SCHUR–WEYL DUALITY OVER FINITE FIELDS

DAVID BENSON AND STEPHEN DOTY

Abstract. We prove a version of Schur–Weyl duality over finite fields. We prove that for any field k, if k has at least r + 1 elements, then Schur– Weyl duality holds for the rth tensor power of a finite dimensional vector space V. Moreover, if the dimension of V is at least r + 1, the natural map k𝔖_{r} → End_{GL(V)}(V^{⊗r}) is an isomorphism. This isomorphism may fail if dim_{k} V is not strictly larger than r.

1. Introduction

Let k be a field and let V be an n-dimensional vector space over k. We identify G = GL(V) with GL(n, k) and V with k^{n} as usual, by fixing a basis {v_{1}, …, v_{n}} of V. G acts on V in the natural way, and thus on the tensor product space V^{⊗r}. Moreover, V^{⊗r} admits an action of the symmetric group 𝔖_{r} permuting the tensor multiplicands. For σ ∈ 𝔖_{r}, the action is given by “place permutation”

σ(x_{1} ⊗ ⋯ ⊗ x_{r}) = x_{σ^{−1}(1)} ⊗ ⋯ ⊗ x_{σ^{−1}(r)},

extended linearly to k𝔖_{r}. One observes that the actions of G and 𝔖_{r} commute, so there are natural algebra homomorphisms

(1)  ρ: k𝔖_{r} → End_{G}(V^{⊗r});  φ: kG → End_{𝔖_{r}}(V^{⊗r}).

In case k = ℂ, Schur [7] proved that both maps are surjective, so the image of each map is the full centralizer algebra for the other action. In particular, the ring of invariants

End_{G}(V^{⊗r}) ≅ (V^{⊗r} ⊗ (V^{∗})^{⊗r})^{G}

is generated by the action of the symmetric group.

Assume that k is infinite. In that case, Theorem 4.1 of De Concini and Procesi [3] shows the surjectivity of ρ; in fact, their result is even more general. In the special case n ≥ r, the surjectivity of ρ was treated earlier in Lemma 3.1 of [2], where it is in fact an isomorphism. The surjectivity of φ for k infinite follows immediately from (2.4b) and (2.6c) of J. A. Green’s monograph [6].

Both authors would like to thank MSRI for its hospitality while this work was in progress.

The purpose of this paper is to discuss the question of the surjectivity of the maps ρ, φ in (1) in case k is a finite field. Our main results are obtained in Theorem 2.1 and Corollary 4.4. Theorem 2.1 states that ρ is an isomorphism provided that dim_{k} V > r, a result that holds independently of the field k. This generalizes a result of Carter and Lusztig. To obtain Corollary 4.4, which extends Schur–Weyl duality to the case where k has at least r + 1 elements, we need only the surjectivity of ρ, φ in case k is algebraically closed. Our approach also gives information about a version of Schur–Weyl duality in which GL(V) is replaced with SL(V). We note that Thrall [8] also obtained the surjectivity of ρ in case k is sufficiently large, although the stated bound on the cardinality of the field is incorrect. The methods of this paper are different than those of [8].

Several remarks are in order. First, in this paper we regard G = GL(n, k) as a linear group and not as a group scheme. To distinguish these objects notationally, we denote by GL_{n,k} the algebraic k-group scheme A ↦ GL(n, A) regarded as a representable functor from commutative k-algebras to groups. If the ground field k is understood we may write GL_{n} instead of GL_{n,k}. When k is infinite the distinction between GL(n, k) and GL_{n,k} makes little difference, since the algebraic group GL(n, \overline{k}) may be identified with the algebraic \overline{k}-group scheme GL_{n,\overline{k}}, and the group GL(n, k) of k-rational points in GL(n, \overline{k}) may be identified with the algebraic k-group scheme GL_{n,k}. However, in case k is finite the algebraic k-group scheme GL_{n,k} is a completely different object from the finite group G = GL(n, k) = GL_{n,k}(k).

Our second remark is that in the case when k is infinite a sensible way to study the kG-module V^{⊗r} is to replace the complicated infinite dimensional group algebra kG by its finite dimensional image ρ(kG). Obviously one does not lose information about the module structure by doing this. The algebra ρ(kG) = End_{𝔖_{r}}(V^{⊗r}) is isomorphic to the Schur algebra S_{k}(n, r) defined in [6]; its module category (when k is infinite) is equivalent to the category of homogeneous polynomial kG-modules of degree r.

