PRESENTING SCHUR ALGEBRAS

STEPHEN DOTY AND ANTHONY GIAQUINTO

Abstract. Motivated by work of R.M. Green, we obtain a presentation of Schur algebras (both the classical and quantized versions) in terms of generators and relations. The presentation is compatible with the usual presentation of the (quantized or classical) enveloping algebra of 𝔤𝔩_{n}. As a result, we obtain a new “integral” basis for Schur algebras which is a subset of Kostant’s basis of the integral form of the enveloping algebra (or its q-analogue). Projection onto an appropriate component gives a new “integral” basis and a presentation for the Hecke algebra, compatible with the basis and presentation for the Schur algebra. Finally, we find a second presentation of Schur algebras which is similar to Luzstig’s modified form of the quantized enveloping algebra.

Introduction

Let R be a commutative ring. The (classical) Schur algebra S_{R}(n, d) may be defined as the algebra End_{Σ_{d}}(V_{R}^{⊗d}) of linear endomorphisms on the dth tensor power of an n-dimensional free R-module V_{R} commuting with the action of the symmetric group Σ_{d}, acting by permutation of the tensor places (see [Gr]). The Schur algebras form an important class of quasi-hereditary algebras, and, when R is an infinite field, the family of Schur algebras {S_{R}(n, d)}_{d≥0} determines the polynomial representation theory of the general linear group GL(V_{R}).

All these algebras, for various R, can be constructed from the integral form S_{ℤ}(n, d) by base change, since S_{R}(n, d) ≅ R ⊗_{ℤ} S_{ℤ}(n, d). Fixing our base field at ℚ (we could use any field of characteristic zero), we henceforth write S(n, d) for S_{ℚ}(n, d), V for V_{ℚ}.

In this paper, we give an alternative construction of Schur algebras, as follows. First we obtain a presentation of S(n, d) by generators and relations (Theorem 1.1). This presentation is compatible with Serre’s presentation of the universal enveloping algebra U = U(𝔤𝔩_{n}) of the Lie algebra 𝔤𝔩_{n} of n × n matrices. Then we construct S_{ℤ}(n, d) as the precise analogue of the Kostant ℤ-form U_{ℤ}, and we obtain new bases for S_{ℤ}(n, d) (Theorem 1.3). We also obtain a second presentation of S(n, d) by generators and relations (Theorem 1.4) which is closely related to Lusztig’s construction (see [Lu3]) of the modified form U̇ of the quantized enveloping algebra U.

Date: March 31, 2002.

Our approach is based on the classic “double-centralizer” theory of Schur [Sc] (and its quantization). The group GL(V) acts on V^{⊗d} by means of the natural action in each tensor factor, and by differentiating this action on tensors we obtain an action of U on V^{⊗d}. These actions obviously commute with the action of the symmetric group Σ_{d}. So we have representations

U ⟶ End(V^{⊗d}) ⟵ ℚΣ_{d}

induced from the commuting actions. Then Schur’s result is that the image of each representation is precisely the commuting algebra for the action of the other algebra. In particular, S(n, d) is the image of the representation U → End(V^{⊗d}). It is a very natural problem to ask for an efficient generating set for the kernel of this representation. By solving this problem we obtain the presentation of Theorem 1.1. We note that, in the quantum case, the analogue of the surjective map U → S(n, d) was studied by R.M. Green [RG1, RG2], who described a basis for the kernel.

In the quantum case one replaces in the above setup ℚ by ℚ(v) (v an indeterminate), V by an n-dimensional ℚ(v)-vector space V, Σ_{d} by the corresponding Hecke algebra H = H(Σ_{d}), and U by the Drinfeld-Jimbo quantized enveloping algebra U = U(𝔤𝔩_{n}). Then the resulting commuting algebra, S(n, d), is known as the q-Schur algebra, or quantized Schur algebra. It appeared first in work of Dipper and James [DJ1, DJ2], and, independently, Jimbo [Ji]. Dipper and James showed that the q-Schur algebras determine the representation theory of the finite general linear groups in non-describing characteristic. (Note that one should replace their parameter q by v^{2} to make the correspondence with our version of S(n, d).) In [BLM] a geometric realization of S(n, d) was given. In [Du] the [BLM] approach was reconciled with the Dipper-James approach; moreover, it was shown in that paper that S(n, d) may be identified with the image of the map

U ⟶ End(V^{⊗d}).

In the quantum situation, one replaces ℤ by A = ℤ[v, v^{−1}]. In this case the analogue of S_{ℤ}(n, d) is a certain A-form S_{A}(n, d) in S(n, d).

Our results in the quantum case are almost exact analogues of the results in the classical case, although the proofs are sometimes more difficult. First we obtain a presentation of S(n, d) by generators and relations (Theorem 2.1). This presentation is compatible with the usual presentation of the quantized enveloping algebra U (over ℚ(v)) corresponding to 𝔤𝔩_{n}. Then we construct S_{A}(n, d) as the analogue of Lusztig’s A-form U_{A}, and we obtain new bases for S_{A}(n, d) (Theorem 2.3). Finally, we have a second presentation of S(n, d) by generators and relations (Theorem 2.4) which is closely related to the algebra U̇. Upon specializing v to 1, the presentation of Theorem 2.4 coincides with the presentation of Theorem 1.4. (This does not apply to Theorem 2.1 in relation to Theorem 1.1.)

In the final section we give some applications of our results to the Borel Schur algebras and Hecke algebras. In particular, in Proposition 11.3 we obtain a simple basis for the Borel Schur algebras which is a subset of the integral basis obtained (in Theorem 2.3) for the entire Schur algebra. We also get a new basis for the Hecke algebra (realized as a subalgebra of the Schur algebra) which is a subset of our basis in Theorem 2.3, and an integral presentation of it. We also write out some examples in the final section.

In rank 1 we have more precise results than in this paper, obtained by different arguments [DG1]. The results in the current paper were summarized in the announcement [DG3].

Acknowledgements. The authors have benefitted from discussions with T. Brüstle, K. Erdmann, R.M. Green, S. König, R. Marsh. We are grateful to the referee for useful suggestions.

1. Main results: classical case

Let Φ be the root system of type A_{n−1}: Φ = {ε_{i} − ε_{j} | 1 ≤ i ≠ j ≤ n}. Here the ε_{i} form the standard orthonormal basis of the euclidean space ℝ^{n}. Let ( , ) denote the inner product on this space and define α_{i} = ε_{i} − ε_{i+1}. Then {α_{1}, …, α_{n−1}} is a base of simple roots and Φ^{+} = {ε_{i} − ε_{j} | i < j} is the corresponding set of positive roots.

We now give a precise statement of our main results in the classical case. The proofs are contained in sections 3–6. The first result describes a presentation by generators and relations of the Schur algebra over the rational field ℚ.

Theorem 1.1. Over ℚ, the Schur algebra S(n, d) is isomorphic with the associative algebra (with 1) on the generators e_{i}, f_{i} (1 ≤ i ≤ n−1), and H_{i} (1 ≤ i ≤ n) with relations

(R1)  H_{i}H_{j} = H_{j}H_{i}
(R2)  e_{i}f_{j} − f_{j}e_{i} = δ_{ij}(H_{j} − H_{j+1})

H_{i}e_{j} − e_{j}H_{i} = (ε_{i}, α_{j})e_{j}, H_{i}f_{j} − f_{j}H_{i} = −(ε_{i}, α_{j})f_{j} (R3)

e_{i}^{2}e_{j} − 2e_{i}e_{j}e_{i} + e_{j}e_{i}^{2} = 0 (|i − j| = 1)
e_{i}e_{j} − e_{j}e_{i} = 0 (otherwise)

(R4)

f_{i}^{2}f_{j} − 2f_{i}f_{j}f_{i} + f_{j}f_{i}^{2} = 0 (|i − j| = 1)
f_{i}f_{j} − f_{j}f_{i} = 0 (otherwise)

(R5)

(R6)  H_{1} + H_{2} + ⋯ + H_{n} = d
(R7)  H_{i}(H_{i} − 1) ⋯ (H_{i} − d) = 0.

Note that the enveloping algebra U = U(𝔤𝔩_{n}) is the algebra on the same generators but subject only to the relations (R1)–(R5), and U(𝔰𝔩_{n}) is isomorphic with the subalgebra of U generated by the e_{i}, f_{i}, H_{i}−H_{i+1} (1 ≤ i ≤ n − 1).

Next we introduce the root vectors x_{α} (α ∈ Φ), which may be defined inductively as follows. Write α = ε_{i} − ε_{j} and assume that i < j. If j − i = 1 then α = α_{i} and we set x_{α} = e_{i}, x_{−α} = f_{i}. If j − i > 1 then we inductively set

x_{α} = e_{i}x_{α−α_{i}} − x_{α−α_{i}}e_{i}, x_{−α} = x_{−α+α_{i}}f_{i} − f_{i}x_{−α+α_{i}}.

The linear span of the set {x_{α}} ∪ {H_{i}} is a subspace of U isomorphic with the Lie algebra 𝔤𝔩_{n} under the Lie bracket given by [x, y] = xy−yx, and the x_{α} correspond to the usual root vectors in 𝔤𝔩_{n}.

