COMPUTATIONS IN GENERIC REPRESENTATION THEORY: MAPS FROM SYMMETRIC POWERS TO COMPOSITE

FUNCTORS

NICHOLAS J. KUHN

July 12, 1996

Abstract. If F_{q} is the finite field of order q and characteristic p, let F(q) be the category whose objects are functors from finite dimensional F_{q}–vector spaces to F_{q}–vector spaces, and with morphisms the natural transformations between such functors. Important families of objects in F(q) include the families S_{n}, S^{n}, Λ^{n}, S̄^{n}, and cT^{n}, with c ∈ F_{q}[Σ_{n}], defined by S_{n}(V) = (V^{⊗n})^{Σ_{n}},S^{n}(V) = V^{⊗n}/Σ_{n}, Λ^{n}(V) = n^{th} exterior power of V, S̄^{∗}(V) = S^{∗}(V)/(p^{th} powers), and cT^{n}(V) = c(V^{⊗n}).

Fixing F, we discuss the problem of computing Hom_{F(q)}(S_{m}, F ∘ G), for all m, given knowledge of Hom_{F(q)}(S_{m}, G) for all m. When q = p, we get a complete answer for any functor F chosen from the families listed above.

Our techniques involve Steenrod algebra technology, and, indeed, our most striking example, when F = S^{n}, arose in recent work on the homology of iterated loopspaces.

1. Introduction

As in our series of papers [K:I, K:II, K:III], if F_{q} is the finite field of order q and characteristic p, let F(q) be the abelian category with objects the functors

F : finite dimensional F_{q}–vector spaces ⟶ F_{q}–vector spaces,

and with morphisms the natural transformations. We like to view an object F ∈ F(q) as a ‘generic representation’ of the general linear groups over F_{q}, as F (V) becomes an F_{q}[GL(V)]–module for all F_{q}–vector spaces V. The tight relationship, explored in [K:II], between F(q) and the categories of F_{q}[GL_{n}(F_{q})]–modules, for all n, makes the study of F(q) of great representation theoretic interest.

We call F ∈ F(q) simple if it has no nontrivial subobjects, finite if it has a finite composition series with simple subquotients^{1} , and locally finite if it is the union of its finite subobjects. The full subcategory consisting of the locally finite functors is denoted by F_{ω}(q).

1991 Mathematics Subject Classification. Primary 20G05; Secondary 55S10,55S12. Partially supported by the NSF 1The finite functors are also characterized as the functors taking finite dimensional values, and

that are polynomial in the sense of Eilenberg–MacLane [EM] (or anyone else, e.g. [MacD]). See [K:I, Appendix A].

Let us introduce some examples of interest. I is the ‘inclusion’ functor: I(V) = V. The families T^{n}, S_{n}, S^{n}, and S̄^{n} are defined, for n ≥ 0, by T^{n}(V) = V^{⊗n}, S_{n}(V) = (V^{⊗n})^{Σ_{n}}, S^{n}(V) = V^{⊗n}/Σ_{n}, and S̄^{∗}(V) = S^{∗}(V)/(p^{th} powers). All of these are finite, and the S̄^{n} are simple, as are the exterior powers Λ^{n}. Note that T^{n} admits an F_{q}[Σ_{n}] action; thus, for any c ∈ F_{q}[Σ_{n}], we have a functor cT^{n}. For a fixed F_{q}–vector space W, let J_{W} (V) = F_{q}^{Hom(V,W)}. The J_{W} are locally finite injectives satisfying Hom_{F(q)}(F, J_{W}) = F (W)^{∗}. (See [K:I, K:II] for further discussion of these points.)

More complex functors can be built out of these by using the operations F ⊗ G and F ∘ G. The point of this paper is to discuss the following problem.

Problem For a fixed F, compute Hom_{F(q)}(S_{∗}, F ∘G), in terms of Hom_{F(q)}(S_{∗}, G).

A number of remarks are in order here.

Firstly, Hom_{F(q)}(S_{∗}, G) means the graded vector space ⨁_{n=0}^{∞} Hom_{F(q)}(S_{n}, G). This has extra structure, to be described shortly.

Secondly, there are natural isomorphisms[K:I]^{2}:

Hom_{F(q)}(S_{∗}, F) ⊗ Hom_{F(q)}(S_{∗}, G) ≃ Hom_{F(q)}(S_{∗}, F ⊗ G).

The results in this paper, together with this ‘Kunneth formula’, suffice to give routine algorithms for computing Hom_{F(q)}(S_{∗}, G), where G is any functor built up from the examples defined above, using tensor products and compositions, under the restriction that q is prime.

Thirdly, there is a duality functor D : F(q)^{op} ⟶ F(q) defined by DF (V) = F (V^{∗})^{∗}, under which S_{n} and S^{n} are dual^{3}. Thus our results about Hom_{F(q)}(S_{∗}, F ∘ G) are equivalent to results about Hom_{F(q)}(F ∘ G, S^{∗}).

