}, α_{μ}) is nonzero then μ^{R} ≤ λ and so μ ≤ λ and therefore Len(λ) ≤ Len(μ).

Theorem 5.3. If Len(λ) > |V | then Q_{λ}(V) = 0.

Proof. Suppose Len(λ) > |V |. Then (Q̃_{λ}, α_{μ}) = 0 for any partition μ with Len(μ) ≤ |V |. According to [7, Lemma 2.1.10, p. 37], the nonzero components of ξ_{τ} in the basis of α_{μ}’s all satisfy τ ≤ μ. If (Q̃_{λ}, ξ_{τ}) ≠ 0, then τ has a nonzero component α_{μ} such that (Q̃_{λ}, α_{μ}) ≠ 0. For this μ we have τ ≤ μ ≤ λ, which implies that Len(λ) ≤ Len(τ). Therefore (Q̃_{λ}, ξ_{τ}) = 0 for any partition τ with Len(τ) ≤ |V |. But according to [9, Theorem 8.1,p. 179] (Q̃_{λ}, ξ_{τ}) equals the dimension of the space of elements of type τ in Q̃_{λ}(V). Therefore (Q̃_{λ}, ξ_{τ}) = 0 for all partitions τ with Len(τ) ≤ |V | implies that Q_{λ}(V) = 0.

Theorem 5.4. Let k(Σ_{n}) = ∑ d_{λ}Q_{λ} be the decomposition of the group ring into indecomposable projectives. Then V^{⊗n} = ∑_{Len(λ)≤|V |} d_{λ}Q_{λ}(V) is the complete natural decomposition of V^{⊗n} into indecomposable pieces.

Proof. k(Σ_{n}) = ∑ d_{λ}Q_{λ} implies that for any vector space V, we have a natural decomposition V^{⊗n} = ∑ d_{λ}Q_{λ}(V) and so the preceding theorem implies that there is a natural decomposition V^{⊗n} = ∑_{Len(λ)≤|V |} d_{λ}Q_{λ}(V).

For any λ, f_{λ} is primitive (not a nontrivial sum of orthogonal idempotents) so by the definition of a functorial decomposition (section 6.1) there is no natural decomposition of Q_{λ}(V). Therefore V^{⊗n} = ∑_{Len(λ)≤|V |} d_{λ}Q_{λ}(V) is the complete functorial decomposition of V^{⊗n}.

Proof of Theorem 1.1. By Theorems 5.3 and 5.4, we have

H̄_{∗}(Q_{λ}(Y)) = 0

if and only if Len(λ) > dim H̄_{∗}(X). Since H̄_{∗}(X; ℤ_{(p)}) is a finitely generated module over ℤ_{(p)}, H̄_{∗}(Q_{λ}(Y); ℤ_{(p)}) is a finitely generated module over ℤ_{(p)}. It follows that H̄_{∗}(Q_{λ}(Y)) = 0 if and only if H̄_{∗}(Q_{λ}(Y); ℤ_{(p)}) = 0 if and only if Q_{λ}(Y) is contractible. The assertion follows.

Lemma 5.5. Let V be a graded module with a basis {x_{1}, ⋯, x_{k}}. Let λ = (λ_{1}, ⋯, λ_{s}) be a p-regular partition with s ≤ k and let

t_{λ} = λ_{1}|x_{k}| + λ_{2}|x_{k−1}| + ⋯ + λ_{s}|x_{k−s+1}|.

Suppose that 1) |x_{1}| < |x_{2}| < ⋯ < |x_{k}| and

2) p = 2 or V^{odd} = 0. Then Q_{λ}(V)_{j} = 0 for j > t_{λ} and dim Q_{λ}(V)_{t_{λ}} = 1.

Proof. Let Ṽ be an module over ℚ with dim V̄ = k. According to [7, Chapter 8], the functorial indecomposable factors are classified by the Weyl modules W_{μ}(Ṽ). By the construction of the Weyl modules [7, Section 8.2], the top degree of W_{μ}(Ṽ) is t_{μ} for any partition μ = (μ_{1}, ⋯, μ_{t}) with t ≤ k and dim W_{μ}(Ṽ)_{t_{μ}} = 1.

As above [9, Theorem 8.1], Q̃_{λ}(V) has elements of type τ if and only if (Q̃_{λ}, ξ_{τ}) ≠ 0. Since (Q̃_{λ}, ξ_{λ}) = 1, Q̃_{λ}(V) has elements of type λ and the space of elements of type λ occurs once as a retract. If Q̃_{λ}(V) has elements of type λ′ with λ′ ≠ λ then as above λ′ ≤ λ and therefore t_{λ′} < t_{λ}. The assertion follows.