Our last remark is on the relevance of polynomial functor technology, which in the case of infinite fields yields strong results by the methods of Friedlander and Suslin [5] and Franjou, Friedlander, Scorichenko and Suslin [4]. Referring to Section 2 of [4] and Section 1 of [5], the problem for finite fields is that polynomial functors and strict polynomial functors are different. Indeed, strict polynomial functors of a given degree always have finite projective resolutions, while polynomial functors sometimes do not. The methods of [4, 5] produce results for strict polynomial functors, which gives strong information for infinite fields. But for example if we look at Corollary 3.13 of [5] we might be tempted to suppose that our Theorem 2.1 can be extended to the case n = r, which it cannot; see the discussion preceding the theorem.

2. Generalization of the Carter–Lusztig result

If k is an infinite field, the map ρ described in the first paragraph of the introduction is surjective by the theorem of De Concini and Procesi. However, over a finite field this may no longer be true. For example, if k = 𝔽_{2} and n = r = 2 then G = GL(2, 2) ≅ 𝔖_{3}, V is a projective irreducible 𝔽_{2}G-module, and V^{⊗2} decomposes as a direct sum of V and a two dimensional module with a one dimensional fixed space; the latter is isomorphic to the projective cover of the trivial module. It follows that End_{G}(V^{⊗2}) is three dimensional while k𝔖_{2} is two dimensional, so ρ cannot be surjective, although it is injective. The situation is worse for n < r: we let the reader check that if k = 𝔽_{2}, n = 2 and r = 3 then ρ is neither injective nor surjective.

In Lemma 3.1 of [2] it is proved that for any infinite field k the map ρ is an isomorphism, provided that n ≥ r. However, a slightly weaker statement remains true over finite fields. We prove the following, which does not depend on whether k is finite or infinite.

Theorem 2.1. Let k be an arbitrary field. If n ≥ r then ρ: k𝔖_{r} → End_{G}(V^{⊗r}) is injective. If n ≥ r + 1 then ρ is an isomorphism.

Proof. Let v_{1}, …, v_{n} be a k-basis for V. The injectivity for n ≥ r follows from the fact that the images under 𝔖_{r} of v_{1} ⊗ ⋯ ⊗ v_{r} are linearly independent.

It remains to show that ρ is surjective when n ≥ r+1, so we assume we are in this case. Let ϕ ∈ End_{G}(V^{⊗r}). There exist uniquely determined scalars λ_{i_{1},...,i_{r}} (1 ≤ i_{j} ≤ r) such that

ϕ(v_{1} ⊗ ⋯ ⊗ v_{r}) = ∑_{i_{1},...,i_{r}} λ_{i_{1},...,i_{r}}v_{i_{1}} ⊗ ⋯ ⊗ v_{i_{r}}.

This information determines the value of ϕ on any basis element of the form v_{j_{1}} ⊗ ⋯ ⊗ v_{j_{r}} where there are no repetitions among j_{1}, …, j_{r}.

We claim that λ_{i_{1},...,i_{r}} = 0 unless i_{1}, …, i_{r} is a permutation of 1, …, r. First suppose some i_{j} is strictly greater than r. Consider the element g ∈ G defined by g(v_{i}) = v_{i} if i ≠ i_{j} and g(v_{i_{j}}) = v_{i_{j}} + v_{1}. Then g fixes v_{1} ⊗ ⋯ ⊗ v_{r}. Since ϕ commutes with the action of g, it follows that g fixes ϕ(v_{1} ⊗ ⋯ ⊗ v_{r}). So we have

λ_{i_{1},...,i_{j−1},1,i_{j+1},...,i_{r}} = λ_{i_{1},...,i_{j−1},1,i_{j+1},...,i_{r}} + λ_{i_{1},...,i_{j−1},i_{j},i_{j+1},...,i_{r}}.

Notice that in the left hand side of the above equation, as well as in the first term on the right hand side, not only is i_{j} replaced by 1, but also so is every i_{k} which happens to satisfy i_{k} = i_{j}. It follows from this equation that λ_{i_{1},...,i_{j−1},i_{j},i_{j+1},...,i_{r}} = 0.