Remark 1.2. The defining relation (R6) can be used to rewrite one of the H_{i}’s in terms of the others. Fix an integer i_{0} with 1 ≤ i_{0} ≤ n and set

G = {x_{α} | α ∈ Φ} ∪ {H_{i} | i ≠ i_{0}}

and fix an arbitrary ordering for this set. We conjecture that S(n, d) has a ℚ-basis consisting of all monomials in G (with specified order) of total degree not exceeding d. This basis would be an analogue of the Poincare-Birkhoff-Witt (PBW) basis of U.

Our next result constructs the integral Schur algebra S_{ℤ}(n, d) in terms of the generators given above. We need more notation. For B = (B_{i}) in ℕ^{n}, we write

H_{B} = ∏_{i=1}^{n} \binom{H_{i}}{B_{i}}

where \binom{H_{i}}{m} = H_{i}(H_{i} − 1) … (H_{i} − m + 1)/(m!) (m ≥ 1), \binom{H_{i}}{0} = 1.

Let Λ(n, d) be the subset of ℕ^{n} consisting of those λ ∈ ℕ^{n} satisfying |λ| = d (here |λ| = ∑ λ_{i}); this is the set of n-part compositions of d.

Given λ ∈ Λ(n, d) we set 1_{λ} = H_{λ}. We will show that the collection {1_{λ}} as λ varies over Λ(n, d) forms a set of pairwise orthogonal idempotents in S_{ℤ}(n, d) which sum to the identity element.

For m ∈ ℕ and α ∈ Φ, set x_{α}^{(m)} = x_{α}^{m}/(m!). Any product of elements of the form

x_{α}^{(r)}, \binom{H_{i}}{s} (r, s ∈ ℕ, α ∈ Φ, 1 ≤ i ≤ n),

taken in any order, will be called a Kostant monomial. Note that the set of Kostant monomials is multiplicatively closed. We define a function χ (content function) on Kostant monomials by setting

χ(x_{α}^{(m)}) = m ε_{max(i,j)}, χ(\binom{H_{i}}{m}) = 0

where α = ε_{i} −ε_{j} (i ≠ j), and by declaring that χ(XY) = χ(X)+χ(Y) whenever X, Y are Kostant monomials.

For A ∈ ℕ^{Φ^{+}} we set |A| = ∑_{α∈Φ^{+}} A(α). For A, C ∈ ℕ^{Φ^{+}} we write

e_{A} = ∏_{α∈Φ^{+}} x_{α}^{(A(α))}, f_{C} = ∏_{α∈Φ^{+}} x_{α}^{(C(−α))}

where the products in e_{A} and f_{C} are taken relative to any two fixed orders on Φ^{+}.

The first part of the next result shows that S_{ℤ}(n, d) is the analogue in S(n, d) of Kostant’s ℤ-form U_{ℤ} in U.

Theorem 1.3. The integral Schur algebra S_{ℤ}(n, d) is the subring of S(n, d) generated by all divided powers e_{i}^{(m)}, f_{i}^{(m)}. Moreover, each of the disjoint unions

(a)  Y_{+} = ⋃_{λ} {e_{A}1_{λ}f_{C} | χ(e_{A}f_{C}) ⪯ λ}
(b)  Y_{−} = ⋃_{λ} {f_{A}1_{λ}e_{C} | χ(f_{A}e_{C}) ⪯ λ},

as λ varies over Λ(n, d), and where ⪯ denotes the componentwise partial ordering on ℕ^{n}, is a ℤ-basis of S_{ℤ}(n, d).

Finally, we have another presentation of the Schur algebra by generators and relations. This presentation has the advantage that it possesses a quantization of the same form, in which we can specialize v to 1 to recover the classical version.

Theorem 1.4. The ℚ-algebra S(n, d) is the associative algebra (with 1) given by generators 1_{λ} (λ ∈ Λ(n, d)), e_{i}, f_{i} (1 ≤ i ≤ n − 1) subject to the relations

1_{λ}1_{μ} = δ_{λ,μ}1_{λ}, ∑_{λ∈Λ(n,d)} 1_{λ} = 1 (R1′)

e_{i}1_{λ} = \begin{cases}1_{λ+α_{i}}e_{i} & if λ + α_{i} ∈ Λ(n, d) \\ 0 & otherwise \end{cases}

f_{i}1_{λ} = \begin{cases}1_{λ−α_{i}}f_{i} & if λ − α_{i} ∈ Λ(n, d) \\ 0 & otherwise \end{cases}

1_{λ}e_{i} = \begin{cases}e_{i}1_{λ−α_{i}} & if λ − α_{i} ∈ Λ(n, d) \\ 0 & otherwise \end{cases}

1_{λ}f_{i} = \begin{cases}f_{i}1_{λ+α_{i}} & if λ + α_{i} ∈ Λ(n, d) \\ 0 & otherwise \end{cases}

(R2′)

e_{i}f_{j} − f_{j}e_{i} = δ_{ij} ∑_{λ∈Λ(n,d)}(λ_{j} − λ_{j+1})1_{λ} (R3′)

along with the Serre relations (R4), (R5).

2. Main results: quantum case

Our main results in the quantum case are similar in form to those in the classical case. Proofs are given in sections 7–10. The first result describes a presentation by generators and relations of the quantized Schur algebra over the rational function field ℚ(v).

Theorem 2.1. Over ℚ(v), the q-Schur algebra S(n, d) is isomorphic with the associative algebra (with 1) with generators E_{i}, F_{i} (1 ≤ i ≤ n − 1), K_{i}, K_{i}^{−1} (1 ≤ i ≤ n) and relations

K_{i}K_{j} = K_{j}K_{i}, K_{i}K_{i}^{−1} = K_{i}^{−1}K_{i} = 1 (Q1)

E_{i}F_{j} − F_{j}E_{i} = δ_{ij} \frac{K_{i}K_{i+1}^{−1} − K_{i}^{−1}K_{i+1}}{v − v^{−1}} (Q2)

K_{i}E_{j} = v^{(ε_{i},α_{j})}E_{j}K_{i}, K_{i}F_{j} = v^{−(ε_{i},α_{j})}F_{j}K_{i} (Q3)

E_{i}^{2}E_{j} − (v + v^{−1})E_{i}E_{j}E_{i} + E_{j}E_{i}^{2} = 0 (|i − j| = 1)
E_{i}E_{j} − E_{i}E_{j} = 0 (otherwise)

(Q4)

F_{i}^{2}F_{j} − (v + v^{−1})F_{i}F_{j}F_{i} + F_{j}F_{i}^{2} = 0 (|i − j| = 1)
F_{i}F_{j} − F_{j}F_{i} = 0 (otherwise)

(Q5)

K_{1}K_{2} ⋯ K_{n} = v^{d} (Q6)

(K_{i} − 1)(K_{i} − v)(K_{i} − v^{2}) ⋯ (K_{i} − v^{d}) = 0. (Q7)

We note that the quantized enveloping algebra U = U(𝔤𝔩_{n}) is the algebra on the same set of generators, subject only to the relations (Q1)– (Q5). Moreover, the quantized enveloping algebra U(𝔰𝔩_{n}) is isomorphic with the subalgebra of U generated by all E_{i}, F_{i}, K_{i}K_{i+1}^{−1} (1 ≤ i ≤ n−1).

We have q-analogues of the root vectors in 𝔤𝔩_{n}, which can be defined as follows. For α ∈ Φ^{+}, write α = ε_{i} − ε_{j} for i < j. If j − i = 1 then set X_{α} = E_{i}, X_{−α} = F_{i}. For j − i > 1 we inductively set (following Xi [Xi1, §5.6])

X_{α} = v^{−1}E_{i}X_{α−α_{i}} − X_{α−α_{i}}E_{i}, X_{−α} = vX_{−α+α_{i}}F_{i} − F_{i}X_{−α+α_{i}}.

Our notation differs from that in [Xi1] where the elements X_{α} and X_{−α} are denoted E_{i,j−1} and F_{j,i+1}, resp. Up to scalar multiplication by units in A, the elements X_{α} (α ∈ Φ) first appeared in Jimbo’s paper [Ji].

Remark 2.2. Fix an integer i_{0} with 1 ≤ i_{0} ≤ n and write ℕ_{i_{0}}^{n} for the set of B ∈ ℕ^{n} such that B_{i_{0}} = 0. We conjecture that S(n, d) has a ℚ(v)-basis consisting of all monomials of the form

∏_{α∈Φ^{+}} X_{α}^{A(α)} ∏_{i≠i_{0}} K_{i}^{B_{i}} ∏_{α∈Φ^{+}} X_{−α}^{C(α)} (A, C ∈ ℕ^{Φ^{+}}, B ∈ ℕ_{i_{0}}^{n})

of total degree not exceeding d, where the products of powers of X_{α}, X_{−α} are taken with respect to arbitrary fixed orders on Φ^{+}. This basis would be the analogue of the PBW-type basis of U, given in Lusztig [Lu2, Proposition 1.13].