Theorem 5.6. Let V be an ungraded module with |V | = k. Then

Q_{λ}(V)

is an indecomposable module over U^{V} if Len(λ) ≤ k.

Proof. Let K be an extension field of k such that |K| is infinite. According to [5, Theorem (3.5a)] and [7, Theorem 8.3.8], any irreducible factor of V^{⊗n} is absolutely irreducible. It follows that Q_{λ}(V) is indecomposable over U^{V} if and only if

Q_{λ}(V) ⊗_{k} K

is indecomposable over U^{V} ⊗_{k} K. Hence we may assume that the ground field itself is infinite. By Theorem 3.5, Q_{λ}(V) is indecomposable over U^{V} if and only if Q_{λ}(V) is indecomposable over GL(k, k). If k ≥ n, the assertion follows from [6, Theorem (2.2)]. Now we consider the degenerate case that k < n. Let U be an n-dimensional module such that V is a submodule of U. Then Q_{λ}(U)^{∗} is a summand of U^{∗⊗n} ≅ U^{⊗n} as modules over GL(n, k). Then the socle Soc(Q_{λ}(U)^{∗}) is a direct sum of irreducible submodules of U^{⊗n} as modules over GL(n, k). According to [7, Theorem (2.8)], Soc(Q_{λ}(U)^{∗}) is an irreducible module because the socle of a primitive projective S_{n}- module is simple right ideal of k(S_{n}). Thus the head Hd(Q_{λ}(U)) is an irreducible module over GL(n, k) and so Q_{λ}(U) has a unique maximal proper submodule as modules over GL(n, k). According to [5, pp.87], the restriction

Q_{λ}(V) = Q_{λ}(U) ∩ V^{⊗n}

has a unique maximal proper submodule as modules over GL(k, k). It follows that Q_{λ}(V) is an indecomposable module over GL(k, k) and hence the result.

Theorem 5.7. Let k be of characteristic two and let (a, n−a) be a 2-regular partition of n. Then Q_{(a,n−a)}(V) is a functorial retract of

Q_{2a−n}(V) ⊗ L_{2}(V)^{⊗n−a}.

Proof. Let T_{ij}: V^{⊗n} → V^{⊗n} be the k-linear map that switches positions i and j. Let ϕ: V^{⊗n} → V^{⊗n} be the composite

ϕ = T_{2a−n,n−1} ∘ T_{2a−n,n−3} ∘ ⋯ ∘ T_{2a−n,2a−n+1}.

Let θ: V^{⊗n} → V^{⊗n} be the composite

V^{⊗n} ^{ϕ}^{}V^{⊗n} ^{Q_{(2a−n)}⊗β_{2}^{⊗n−a}}^{}V^{⊗n}.

Let A(V) be the colimit

A(V) = colim_{θ} V^{⊗n}.

Then A(V) is a functorial retract of V^{⊗n}. Since the map θ factors through Q_{(2a−n)}(V)⊗ L_{2}(V)^{⊗n−a}, A(V) is a functorial retract of Q_{(2a−n)}(V) ⊗ L_{2}(V)^{⊗n−a}. Now let V be a two-dimensional module with a basis u, v. Then the element

v^{2a−n}[u, v]^{n−a}

is a fixed point of the map θ. It follows that A is a nontrivial functor. Since A(V) has one top degree element in degree a|v|+(n−a)|u|, the functor Q_{(a,n−a)} is a retract of A by Lemma 5.5. Thus Q_{(a,n−a)} is a retract of the functor Q_{(2a−n)} ⊗ L_{2}^{⊗n−a} and hence the result.

By Theorems 4.5, 5.6 and 5.7, we finish to prove the following theorem.

Theorem 5.8. The decomposition in Proposition 4.3 is a complete functorial decomposition.

6. Other Applications

6.1. Decomposition Numbers. Let V be an ungraded module over ℤ/p. Let V̂ be a free ℤ_{(p)}-module such that V = V̂ ⊗ ℤ/p and let Ṽ = V̂ ⊗ ℚ. Let Q_{λ} be the primitive idempotent in ℤ/p(S_{n}) corresponding a p-regular partition λ. Then we have a p-modular system

Q_{λ}(V) ^{}^{mod p} Q̂_{λ}(V̂) ^{}Q̃_{λ}(Ṽ).

