Algebra Colloquium, 2 (1995), 363–372.

A NOTE ON QUANTIZED WEYL RECIPROCITY AT ROOTS OF UNITY

Jie Du School of Mathematics University of New South Wales Sydney, 2052, Australia

In this note we show that the epimorphism over ℚ(v) from the quantized enveloping algebra U to a q-Schur algebra can be taken up to any field via the quantized Kostant ℤ[v, v^{−1}]-form of U.

0. Introduction

Let n, r be integers ⩾ 1 and let V be a vector space of dimension n over ℂ. Classically, we have natural commuting actions of the general linear group GL(V) and the symmetric group 𝔖_{r} on the tensor space V^{⊗r}, and both actions satisfy the so-called Weyl reciprocity [W] (or the double centralizer property), which has been used by Schur and Weyl to link representations of 𝔖_{r} with representations of GL(V). This reciprocity is also true over any field by taking them up over ℤ in [CL]. That is, if 𝔄_{ℤ} is the Kostant ℤ-form of the universal enveloping algebra 𝔄 = 𝔄(𝔤𝔩_{n}(ℚ)) and V is ℤ-free and of rank n, then the natural homomorphisms

(0.a)  𝔄_{ℤ} → End_{ℤ[𝔖_{r}]}(V^{⊗r}), ℤ[𝔖_{r}] → End_{𝔄_{ℤ}}(V^{⊗r})

are surjective (see [CL, Theorem 3.1] where n ≥ r for the second surjection and [CP, 4.1] or [D,§2 Corollary] in general). The algebra End_{ℤ[𝔖_{r}]}(V^{⊗r}) and its extensions to fields are usually called Schur algebras.

Let A = ℤ[v, v^{−1}] be the Laurent polynomial ring in an indeterminate v, and let U be the quantized deformation of 𝔄 over ℚ(v) and U the quantized Kostant

Typeset by AMS-TEX 2 JIE DU

1991 Mathematics Subject Classification. 17B37, 16S50.

Key words and phrases. quantized Weyl reciprocity, quantized enveloping algebras, q-Schur algebras.

Research supported by ARC Large Grant.

A-form of U (defined in §2). For a specialization A → ℂ with q being the image of v, we put U_{q} = U ⊗ ℂ. Replacing the commuting actions of GL(V) and 𝔖_{r} on V^{⊗r} by commuting actions of the quantized enveloping algebra U_{q} and of the Hecke algebra H_{q} corresponding to 𝔖_{r}, we obtain the quantum version over ℂ of (0.a) with Schur algebras replaced by the so-called q-Schur algebras. M. Jimbo proved in [Ji] a quantized Weyl reciprocity for U_{q} and H_{q} over ℂ in which the involved parameter q is not a root of 1. That is, the quantum version over ℂ of (0.a) holds if q is not a root of 1. It is natural to expect that this quantized Weyl reciprocity is also true when the parameter q is a root of 1. In this note, we will prove that the quantum version over any field F of the first epimorphism in (0.a) is true for any q ≠ 0 in F (Theorem 3.6). To this end, it suffices to take them up over A as is in (0.a) and prove the homomorphism (1.2) from U to a q-Schur algebra over ℚ(v) sends the quantized Kostant A-form U of U onto the q-Schur algebra over A.

We shall use the explicit construction ([BLM]) of a surjective map α_{r} from U to the q-Schur algebra S_{n,r} over the field ℚ(v). This will be described in section 1. In section 2, the quantized Kostant A-form U of U is defined, based on the integral quantum group ′U of type A defined in [L1]. We shall see that U contains ′U and is free over ′U. The main result of this note will be proved in section 3.

We remark that the quantum integral version of the second epimorphism in (0.a) follows from our result and the result in [DJ,6.6] for n ≥ r. It is very likely that the result holds in general.

This note was written up while the author was visiting the University of Sydney in 1991. He would like to thank the University of Sydney for its hospitality. The author was told that I. Grojnowski has some unpublished results similar to ours.