Our next result constructs the A-form S_{A}(n, d) in terms of the gen-

erators given above. We write [\begin{matrix}K_{i} \\ t \end{matrix}] short for [\begin{matrix}K_{i}; 0 \\ t \end{matrix}], where (following

Lusztig) we define

[\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}} , [\begin{matrix}K_{i}; c \\ 0 \end{matrix}] = 1

for t ≥ 1, c ∈ ℤ. For B in ℕ^{n}, we write

K_{B} = ∏_{i=1}^{n} [\begin{matrix}K_{i} \\ B_{i} \end{matrix}] .

Given λ ∈ Λ(n, d) we set 1_{λ} = K_{λ}. Just as in the classical case, the collection {1_{λ}} forms a set of pairwise orthogonal idempotents in S_{A}(n, d) which sum to the identity element.

For m ∈ ℤ let [m] denote the quantum integer [m] = (v^{m}−v^{−m})/(v− v^{−1}) and set

[m]! = [m][m − 1] ⋯ [1], [0]! = 1

[\begin{matrix}c \\ m \end{matrix}] = \frac{[c][c − 1] ⋯ [c − m + 1]}{[m]!}, [\begin{matrix}c \\ 0 \end{matrix}] = 1

for c ∈ ℤ, m ≥ 1. The q-analogues of the divided powers of root vectors are defined by X_{α}^{(m)} = X_{α}/[m]!. The Kostant monomials in this situation are products of elements of the form

X_{α}^{(r)}, [\begin{matrix}K_{i} \\ s \end{matrix}] , K_{i}^{±1} (r, s ∈ ℕ, α ∈ Φ, 1 ≤ i ≤ n),

taken in any order. As before, the set of Kostant monomials is multiplicatively closed. By analogy with the classical case, χ is defined by

χ(X_{α}^{(m)}) = m ε_{max(i,j)}, χ([\begin{matrix}K_{i} \\ m \end{matrix}]) = χ(K_{i}^{±1}) = 0

where α = ε_{i} − ε_{j} ∈ Φ, and by declaring that χ(XY) = χ(X) + χ(Y) whenever X, Y are Kostant monomials. For A, C ∈ ℕ^{Φ^{+}} we write

E_{A} = ∏_{α∈Φ^{+}} X_{α}^{(A(α))}, F_{C} = ∏_{α∈Φ^{+}} X_{−α}^{(C(α))}

where the products in E_{A} and F_{C} are taken relative to any two specified orderings on Φ^{+}.

Theorem 2.3. The integral q-Schur algebra S_{A}(n, d) is the subring of S(n, d) generated by all quantum divided powers E_{i}^{(m)}, F_{i}^{(m)}, along with

the elements [\begin{matrix}K_{i} \\ m \end{matrix}]. Moreover, each of the sets

(a)  Y_{+} = ⋃_{λ} {E_{A}1_{λ}F_{C} | χ(E_{A}F_{C}) ⪯ λ}
(b)  Y_{−} = ⋃_{λ} {F_{A}1_{λ}E_{C} | χ(F_{A}E_{C}) ⪯ λ},

as λ ranges over Λ(n, d), forms an A-basis of S_{A}(n, d).

We conjecture that the elements [\begin{matrix}K_{i} \\ m \end{matrix}] lie within the subring gener-

ated by the E_{i}^{(m)}, F_{i}^{(m)}, in which case we would obtain the more precise analogue of Theorem 1.3.

Finally, we have another presentation of the q-Schur algebra by generators and relations. These relations are similar to relations that hold for the modified form U̇ of U (see [Lu3, Chap. 23]). This presentation has the advantage that upon specializing v to 1, we recover the classical version given in Theorem 1.4.

Theorem 2.4. The algebra S(n, d) is the associative algebra (with 1) given by generators 1_{λ} (λ ∈ Λ(n, d)), E_{i}, F_{i} (1 ≤ i ≤ n − 1) subject to the relations

1_{λ}1_{μ} = δ_{λ,μ}1_{λ}, ∑_{λ∈Λ(n,d)} 1_{λ} = 1 (Q1′)

E_{i}1_{λ} = \begin{cases}1_{λ+α_{i}}E_{i} & if λ + α_{i} ∈ Λ(n, d) \\ 0 & otherwise \end{cases}

F_{i}1_{λ} = \begin{cases}1_{λ−α_{i}}F_{i} & if λ − α_{i} ∈ Λ(n, d) \\ 0 & otherwise \end{cases}

1_{λ}E_{i} = \begin{cases}E_{i}1_{λ−α_{i}} & if λ − α_{i} ∈ Λ(n, d) \\ 0 & otherwise \end{cases}

1_{λ}F_{i} = \begin{cases}F_{i}1_{λ+α_{i}} & if λ + α_{i} ∈ Λ(n, d) \\ 0 & otherwise \end{cases}

(Q2′)

E_{i}F_{j} − F_{j}E_{i} = δ_{ij} ∑_{λ∈Λ(n,d)}[λ_{j} − λ_{j+1}]1_{λ} (Q3′)

along with the q-Serre relations (Q4), (Q5).

3. The algebra T

From now on we hold n and d fixed, and set S = S(n, d). We define an algebra T = T (n, d) (over ℚ) by the generators and relations of Theorem 1.1. Since U is the algebra on the same generators but subject only to relations (R1)–(R5), we have a surjective quotient map U → T (mapping generators onto generators). Eventually we shall show that T ≃ S, which will prove Theorem 1.1.

Lemma 3.1. Under the representation U → End(V^{⊗d}) the images of the H_{i} satisfy the relations (R6) and (R7). Moreover, the relation (R7) is the minimal polynomial of (the image of) H_{i} in End(V^{⊗d}).

Proof. Relation (R6) is trivial in the case d = 1, from which the general case follows since each H_{i} acts as a derivation of V^{⊗d}. The relation (R7) follows from the fact (which can be verified by induction on d) that the eigenvalues of the diagonal operators H_{i} are 0, 1, …, d. The proof is complete.

As we know, S = S(n, d) is the image of the representation U → End(V^{⊗d}) mentioned in the introduction. From the above lemma it follows that this surjection U → S factors through T. Because T, S are homomorphic images of U, any relations between generators holding in U will automatically carry over to T, S. We will not distinguish notationally between the generators or root vectors for U, T, or S.

Recall the triangular decomposition of U, that the multiplication map U^{−} ⊗U^{0} ⊗U^{+} ⟶^{≈} U is an isomorphism of vector spaces, where U^{+} (resp., U^{−}) is the subalgebra of U generated by the e_{i} (resp., f_{i}) and U^{0} is the subalgebra of U generated by all H_{i}. Thus U = U^{−}U^{0}U^{+}. From this we obtain a similar triangular decomposition of T:

T = T^{−}T^{0}T^{+} (3.2)

where T^{+}, T^{−}, T^{0} are defined to be the images of U^{+}, U^{−}, U^{0} under the quotient mapping U → T.

We also have similar factorizations over ℤ. Setting U_{ℤ}^{+}, U_{ℤ}^{−}, U_{ℤ}^{0} to be, respectively, the intersection of U^{+}, U^{−}, U^{0} with the Kostant ℤ-

form U_{ℤ} (the subring of U generated by all e_{i}^{(m)}, f_{i}^{(m)}, \binom{H_{i}}{m}), we have

the factorization U_{ℤ} = U_{ℤ}^{−}U_{ℤ}^{0}U_{ℤ}^{+}, which immediately induces similar equalities

T_{ℤ} = T_{ℤ}^{−}T_{ℤ}^{0}T_{ℤ}^{+} (3.3)

where the various subalgebras are defined in the obvious way as appropriate homomorphic images of U_{ℤ}^{+}, U_{ℤ}^{−}, U_{ℤ}^{0}.

Since U_{ℤ}^{+} (resp., U_{ℤ}^{−}) is the ℤ-subalgebra of U generated by the x_{α}^{(m)} for α ∈ Φ^{+} (resp., α ∈ Φ^{−}) and m ∈ ℕ, the same statement applies to T_{ℤ}^{+} (resp., T_{ℤ}^{−}) in relation to T. Moreover, U_{ℤ}^{0} is the ℤ-subalgebra

of U generated by the \binom{H_{i}}{m} for 1 ≤ i ≤ n and m ∈ ℕ, so T_{ℤ}^{0} is the

ℤ-subalgebra of T generated by the same elements.

Now we investigate the structure of the algebra T^{0}. We start with the algebra U^{0}, which is isomorphic with the polynomial ring ℚ[H_{1}, …, H_{n}] in n commuting indeterminates H_{1}, …, H_{n}. By the remarks following Lemma 3.1 we have surjections U → T → S. Let S^{0} be the image in S of U^{0} under the map U → S. Clearly we have surjections U^{0} → T^{0} → S^{0} obtained from U → T → S by restriction.

Proposition 3.4. Define an algebra T′ = U^{0}/I^{0} where I^{0} is the ideal in U^{0} generated by elements H_{i}(H_{i} − 1) ⋯ (H_{i} − d) (1 ≤ i ≤ n) and H_{1} + ⋯ + H_{n} − d.