Next, we suppose that all the indices i_{j} satisfy 1 ≤ i_{j} ≤ r, and we show that if two of the indices are equal then λ_{i_{1},...,i_{r}} = 0. By assumption, we have {i_{1}, …, i_{r}} ⊆ {1, …, r}. So if two of the indices are equal then we may choose j such that 1 ≤ j ≤ r and j ∉ {i_{1}, …, i_{r}}. Consider the element g ∈ G defined by g(v_{i}) = v_{i} if i ≠ j, and g(v_{j}) = v_{j} −v_{r+1}. This makes sense because n ≥ r+1. Then

g(v_{1} ⊗ ⋯ ⊗ v_{r}) = v_{1} ⊗ ⋯ ⊗ v_{r} − v_{1} ⊗ ⋯ ⊗ v_{j−1} ⊗ v_{r+1} ⊗ v_{j+1} ⊗ ⋯ ⊗ v_{r}.

The coefficient of v_{i_{1}} ⊗ ⋯ ⊗ v_{i_{r}} in g(ϕ(v_{1} ⊗ ⋯ ⊗ v_{r})) is λ_{i_{1},...,i_{r}} whereas the coefficient in ϕ(g(v_{1} ⊗⋯⊗v_{r})) is λ_{i_{1},...,i_{r}} −λ_{i_{1},...,i_{r}} = 0. Since these are supposed to be equal, it follows that λ_{i_{1},...,i_{r}} = 0.

Next, we show that the value of ϕ on v_{1} ⊗ ⋯ ⊗ v_{r} determines the value on v_{j_{1}} ⊗ ⋯ ⊗ v_{j_{r}}, where we allow repetitions among j_{1}, …, j_{r}. We do this by induction on m = r − |{j_{1}, …, j_{r}}|, the case m = 0 being clear. Permuting the indices if necessary, without loss of generality we may suppose that j_{r−1} = j_{r}. Choose ℓ such that 1 ≤ ℓ ≤ n and ℓ ∉ {j_{1}, …, j_{r}}. Consider the element g ∈ G defined by g(v_{i}) = v_{i} if i ≠ ℓ and g(v_{ℓ}) = v_{ℓ} + v_{j_{r}}. Then

g(v_{1} ⊗ ⋯ ⊗ v_{j_{r−1}} ⊗ v_{ℓ}) = v_{1} ⊗ ⋯ ⊗ v_{j_{r−1}} ⊗ v_{ℓ} + v_{1} ⊗ ⋯ ⊗ v_{j_{r−1}} ⊗ v_{j_{r}}.

Hence

ϕ(v_{1} ⊗ ⋯ ⊗ v_{j_{r}}) = g(ϕ(v_{1} ⊗ ⋯ ⊗ v_{j_{r−1}} ⊗ v_{ℓ})) − ϕ(v_{1} ⊗ ⋯ ⊗ v_{j_{r−1}} ⊗ v_{ℓ})

which is determined by the inductive hypothesis.

The conclusion of this argument is that the map ϕ is determined by the constants λ_{i_{1},...,i_{r}} with i_{1}, …, i_{r} a permutation of 1, …, r. Since every such set of constants is the effect of some element of k𝔖_{r}, ρ is surjective. □

3. Schur–Weyl duality for Schur algebras

In this expository section we summarize the known results concerning Schur– Weyl duality between the Schur algebra and symmetric group. Let us briefly recall the definition of the Schur algebra S_{k}(n, r), which makes sense for any commutative ring k. First, consider the polynomial algebra k[x_{ij}] in n^{2} commuting variables x_{ij}, 1 ≤ i, j ≤ n. This is naturally graded by regarding each generator as having degree 1; denote by A_{k}(n, r) the rth graded component. Thus k[x_{ij}] = ⨁_{r≥0} A_{k}(n, r). Now, k[x_{ij}] has a natural coalgebra structure with comultiplication Δ the algebra homomorphism given on generators by x_{ij} → ∑_{l=1}^{n} x_{il} ⊗ x_{lj}, and counit given by x_{ij} → δ_{ij} (Kronecker’s delta). Since comultiplication is an algebra homomorphism, k[x_{ij}] is a bialgebra. One easily checks that A_{k}(n, r) is a subcoalgebra of k[x_{ij}], for every r ≥ 0.