Finally, our methods are based on modern versions of algebraic topologists’ ‘Steenrod algebra technology’. We are dubious that our results can be attained without using these.

To state our main results, we need to explain how the Steenrod algebra enters. Let S^{∗} be the full subcategory of F(q) with objects S^{n}, and then let U(q) = Rep(S^{∗}), the category of additive functors from S^{∗} to F_{q} vector spaces. Thus M ∈ U(q) is a graded vector space equipped with suitably compatible maps a : M_{m} ⟶ M_{n} for all a : S^{m} ⟶ S^{n}. An observation in [K:I] is that all such a : S^{m} ⟶ S^{n} arise though the action of Steenrod operations on S^{∗}(V), and that U(q) is precisely the category of unstable modules over the (Hopf) algebra A(q) of Steenrod reduced q^{th} powers^{4}.

For any M ∈ U(q), the q^{th} power maps Φ : S^{n} ⟶ S^{qn} induce a map Φ : M ⟶ M, which multiplies degree by q. We let U_{Φ}(q) denote the category of graded F_{q}–vector spaces having such a self map, and we let U_{∅}(q) be the category of graded F_{q}–vector spaces. Then there are forgetful functors U(q) ⟶ U_{Φ}(q) ⟶ U_{∅}(q).

Let M_{G} = Hom_{F(q)}(S_{∗}, G). The ‘extra structure’ on M_{G} arises from noting that M_{G} ∈ U(q). This is perhaps a good place to recall the basic computation : M_{I} = ⟨x_{k} | k = 0, 1, … ⟩, where x_{k} has degree q^{k} and corresponds to Φ^{k} : I = S^{1} ⟶ S^{q^{k}} (see [K:I]).

2This extends to Ext_{F(q)}^{∗}. See [K:III].

3Less obviously, simple functors are self dual. See [K:II, §7].

4A module M is unstable if it satisfies ax = 0 whenever the excess of a is greater than the degree of x. For example, when q = 2, this means that Sq^{i}x = 0 if i > |x|.

Our first observation gives content to the problem posed above.

Theorem 1.1. Given F ∈ F_{ω}(q) taking finite dimensional values, there exists a functor U_{F} : U(q) ⟶ U(q) and natural isomorphisms of unstable A(q)–modules, U_{F} (M_{G}) ≃ M_{F ∘G}.

The naturality is over G ∈ F_{ω}(q).

Our remaining results amount to explicit descriptions of U_{F} ’s, for various types of functors F ∈ F(q), sometimes with the restriction that q be prime.

All of the functors Λ^{n}, S_{n}, S^{n}, S̄^{n}, cT^{n} extend in an obvious way to U(q).

Theorem 1.2. There are natural isomorphisms in U(q), F (M_{G}) ≃ M_{F ∘G}, in the following cases:

(1) F = S_{n},

(2) F = Λ^{n},

(3) F = S̄^{n},

(4) F = cT^{n}, with c ∈ F_{q}[Σ_{n}].

Corollary 1.3. In these cases, M_{F ∘G} just depends on M_{G} as a graded vector space.

Remarks 1.4.

(1) Note that we have omitted the family F = S^{n}. For good reason: see Theorem 1.8, below.

(2) Cases (2) and (3) are special cases of (4). However, they have technically simpler proofs.

(3) When q is prime, case (4) includes all the simple functors. In particular, if F_{λ} is simple, then F_{λ}(M_{I}) computes the occurrences of F_{λ} in the socle of S^{n}, for all n. This was already noted in [KK, §3], where the nonprime field case was also briefly discussed. See also [FS] for a somewhat different proof when q = 2, and our discussion in §5.

Let K(q) be the category of commutative, unital algebras K in U(q) (a category with a tensor product) satisfying the ‘restriction axiom’: Φ(x) = x^{q} for all x ∈ K. Let U : U(q) ⟶ K(q) be the free functor, left adjoint to the forgetful functor. Explicitly, U(M) = S^{∗}(M)/(Φ(x) − x^{q}).

Theorem 1.5. M_{J_{W} ∘G} is naturally an object in K(q), with algebra structure induced from that on the ring of functions J_{W} (V). If q is prime, there are natural isomorphisms in K(q), U(M_{G} ⊗ W^{∗}) ≃ M_{J_{W} ∘G}.

Here W^{∗} is to be viewed as an unstable A(q)–module concentrated in degree 0.

Corollary 1.6. If q is prime, as a graded algebra, M_{J_{W} ∘G} only depends on M_{G} as an object in U_{Φ}(q), and is polynomial. The Poincaré series of M_{J_{W} ∘G} only depends on the Poincaré series of M_{G}.

Remarks 1.7.

(1) If q is not a prime, U(M_{G} ⊗ W^{∗}) need not equal M_{J_{W} ∘G}, though it does when G = I.