(a) We have an algebra isomorphism T′ ≅ T^{0}.

(b) The set {1_{λ} | λ ∈ Λ(n, d)} is a ℚ-basis for T^{0} and a ℤ-basis for T_{ℤ}^{0}; moreover, this set is a set of pairwise orthogonal idempotents which add up to 1.

(c) H_{B} = 0 for any B ∈ ℕ^{n} such that |B| > d.

Proof. Consider first the algebra T̃′ defined to be the quotient of U^{0} by the ideal generated only by the elements H_{i}(H_{i} − 1) ⋯ (H_{i} − d) (1 ≤ i ≤ n). Since each of the relations in T̃′ is a polynomial in just one of the variables, we have the factorization

T̃′ ≅ ℚ[H_{1}]/(p(H_{1})) ⊗ ⋯ ⊗ ℚ[H_{n}]/(p(H_{n}))

where p(X) = X(X − 1) ⋯ (X − d). By the Chinese Remainder Theorem applied to each tensor factor we obtain from the above isomorphisms (products denote direct products)

T̃′ ≅ ∏_{i=0}^{d} (ℚ[H_{1}]/(H_{1} − i)) ⊗ ⋯ ⊗ ∏_{i=0}^{d} (ℚ[H_{n}]/(H_{n} − i))

and by rearranging the order of factors we obtain

≅ ∏_{0≤μ_{1},... ,μ_{n}≤d} (ℚ[H_{1}]/(H_{1} − μ_{1}) ⊗ ⋯ ⊗ ℚ[H_{n}]/(H_{n} − μ_{n}))

≅ ∏_{0≤μ_{1},... ,μ_{n}≤d} ℚ[H_{1}, …, H_{n}]/(H_{1} − μ_{1}, …, H_{n} − μ_{n}).

The isomorphism is realized by the map which sends a polynomial f(H_{1}, …, H_{n}) in T̃′ to the element (f(μ_{1}, …, μ_{n}))_{0≤μ_{1},... ,μ_{n}≤d} of the direct product. Since T′ is isomorphic with T̃′/(H_{1} + ⋯ + H_{n} − d), we deduce from the above that

T′ ≅ ∏_{μ∈Λ(n,d)} ℚ[H_{1}, …, H_{n}]/(H_{1} − μ_{1}, …, H_{n} − μ_{n})

and this isomorphism (which we denote by ϕ) is realized by the map sending f(H_{1}, …, H_{n}) to the element (f(μ_{1}, …, μ_{n}))_{μ∈Λ(n,d)}.

Thus, given any λ ∈ Λ(n, d), we have

ϕ(1_{λ}) = ( \binom{μ_{1}}{λ_{1}} ⋯ \binom{μ_{n}}{λ_{n}} )_{μ∈Λ(n,d)} = (δ_{λμ})_{μ∈Λ(n,d)}

Since the vector on the right of the above equality consists of zeros and ones, with precisely one nonzero entry, and since ϕ is an isomorphism, it follows that the various 1_{λ} are orthogonal idempotents which add up to the identity of T′.

Now let I be the ideal in U generated by elements H_{i}(H_{i}−1) ⋯ (H_{i}− d) (1 ≤ i ≤ n) and H_{1} + ⋯ + H_{n} − d. Then by definition T ≅ U/I. The canonical quotient map U → U/I induces, upon restriction to U^{0}, a map U^{0} → U/I. The image of this map is T^{0} and its kernel is U^{0} ∩I, so T^{0} ≅ U^{0}/(U^{0}∩I). Clearly I^{0} ⊂ U^{0}∩I. Thus we obtain the following sequence of algebra surjections

T′ = U^{0}/I^{0} ⟶ U^{0}/(U^{0} ∩ I) ⟶^{≈} T^{0} ⟶ S^{0} (3.5)

where the last map is obtained by Lemma 3.1, and the middle map is actually an isomorphism. By the above we see that the dimension of T′ is the cardinality of the set Λ(n, d). This is well-known to equal the dimension of the zero part S^{0} of the Schur algebra S = S(n, d). It follows that all the surjections above are algebra isomorphisms. This proves parts (a) and (b).

To prove part (c), suppose that B ∈ ℕ^{n} such that |B| > d. Then for each μ ∈ Λ(n, d), there exists i with μ_{i} < b_{i}. Thus ϕ(H_{B}) = 0 and H_{B} must be 0 since the map ϕ is an isomorphism. The proof is complete.

The next result is obtained by a similar argument.

Proposition 3.6. Let 1 ≤ i ≤ n, b ∈ ℕ, λ ∈ Λ(n, d), and B ∈ ℕ^{n}. We have the following identities in the algebra T^{0}:

H_{i}1_{λ} = λ_{i}1_{λ}, \binom{H_{i}}{b}1_{λ} = \binom{λ_{i}}{b}1_{λ} (a)

H_{B} 1_{λ} = λ_{B} 1_{λ}, where λ_{B} = ∏_{i} \binom{λ_{i}}{B_{i}} (b)

H_{B} = ∑_{λ} λ_{B} 1_{λ} (c)

where the sum in part (c) is carried out over all λ ∈ Λ(n, d).

Proof. We apply the isomorphism ϕ from the proof of the preceding proposition to the product on the left-hand-side of (a), obtaining

ϕ(\binom{H_{i}}{b} 1_{λ}) = ϕ(\binom{H_{i}}{b}\binom{H_{1}}{λ_{1}} ⋯ \binom{H_{n}}{λ_{n}}).

The right-hand side of the above equality gives the vector

( \binom{μ_{i}}{b}\binom{μ_{1}}{λ_{1}} ⋯ \binom{μ_{n}}{λ_{n}} )_{μ∈Λ(n,d)} = ( \binom{μ_{i}}{b}δ_{μλ} )_{μ∈Λ(n,d)}

which is the same as \binom{λ_{i}}{b}ϕ(1_{λ}) since from the preceding proof ϕ(1_{λ}) =

(δ_{μλ})_{μ}. Since ϕ is an isomorphism, part (a) is proved.

Part (b) follows immediately from part (a). Then by the result of part (b) we obtain the equalities H_{B} = H_{B} ·1 = H_{B} ∑_{λ} 1_{λ} = ∑_{λ} λ_{B}1_{λ}, proving part (c).

We write ℕ_{i_{0}}^{n} for the set of B = (B_{i}) ∈ ℕ^{n} such that B_{i_{0}} = 0.

Corollary 3.7. For any fixed choice of i_{0} (1 ≤ i_{0} ≤ n) the set {H_{B} | B ∈ ℕ_{i_{0}}^{n}, |B| ≤ d} is a ℚ-basis for T^{0} and a ℤ-basis for T_{ℤ}^{0}.

Proof. The sets Λ(n, d) and {B ∈ ℕ_{i_{0}}^{n}, |B| ≤ d} have the same cardinality. For instance, the map

λ → λ − λ_{i_{0}}ε_{i_{0}}

is bijective between the sets in question, with inverse map

B → B + (d − |B|)ε_{i_{0}}.

Thus, to prove the result it is enough to show that the set {H_{B} | B ∈ ℕ_{i_{0}}^{n}, |B| ≤ d} spans T_{ℤ}^{0}. This can be deduced by considering a certain order (depending on i_{0}) on each of these sets.

Fixing 1 ≤ i_{0} ≤ n, we order the set Λ(n, d) by declaring that λ precedes λ′ if λ_{i_{0}} < λ′_{i_{0}}, or if λ_{i_{0}} = λ′_{i_{0}} and there exists an index l ≠ i_{0} such that λ_{j} ≥ λ′_{j} for all j ∈ {1, … l} − {i_{0}}. Similarly, we order the set {B ∈ ℕ_{i_{0}}^{n}, |B| ≤ d} by declaring that B precedes B′ if there exists an index l ≠ i_{0} such that b_{j} ≥ b′_{j} for all j ∈ {1, … l}−{i_{0}}. With these orderings, it follows from part (c) of the preceding proposition that the matrix of coefficients obtained by expressing the H_{B}’s in terms of the 1_{λ}’s is lower unitriangular. To see this, observe that for any given B, with corresponding λ = B +(d−|B|)ε_{i_{0}}, any μ which succeeds λ in the above order satisfies μ_{B} = 0, and moreover λ_{B} = 1. It follows that these equations can be inverted over ℤ and so every 1_{λ} is expressible as a ℤlinear combination of elements from the set {H_{B} | B ∈ ℕ_{i_{0}}^{n}, |B| ≤ d}. This proves that this set spans T_{ℤ}^{0} and, by our remarks above, it must therefore be a ℚ-basis, and thus is linearly independent over ℤ, and hence also a ℤ-basis. The proof is complete.

The next step is to find spanning sets for the plus part T^{+} and minus part T^{−} of T. For this we use the following result.