Definition 3.1. The Schur algebra is the algebra S_{k}(n, r) := A_{k}(n, r)^{∗}.

Note that it is a well known general fact that the linear dual C^{∗} = Hom_{k}(C, k) of any k-coalgebra C is an algebra in a natural way. The multiplication on C^{∗} comes from the comultiplication on C, by the rule ff′ = (f\overline{⊗}f′)Δ, for all f, f′ ∈ C^{∗}. (As usual, f\overline{⊗}f′ stands for the map a ⊗ a′ ↦ f(a)f′(a′).)

Now let I(n, r) denote the set {1, …, n}^{r} of r-tuples of elements of {1, …, n}. For i = (i_{1}, …, i_{r}), j = (j_{1}, …, j_{r}) in I(n, r) define

x_{i, j} := x_{i_{1}, j_{1}} ⋯ x_{i_{r−1}, j_{r−1}} x_{i_{r}, j_{r}}.

The symmetric group 𝔖_{r} acts naturally by place-permutation on I(n, r), and thus acts on I(n, r) × I(n, r), and x_{i, j} = x_{i′, j′} if and only if (i, j) and (i′, j′) lie in the same orbit. Thus if Ω is a set of representatives for the orbits then {x_{i, j} : (i, j) ∈ Ω} is a basis of A_{k}(n, r). Write (i, j) ∼ (i′, j′) if (i, j) and (i′, j′) lie in the same orbit. For i, j ∈ I(n, r) define ξ_{i, j} ∈ S_{k}(n, r) by

ξ_{i, j}(x_{i′, j′}) = \begin{cases}1 & if (i, j) ∼ (i′, j′) \\ 0 & otherwise. \end{cases}

Then of course ξ_{i, j} = ξ_{i′, j′} if and only if (i, j) ∼ (i′, j′), and {ξ_{i, j} : (i, j) ∈ Ω} is a basis of S_{k}(n, r). Now we define an action of S_{k}(n, r) on V^{⊗r} by the rule

ξ · v_{j} = ∑_{i∈I(n,r)} ξ(x_{i, j}) v_{i}

for any ξ ∈ S_{k}(n, r), where v_{i} := v_{i_{1}} ⊗ ⋯ ⊗ v_{i_{r}} for any i ∈ I(n, r). It is straightforward to check that this action commutes with the action of 𝔖_{r}. Then an elementary direct computation [6, Theorem (2.6c)] shows that

(2)  S_{k}(n, r) ≅ End_{𝔖_{r}}(V^{⊗r}).

The argument given there is valid for any commutative ring k.

Since the actions of S_{k}(n, r) and 𝔖_{r} on V^{⊗r} commute, one has also a natural algebra homomorphism

(3)  k𝔖_{r} → End_{S_{k}(n,r)}(V^{⊗r}).

Assume now that k is a field. If k is infinite then of course the map (3) is surjective, because φ(kG) ≅ S_{k}(n, r) and the action of kG induces the action of S_{k}(n, r), so End_{S_{k}(n,r)}(V^{⊗r}) = End_{G}(V^{⊗r}) and thus the map in (3) may be identified with the map ρ. But the surjectivity statement also holds in case k is finite, even though the isomorphism φ(kG) ≅ S_{k}(n, r) may fail. This has been shown by an elementary argument of Bryant [1, Lemma 2.4]. So the map (3) is surjective for any field k, finite or infinite. This is needed in the proof of Theorem 4.3. We note that in Bryant’s argument it is enough to assume the surjectivity of ρ in case k is algebraically closed.

4. Large enough finite fields

In this section we prove Schur–Weyl duality over finite fields which are sufficiently large, assuming the result is known in case the field is algebraically closed. The central issue is whether or not the Schur algebra S_{k}(n, r) is isomorphic to the subalgebra of End_{k}(V^{⊗r}) generated by the image of the G = GL(V)- action. This holds when k is infinite, but in fact we will show it holds whenever k has strictly more than r elements, as an application of the Chevalley group construction. Our argument is a refinement of an argument given in Section 3.1 of [2].

We will need to pay close attention to change of scalars in this section; in particular, we are going to write V_{k} instead of V in order to emphasize the field of scalars.