(2) The functor U has long been in the topology literature (see e.g. [SE]). One learns that, at least at the prime 2, the cohomology of Eilenberg MacLane spaces has a representation theoretic description: there are isomorphisms in K(2),

H^{∗}(K(W, n); F_{2}) ≃ Hom_{F(2)}(S_{∗}, J_{W} ∘ S_{n}),

natural in both W and n.

To describe how M_{S^{n}∘G} depends on M_{G}, we need variants of the categories U(q) and K(q). As in [K:III], let U^{2}(q) be Rep(S^{∗} ⊗ S^{∗}), i.e., the category of bigraded modules over the bigraded (Hopf) algebra A(q) ⊗ A(q), unstable in each grading. Note that, given M, N ∈ U(q), M ⊗ N can be regarded as an object in U^{2}(q). Given M ∈ U^{2}(q), there are natural maps Φ_{1} : M_{m,∗} ⟶ M_{qm,∗} and Φ_{2} : M_{∗,n} ⟶ M_{∗,qn}, and we let Φ = Φ_{1} ⊗ Φ_{2} : M_{m,n} ⟶ M_{qm,qn}. Let K^{2}(q) denote the category of commutative, unital algebras K in U^{2}(q) satisfying the ‘restriction axiom’: Φ(x) = x^{q} for all x ∈ K. Let U_{2} : U^{2}(q) ⟶ K^{2}(q) be the free functor, left adjoint to the forgetful functor. Explicitly, U_{2}(M) = S^{∗}(M)/(Φ(x) − x^{q}).

Theorem 1.8. M_{S^{∗}∘G} is naturally an object in K^{2}(q), with algebra structure induced from that on S^{∗}(V). If q is prime, there are natural isomorphisms in K^{2}(q), U_{2}(M_{G} ⊗ M_{I}) ≃ M_{S^{∗}∘G}.

Corollary 1.9. If q is prime, as a bigraded algebra, M_{S^{∗}∘G} only depends on M_{G} as an object in U_{Φ}(q), and is polynomial. The Poincaré series of M_{S^{n}∘G} only depends on the Poincaré series of M_{G}.

Remarks 1.10.

(1) As in the previous remark, if q is not a prime, U_{2}(M_{G} ⊗ M_{I}) need not equal M_{S^{∗}∘G}, though it does in the when G = I [K:III].

(2) Starting from iterated loopspaces on spheres, the author has recently constructed [K2] a bigraded family of topological spectra X(n, j), such that there are A(2)–module isomorphisms

H^{∗}(X(n, j); F_{2}) ≃ Hom_{F(2)}(S_{∗}, (Φ^{−1}S^{j}) ∘ S_{n}) = Φ_{2}^{−1} Hom_{F(2)}(S_{∗}, S^{j} ∘ S_{n}).

Our proof, and then use, of this isomorphism depends on Theorem 1.8, and was the initial motivation for our investigations here.

As remarked above, both Theorem 1.5 and Theorem 1.8 do not hold without modification if q is not prime. Our last result illustrates the added complexity of the nonprime field case.

As in [K:II, §5], if q = p^{s} with s > 1, let G_{ξ} denote the functor G twisted by the Frobenius. Noting that G_{ξ} = I_{ξ} ∘ G, we see that G ⟼ G_{ξ} is a composition operator which iterated s times is the identity. In §5, we will define a functor U_{ξ} : U(q) ⟶ U(q). As graded vector spaces,

U_{ξ}(M)_{n} = ⋂_{i≥1} Ker{Q^{i} : M_{pn} ⟶ M_{pn+q^{i}−1}},

where Q^{i} ∈ A(q) is the q–analogue of the operation Q^{i} ∈ A(p) studied by Adams and Wilkerson in [AW].

Theorem 1.11. There are natural isomorphisms in U(q), U_{ξ}(M_{G}) ≃ M_{G_{ξ^{−1}}} .

Thus it appears that, unlike all of our previous examples, M_{G_{ξ}} can only be computed if one knows a lot about the A(q)–module stucture on M_{G}.

In §2 we recall needed results about F(q) and U(q). In particular, central to our arguments are the Embedding and Vanishing Theorems of [K:I, K:III], and some of their consequences. Theorem 1.1 then becomes obvious, and a general strategy for explicit computation emerges. Using this, Theorem 1.2 is proved in §3, and Theorems 1.5,1.8 in §4. Theorem 1.11 is proved more directly in §5.

2. Prerequisites about U(q) and F(q)

Recalling that U(q) = Rep(S^{∗}), very formally, there are adjoint functors

r : F(q) ⟶ U(q),

and

l : U(q) ⟶ F(q),

with r(F) = Hom_{F(q)}(S_{∗}, F) = M_{F}, and l left adjoint to r. Since r is a right adjoint, it is left exact, and commutes with limits. Furthermore, the finiteness of the family S_{n} implies that r commutes with filtered colimits.