Proposition 3.8. For any α ∈ Φ, λ ∈ Λ(n, d) we have the commutation formulas

x_{α}1_{λ} = \begin{cases}1_{λ+α}x_{α} & if λ + α ∈ Λ(n, d) \\ 0 & otherwise \end{cases}

and similarly

1_{λ}x_{α} = \begin{cases}x_{α}1_{λ−α} & if λ − α ∈ Λ(n, d) \\ 0 & otherwise. \end{cases}

Proof. Write α = ε_{i} − ε_{j} with i ≠ j. From the defining relation (R3) and the definition (see §1) of x_{α} we have

H_{l}x_{α} = x_{α}(H_{l} + (ε_{l}, α)). (3.9)

From this we obtain equalities

x_{α}1_{λ} = x_{α}\binom{H_{1}}{λ_{1}} ⋯ \binom{H_{n}}{λ_{n}} = (\binom{H_{i} − 1}{λ_{i}}\binom{H_{j} + 1}{λ_{j}} ∏_{l≠i,j} \binom{H_{l}}{λ_{l}}) x_{α}.

Multiplying on the left by H_{i}/(λ_{i} + 1) and then commuting with x_{α} yields the equality

x_{α} \frac{H_{i} + 1}{λ_{i} + 1}1_{λ} = \frac{H_{i}}{λ_{i} + 1} (\binom{H_{i} − 1}{λ_{i}}\binom{H_{j} + 1}{λ_{j}} ∏_{l≠i,j} \binom{H_{l}}{λ_{l}}) x_{α}

which by Proposition 3.6(a) simplifies to give

x_{α}1_{λ} = (\binom{H_{i}}{λ_{i} + 1}\binom{H_{j} + 1}{λ_{j}} ∏_{l≠i,j} \binom{H_{l}}{λ_{l}}) x_{α}

which (if λ_{j} > 0) can be rewritten in the form

x_{α}1_{λ} = \binom{H_{i}}{λ_{i} + 1} (\binom{H_{j}}{λ_{j}} + \binom{H_{j}}{λ_{j} − 1}) ∏_{l≠i,j} \binom{H_{l}}{λ_{l}}x_{α}.

The first summand on the right-hand-side of the preceding equality vanishes, by Proposition 3.4(c). This proves the first part of the proposition in the case λ_{j} > 0. In case λ_{j} = 0 the right-hand-side is zero. This proves the first part of the proposition. The proof of the second part is similar.

Corollary 3.10. We have in T the equalities x_{α}^{d+1} = 0 for all α ∈ Φ.

Proof. By iterating the result of the preceding proposition we see that x_{α}^{d+1}1_{λ} = 0 for any α ∈ Φ and any λ ∈ Λ(n, d), since it is clear that λ + (d + 1)α does not belong to Λ(n, d). Thus we have equalities

x_{α}^{d+1} = x_{α}^{d+1} · 1 = x_{α}^{d+1} ∑_{λ∈Λ(n,d)} 1_{λ} = ∑_{λ∈Λ(n,d)} x_{α}^{d+1}1_{λ} = 0

and this proves the claim.

4. Straightening

We need the following variants of the notion of content. We define functions χ_{L} and χ_{R} (left and right content) on Kostant monomials by setting

\begin{matrix}χ_{R}(x_{α}^{(m)}) = m ε_{j}, χ_{L}(x_{α}^{(m)}) = m ε_{i}, \\ χ_{R}(\binom{H_{i}}{m}) = χ_{L}(\binom{H_{i}}{m}) = 0 \end{matrix} (4.1)

where α = ε_{i} − ε_{j}, and again using the rule χ_{L}(XY) = χ_{L}(X) + χ_{L}(Y) (similarly for χ_{R}) whenever X and Y are Kostant monomials. Note that for A, C ∈ ℕ^{Φ^{+}} we have

χ(e_{A}) = χ_{R}(e_{A}), χ(f_{C}) = χ_{L}(f_{C}). (4.2)

From this it follows immediately that

χ(e_{A}f_{C}) = χ_{R}(e_{A}) + χ_{L}(f_{C}). (4.3)

From Proposition 3.8 it follows that for any A, C ∈ ℕ^{Φ^{+}}, λ ∈ Λ(n, d) we have equalities

e_{A}1_{λ}f_{C} = 1_{λ′}e_{A}f_{C} = e_{A}f_{C}1_{λ′′} (4.4)

where λ′ = λ + ∑_{α} A(α) α, λ′′ = λ + ∑_{α} C(α) α (both sums over Φ^{+}). Moreover, we have equalities

∑_{α∈Φ^{+}} A(α) α = ∑_{α=ε_{i}−ε_{j}; i<j} A(α)(ε_{i} − ε_{j})
= χ_{L}(e_{A}) − χ_{R}(e_{A})
= −χ_{L}(f_{A}) + χ_{R}(f_{A})

from which it follows that

λ′ = λ − χ_{R}(e_{A}) + χ_{L}(e_{A}), λ′′ = λ + χ_{R}(f_{C}) − χ_{L}(f_{C}). (4.5)

Lemma 4.6. χ(e_{A}1_{λ}f_{C}) ⪯ λ ⇔ χ_{L}(1_{λ′}e_{A}f_{C}) ⪯ λ′ ⇔ χ_{R}(e_{A}f_{C}1_{λ′′}) ⪯ λ′′.

Proof. By the definitions we have χ(e_{A}1_{λ}f_{C}) = χ(e_{A}f_{C}), with similar equalities for χ_{R}, χ_{L}. From equation (4.3) and the above we have the following equivalences

χ(e_{A}1_{λ}f_{C}) ⪯ λ ⇔ χ_{R}(e_{A}) + χ_{L}(f_{C}) ⪯ λ
⇔ χ_{L}(f_{C}) ⪯ λ − χ_{R}(e_{A})
⇔ χ_{L}(f_{C}) + χ_{L}(e_{A}) ⪯ λ − χ_{R}(e_{A}) + χ_{L}(e_{A})
⇔ χ_{L}(1_{λ′}e_{A}f_{C}) ⪯ λ′.

This establishes the first equivalence of the lemma. The second equivalence is established by a similar argument.

By similar reasoning one can obtain similar equivalences in which the f_{A} precede the e_{C}.

It follows from the preceding lemma that the set Y_{+} (see Theorem 1.3) can be rewritten in either of the forms

\begin{aligned}⋃_{λ′} \{1_{λ′}e_{A}f_{C} & | χ_{L}(e_{A}f_{C}) ⪯ λ′\} \\ & = ⋃_{λ′′} \{e_{A}f_{C}1_{λ′′} | χ_{R}(e_{A}f_{C}) ⪯ λ′′\} \end{aligned} (4.7)

with a similar statement applying to the set Y_{−}.

The subspace of U spanned by the H_{i} and x_{α} is isomorphic with the Lie algebra 𝔤𝔩_{n} of n × n matrices. The isomorphism is determined by sending e_{i} → e_{i,i+1}, f_{i} → e_{i+1,i}. Here the notation e_{ij} stands for the matrix with all entries 0, except for the (i, j)th entry, which is 1. It follows from the definition of x_{α} (see §1) that the isomorphism carries H_{i} to e_{ii} and x_{α} to e_{ij} when α = ε_{i} − ε_{j} ∈ Φ. From this one can now easily verify that (for α = ε_{i} − ε_{j}, β = ε_{k} − ε_{ℓ} for i ≠ j, k ≠ ℓ)

(4.8) x_{α}x_{β} − x_{β}x_{α} = \begin{cases}H_{α} & if α + β = 0 \\ c_{α,β}x_{α+β} & if α + β ∈ Φ \\ 0 & otherwise \end{cases}

where H_{α} = H_{i} − H_{j} and where for α + β ∈ Φ we have

c_{α,β} = \begin{cases}1 & if j = k and i ≠ ℓ \\ −1 & if i = ℓ and j ≠ k. \end{cases}

The above relations hold in the enveloping algebra U, since we use only the defining relations (R1)–(R5) in their verification. Hence they are valid as well as in the quotient T.

For r, s ∈ ℕ, α, β ∈ Φ, one deduces the following commutation relations from the above by induction (compare with Kostant [Ko, Lemma 1]):

x_{α}^{(r)}x_{−α}^{(s)} = ∑_{j=0}^{min(r,s)} x_{−α}^{(s−j)}\binom{H_{α} − r − s + 2j}{j}x_{α}^{(r−j)}, (4.8a)

and, for α + β ≠ 0, α + β ∈ Φ

x_{α}^{(r)}x_{β}^{(s)} = x_{β}^{(s)}x_{α}^{(r)} + ∑_{j=1}^{min(r,s)} c_{α,β}^{j}x_{β}^{(s−j)}x_{α+β}^{(j)}x_{α}^{(r−j)}. (4.8b)

For α + β ≠ 0, α + β ∉ Φ

x_{α}^{(r)}x_{β}^{(s)} = x_{β}^{(s)}x_{α}^{(r)}. (4.8c)

From these formulas it follows that we can always interchange the order of a product of two divided powers of root vectors, up to a ℤ-linear combination of terms of strictly lower degree and (right or left) content.