Consider the complex Lie group GL(n, ℂ) and its Lie algebra 𝔤 := 𝔤𝔩_{n}(ℂ). Let U = U_{ℂ}(𝔤) be the universal enveloping algebra (over ℂ) of 𝔤. One may regard U as the associative ℂ-algebra given by generators e_{ij}, 1 ≤ i, j ≤ n with the relations

e_{ij}e_{ab} − e_{ab}e_{ij} = δ_{aj}e_{ib} − δ_{ib}e_{aj}

for all 1 ≤ i, j, a, b ≤ n. (The e_{ij} correspond to the matrix units in the Lie algebra.) Let U′ be the subalgebra of U generated by the e_{ij} for i ≠ j. Then U′ may be identified with the universal enveloping algebra of 𝔰𝔩_{n}(ℂ).

Let V be an n-dimensional complex vector space, with basis {v_{1}, …, v_{n}}. Define an action of 𝔤 on V by e_{ij}v_{a} = δ_{aj}v_{i}. Identifying V with ℂ^{n} by means of the basis, one sees that the action is just by matrix multiplication. Thus V^{⊗r} is for any r ≥ 0 a 𝔤-module and hence a U-module. We regard V^{⊗r} as a U′-module by restriction. Let U′_{ℤ} be the subring of U generated by all divided powers

\frac{e_{ij}^{m}}{m!} (1 ≤ i ≠ j ≤ n, m ≥ 0)

and let U_{ℤ} be the subring generated by U′_{ℤ} and all elements of the form

\binom{e_{ii}}{m} := \frac{e_{ii}(e_{ii} − 1) ⋯ (e_{ii} − m + 1)}{m!} (1 ≤ i ≤ n, m ≥ 0).

Then U_{ℤ} (resp., U′_{ℤ}) is the Kostant ℤ-form of U (resp., U′). Let V_{ℤ} = ∑ ℤv_{i} and note that V_{ℤ} is stable under the action of U_{ℤ}. It follows that we have for any r ≥ 0 a U_{ℤ}-module structure on V_{ℤ}^{⊗r}, and hence also a U′_{ℤ}-module structure. Thus for any commutative ring k we have a k⊗_{ℤ}U_{ℤ}-module structure on k ⊗_{ℤ} V_{ℤ}^{⊗r}. Set U_{k} = k ⊗_{ℤ} U_{ℤ}, U′_{k} = k ⊗_{ℤ} U′_{ℤ}. Then U_{k} (resp., U′_{k}) is the algebra of distributions on the algebraic group scheme GL_{n,k} (resp., SL_{n,k}). We identify k ⊗_{ℤ} V_{ℤ}^{⊗r} with V_{k}^{⊗r} as usual, where V_{k} := k ⊗_{ℤ} V_{ℤ}. Note that U_{ℂ} ≅ U, U′_{ℂ} ≅ U′ as algebras, and V_{ℂ} ≅ V as vector spaces. Moreover, the U-module structure on V is isomorphic with the U_{ℂ}-module structure on V_{ℂ}, so that when k = ℂ these constructions recover the original objects.

So we have a U_{k}-module structure on V_{k}^{⊗r}. It is easy to check that on V_{k}^{⊗r} the action of U_{k} commutes with the place-permutation action of the symmetric group 𝔖_{r}. Thus we have algebra homomorphisms

(4)  ψ: U_{k} → End_{𝔖_{r}}(V_{k}^{⊗r});  ψ′: U_{k} → End_{𝔖_{r}}(V_{k}^{⊗r})

where the second map is just the restriction of the first. For our purposes, the following simple observation is key.

Lemma 4.1. For any m > r the elements

\frac{e_{ij}^{m}}{m!}, \binom{e_{ii}}{m} ∈ U (1 ≤ i ≠ j ≤ n, m ≥ 0)

act as zero on V^{⊗r}; hence for any m > r the elements

1 ⊗ \frac{e_{ij}^{m}}{m!}, 1 ⊗ \binom{e_{ii}}{m} ∈ U_{k} (1 ≤ i ≠ j ≤ n, m ≥ 0)

act as zero on V_{k}^{⊗r}.