In [K:I, §7], it is observed that both the existence of a tensor product in U(q), and the Kunneth formula of the introduction, are consequences of the two natural isomorphisms:

S^{∗}(V) ⊗ S^{∗}(W) ≃ S^{∗}(V ⊕ W),
J_{V} ⊗ J_{W} ≃ J_{V ⊕W} .

The map of the Kunneth isomorphism, M_{F} ⊗ M_{G} ⟶ M_{F ⊗G}, is explicitly defined as follows: given α : S_{m} ⟶ F and β : S_{n} ⟶ G, α ⊗ β is sent to the composite

S_{m+n} ⟶^{Ψ} S_{m} ⊗ S_{n} ⟶^{α⊗β} F ⊗ G.

Here Ψ : S_{∗}(V) ⟶ S_{∗}(V) ⊗ S_{∗}(V) is dual to the multiplication on S^{∗}(V).

Papers [K:I, K:III] featured two fundamental theorems about the family S^{n} (and, via duality, the family S_{n}). The first of these is the following ‘Embedding Theorem’.

Theorem 2.1. [K:I] Every finite functor F ∈ F(q) embeds in a sum of the form ⊕_{i}S^{n_{i}}. Equivalently, the family S_{n}, n ≥ 0, generates F_{ω}(q).

Via an appropriate version of the Gabriel-Popescu Theorem [P] (a categorical theorem about one-sided Morita equivalence), this theorem is equivalent to the following corollary.

Corollary 2.2. [K:I] l : U(q) ⟶ F(q) is exact, and the natural map l(r(F)) ⟶ F is an isomorphism for all F ∈ F_{ω}(q).

The Kunneth formula above asserts that r preserves tensor products. With this and the last corollary, one can deduce

Corollary 2.3. [K:I] l : U(q) ⟶ F(q) preserves tensor products.

(The original proofs of these last corollaries, in the prime field case, are due to Lannes [L], and Henn-Lannes-Schwartz [HLS]. They work heavily in the category U(q).)

Let Φ^{−1}S^{n} denote the locally finite functor

Φ^{−1}S^{n} = colim {S^{n} ⟶^{Φ} S^{qn} ⟶^{Φ} S^{q^{2}n} ⟶ … }.

The ‘Vanishing Theorem’ reads

Theorem 2.4. [K:III] For all n, Φ^{−1}S^{n} is injective in F_{ω}(q).

This again is equivalent to a corollary about U(q). Call M ∈ U(q) nilpotent if

colim {M ⟶^{Φ} M ⟶^{Φ} M ⟶^{Φ} … } = 0^{5}.

Corollary 2.5. [K:III] l(M) ≃ 0 if and only if M is nilpotent.

(The original Steenrod algebra proof of this appeared in [LS].)

Let N il(q) ⊂ U(q) be the full subcategory of nilpotent modules. As a consequence of Corollary 2.2 and Corollary 2.5, l and r induce an equivalence of abelian categories

U(q)/N il(q) ≃ F_{ω}(q),

and M ⟶ r(l(M)) can be thought of as ‘localization away from N il(q)’. Thus, following standard terminology [G], we call M ∈ U(q) nilclosed if it is isomorphic to a module of the form r(F).

Theorem 1.1 is an immediate consequence of Corollary 2.2.

Proof of Theorem 1.1. Let U_{F} : U(q) ⟶ U(q) be defined by U_{F} (M) = r(F ∘ l(M)). Then, for all G ∈ F_{ω}(q), U_{F} (M_{G}) = r(F ∘ l(r(G))) ≃ r(F ∘ G) = M_{F ∘G}.

We remark that a functor U_{F} : U(q) ⟶ U(q) is not uniquely determined by the requirement that U_{F} (M_{G}) ≃ M_{F ∘G}. (The functor defined in this last proof is just the terminal example.) In our most interesting examples, those occurring in Theorem 1.5 and Theorem 1.8, our strategy for finding explicit U_{F} ’s will be based on the following consequence of the above.

Theorem 2.6. Suppose a functor U_{F} : U(q) ⟶ U(q) satisfies the two conditions:

(1) U_{F} preserves nilclosed modules.

(2) For all M ∈ U(q), l(U_{F} (M)) ≃ F ∘ l(M). Then, for all G ∈ F_{ω}(q), U_{F} (M_{G}) ≃ M_{F ∘G}.

The point of this theorem is that condition (2) is often easy to verify, due to the good properties of l.

We end this section by recalling from [K1] the q-analogue of what in the topology literature is known as the ‘doubling construction’. Let Φ : U(q) ⟶ U(q) be defined as follows:

ΦM = Im {S_{q}(M) ↪ M^{⊗q} → S^{q}(M)}.

5Note that for K ∈ K(q), K is nilpotent in our sense if and only if it is a nilpotent algebra in the usual algebraic sense.