In the following argument we make use of the fact that \binom{H_{α} − t}{s}

(s, t ∈ ℤ, α ∈ Φ) belongs to the subring of U generated by the divided powers of all root vectors, and thus belongs to T_{ℤ}^{0}. This follows from [Ko, Lemma 1] by an inductive argument. We will also use the identity

x_{γ}^{(a)}x_{γ}^{(b)} = \binom{a + b}{a}x_{γ}^{(a+b)} (γ ∈ Φ). (4.9)

which follows immediately from the definitions.

Proposition 4.10. The sets Y_{+} and Y_{−} span the algebra T_{ℤ}.

Proof. We prove just the claim about Y_{+}, as the other case is similar. We use the second formulation of Y_{+} given in (4.7). The algebra T_{ℤ} is spanned by the set of all products formed from divided powers of root vectors and idempotents 1_{λ}. By Proposition 3.8, in any such product, one may always commute the idempotents all the way to the right. Hence, T_{ℤ} is spanned (over ℤ) by the set of all monomials of the form

M = x_{γ_{1}}^{(ψ_{1})} ⋯ x_{γ_{m}}^{(ψ_{m})}1_{μ} (4.11)

for various m ∈ ℕ, γ ∈ Φ^{m}, ψ ∈ ℕ^{m}, μ ∈ Λ(n, d). We can assume that γ_{i} ≠ γ_{i+1} for all i ≤ m − 1, for otherwise the monomial is an integral multiple of a monomial having that property, by (4.9). Call monomials of such form special.

Let χ = χ_{R}(M). We define the deviation of M by δ = ∑_{χ_{i}>μ_{i}}(χ_{i} − μ_{i}). Note that if

M′ = x_{γ′_{1}}^{(ψ′_{1})} ⋯ x_{γ′_{m′}}^{(ψ′_{m′})}1_{μ}

is another special monomial with the same idempotent part 1_{μ}, then χ_{R}(M) ⪯ χ_{R}(M′) ⇒ δ(M) ≤ δ(M′) and χ_{R}(M) ⪯ μ ⇔ δ(M) = 0.

Now we claim that all special monomials M of deviation 0 lie in the ℤ-span of Y_{+}. We argue by induction on the degree r = ∑_{i} ψ_{i} of M. If r = 0 then M = 1_{μ}, which belongs to Y_{+}. Suppose that r > 0. If M ∈ Y_{+} then we are done. Otherwise, we can apply the commutation relations (4.8) to reorder the factors in M, obtaining an equality of the form

M = x_{γ_{1}}^{(ψ_{1})} ⋯ x_{γ_{m}}^{(ψ_{m})} = c e_{A}f_{C} + lower terms

for some c ∈ ℤ, A, C ∈ ℕ^{Φ^{+}}. Here, the lower terms are integral multiples of Kostant monomials of strictly lower degree and content. The constant c ∈ ℤ arises not from the commutation formulas but when two divided powers are combined via the equality (4.9). Note that, in the

lower order terms, whenever a factor of the form \binom{H_{α} − t}{s} (s, t ∈ ℤ)

appears, we express it in terms of a ℤ-linear combination of 1_{λ}’s, and then apply Proposition 3.8 to commute the idempotents as far to the right as possible. Upon right multiplication of the above equality by 1_{μ} we obtain the equality

M = c e_{A}f_{C}1_{μ} + lower terms

where the lower terms are integral multiples of terms of the same form as M (each having a factor 1_{μ} on the right), but again, of strictly lower degree and content than that of M. Now, in the above equality the right content of e_{A}f_{C}1_{μ} is equal to the right content of M, and thus e_{A}f_{C}1_{μ} ∈ Y_{+}. By induction the lower terms belong to the ℤ-span of Y_{+}. The claim is proved.

Now we proceed by induction on deviation. (The above claim forms the base step in the induction.) Let M be a special monomial of the form (4.11) of deviation δ = δ(M) > 0. Set χ = χ_{R}(M). Since χ ⪯̸ μ, there exists an index j for which χ_{j} > μ_{j}. Fixing this index j, we call β ∈ Φ bad if β = ε_{i} − ε_{j} for some i ≠ j and call β good otherwise. We extend this terminology to the factors x_{γ}^{(ψ_{γ})} of M. We define the defect D of the monomial M by the equality D(M) = ∑ ψ_{α} where the sum is taken over the set {x_{α}^{(ψ_{α})}} of good factors in M which appear to the right of some bad factor. Note that D = 0 if and only if all the bad factors in M appear as far to the right as possible. From Proposition 3.8 it follows immediately that M = 0 in T whenever D(M) = 0.

Now suppose that D(M) > 0. So there exists at least one good factor x_{α}^{(a)} in M appearing to the right of some bad factor x_{β}^{(b)}. We may assume that x_{α}^{(a)} is the leftmost such good factor in M. It has one or more bad factors immediately to its left. We successively commute x_{α}^{(a)} with each bad factor to its left, using relations (4.8). As before,

we express factors of the form \binom{H_{α} − t}{s} in terms of idempotents 1_{λ},

and commute these all the way to the right using Proposition 3.8. By Proposition 3.6 we know that such factors act as integral scalars on 1_{μ}. The result of all this is thus, up to a ℤ-linear combination L of Kostant monomials (all involving the same idempotent 1_{μ} on the right) of strictly lower right content, an integral multiple of a monomial M′ of the same right content as M but of strictly lower defect. By induction (on defect) M′ lies in the ℤ-span of Y_{+}, and, since the monomials in L have lower deviation than M does, L must also lie within the ℤ-span of Y_{+}. This proves that M lies in the ℤ-span of Y_{+}, and hence that Y_{+} spans T_{ℤ}. The argument for Y_{−} is similar (interchange right and left, + and − in the above argument).

5. Proof of Theorems 1.1 and 1.3

Since S = S(n, d) is the image of the map U → End(V^{⊗d}), it follows immediately from Lemma 3.1 that the images of the H_{i} in the Schur algebra S satisfy the defining relations for the algebra T, so the surjection U → S factors through the algebra T. In particular, this gives a surjection T → S. It follows that dim T ≥ dim S. In order to produce the opposite inequality, which will complete the proof of Theorem 1.1, it is enough to produce a spanning set in T of cardinality equal to the dimension of S. We know from Proposition 4.10 that the sets Y_{+}, Y_{−} span T, so the proof of Theorem 1.1 is completed by the following lemma.

Lemma 5.1. The cardinality of Y_{+} and Y_{−} is equal to the dimension of S = S(n, d).

Proof. By symmetry, it is enough to prove this for Y_{+}. It is well known (see [Gr]) that the dimension of S(n, d) is given by the number of monomials of total degree d in n^{2} variables. This is the same as the number of monomials in n^{2} − 1 variables of total degree not exceeding d; in other words, the dimension of S(n, d) is the same as the cardinality of the set

P = {e_{A}H_{B}f_{C} | B ∈ ℕ_{1}^{n}, A, C ∈ ℕ^{Φ^{+}}, |A| + |B| + |C| ≤ d}.

Thus, to prove the result it suffices to give a bijective correspondence between P and Y_{+}. One such is given by the map

e_{A}H_{B}f_{C} → e_{A}1_{λ}f_{C}

where λ = (d − |A| − |B| − |C|)ε_{1} + B + χ(e_{A}f_{C}). The inverse map is given by

e_{A}1_{λ}f_{C} → e_{A}H_{B}f_{C}

where B = λ − χ(e_{A}f_{C}) − λ_{1}ε_{1}. The lemma is proved.

It remains to prove Theorem 1.3. It follows from the preceding arguments that the quotient map T → S is an isomorphism of algebras, and from Proposition 4.10 we conclude that Y_{+} and Y_{−} are bases for T (over ℚ). Hence these sets are linearly independent over ℚ, and thus also over ℤ. Thus they are ℤ-bases for T_{ℤ}. Carter and Lusztig [CL, Thm. 3.1] showed that the restriction to U_{ℤ} of the map U → S gives a surjection U_{ℤ} → S_{ℤ} = S_{ℤ}(n, d). It follows that the restriction map T_{ℤ} → S_{ℤ} is an isomorphism, and that the sets Y_{+}, Y_{−} are integral bases for the Schur algebra S_{ℤ}. Moreover, the restriction of the map U_{ℤ} → S_{ℤ} to U_{ℤ}(𝔰𝔩_{n}) is still surjective, according to [Do, p. 44], and thus the image is generated by all e_{i}^{(m)}, f_{i}^{(m)}. This proves Theorem 1.3.

Remark 5.2. We conjecture that the set P appearing in the proof of Lemma 5.1 is actually another integral basis for S_{ℤ}(n, d). More generally, for any fixed i_{0}, (1 ≤ i_{0} ≤ n), either of the sets

{e_{A}H_{B}f_{C}}, {f_{A}H_{B}e_{C}}

(B ∈ ℕ_{i_{0}}^{n}, A, C ∈ ℕ^{Φ^{+}}, |A| + |B| + |C| ≤ d) should be an integral basis of S_{ℤ}(n, d). This would be a truncated form of Kostant’s well-known basis for U_{ℤ}. The conjecture is true when n = 2; see [DG1].