Then Q̃_{λ}(Ṽ) is a module over U Ṽ and so there is a U Ṽ -isomorphism

Q̃_{λ}(Ṽ) ≅ ⨁_{μ} W^{μ}(Ṽ)^{⊕d_{μλ}},

where W^{μ}(Ṽ) is the Weyl module corresponding to a partition λ. Suppose that dim V = k ≤ n. By Theorems 5.4 and 5.6, the number d_{μλ} is the entries in position (μ, λ) of the decomposition matrix when Len(λ) ≤ k and Len(μ) ≤ k (See [7]).

Now we specialize at the case p = 2 and dim V = 2. We are able to compute d_{λμ} for Len(λ) ≤ 2 and Len(μ) ≤ 2 using our explicit decomposition for V^{⊗n}. From now on, V is assumed to be of dimension 2. Let μ = (μ_{1}, μ_{2}) be a partition of n and let n/2 < a ≤ n. Let

μ^{a} = \{ \begin{matrix}0 & if & μ_{1} > a \\ (μ_{1} − (n − a), μ_{2} − (n − a)) & if & μ_{1} ≤ a \end{matrix}

be a partition of 2a − n.

Theorem 6.1. Let μ = (μ_{1}, μ_{2}) be partition of n. Then there is an formula

d_{(a,n−a)μ} = d_{(2a−n)μ^{a}}.

Proof. By Theorem 5.7, there is a U^{V} -isomorphism

Q_{(a,n−a)}(V) ≅ Q_{2a−n}(V) ⊗ L_{2}(V)^{⊗n−a}.

The assertion follows.

Now we are going to determine d_{(n)μ} for Len(μ) ≤ 2.

Lemma 6.2. If 2^{k} ≤ n, then there is a short exact sequence of modules over U^{V}

S_{n−2^{k}}(V) ⊗ L_{2}(V)^{⊗2^{k}} ^{⊂} ^{}S_{n}(V) ⊗ P_{2^{k}}(V) [figure] S_{n+2^{k}}(V).

Proof. Let j: P_{2^{k}}(V) → S_{2^{k}}(V) be the inclusion and let q: S_{n}(V) ⊗ P_{2^{k}}(V) → S_{n+2^{k}}(V) be the composite

S_{n}(V) ⊗ P_{2^{k}}(V)^{⊂} ^{}S_{n}(V) ⊗ S_{2^{k}}(V) ^{mult}^{}S_{n+2^{k}}.

Then q is an epimorphism over U^{V}. Let

g: S_{n−2^{k}}(V) ⊗ L_{2}(V)^{⊗2^{k}} → S_{n}(V) ⊗ P_{2^{k}}(V)

be the k-linear map defined by

g(x ⊗ [u, v]^{2^{k}}) = xu^{2^{k}} ⊗ v^{2^{k}} − xv^{2^{k}} ⊗ u^{2^{k}}.

Then q ∘ g = 0 and so we have a U^{V} -map g: S_{n−2^{k}}(V) ⊗ L_{2}(V)^{⊗2^{k}} → Ker(q), which is an isomorphism. The assertion follows.

Given a positive integer n, let

n = 2^{k} + 2^{k−1} + ⋯ + 2 + 1 + 2^{a_{1}} + ⋯ + 2^{a_{s}},

where 0 ≤ a_{1} < a_{2} < ⋯ < a_{s} ≤ k.

Theorem 6.3. There is an isomorphism of modules over U Ṽ

Q̃_{(n)}(Ṽ) ≅ ⨁_{1≤i_{1}<i_{2}<···<i_{t}≤s} L_{2}(Ṽ)^{2^{a_{i_{1}}} +···+2^{a_{i_{t}}}} ⊗ W^{(n−2^{a_{i_{1}}} −···−2^{a_{i_{t}}})}(Ṽ).

Proof. By Theorem 5.8, we have

Q_{(n)}(V) ≅ P_{2^{k}}(V) ⊗ P_{2^{k−1}}(V) ⋯ P_{1}(V) ⊗ P_{2^{a_{1}}}(V) ⊗ ⋯ ⊗ P_{2^{as}}(V)
≅ S_{2^{k+1}−1}(V) ⊗ P_{2^{a_{1}}}(V) ⊗ ⋯ ⊗ P_{2^{as}}(V).

The assertion follows by induction on s and by using Lemma 6.2.

This gives the following theorem.