Then there are natural projections π_{M} : S_{qj}(M) ⟶ S_{j}(ΦM), natural inclusions i_{M} : S^{j}(ΦM) ⟶ S^{qj}(M), and natural isomorphisms of vector spaces

(ΦM)_{n} ≃ \begin{cases}M_{n/q} & if q divides n, \\ 0 & otherwise. \end{cases}

Using this isomorphism, a natural map λ_{M} : ΦM ⟶ M can be defined by λ_{M}(x) = Φ(x). Then λ_{M} is A(q)–linear, and is monic if M is nilclosed.

3. Proof of Theorem 1.2

The first cases of Theorem 1.2 follow from just the Kunneth formula of the introduction, and the fact that r is left exact.

To prove case (1), we start with the observation that there is an exact sequence

0 ⟶ S_{n} ⟶ T^{n} \underrightarrow{∏_{i=1}^{n−1}(1−w_{i})} ∏_{i=1}^{n−1} T^{n},

where w_{i} ∈ Σ_{n}, i = 1, …, n − 1, are transpositions generating the n^{th} symmetric group. Precomposing with G yields

0 ⟶ S_{n} ∘ G ⟶ G^{⊗n} \underrightarrow{∏_{i=1}^{n−1}(1−w_{i})} ∏_{i=1}^{n−1} G^{⊗n}.

Applying r and the Kunneth formula to this yields the exact sequence in U(q),

0 ⟶ M_{S_{n}∘G} ⟶ M_{G}^{⊗n} \underrightarrow{∏_{i=1}^{n−1}(1−w_{i})} ∏_{i=1}^{n−1} M_{G}^{⊗n},

which shows that M_{S_{n}∘G} ≃ S_{n}(M_{G}).

If char F_{q} is not 2, case (2) follows from a similar proof, with ∏_{i=1}^{n−1}(1 − w_{i}) replaced by ∏_{i=1}^{n−1}(1 + w_{i}). If char F_{q} = 2, Λ^{n} = S̄^{n}, so is included in case (3).

To prove case (3), we start by recalling [K:II, Example 7.6] that S̄^{n} is the image of the ‘norm’ N : S^{n} ⟶ S_{n}, and thus is self dual (and simple). One sees also that case (3) is a special subcase of case (4), but if q = p (i.e. is prime) one can continue as follows.

There is an exact sequence in F(p),

⨁_{j≥1} S^{j} ⊗ S^{n−pj} ⟶ S^{n} ⟶ S̄^{n} ⟶ 0,

where S^{j} ⊗ S^{n−pj} ⟶ S^{n} is the composite

S^{j} ⊗ S^{n−pj} ⟶^{Φ⊗1} S^{pj} ⊗ S^{n−pj} ⟶ S^{n}.

Dualizing yields the exact sequence in F(p),

0 ⟶ S̄^{n} ⟶ S_{n} ⟶ ∏_{j≥1} S_{j} ⊗ S_{n−pj}. (3.1)

Precomposing with G, and then applying r, yields an isomorphism in U(p)

0 ⟶ M_{S̄^{n}∘G} ⟶ S_{n}(M_{G}) ⟶ ∏_{j≥1} S_{j}(M_{G}) ⊗ S_{n−pj}(M_{G}), (3.2)

where S_{n}(M_{G}) ⟶ S_{j}(M_{G}) ⊗ S_{n−pj}(M_{G}) is the composition

S_{n}(M_{G}) ⟶ S_{pj}(M_{G}) ⊗ S_{n−pj}(M_{G}) ⟶^{r(Φ)⊗1} S_{j}(M_{G}) ⊗ S_{n−pj}(M_{G}).

(Here, by abuse of notation, we denote the dual of Φ : S^{j} ⟶ S^{jp} by ‘Φ’.) Meanwhile, using (3.1), one can deduce an exact sequence in U(p),

0 ⟶ S̄^{n}(M) ⟶ S_{n}(M) ⟶ ∏_{j≥1} S_{j}(ΦM) ⊗ S_{n−pj}(M), (3.3)

where S_{n}(M) ⟶ S_{j}(ΦM) ⊗ S_{n−pj}(M) is the composition

S_{n}(M) ⟶ S_{pj}(M) ⊗ S_{n−pj}(M) ⟶^{π_{M} ⊗1} S_{j}(ΦM) ⊗ S_{n−pj}(M).

Equation (3.3) can be then compared with (3.2), yielding case (3) in the q = p subcase, using the following addendum to case (1).

Proposition 3.1. Under the isomorphism S_{n}(M_{G}) ≃ M_{S_{n}∘G}, the map r(Φ) :

M_{S_{qj}∘G} ⟶ M_{S_{J} ∘G} corresponds to the composite S_{qj}(M_{G}) ⟶^{π_{M}} S_{j}(ΦM_{G}) ⟶^{S_{j}(λ)}

S_{j}(M_{G}).

To prove this, one can immediate reduce to the case j = 1, which can be easily checked directly. (Compare with [K1, Proposition 6.7 and its proof].)

We end this section with the more subtle proof of case (4) of Theorem 1.2. Restated, this says the following.