1. Notation and Recollections

1.1 Let A be the quotient field ℚ(v) of A. Following Drinfeld and Jimbo, we define the algebra U to be the A-algebra with generators

E_{i,i+1}, E_{i+1,i}, K_{j}, K_{j}^{−1}(1 ≤ i ≤ n − 1, 1 ≤ j ≤ n)

A NOTE ON QUANTIZED WEYL RECIPROCITY AT ROOTS OF UNITY 3

and relations

K_{i}K_{j} = K_{j}K_{i}, K_{i}K_{i}^{−1} = 1
K_{i}E_{j,k} = v^{ϵ(i,j,k)}E_{j,k}K_{i},

where ϵ(i, j, k) = 1 if (i, j, k) = (a, a, a + 1) or (a, a, a − 1), and −1 if (i, j, k) = (a, a − 1, a) or (a, a + 1, a),

E_{ij}E_{kh} = E_{kh}E_{ij} for j ≠ k and i ≠ h

E_{i,i+1}E_{i+1,i} − E_{i+1,i}E_{i,i+1} = \frac{K_{i,i+1} − K_{i,i+1}^{−1}}{v − v^{−1}}

E_{ij}^{2}E_{jk} − (v + v^{−1})E_{ij}E_{jk}E_{ij} + E_{jk}E_{ij}^{2} = 0

where (i, j, k) = (a, a + 1, a + 2) or (a, a − 1, a − 2), and K_{i,i+1} = K_{i}K_{i+1}^{−1}.

The algebra U is the quantized deformation of the universal enveloping algebra of 𝔤𝔩_{n}. It is a Hopf algebra with comultiplication Δ defined on the generators by

\begin{aligned}Δ(E_{i,i+1}) & = E_{i,i+1} ⊗ K_{i,i+1} + 1 ⊗ E_{i,i+1}; \\ Δ(E_{i+1,i}) & = E_{i+1,i} ⊗ 1 + K_{i,i+1}^{−1} ⊗ E_{i+1,i}; \\ Δ(K_{i}) & = K_{i} ⊗ K_{i}. \end{aligned}

The algebra U contains the A-subalgebra ′U generated by

E_{i,i+1}, E_{i+1,i}, K_{i,i+1}, K_{i,i+1}^{−1} (1 ≤ i ≤ n − 1).

Note that ′U is the quantized deformation of the universal enveloping algebra of 𝔰𝔩_{n}. We denote by ′U ([L1]) the A-(lattice) subalgebra of ′U generated by

E_{ij}^{(a_{ij})} (a_{ij} ∈ ℕ, |i − j| = 1), K_{i,i+1}, K_{i,i+1}^{−1} (1 ≤ i ≤ n − 1),

where E_{ij}^{(m)} = E_{ij}^{m}/[m]! and [m]! = ∏_{i=1}^{m} \frac{v^{i}−v^{−i}}{v−v^{−1}} .

1.2 Let E be an A-free module with a given basis e_{1}, ⋯, e_{n}. We shall regard E = E ⊗ A as a U-module via the representation

ρ : U → End(E)

defined by ρ(E_{ij})e_{h} = δ_{jh}e_{i} if |i−j| = 1, and K_{j}e_{h} = v^{δ_{jh}}e_{h}. Thus, the tensor space E^{⊗r} is a U-module via ρ^{⊗r} ∘ Δ^{(r)} where Δ^{(r)} : U → U^{⊗r} is defined inductively by

4 JIE DU

Δ^{(2)} = Δ and Δ^{(i)} = ( id_{U}^{⊗(i−2)} ⊗ Δ) ∘ Δ^{(i−1)}. We denote by ρ_{r} : U → End(E^{⊗r}) the corresponding representation.