6. Proof of Theorem 1.4

Let B be the ℚ-algebra given by the generators and relations of Theorem 1.4. In B we define elements H_{j} for j = 1, …, n by setting H_{j} = ∑_{λ} λ_{j}1_{λ}, where the sum is carried over all λ ∈ Λ(n, d).

Since the various 1_{λ}’s commute it follows that the H_{j}’s must also commute, so relation (R1) holds for the elements H_{j}. Relation (R2) follows immediately from the defining relations (R3′) and the definition of the H_{j}.

From the defining relations (R1′) and the definition of the H_{j} it follows that

∑_{j=1}^{n} H_{j} = ∑_{j=1}^{n} ∑_{λ} λ_{j}1_{λ} = ∑_{λ} (∑_{j=1}^{n} λ_{j}) 1_{λ} = ∑_{λ} d 1_{λ} = d · 1 = d.

This proves that the H_{j} satisfy relation (R6). Moreover, we also have equalities

1_{λ} H_{j} = H_{j} 1_{λ} = ∑_{μ} μ_{j}1_{μ} 1_{λ} = λ_{j}1_{λ}

for each λ, j, from which we obtain the equalities

\begin{aligned}H_{j}(H_{j} − 1) ⋯ (H_{j} − b) & = (∑_{λ} λ_{j} 1_{λ}) (H_{j} − 1) ⋯ (H_{j} − b) \\ & = ∑_{λ} λ_{j}(λ_{j} − 1) ⋯ (λ_{j} − b) 1_{λ} \end{aligned} (6.1)

for any b ∈ ℕ. This is 0 when b = d, since λ is a composition of d, and thus each part λ_{j} of λ is an integer in the interval 0, …, d. This proves that the H_{j} satisfy relation (R7).

We now want to show that the H_{j} also satisfy relation (R3). For this we will use the defining relations (R2′). For convenience, we extend the definition of the symbol 1_{λ} to all λ ∈ ℤ^{n} such that |λ| = ∑ λ_{i} = d, defining it to have the value 0 if any part of λ is negative. With this convention we have

H_{j}e_{i} = ∑_{λ} λ_{j} 1_{λ}e_{i} = ∑_{λ} λ_{j} e_{i}1_{λ−α_{i}} (6.2)

where the sums are taken over all λ ∈ ℤ^{n} such that |λ| = d. Replacing λ − α_{i} by μ and noting that λ_{j} = μ_{j} if j ≠ i, i + 1, λ_{j} = μ_{j} + 1 if j = i, and λ_{j} = μ_{j} − 1 if j = i + 1 we obtain

H_{j}e_{i} = ∑_{μ} λ_{j} e_{i}1_{μ} = e_{i}H_{j} + (δ_{ij} − δ_{i+1,j})e_{i}. (6.3)

where again the sum is over all μ ∈ ℤ^{n} satisfying |μ| = d. This proves that the H_{j}, e_{i} satisfy relation (R3); a similar argument shows the same for the H_{j}, f_{i}.

From (6.1) it follows (upon replacing b by b − 1 and dividing by b!) that

\binom{H_{j}}{b} = ∑_{λ} \binom{λ_{j}}{b} 1_{λ} (6.4)

where the sum is over Λ(n, d). It then follows from relations (R1′) and the above, by a simple calculation, that for any μ ∈ Λ(n, d) we have

∏_{j=1}^{n} \binom{H_{j}}{μ_{j}} = 1_{μ}

and thus the H_{j}, (1 ≤ j ≤ n) together with the e_{i}, f_{i} (1 ≤ i ≤ n − 1) generate the algebra B. Since we have proved that these generators satisfy the defining relations for the algebra T, it follows that B is a homomorphic image of T.

On the other hand, by Proposition 3.8 we know that the elements e_{i}, f_{i}, 1_{λ} of T satisfy relations (R2′). They also satisfy relation (R1′), clearly, and relation (R3′), by Proposition 3.6. Moreover, T is generated by the 1_{λ}, e_{i}, f_{i} since by Proposition 3.6(c) we know that H_{j} is expressible as a linear combination of the 1_{λ}. Thus T is a homomorphic image of B. Combining this with the conclusion of the preceding paragraph, we see that B ≅ T ≅ S(n, d). Theorem 1.4 is proved.

7. The algebra T.

We turn now to the quantum case. Fix n and d, and set S = S(n, d). We define an algebra T = T(n, d) (over ℚ(v)) by the generators and relations of Theorem 2.1. Since U is the algebra on the same generators but subject only to relations (Q1)–(Q5), we have a surjective quotient map U → T. Eventually we shall show that T ≃ S, which will prove Theorem 2.1.

The q-analogue of Lemma 3.1 is the following. The proof is similar to the proof in the classical case, except that it is multiplicative where the classical argument is additive.

Lemma 7.1. Under the representation U → End(V^{⊗d}) the images of the K_{i} satisfy the relations (Q6) and (Q7). Moreover, the relation (Q7) is the minimal polynomial of K_{i} in End(V^{⊗d}).

Since S = S(n, d) is the image of the representation U → End(V^{⊗d}), we have a surjection U → S. From the lemma it follows that the surjection U → S factors through T. Because T, S are homomorphic images of U, any relations between generators in U will carry over to T, S. We do not distinguish notationally between the generators or root vectors for U, T, or S.

Rosso [Ro] has shown that multiplication defines a ℚ(v)-linear isomorphism U^{−}⊗U^{0}⊗U^{+} ⟶^{≈} U, where U^{+} (resp., U^{−}) is the subalgebra of U generated by the E_{i} (resp., F_{i}), and U^{0} is the subalgebra of U generated by all K_{i}, K_{i}^{−1}. It follows that U = U^{−}U^{0}U^{+}. From this we obtain a similar triangular decomposition of T:

T = T^{−}T^{0}T^{+} (7.2)

where T^{+}, T^{−}, T^{0} are defined to be the images of U^{+}, U^{−}, U^{0} under the quotient mapping U → T.

There are similar factorizations over A. Setting U_{A}^{+}, U_{A}^{−}, U_{A}^{0} to be, respectively, the intersection of U^{+}, U^{−}, U^{0} with Lusztig’s A-form U_{A}

(the A-subalgebra of U generated by the E_{i}^{(m)}, F_{i}^{(m)}, K_{i}^{±1}, [\begin{matrix}K_{i} \\ m \end{matrix}]), Du

[Du, §2] has shown (using results of Lusztig) that U_{A} = U_{A}^{−}U_{A}^{0}U_{A}^{+}; thus

T_{A} = T_{A}^{−}T_{A}^{0}T_{A}^{+} (7.3)

where the various subalgebras are defined in the obvious way as appropriate homomorphic images of U_{A}^{+}, U_{A}^{−}, U_{A}^{0}.

Since U_{A}^{+} (resp., U_{A}^{−}) is the A-subalgebra of U generated by the E_{α}^{(m)} (resp., F_{α}^{(m)}) for α ∈ Φ^{+} and m ∈ ℕ, the same statement applies to T_{A}^{+} (resp., T_{A}^{−}) in relation to T. Moreover, U_{A}^{0} is the A-subalgebra

of U generated by the K_{i}^{±1}, [\begin{matrix}K_{i} \\ m \end{matrix}] for 1 ≤ i ≤ n, m ∈ ℕ, so T_{A}^{0} is the

A-subalgebra of T generated by the same elements.

Now we determine the structure of the algebra T^{0}. As we shall see, the structure turns out to be essentially the same as that in the classical case. Consider first the algebra U^{0}, which may be identified with the commutative polynomial algebra ℚ(v)[K_{1}^{±1}, …, K_{n}^{±1}]. We define S^{0} to be the image of U^{0} under the quotient map U → S. As in the classical case, this map factors through the algebra T^{0}.

Proposition 7.4. Define an algebra T′ = U/I^{0} where I^{0} is the ideal in U^{0} generated by the elements (K_{i}−1)(K_{i}−v) ⋯ (K_{i}−v^{d}) (1 ≤ i ≤ n) and K_{1}K_{2} ⋯ K_{n} − v^{d}.

(a) We have an algebra isomorphism T′ ≅ T^{0}.

(b) The set {1_{λ} | λ ∈ Λ(n, d)} is a ℚ(v)-basis for T^{0} and an A-basis for T_{A}^{0}; moreover, this set is a set of pairwise orthogonal idempotents which add up to 1.

(c) K_{B} = 0 for any B ∈ ℕ^{n} such that |B| > d.

Proof. The argument is similar to the proof of Proposition 3.4. Consider first the algebra T̃′ defined to be the quotient of U^{0} by the ideal generated only by the p(K_{i}) = (K_{i} − 1)(K_{i} − v) ⋯ (K_{i} − v^{d}) for i = 1, …, n. Since p(K_{i}) is a non-constant polynomial with nonzero constant term, each K_{i}^{−1} already lies in the ring T̃′. Thus it follows that

T̃′ ≅ ℚ(v)[K_{1}]/(p(K_{1})) ⊗ ⋯ ⊗ ℚ(v)[K_{n}]/(p(K_{n})).