Proposition 3.2. For any c ∈ F_{q}[Σ_{n}], and locally finite G ∈ F(q), the natural inclusion

c Hom_{F(q)}(S_{∗}, G^{⊗n}) ⊆ Hom_{F(q)}(S_{∗}, cG^{⊗n})

is an isomorphism.

Via duality (and the fact that r commutes with filtered direct limits), this is equivalent to

Proposition 3.3. For any c ∈ F_{q}[Σ_{n}], and finite G ∈ F(q), the natural inclusion

c Hom_{F(q)}(G^{⊗n}, S^{∗}) ⊆ Hom_{F(q)}(G^{⊗n}c, S^{∗})

is an isomorphism.

This we now prove, following the line of argument in [KK, §3]. Noting that Hom_{F(q)}(G^{⊗n}c, S^{∗}) ⊆ Hom_{F(q)}(G^{⊗n}, S^{∗}) = M_{DG}^{⊗n}, Proposition 3.3 will follow from the next two lemmas.

Lemma 3.4. In the situation of Proposition 3.3, for all x ∈ Hom_{F(q)}(G^{⊗n}c, S^{j}), there exists a k so that Φ^{k}(x) ∈ c Hom_{F(q)}(G^{⊗n}, S^{q^{k}j}).

Lemma 3.5. Given c ∈ F_{q}[Σ_{n}] and M ∈ U_{Φ}(q), if Φ^{k}(x) ∈ cM^{⊗n}, then x ∈ cM^{⊗n}.

Proof of Lemma 3.4. The Vanishing Theorem, Theorem 2.4, asserts the injectivity of Φ^{−1}S^{j}. It follows that, given x : G^{⊗n}c ⟶ S^{j}, there is an extension

[figure: 0; G^{⊗n}c; G^{⊗n}; S^{j}; Φ^{−1}S^{j}; x; y]

The image of y will be contained in S^{q^{k}j} for k large enough. But this means that Φ^{k}(x) ∈ c Hom_{F(q)}(G^{⊗n}, S^{q^{k}j}), as needed.

Proof of Lemma 3.5. The lemma follows from the claim that

0 ⟶ Φ^{k}(M^{⊗n}) ⟶ M^{⊗n} ⟶ M^{⊗n}/Φ^{k}(M^{⊗n}) ⟶ 0

splits as a short exact sequence of F_{q}[Σ_{n}]–modules. (Here Φ^{k}(M^{⊗n}) means Im {Φ^{k} : M^{⊗n} ⟶ M^{⊗n}}.) To see that this claim holds, first note that

Φ^{k}(M^{⊗n}) ≃ (Φ^{k}(M))^{⊗n},

and then that, for any two (graded) vector spaces V, W, there is an isomorphism of F_{q}[Σ_{n}]–modules

(V ⊕ W)^{⊗n} ≃ ⨁_{s+t=n} Ind_{Σ_{s}×Σ_{t}}^{Σ_{n}}(V^{⊗s} ⊗ W^{⊗t}),

so that, in particular, V^{⊗n} ⊆ (V ⊕ W)^{⊗n} splits as F_{q}[Σ_{n}]–modules.

4. Proofs of Theorems 1.5 and 1.8

Proof of Theorem 1.5. We begin by explaining why M_{J_{W} ∘G} is naturally an object in K(q), a.k.a. an unstable A(q) algebra. The point is that, since J_{W} (G(V)) is the ring of functions F_{q}^{Hom(G(V),W)}, J_{W} ∘ G takes values in q–Boolean algebras: commutative F_{q}–algebras B satisfying x^{q} = x, for all x ∈ B. Theorem 6.9 of [K1] then says that r takes q–Boolean algebra valued functors to unstable A(q) algebras (and l does the opposite).

We now show that U(M_{G} ⊗ W^{∗}) ≃ M_{J_{W} ∘G}, in K(q), if q is prime, by verifying that U satisfies the two conditions of Theorem 2.6.

Recall that U(M) = S^{∗}(M)/(Φ(x) − x^{q}). Filtering U(M) by polynomial degree shows that U(M) has an associated graded object isomorphic in U(q) to S^{∗}(M)/(x^{q}). If q is prime, this identifies with S̄^{∗}(M), which is nilclosed if M is nilclosed, by case (3) of Theorem 1.2. Thus U(M) is nilclosed if M is, verifying the first condition in Theorem 2.6.

By construction, U(M) is the coequalizer in the category of commutative, unital algebras in U(q):

S^{∗}(ΦM) \begin{matrix}λ_{M} \\ ⟶ \\ ⟶ \\ i_{M} \end{matrix} S^{∗}(M) ⟶ U(M) → 0.

Since l is exact and preserves tensor products, we deduce that l(U(M ⊗W^{∗})) is the coequalizer in the category of commutative algebras in F(q):

S^{∗}(l(ΦM) ⊗ W^{∗}) \begin{matrix}l(λ_{M}) \\ ⟶ \\ ⟶ \\ l(i_{M}) \end{matrix} S^{∗}(l(M) ⊗ W^{∗}) ⟶ l(U(M ⊗ W^{∗})) → 0.