Let H be the Hecke algebra over A associated with the symmetric group 𝔖_{r} on r letters. We denote by {T_{w}}_{w∈𝔖_{r}} the standard basis for H. Define a right action of H on E^{⊗r} (see e.g. [GL,1.12] or [DPW,§5]) by

(e_{i_{1}}⊗⋯⊗e_{i_{r}})T_{(j,j+1)} =\{ \begin{matrix}ve_{i_{1}} ⊗⋯⊗e_{i_{j+1}} ⊗e_{i_{j}} ⊗⋯⊗e_{i_{r}}, & if i_{j} < i_{j+1}; \\ v^{2}e_{i_{1}} ⊗⋯⊗e_{i_{r}}, & if i_{j} = i_{j+1}; \\ (v^{2}−1)e_{i_{1}}⊗⋯⊗e_{i_{r}}+ \\ +ve_{i_{1}}⊗⋯⊗e_{i_{j+1}} ⊗e_{i_{j}} ⊗⋯⊗e_{i_{r}}, & if i_{j} > i_{j+1}; \end{matrix}

and put H = H ⊗ A. Let π be the corresponding representation. The following result is due to Jimbo [Ji] with a slight modification of the action for H (compare [Du, (2.c)] and [DPW,5.8]).

Theorem. E^{⊗r} is a U-H-bimodules and the quantized Weyl reciprocity holds: ρ_{r}(U) = End_{H}(E^{⊗r}), π(H) = End_{U}(E^{⊗r}).

1.3 A geometric setting for U has been discovered in [BLM] by embedding U into a “limit” algebra of certain finite dimensional algebras S_{n,r} which were realized in terms of the geometry of relative positions of pairs of n-step filtrations in an rdimensional space. For completeness, we restate the definition of S_{n,r} as follows.

Let V be a vector space of dimension r over a field F, and let F be the set of all n-step flags

V_{1} ⊂ V_{2} ⊂ ⋯ ⊂ V_{n} = V.

The group G = GL(V) acts naturally on F, and hence diagonally on X = F × F. For (f, f′) ∈ X,

(1.3.a)  f = (V_{1} ⊂ V_{2} ⊂ ⋯ ⊂ V_{n}), f′ = (V′_{1} ⊂ V′_{2} ⊂ ⋯ ⊂ V′_{n}),

let V_{0} = V′_{0} = 0 and

a_{ij} = dim(V_{i−1} + (V_{i} ∩ V′_{j})) − dim(V_{i−1} + (V_{i} ∩ V′_{j−1})).

One sees easily that the map (f, f′) ↦ (a_{ij}) induces a bijection from the set of G-orbits on X to the set Θ_{r} of n × n matrices with non-negative integer entries whose sum is r. We denote by O_{A} the G-orbit corresponding to A ∈ Θ_{r}, and let

co(A) = (∑_{i} a_{i1}, ⋯, ∑_{i} a_{in}) , ro(A) = (∑_{i} a_{1i}, ⋯, ∑_{i} a_{ni}) .

A NOTE ON QUANTIZED WEYL RECIPROCITY AT ROOTS OF UNITY 5

Clearly, if (f, f′) ∈ O_{A} as in (1.3.a) then

ro(A) = (dimV_{1}, dimV_{2}/V_{1}, ⋯, dimV_{n}/V_{n−1})

and

co(A) = (dimV′_{1}, dimV′_{2}/V′_{1}, ⋯, dimV′_{n}/V′_{n−1}).

For A, A′, A′′ ∈ Θ_{r}, there exist integers c_{0}, c_{1}, ⋯, c_{m} such that for any finite field F = 𝔽_{q} of q elements, we have (see [BLM, Proposition 1.2])

g_{A,A′,A′′,q} = #{f ∈ F | (f_{1}, f) ∈ O_{A}, (f, f_{2}) ∈ O_{A′}} = c_{0} + c_{1}q + ⋯ + c_{m}q^{m},

where (f_{1}, f_{2}) ∈ O_{A′′}. Put

g_{A,A′,A′′} = c_{0} + c_{1}v^{2} + ⋯ + c_{m}v^{2m}.