Proof. The weight of a simple tensor v_{j} (for j = (j_{1}, …, j_{r}) ∈ I(n, r)) is the composition λ = (λ_{1}, …, λ_{n}) where λ_{i} counts the number of a ∈ [1, r] such that j_{a} = i, for each i. The weight λ is an n-part composition of r (i.e., λ_{1} + ⋯ + λ_{n} = r). One checks that e_{ij} for i ≠ j sends a simple tensor of weight λ onto

\begin{cases}a simple tensor of weight λ + ε_{i} − ε_{j} & if λ_{j} ≥ 1 \\ 0 & otherwise. \end{cases}

Here {ε_{1}, …, ε_{n}} is the standard basis of ℤ^{n}. Now by induction on m it follows that \frac{e_{ij}^{m}}{m!} takes a simple tensor of weight λ onto

\begin{cases}a simple tensor of weight λ + mε_{i} − mε_{j} & if λ_{j} ≥ m \\ 0 & otherwise. \end{cases}

When m > r, no composition λ of r satisfies λ_{j} ≥ m, no matter what j we look at. Hence \frac{e_{ij}^{m}}{m!} acts as zero on all simple tensors in V^{⊗r}. This proves the claim for i ≠ j.

It remains to show the \binom{e_{ii}}{m} also act as zero. For this note that e_{ii} acts as the scalar δ_{i, j_{1}} + ⋯ + δ_{i, j_{r}} on the simple tensor v_{j} = v_{j_{1}} ⊗ ⋯ ⊗ v_{j_{r}}. This scalar is an integer in the interval [0, r], so for any m > r the element \binom{e_{ii}}{m} acts as zero. □

Assume from now on that k is a field. For any t ∈ k, 1 ≤ i, j ≤ n we define elements E_{ij}(t) ∈ End(V_{k}) by the rule

E_{ij}(t) = 1 + tψ(1 ⊗ e_{ij}).

Then from the lemma it follows that for i ≠ j, E_{ij}(s)E_{ij}(t) = E_{ij}(s + t) for all s, t ∈ k. Thus E_{ij}(t) is for i ≠ j invertible, i.e., E_{ij}(t) ∈ GL(V_{k}). One easily checks that E_{ii}(t) ∈ GL(V_{k}) provided that t ≠ −1. Now it is clear that GL(V_{k}) is generated by the elements E_{ij}(t) (1 ≤ i ≠ j ≤ n, t ∈ k) along with the elements E_{ii}(t) (1 ≤ i ≤ n, −1 ≠ t ∈ k). Moreover, SL(V_{k}) is generated by the E_{ij}(t) (1 ≤ i ≠ j ≤ n, t ∈ k). Of course, GL(V_{k}) acts naturally on V_{k}, so GL(V_{k}) acts on V_{k}^{⊗r}; this is the action already considered in the introduction. By restriction, we also have an action of SL(V_{k}) on V_{k}^{⊗r}. As already noted, these actions commute with the place-permutation action of 𝔖_{r}, so we have algebra homomorphisms

(5)  φ: kGL(V_{k}) → End_{𝔖_{r}}(V_{k}^{⊗r});  φ′: kSL(V_{k}) → End_{𝔖_{r}}(V_{k}^{⊗r}).

Lemma 4.2. Let k be a field.

(i) If the order of k is strictly larger than r + 1, then φ is surjective if and only if ψ is surjective.

(ii) If the order of k is strictly larger than r, then φ′ is surjective if and only if ψ′ is surjective.

Proof. First, we claim that as operators on V_{k}^{⊗r} the maps ψ and φ are related by the following, for t ∈ k and i ≠ j:

(6)  φ(E_{ij}(t)) = ∑_{m=0}^{r} t^{m}ψ(1 ⊗ \frac{e_{ij}^{m}}{m!}).

This may be checked directly; see formula (25) of [2].