By Proposition 6.7 of [K1], l(λ_{M}) : l(ΦM) ⟶ l(M) can be identified with 1 : l(M) ⟶ l(M), and, under this identification, l(i_{M}) : S^{j}(l(ΦM)) ⟶ S^{qj}(M) corresponds to Φ : S^{j}(l(M)) ⟶ S^{qj}(l(M)). Thus l(U(M ⊗ W^{∗})) is the coequalizer in the category of commutative algebras in F(q):

S^{∗}(l(M) ⊗ W^{∗}) \begin{matrix}1 \\ ⟶ \\ ⟶ \\ Φ \end{matrix} S^{∗}(l(M) ⊗ W^{∗}) ⟶ l(U(M ⊗ W^{∗})) → 0.

The next lemma then allows us to conclude that l(U(M ⊗ W^{∗})) ≃ J_{W} ∘ l(M), thus verifying the second condition in Theorem 2.6, and finishing the proof of Theorem 1.5.

Lemma 4.1. J_{W} (V) is naturally the coequalizer in the category of F_{q}–algebras:

S^{∗}(V ⊗ W^{∗}) \begin{matrix}1 \\ ⟶ \\ ⟶ \\ Φ \end{matrix} S^{∗}(V ⊗ W^{∗}) ⟶ J_{W} (V) → 0.

This is just saying that J_{W} (V) = S^{∗}(V ⊗ W^{∗})/(x^{q} − x), an elementary, but useful, observation made in [K:I, Lemma 4.12].

Remark 4.2. If q is not prime, U(M) need not be nilclosed if M is. For a simple example when q = 4, let M = M_{I_{ξ}}, which has a generator in degrees 2^{2k+1}, k ≥ 0. Thus U(M) is concentrated in even degrees.

We claim that the nilclosure of U(M) has a nonzero class in degree 1. Our argument above shows that l(U(M)) = (J_{F_{4}})_{ξ}, so we are asserting that there is a nonzero map I ⟶ (J_{F_{4}})_{ξ}, or equivalently, a nonzero map I_{ξ} ⟶ J_{F_{4}}. But I_{ξ} ⊂ S^{2}, and S^{2} ⊂ J_{F_{4}}, so this is obvious.

Proof of Theorem 1.8. This is an argument similar to the proof of Theorem 1.5 above.

That M_{S^{∗}∘G} is an object in K^{2}(q), when G = I, was shown in [K:III, §5]. The same argument shows this is true for arbitrary G.

Now consider the natural map

M_{G} ⊗ M_{I} = Hom_{F(q)}(S_{∗}, G) ⊗ Hom_{F(q)}(I, S^{∗}) ⟶ Hom_{F(q)}(S_{∗}, S^{∗} ∘ G) = M_{S^{∗}∘G}

that sends (S_{m} ⟶^{x} G) ⊗ (I ⟶^{Φ^{k}} S^{q^{k}}) to the composite S_{m} ⟶^{x} G ⟶^{Φ^{k}∘1} S^{q^{k}} ∘ G. This is a map in U^{2}(q), and thus extends uniquely to a map in K^{2}(q),

U_{2}(M_{G} ⊗ M_{I}) ⟶ M_{S^{∗}∘G}.

We show that this is an isomorphism, when q is prime, by checking the two conditions in Theorem 2.6.

Firstly, to see that U_{2}(M ⊗ M_{I}) is nilclosed (with respect to the first grading) if M is nilclosed, one can filter U_{2}(M ⊗ M_{I}) = S^{∗}(M ⊗ M_{I})/(Φ(x) − x^{q}), obtaining an associated graded object isomorphic in U^{2}(q) to S̄^{∗}(M ⊗ M_{I}) (since q is prime). Ignoring the second grading, Theorem 1.2 shows this is nilclosed if M is.

By construction, U_{2}(M ⊗ M_{I}) is the coequalizer in the category of commutative, unital algebras in U^{2}(q):

S^{∗}(ΦM ⊗ ΦM_{I}) \begin{matrix}λ_{M} ⊗ λ_{M_{I}} \\ ⟶ \\ ⟶ \\ i_{M} ⊗ i_{M_{I}} \end{matrix} S^{∗}(M ⊗ M_{I}) ⟶ U_{2}(M ⊗ M_{I}) → 0.

Since l is exact and preserves tensor products, we deduce that l(U_{2}(M ⊗ M_{I})) is the coequalizer in the category of graded commutative algebras in F(q):

S^{∗}(l(M) ⊗ ΦM_{I}) \begin{matrix}λ_{M_{I}} \\ ⟶ \\ ⟶ \\ i_{M_{I}} \end{matrix} S^{∗}(l(M) ⊗ M_{I}) ⟶ l(U(M ⊗ M_{I})) → 0.