We now define the A-algebra S_{n,r} to be the free A-module with basis {e_{A}|A ∈ Θ_{r}} satisfying

e_{A}e_{A′} = ∑_{A′′∈Θ_{r}} g_{A,A′,A′′}e_{A′′},

and define S_{n,r} = S_{n,r} ⊗ A.

Let I(n, r) (resp. S_{r}) be the set of n × r matrices (resp. r × r matrices) with entries 0, 1 such that each column (resp. each column and each row) contains exactly one entry equal to 1. Define a q-tensor space T , following [GL] or [DJ], to be the A-submodule of S_{n,r} if n ≥ r or of S_{r,r} if n < r spanned by all e_{A}, A ∈ I(n, r). Here we identify the set I(n, r) with a subset of Θ_{r} by adding zero columns or rows to the matrices in I(n, r). Thus, T is an S_{n,r}-H-bimodule via the multiplication in S_{n,r} if n ≥ r, or in S_{r,r} if n < r. In the latter case (respectively, both cases) we identify S_{n,r} (resp. H) with a subalgebra of S_{r,r} (resp. the subalgebra generated by all e_{A}, A ∈ S_{r}) in a natural way. 1.4 Remark. The algebra S_{n,r} is isomorphic to the q-Schur algebra S_{q}(n, r) with q = v^{2} defined by Dipper and James ([DJ]). The basis {e_{A}} corresponds to the standard basis {ϕ_{λμ}^{d}} of S_{q}(n, r) ([loc.cit., 1.4]) in the following way: for λ, μ ∈ Λ(n, r) (the set of all compositions of r into n parts), and d ∈ D_{λ,μ} (the set of distinguished W_{λ} − W_{μ} double coset representatives for the Young subgroups W_{λ}, W_{μ} of W = 𝔖_{r}), the double coset W_{λ}dW_{μ} determines uniquely a matrix A = (a_{ij}) ∈ Θ_{r} (see [KJ, (1.3.10)]) with the properties:

6 JIE DU

(1) ro(A) = λ and co(A) = μ, and

(2) W_{λ}^{d} ∩ W_{μ} = W_{ν} where ν = (ν^{(1)}, ⋯, ν^{(n)}) with ν^{(i)} = (a_{1i}, ⋯, a_{ni}). Then the linear isomorphism sending e_{A} to ϕ_{λμ}^{d} gives the required isomorphism of algebras from S_{n,r} to S_{q}(n, r).

1.5 Let Θ be the set of n×n matrices with non-negative integer entries, σ : Θ → ℕ the map sending A = (a_{ij}) to σ_{A} = ∑ a_{ij}. Thus, σ^{−1}(r) = Θ_{r}. Denote by 𝔻 the set of all diagonal matrices in Θ and 𝔻_{r} = 𝔻 ∩ Θ_{r}.

Let [A] = v^{− dim O_{A}+dim pr_{1}(O_{A})}e_{A}, where pr_{1} : X → F is the projection on the first component. We define a homomorphism α_{r} : U → S_{n,r} = S_{n,r} ⊗ A, following [BLM], by the following rules:

\begin{matrix}E_{ij} & ↦ ∑_{D∈𝔻_{r−1}}[E_{ij} + D], |i − j| = 1, \\ K_{i} & ↦ ∑_{D∈𝔻_{r}} v^{d_{i}}[D], 1 ≤ i ≤ n \end{matrix}

where E_{ij} in the right hand side stands for the n × n matrix with 1 in the (i, j)- position and 0 elsewhere, and d_{i} is the i-th entry of D on the diagonal. Since [D][D′] = δ_{D,D′}[D] for any D, D′ ∈ 𝔻_{r}, any product K_{1}^{j_{1}} ⋯ K_{n}^{j_{n}} of the K_{i}’s is mapped to ∑_{D∈𝔻_{r}} v^{d_{1}j_{1}+···+d_{n}j_{n}}[D].