Choose r +1 distinct values t_{0}, t_{1}, …, t_{r} for t in k. (This is possible because k has more than r elements, by hypothesis.) This results in a system of equations

φ(E_{ij}(t_{0})) = 1 + t_{0}ψ(1 ⊗ e_{ij}) + t_{0}^{2}ψ(1 ⊗ \frac{e_{ij}^{2}}{2!} ) + ⋯ + t_{0}^{r}ψ(1 ⊗ \frac{e_{ij}^{r}}{r!} )

φ(E_{ij}(t_{1})) = 1 + t_{1}ψ(1 ⊗ e_{ij}) + t_{1}^{2}ψ(1 ⊗ \frac{e_{ij}^{2}}{2!} ) + ⋯ + t_{1}^{r}ψ(1 ⊗ \frac{e_{ij}^{r}}{r!} )
⋮  ⋮  ⋮
φ(E_{ij}(t_{r})) = 1 + t_{r}ψ(1 ⊗ e_{ij}) + t_{r}^{2}ψ(1 ⊗ \frac{e_{ij}^{2}}{2!} ) + ⋯ + t_{r}^{r}ψ(1 ⊗ \frac{e_{ij}^{r}}{r!} )

whose coefficient matrix is a Vandermonde matrix with nonzero determinant, hence invertible. Thus there exist scalars a_{ml} (0 ≤ m, l ≤ r) in k such that

(7)  ψ(1 ⊗ \frac{e_{ij}^{m}}{m!}) = ∑_{l=0}^{r} a_{ml}φ(E_{ij}(t_{l}))

for each m. Thus the generators ψ(1⊗ \frac{e_{ij}^{m}}{m!} ) of the image of ψ′ all lie in the image of φ on SL(V_{k}), proving that im φ′ ⊆ im ψ′. The reverse containment is clear from (6), so we have proved part (ii).

It remains to consider part (i). For this we must consider in addition the operators E_{ii}(t). We claim that

(8)  φ(E_{ii}(t)) = ∑_{m=0}^{r} t^{m}ψ(1 ⊗ \binom{e_{ii}}{m})

for any −1 ≠ t ∈ k, and any i. Again this may be checked directly, as in (25) of [2]. We can now repeat the above argument, and conclude that so long as k contains at least r + 1 elements different from −1, the generators ψ(1 ⊗ \binom{e_{ii}}{m}) of the image of ψ all lie in the image of φ on GL(V_{k}), so we have im ψ ⊆ im φ. The opposite inclusion is clear from (6) and (8), so part (i) is proved. □

Theorem 4.3. If the order of the field k is strictly larger than r, then both φ and φ′ are surjective.

Proof. For k = \overline{k} algebraically closed the statement is known. In that case, the surjectivity of φ′ follows from the surjectivity of φ, since (for k algebraically closed) GL(V_{k}) is generated by SL(V_{k}) and the scalar operators c id_{V_{k}} (0 ≠ c ∈ k). As pointed out by Carter and Lusztig [2, p. 209] this implies that the natural maps

(9)  ψ_{ℤ}: U_{ℤ} → End_{𝔖_{r}}(V_{ℤ}^{⊗r});  ψ′_{ℤ}: U′_{ℤ} → End_{𝔖_{r}}(V_{ℤ}^{⊗r})

are both surjective. The argument is as follows. First, apply the preceding lemma to conclude that ψ, ψ′ are surjective. Then apply the fact that if f: A → A′ is a homomorphism between two free ℤ-modules of finite rank, then f is surjective if and only if 1 ⊗ f: k ⊗_{ℤ}A → k ⊗_{ℤ}A′ is surjective for every algebraically closed field k.

Then we obtain the surjectivity of ψ = 1⊗ψ_{ℤ}, ψ′ = 1⊗ψ′_{ℤ} by right exactness of tensor product. Thus it follows that ψ, ψ′ are surjective for any field k. (In fact, this holds for any commutative ring.) Now we again apply the preceding lemma to conclude that φ′ is surjective provided |k| > r. Since φ′ is the restriction of φ this implies the surjectivity of φ as well. □

Corollary 4.4. If the field k has strictly more than r elements, then the natural maps φ: kGL(V_{k}) → End_{𝔖_{r}}(V_{k}^{⊗r}), ρ: k𝔖_{r} → End_{GL(V_{k})}(V_{k}^{⊗r}) are surjective. The same statement holds if GL(V_{k}) is replaced by SL(V_{k}).

Proof. The surjectivity of φ is the statement of the preceding theorem. It follows (setting G = GL(V_{k})) that End_{G}(V_{k}^{⊗r}) = End_{φ(kG)}(V_{k}^{⊗r}) = End_{S_{k}(n,r)}(V_{k}^{⊗r}) = ρ(k𝔖_{r}), where the last equality is by the surjectivity statement of (3). This proves the surjectivity of ρ, so the first claim is proved. The proof of the second claim is similar. □

We note the following improvement of [2, Theorem 3.1(i)].