Theorem 6.4. Let μ = (μ_{1}, μ_{2}) be a partition of n and let

n = 2^{k} + 2^{k−1} + ⋯ + 1 + 2^{a_{1}} + 2^{a_{2}} + ⋯ + 2^{a_{s}}

with 0 ≤ a_{1} < a_{2} < ⋯ < a_{s} ≤ k. Then

d_{(n)μ} = \begin{cases}1 & if μ_{2} = 2^{a_{i_{1}}} + ⋯ + 2^{a_{i_{t}}} for some 1 ≤ i_{1} < i_{2} < ⋯ < i_{t} ≤ s \\ 0 & otherwise \end{cases}

6.2. Irreducible GL(2; k)-modules. Let k be an infinite field of characteristic 2. Let V be a two-dimensional module with a basis u, v. The irreducible GL(2; k)- modules are classified as the head of Weyl modules (See [5]). We are going to show that irreducible GL(2; k)-modules have tensor product decompositions. Let (a, n−a) be a partition of n. According to the definition of the Weyl modules [7], we have

W^{(a,n−a)}(V) ≅ L_{2}(V)^{⊗n−a} ⊗ W^{2a−n}(V).

Since L_{2}(V) is of dimension one, it suffices to look at the head of W^{(n)}(V) for each n.

Theorem 6.5. Let n = 2^{a_{1}} +2^{a_{2}} +⋯+2^{a_{t}} with 0 ≤ a_{1} < a_{2} < ⋯ < a_{t} be the 2-adic resolution. Then

Hd(W^{(n)}(V)) ≅ P_{2^{a_{1}}}(V) ⊗ ⋯ P_{2^{a_{t}}}(V).

Proof. According to [7, Theorem 8.3.1], W^{(n)}(V) is a cyclic A^{V} -submodule of V^{⊗n} generated by the bottom cell u^{n}. Let ϕ: W^{(n)}(V) → S_{n}(V) be the composite

W^{(n)}(V)^{⊂} ^{}T_{n}(V) = V^{⊗n} ^{}S_{n}(V)

and let F (V) be the image of ϕ in S_{n}(V). Let

θ: P_{2^{a_{1}}}(V) ⊗ ⋯ P_{2^{a_{t}}}(V) → S(V)

be the k-linear map defined by

θ(x_{1} ⊗ ⋯ ⊗ x_{t}) = x_{1} ⋯ x_{t}

for x_{j} ∈ P_{2^{a_{j}}} (V). It is a routine work to show that the image of θ lies in F (V) and the map

θ: P_{2^{a_{1}}}(V) ⊗ ⋯ ⊗ P_{2^{a_{t}}}(V) → F (V)

is an isomorphism of modules over U^{V}. By looking at Steenrod operations, it is easy to show that

P_{2^{a_{1}}}(V) ⊗ ⋯ ⊗ P_{2^{a_{t}}}(V)

is an irreducible bi-module over the Steenrod algebra. Since k is an infinite field, ⊗_{j=1}^{t}P_{2^{a_{j}}} (V) is an irreducible GL(2, k)-module. According to Theorem [7, Theorem 8.3.2], W^{(n)} has a unique irreducible quotient. The assertion follows.

6.3. Decompositions of Self Smashes of suspensions of ℝP^{2} ∧ ℂP^{2}. Let Y = Σ^{k}ℝP^{2} ∧ℂP^{2} with k ≥ 3. The space Y is important for v_{1}-periodic homotopy theory. By the following theorem, we can apply Theorem 1.2 to give splittings of self smashes of Y. Theorem 6.6. [2, 13] There is a homotopy decomposition

(ΣℝP^{2})^{(3)} ≃ Σ^{6}ℝP^{2} ∨ Σ^{6}ℝP^{2} ∨ Σ^{3}ℝP^{2} ∧ ℂP^{2}.

Let P^{n}(2) = Σ^{n−2}ℝP^{2} for n ≥ 2.

Theorem 6.7. Let ϕ: H̄_{∗}(Y^{(n)}) → H̄_{∗}(Y^{(n)}) be an idempotent over the Steenrod algebra. Then there is a homotopy idempotent f: Y^{(n)} → Y^{(n)} such that f_{∗} = ϕ.

Proof. Since Y is a retract of a suspension of (P^{3}(2))^{(3)}, Y^{(n)} is a retract of a suspension of (P^{3}(2))^{(3n)}. The assertion follows from Theorem 1.2.

Let V = H̄_{∗}(ℝP^{2}).

Corollary 6.8. The homology of a complete homotopy decomposition of Y^{(n)} is a suspension of the complete decomposition of

(P_{2}(V))^{⊗n} ⊗ V^{⊗n}

over the Steenrod algebra.

Proposition 4.3 gives a way to compute the homology of each homotopy indecomposable retract of Y^{(n)}.

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[13] Jie Wu, Thesis, University of Rochester (1995).

Department of Mathematics, Unoversity of Toronto, Toronto, Ontario M5G 3G3, Canada, selick@math.toronto.edu

Department of Mathematics, National University of Singapore, Lower Kent Ridge Road, Singapore 11926, jiewu@math.upenn.edu