The homomorphism α_{r} (over the field A) is surjective by [BLM, 5.7]. Via α_{r}, T = T ⊗ A is an U-module. As pointed out in [GL, 1.13], T is isomorphic to E^{⊗r} as a U-H-bimodule via

(1.5.a)  v^{#{(s,s′)|s<s′,i_{s}<i_{s′} }}e_{(i_{1},··· ,i_{r})} ↦ e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{r}}.

Here we have identified the set I(n, r) with the set of sequences (i_{1}, ⋯, i_{r}) with i_{k} ∈ [1, n]: such a sequence corresponds to the matrix with 1 in the (i_{k}, k)-position and 0 elsewhere. In particular, we see as H-module that T is isomorphic to E^{⊗r}. Hence we have an algebra isomorphism

(1.5.b)  β_{r} : End_{H}(T ) → End_{H}(E^{⊗r}).

This fact has also been noticed in [Du, (2.d)].

2. The A-subalgebra U

2.1 Let U^{0} be the A-subalgebra of U generated by K_{i}, K_{i}^{−1}, and U^{0} the Asubalgebra of U^{0} generated by

K_{i}, K_{i}^{−1}, [ \begin{matrix}K_{i}; c \\ t \end{matrix} ] ,

where

[ \begin{matrix}K_{i}; c \\ t \end{matrix} ] = ∏_{s=1}^{t} \frac{K_{i}v^{c−s+1} − K_{i}^{−1}v^{−c+s−1}}{v^{s} − v^{−s}}

for 1 ≤ i ≤ n, t ∈ ℕ, c ∈ ℤ. One checks easily that these generators satisfy the relations (g1)–(g5) in [Lu3]. By [loc. cit.] we have the following.

Lemma. U^{0} is a free A-module with basis

K_{1}^{δ_{1}} ⋯ K_{n}^{δ_{n}} [ \begin{matrix}K_{1}; 0 \\ t_{1} \end{matrix} ] ⋯ [ \begin{matrix}K_{n}; 0 \\ t_{n} \end{matrix} ] ,

where δ_{i} ∈ {0, 1} and t_{i} ∈ ℕ. Moreover, U^{0} ⊗_{A} A = U^{0}.

2.2 From the lemma above and [L2, 4.5], an A-basis for U is given by

E^{(A,δ)} = ∏_{1≤j<i≤n} E_{ij}^{(a_{ij})} ∏_{i=1}^{n} (K_{i}^{δ_{i}} [ \begin{matrix}K_{i}; 0 \\ a_{ii} \end{matrix} ]) ∏_{1≤i<j≤n} E_{ij}^{(a_{ij})},

where A = (a_{ij}) runs through all n × n matrices with non-negative integer entries, δ = (δ_{i}) with δ_{i} ∈ {0, 1}, and E_{ij}^{(t)} for |i − j| ≥ 2 is defined by using the braid group action (see [L2, §4]).

We define U to be the A-free module spanned by all E^{(A,δ)}.

2.3 Proposition. U is an A-subalgebra of U.

Proof. From the commutation relation [L1, 4.1(a)], one needs only to show that

(2.3.a)  [ \begin{matrix}K_{i,i+1}^{−1}; 0 \\ t \end{matrix} ] ∈ U

for all i and t.

To prove this, we apply induction on t. One checks directly for t = 1 that

[ \begin{matrix}K_{i,i+1}^{−1}; 0 \\ 1 \end{matrix} ] = K_{i+1}^{−1} [ \begin{matrix}K_{i}; 0 \\ 1 \end{matrix} ] − K_{i}^{−1} [ \begin{matrix}K_{i+1}; 0 \\ 1 \end{matrix} ] ∈ U.

Assume now that (2.3.a) is true for all t < m. By [L1, (2.3)(g9)–(g10)], we have

[ \begin{matrix}K_{i,i+1}^{−1}; c \\ t \end{matrix} ] ∈ U

for all c ∈ ℤ and t < m. With this and using the commutation relation again, we see that (2.3.a) is true for t = m. So the result is proved. □

2.4 Corollary. Let ′U be defined as in (1.1). Then ′U is a subalgebra of U. □

2.5 Corollary. For any i, 1 ≤ i ≤ n, and t ∈ ℕ, we have

[ \begin{matrix}K_{i,i+1}^{−1}; 0 \\ t \end{matrix} ] ∈ U^{0}.