Thus, for any vector space V, l(U_{2}(M ⊗ M_{I}))(V) is naturally isomorphic to U(l(M)(V) ⊗ M_{I}) (where now we are using the U(q) stucture on M_{I}).

The next lemma then allows us to conclude that l(U_{2}(M ⊗ M_{I})) ≃ S^{∗} ∘ l(M), thus verifying the second condition in Theorem 2.6, and finishing the proof of Theorem 1.8.

Lemma 4.3. For any vector space W, U(W ⊗ M_{I}) is naturally isomorphic as an algebra to S^{∗}(W).

This is easily checked (and well known).

Remark 4.4. Analogous to the situation in Remark 4.2, if q is not prime, U_{2}(M ⊗ M_{I}) need not be nilclosed if M is. With q = 4 and M = M_{I_{ξ}}, U_{2}(M ⊗ M_{I}) is concentrated in bidegrees (m, n) with m even. However its nilclosure has a nonzero class in bidegree (1,2). This is equivalent to asserting that there is a nonzero map I ⟶ S_{ξ}^{2}, or equivalently, a nonzero map I_{ξ} ⟶ S^{2}. As noted before, I_{ξ} ⊂ S^{2}, and so this is obvious.

5. Some comments on the Frobenious twist

Let q = p^{s}, with p prime. Let A(q)^{∗} be the Hopf algebra dual to A(q). As noted in [K:III, §8], the q analogue of Milnor’s calculation [Mn] holds: as graded algebras

A(q)^{∗} ≃ F_{q}[ξ_{1}, ξ_{2}, …]

where the degree of ξ_{i} is q^{i} − 1. Using the monomial basis for A(q)^{∗}, let Q^{i} ∈ A(q) be dual to ξ_{i} ∈ A(q)^{∗}. By their construction, one sees that, on S^{∗}(F_{q}) = F_{q}[x], where x has degree 1, Q^{i}x = x^{q^{i}}, and also that Q^{i} acts as a derivation on A(q) algebras. Following the discussion in [AW, §5], we thus conclude that if V_{l} has basis x_{1}, …, x_{l}, then

Q^{i} = ∑_{j=1}^{l} a_{ij}∂/∂x_{j} : S^{∗}(V_{l}) ⟶ S^{∗}(V_{l}),

where a_{ij} = (x_{j})^{q^{i}}.

The proof of Lemma 5.9 of [AW] works with p replaced by q, to show that the matrix (a_{ij}), 1 ≤ i, j ≤ l, is nonsingular. It follows that, for any F_{q}–vector space V,

⋂_{i≥1} Ker {Q^{i} : S^{∗}(V) ⟶ S^{∗}(V)}

is the collection of all polynomials with all partial derivatives vanishing. But this is precisely S^{∗}(V)_{ξ}, with grading multiplied by p, embedded in S^{∗}(V) as the p^{th} powers. We have proved

Lemma 5.1. There is an exact sequence in F(q):

0 ⟶ S_{ξ}^{∗} ⟶ S^{∗} \underrightarrow{∏_{i} Q^{i}} S^{∗}.

There is, of course, a dual sequence involving S_{∗}.

This suggests the definition of a functor U_{ξ} : U(q) ⟶ U(q). As graded vector spaces,

U_{ξ}(M)_{n} = ⋂_{i≥1} Ker{Q^{i} : M_{pn} ⟶ M_{pn+q^{i}−1}}.

To see that this has a natural unstable A(q)–module structure, we again follow the lead of [AW].

The p^{th} power map A(q)^{∗} ⟶ A(q)^{∗} is a monomorphism of Hopf algebras. Dual to this is an epic map ϕ : A(q) ⟶ A(q), which divides degree by p in degrees divisible by p, and is 0 otherwise.

Lemma 5.2. Ker ϕ = A(q){Q^{1}, Q^{2}, … }, which is thus a two sided ideal.

Adams and Wilkerson’s proof of the q = p case, Lemma 2.5 of [AW], generalizes immediately.

The A(q) module structure on U_{ξ}(M) is then defined to correspond to the evident action of A(q)/A(q){Q^{1}, Q^{2}, … } on ⋂_{i≥1} Ker{Q^{i} : M ⟶ M}.

Since M_{G_{ξ^{−1}}} = Hom_{F(q)}(S_{∗}, G_{ξ^{−1}}) ≃ Hom_{F(q)}((S_{∗})_{ξ}, G), Lemma 5.1 immediately implies Theorem 1.11: U_{ξ}(M_{G}) ≃ M_{G_{ξ^{−1}}} .

Remark 5.3. One can imagine extensions of some of our results, where one inputs knowledge of M_{G_{ξ^{i}}}, for 0 ≤ i ≤ s − 1, to compute M_{F ∘G_{ξ^{i}}}, for all i. By the generic Steinberg Tensor Product Theorem [K:II, Theorem 5.23], case (4) of Theorem 1.2 here gives such a result, for any simple functor F ∈ F(q).

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Department of Mathematics, University of Virginia, Charlottesville, VA 22903