Corollary 4.5. The natural map ℤ𝔖_{r} → End_{U_{ℤ}}(V_{ℤ}^{⊗r}) is surjective (for any n, r). The same statement holds with U′_{ℤ} in place of U_{ℤ}.

Proof. Let k = \overline{k} be algebraically closed. Then the corresponding map k𝔖_{r} → End_{U_{k}}(V_{k}^{⊗r}) is surjective, by (3), since

End_{U_{k}}(V_{k}^{⊗r}) = End_{ψ(U_{k})}(V_{k}^{⊗r}) ≅ End_{S_{k}(n,r)}(V_{k}^{⊗r}).

This implies the claim about U_{ℤ}. The argument with U′_{ℤ} in place of U_{ℤ} is similar, since we now know that ψ(U_{k}) = ψ(U′_{k}). □

Remarks 4.6. (a) From the preceding result and the first surjectivity statement in (9) it follows by the right exactness of tensor product that the natural maps

U_{k} → End_{𝔖_{r}}(V_{k}^{⊗r}); k𝔖_{r} → End_{U_{k}}(V_{k}^{⊗r})

are surjective, for any commutative ring k. Similarly one sees immediately that this holds with U′_{k} in place of U_{k}.

(b) Our proof of Theorem 4.3 is valid assuming only the surjectivity of the maps ρ, φ is known in case k is algebraically closed. That surjectivity is known by [3] and [6].

(c) If k = 𝔽_{2} and V_{k} has dimension 2, the dimension of kGL(V_{k}) is 6 while the dimension of End_{𝔖_{2}}(V_{k} ⊗ V_{k}) is 10. This shows that the bound |k| > r in Theorem 4.3 cannot be improved. It also shows that the bound in Theorem III of [8] is incorrect. Note that, apart from the bound, the first assertion in Theorem 4.3 is the same as Theorem III of [8].

5. An example

In this section, we take k = 𝔽_{q} with q = p^{e}, and once again write V for the n-dimensional space V_{k}. We will show that some restriction on q with respect to r is necessary in order for Schur–Weyl duality to hold.

Theorem 5.1. In case n = dim_{k} V = 2 and q = p is a prime, for r sufficiently large the map ρ: k𝔖_{r} → End_{GL(V)}(V^{⊗r}) is not surjective.

Proof. First we note that GL(V) = GL(2, p) is a finite group whose Sylow p-subgroups are cyclic. It follows that there are a finite number of isomorphism classes of indecomposable 𝔽_{p}GL(V)-modules. Let d be the sum of the dimensions of the indecomposable 𝔽_{p}GL(V)-modules, one from each isomorphism class. Then by a version of the pigeon hole principle, for all finite dimensional 𝔽_{p}GL(V)-modules M there is a summand consisting of at least (dim_{𝔽_{p}} M/d) − 1 copies of a single indecomposable. Thus dim_{𝔽_{p}} End_{GL(V)}(M) ≥ ((dim_{𝔽_{p}} M/d) − 1)^{2}. Applying this with M = V^{⊗r} we have

(10)  dim_{𝔽_{p}} End_{GL(V)}(V^{⊗r}) ≥ (\frac{2^{r}}{d} − 1)^{2} .

On the other hand, the image of 𝔽_{p}𝔖_{r} in End_{GL(V)}(V^{⊗r}) is the Temperley–Lieb algebra T_{r}, whose dimension is the Catalan number \binom{2r}{r}/(r+1). Using Stirling’s formula, asymptotically we get

(11)  dim_{𝔽_{p}} T_{r} ∼ \frac{2^{2r}}{\sqrt{π} r^{3/2}} .

Comparing (10) with (11), we see that for r large we have

dim_{𝔽_{p}} End_{GL(V)}(V^{⊗r}) > dim_{𝔽_{p}} T_{r}.  □

References

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Department of Mathematics, University of Aberdeen, Aberdeen AB24 3EA, Scotland

  E-mail address: bensondj@maths.abdn.ac.uk

Department of Mathematics, Loyola University Chicago, Chicago IL 60626, USA

  E-mail address: doty@math.luc.edu