In particular, the subalgebra ′U^{0} of ′U generated by K_{i,i+1}, [ \begin{matrix}K_{i,i+1}^{−1}; 0 \\ t \end{matrix} ] is a subalgebra of U^{0}.

Proof. By displaying the bases for U^{0} and U, one sees easily that U ∩ U^{0} = U^{0}, and therefore, our conclusion holds. □

Note that a direct proof of (2.5) has been given by Takeuchi ([T,§3]). We also note that, from the commutative algebra point of view, (2.5) shows a more general result.

2.6 Lemma. Let T_{1}, T_{2} be two indeterminates and let U^{0}(X_{1}, X_{2}), for X_{1}, X_{2} ∈ {T_{1}, T_{2}, T_{1}T_{2}^{−1}}, be the A-subalgebra of A[T_{1}, T_{1}^{−1}; T_{2}, T_{2}^{−1}] generated by

X_{i}, X_{i}^{−1}, [ \begin{matrix}X_{i}; c \\ t \end{matrix} ] (i = 1, 2, t ∈ ℕ, c ∈ ℤ).

Then U^{0}(X_{1}, X_{2}) = U^{0}(T_{1}, T_{2}) for any X_{1}, X_{2} ∈ {T_{1}, T_{2}, T_{1}T_{2}^{−1}}, X_{1} ≠ X_{2}. □

This lemma implies the following.

2.7 Proposition. The algebra U is free over ′U. Proof. By an inductive argument, the previous lemma implies that U^{0} is the algebra generated by ′U^{0} and K_{1}, K_{1}^{−1}, [ \begin{matrix}K_{1}; 0 \\ t \end{matrix} ]. Now, applying Lemma 2.1 to K_{1}, K_{i,i+1}^{−1}, 1 ≤ i ≤ n − 1, we see that U^{0} is free over ′U^{0}. Since U = U^{−}U^{0}U^{+} and ′U = U^{−}′U^{0}U^{+}, where U^{+} (resp. U^{−}) is the subalgebra generated by the divided powers E_{h,h+1}^{(s)} (resp. E_{h+1,h}^{(t)}), the freeness of U over ′U follows immediately. □

A NOTE ON QUANTIZED WEYL RECIPROCITY AT ROOTS OF UNITY 9

2.8 Remark. (1) As we define′U in (1.1), we can define naturally an A-subalgebra ′′U of U to be the algebra generated by E_{ij}^{(t)}, |i − j| = 1, and K_{i}, K_{i}^{−1}. Thus, we have

′U ⊊ ′′U ⊊ U.

(2) The restrictions to U of the maps ρ_{r} (1.2) and α_{r} (1.5) give rise to two A-algebra maps from U to End_{H}(E^{⊗r}) and from U to S_{n,r}. The former is the quantized version of the first map in (0.a). We shall prove that the latter is surjective in (3.4), and consequently, so is the former as ρ_{r} = β_{r} ∘ α_{r} by (1.5.a-b).

3. The Main Result

Now, we are ready to establish the quantum version of the first epimorphism of (0.a).

3.1 Let Θ^{0} be the set of all A ∈ Θ such that all diagonal entries of A are zero. For each A ∈ Θ^{0}, let

(3.1.a)  A(0) = ∑_{\begin{matrix}D∈𝔻 \\ σ_{A}+σ_{D}=r \end{matrix}}[A + D],

where σ is defined in (1.5). Let α_{r} : U → S_{n,r} be as in (2.8). Thus, the image α_{r}(U) is a subalgebra of S_{n,r}, and E_{ij}(0) = α_{r}(E_{ij}) ∈ α_{r}(U).

3.2 Lemma. We have A(0) ∈ α_{r}(U) for A ∈ Θ^{0} ∩ Θ_{m} with m ≤ r.

Proof. Suppose A = (a_{ij}) and let

E^{(A)} = ∏_{1≤i≤h<j≤n} E_{h,h+1}^{(a_{ij})} ∏_{1≤j≤h<i≤n} E_{h+1,h}^{(a_{ij})}.

Then E^{(A)} ∈ U and, by [BLM, 3.9,5.5(c)], α_{r}(E^{(A)}) = A(0) + f, where f is an Alinear combinations of “lower terms” of the form K^{j}A′(0), where K^{j} = K_{1}^{j_{1}} ⋯ K_{n}^{j_{n}} for j = (j_{1}, ⋯, j_{n}). As α_{r}(K^{j}) = ∑_{D∈𝔻_{r}} v^{j_{1}d_{1}+···+j_{n}d_{n}}[D] (1.5), an inductive argument as in [loc.cit., p.674] shows that f ∈ α_{r}(U), and so A(0) ∈ α_{r}(U). □

Observing (3.1.a), we have obviously

ro(A + D) ≠ ro(A + D′) and co(A + D) ≠ co(A + D′)

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for any D ≠ D′ with σ_{A} + σ_{D} = σ_{A} + σ_{D′} = r. Note also that O(0) = ∑_{D∈𝔻_{r}}[D], where O is the zero matrix, is the identity element of S_{n,r}. These observations have proved the following lemma.

3.3 Lemma. Let A = (a_{ij}) ∈ Θ_{r} and A^{0} = A − diag(a_{11}, ⋯, a_{nn}) ∈ Θ^{0}. Then [A] = [D]A^{0}(0) = A^{0}(0)[D′] in S_{n,r} for some D, D′ ∈ 𝔻_{r} with co(D) = ro(A) and ro(D′) = co(A). □

3.4 Theorem. For each r > 0, the A-algebra homomorphism α_{r} : U → S_{n,r} is surjective.

Proof. By the previous lemmas, it suffices to prove [D] ∈ α_{r}(U) for all D ∈ 𝔻_{r}. Given a_{1}, ⋯, a_{m} ∈ ℕ with s_{m} = ∑_{j=1}^{m} a_{j} ≤ r, we define

D(a_{1}, ⋯, a_{m}) = ∑_{\begin{matrix}d_{m+1}+···+d_{n}=r−s_{m} \\ d_{i}∈ℕ \end{matrix}}[diag(a_{1}, ⋯, a_{m}, d_{m+1}, ⋯, d_{n})].

We are going to apply induction on m to show that D(a_{1}, ⋯, a_{m}) ∈ α_{r}(U), for all m, 1 ≤ m ≤ n, and a_{1}, ⋯, a_{m} as above. In particular, when m = n, D(a_{1}, ⋯, a_{n}) = [diag(a_{1}, ⋯, a_{n})] ∈ α_{r}(U) for all a_{1}, ⋯, a_{n} ∈ ℕ with s_{n} = r. Consequently, α_{r}(U) = S_{n,r}.

For any i, j, 1 ≤ i ≤ n, 0 ≤ j ≤ r, let

D_{i}(j) = ∑_{D∈𝔻_{r},d_{i}=j}[D].

(Here, as before, d_{i} is the i-th entry of D on the diagonal.) Note that D_{1}(j) = D(j) by the definition. By (1.5), we have

α_{r}(K_{i}) = ∑_{j=0}^{r} v^{j}D_{i}(j).

Note also that [D][D′] = δ_{D,D′}[D] for all D, D′ ∈ 𝔻_{r}. Thus, we have

(3.4.a)  \begin{aligned}α_{r} [ \begin{matrix}K_{i}; 0 \\ t \end{matrix} ] & = α_{r} (∏_{s=1}^{t} \frac{K_{i}v^{−s+1} − K_{i}^{−1}v^{s−1}}{v^{s} − v^{−s}} ) \\ & = ∏_{s=1}^{t} (∑_{j=0}^{r} \frac{v^{j−s+1} − v^{−(j−s+1)}}{v^{s} − v^{−s}} D_{i}(j)) \\ & = ∑_{j=0}^{r} (∏_{s=1}^{t} \frac{v^{j−s+1} − v^{−(j−s+1)}}{v^{s} − v^{−s}} ) D_{i}(j) \\ & = ∑_{j=0}^{r} [ \begin{matrix}j \\ t \end{matrix} ] D_{i}(j), \end{aligned}

where [ \begin{matrix}j \\ t \end{matrix} ] = ∏_{s=1}^{t} \frac{v^{j−s+1}−v^{−(j−s+1)}}{v^{s}−v^{−s}} is the Gaussian binomial coefficient. Now, we

are ready to prove the case when m = 1. Note first that [ \begin{matrix}j \\ t \end{matrix} ] = 0 for 0 ≤ j < t. By (3.4.a), we have

\{ \begin{matrix}α_{r}(1) = O(0) = D_{i}(0) + D_{i}(1) + ⋯ + D_{i}(r) \\ α_{r} [ \begin{matrix}K_{i}; 0 \\ t \end{matrix} ] = ∑_{j=t}^{r} [ \begin{matrix}j \\ t \end{matrix} ] D_{i}(j), (1 ≤ t ≤ r). \end{matrix}

Solving these equations for D_{i}(j), for j = 0, 1, ⋯, r, we obtain that D_{i}(j) ∈ α_{r}(U). In particular, D(j) = D_{1}(j) ∈ α_{r}(U). So the assertion for m = 1 is proved.

Assume now that (r ≥)m > 1. For given a_{1}, ⋯, a_{m−1} ∈ ℕ with s = s_{m−1} ≤ r, we have by induction

D(a_{1}, ⋯, a_{m−1}) ∈ α_{r}(U),

and

(3.4.b)  D(a_{1}, ⋯, a_{m−1}) = ∑_{j=0}^{r−s} D(a_{1}, ⋯, a_{m−1}, j).

It follows that

D(a_{1}, ⋯, a_{m−1})D_{m}(j) = \{ \begin{matrix}D(a_{1}, ⋯, a_{m−1}, j) & if 0 ≤ j ≤ r − s, \\ 0, & if j > r − s. \end{matrix}

Therefore, by (3.4.a) (3.4.c)

D(a_{1}, ⋯, a_{m−1}) α_{r}[ \begin{matrix}K_{m}; 0 \\ t \end{matrix} ] = ∑_{j=0}^{r−s} [ \begin{matrix}j \\ t \end{matrix} ] D(a_{1}, ⋯, a_{m−1}, j), (1 ≤ t ≤ r − s).

Solving the simultaneous equations (3.4.b) and (3.4.c), and noting that the expres-

sion on the L. H. S. of (3.4.b) and (3.4.c) are in α_{r}(U), we obtain that

D(a_{1}, ⋯, a_{m−1}, j) ∈ α_{r}(U)

for all j, 0 ≤ j ≤ r − s. □

As mentioned in (2.8), this result together with (1.5.b) and the isomorphism

S_{n,r} ≅ End_{H}(T ) imply the following.

3.5 Corollary. The algebra homomorphism ρ_{r} : U → End_{H}(E^{⊗r}) is surjective.

By now, we have almost obtained our main result. The notation M_{F} below

stands for M ⊗_{A} F.

3.6 Theorem. Let F be a field, and A → F a specialization such that the image

of v is nonzero in F. Put ρ_{r,F} = ρ_{r} ⊗1_{F}. Then ρ_{r,F} (U_{F}) = End_{H_{F}} (E_{F}^{⊗r}). If n ≥ r,

then H_{F} = End_{U_{F}} (E_{F}^{⊗r}).

Proof. The first equality follows from (3.5) and the second one follows from the

first equality and [DJ, 6.6(i)]. □

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E-mail: j.du@unsw.edu.au