From the Chinese Remainder Theorem we obtain, as in the proof of 3.4, an isomorphism (products denote direct products)

T̃′ ≅ ∏_{0≤μ_{1},...μ_{n}≤d} ℚ(v)[K_{1}, …, K_{n}]/(K_{1} − v^{μ_{1}}, …, K_{n} − v^{μ_{n}}).

Since T′ = T̃′/(K_{1} ⋯ K_{n} − v^{d}), we deduce that

T′ ≅ ∏_{μ∈Λ(n,d)} ℚ(v)[K_{1} …, K_{n}]/(K_{1} − v^{μ_{1}}, …, K_{n} − v^{μ_{n}}).

The preceding isomorphism, which we denote by ϕ, is given by the map sending f(K_{1}, … K_{n}) to the vector (f(v^{μ_{1}}, …, v^{μ_{n}}))_{μ∈Λ(n,d)}. In particular, if λ ∈ Λ(n, d), then

ϕ(1_{λ}) = ([\begin{matrix}μ_{1} \\ λ_{1} \end{matrix}] ⋯ [\begin{matrix}μ_{n} \\ λ_{n} \end{matrix}])_{μ∈Λ(n,d)} = (δ_{λμ})_{μ∈Λ(n,d)}.

Thus the various 1_{λ} are orthogonal idempotents whose sum is the identity.

Now let I be the ideal in U generated by elements (K_{i} − 1)(K_{i} − v) ⋯ (K_{i} − v^{d}) (1 ≤ i ≤ n) and K_{1} ⋯ K_{n} − v^{d}. Then by definition T ≅ U/I. The canonical quotient map U → U/I induces, upon restriction to U^{0}, a map U^{0} → U/I. The image of this map is T^{0} and its kernel is U^{0} ∩ I, so T^{0} ≅ U^{0}/(U^{0} ∩ I). Clearly I^{0} ⊂ U^{0} ∩ I. Thus we obtain the following sequence of algebra surjections

T′ = U^{0}/I^{0} ⟶ U^{0}/(U^{0} ∩ I) ⟶^{≈} T^{0} ⟶ S^{0} (7.5)

where the last map is obtained by Lemma 7.1, and the middle map is actually an isomorphism. By the above we see that the dimension of T′ is the cardinality of the set Λ(n, d). This is the same as dim S^{0}, so all the surjections above are algebra isomorphisms. This proves parts (a) and (b).

Part (c) is proved in exactly the same way as part (c) of Proposition 3.4.

Proposition 7.6. Suppose 1 ≤ i ≤ n, c ∈ ℤ, t ∈ ℕ, λ ∈ Λ(n, d), and B ∈ ℕ^{n}. Then we have the following identities in the algebra T^{0}:

K_{i}^{±1}1_{λ} = v^{±λ_{i}}1_{λ}; [\begin{matrix}K_{i}; c \\ t \end{matrix}] 1_{λ} = [\begin{matrix}λ_{i} + c \\ t \end{matrix}] 1_{λ} (a)

K_{B} 1_{λ} = λ_{B} 1_{λ}, where λ_{B} = ∏_{i} [\begin{matrix}λ_{i} \\ B_{i} \end{matrix}] (b)

K_{B} = ∑_{λ} λ_{B} 1_{λ}, (c)

where the sum in part (c) is carried out over all λ ∈ Λ(n, d).

Proof. We prove part (a). Apply the isomorphism ϕ from the proof of Proposition 7.4. We have ϕ(K_{i}^{±1}1_{λ}) = (v^{±μ_{i}}δ_{λμ})_{μ} = v^{±λ_{i}}ϕ(1_{λ}). It follows that K_{i}^{±1}1_{λ} = v^{±λ_{i}}1_{λ}. Similarly, we have

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

= (∏_{s=1}^{t} \frac{v^{μ_{i}}v^{c−s+1} − v^{−μ_{i}}v^{−c+s−1}}{v^{s} − v^{−s}} δ_{λμ})_{μ}

= ([\begin{matrix}μ_{i} + c \\ t \end{matrix}] δ_{λμ})_{μ}

= [\begin{matrix}λ_{i} + c \\ t \end{matrix}] ϕ(1_{λ})

which proves the second equality in (a). Note that the equality

[\begin{matrix}K_{i} \\ t \end{matrix}] 1_{λ} = [\begin{matrix}λ_{i} \\ t \end{matrix}] 1_{λ} (7.7)

is the case c = 0 of the above.

The rest of the proof is similar to the proof of Proposition 3.6.

By essentially the same argument as in the classical case we obtain

Corollary 7.8. For any fixed choice of i_{0} (1 ≤ i_{0} ≤ n) the set {K_{B} | B ∈ ℕ_{i_{0}}^{n}, |B| ≤ d} is a ℚ(v)-basis for T^{0} and an A-basis for T_{A}^{0}.

We also have the following exact analogue of Proposition 3.8.

Proposition 7.9. For any α ∈ Φ and any λ ∈ Λ(n, d) we have the commutation formulas

X_{α}1_{λ} = \begin{cases}1_{λ+α}X_{α} & if λ + α ∈ Λ(n, d) \\ 0 & otherwise \end{cases}

and similarly

1_{λ}X_{α} = \begin{cases}X_{α}1_{λ−α} & if λ − α ∈ Λ(n, d) \\ 0 & otherwise. \end{cases}

Proof. We will need the following identities (see [Lu2, §2.3 (g3), (g4)]):

[\begin{matrix}K_{i}; 0 \\ t \end{matrix}] [\begin{matrix}K_{i}; −t \\ t′ \end{matrix}] = [\begin{matrix}t + t′ \\ t \end{matrix}] [\begin{matrix}K_{i}; 0 \\ t + t′ \end{matrix}] (t, t′ ∈ ℕ) (7.10)

[\begin{matrix}K_{i}; c + 1 \\ t \end{matrix}] = v^{t} [\begin{matrix}K_{i}; c \\ t \end{matrix}] + v^{t−c−1}K_{i}^{−1} [\begin{matrix}K_{i}; c \\ t − 1 \end{matrix}] (t ≥ 1). (7.11)

From the defining relation (Q3) and the definition (see §2) of the root vector X_{α} one proves that

K_{l}X_{α} = v^{(ε_{l},α)}X_{α}K_{l} (α ∈ Φ). (7.12)

From this it follows by a simple calculation that

[\begin{matrix}K_{l}; c \\ t \end{matrix}] X_{α} = X_{α} [\begin{matrix}K_{l}; c + (ε_{l}, α) \\ t \end{matrix}] (α ∈ Φ, c ∈ ℤ, t ∈ ℕ). (7.13)

From this last equality it follows that for λ ∈ Λ(n, d), α = ε_{i} − ε_{j} ∈ Φ (i ≠ j) we have

X_{α}1_{λ} = [\begin{matrix}K_{i}; −1 \\ λ_{i} \end{matrix}] [\begin{matrix}K_{j}; 1 \\ λ_{j} \end{matrix}] (∏_{l≠i,j} [\begin{matrix}K_{l} \\ λ_{l} \end{matrix}]) X_{α}.

Multiply both sides of the preceding equality by [\begin{matrix}K_{i} \\ 1 \end{matrix}] = [\begin{matrix}K_{i}; 0 \\ 1 \end{matrix}] and

use (7.10) to simplify the right-hand-side and use (7.13) to simplify the left-hand-side. The result is the equality

X_{α} [\begin{matrix}K_{i}; 1 \\ 1 \end{matrix}] 1_{λ} = [\begin{matrix}λ_{i} + 1 \\ 1 \end{matrix}] [\begin{matrix}K_{i} \\ λ_{i} + 1 \end{matrix}] [\begin{matrix}K_{j}; 1 \\ λ_{j} \end{matrix}] (∏_{l≠i,j} [\begin{matrix}K_{l} \\ λ_{l} \end{matrix}]) X_{α}.

By Proposition 7.6(a) the left-hand-side of this equality simplifies to give the equality

[\begin{matrix}λ_{i} + 1 \\ 1 \end{matrix}] X_{α}1_{λ} = [\begin{matrix}λ_{i} + 1 \\ 1 \end{matrix}] [\begin{matrix}K_{i} \\ λ_{i} + 1 \end{matrix}] [\begin{matrix}K_{j}; 1 \\ λ_{j} \end{matrix}] (∏_{l≠i,j} [\begin{matrix}K_{l} \\ λ_{l} \end{matrix}]) X_{α}

and after cancelling the common scalar factor and expanding by means of (7.11) (assuming λ_{j} > 0) this becomes

X_{α}1_{λ} = [\begin{matrix}K_{i} \\ λ_{i} + 1 \end{matrix}] (v^{λ_{j}} [\begin{matrix}K_{j} \\ λ_{j} \end{matrix}] + v^{λ_{j}−1}K_{j}^{−1} [\begin{matrix}K_{j} \\ λ_{j} − 1 \end{matrix}]) (∏_{l≠i,j} [\begin{matrix}K_{l} \\ λ_{l} \end{matrix}]) X_{α}.

In case λ_{j} > 0 after multiplying through in the above expression and applying Proposition 7.4(c) we see that the first summand must be zero, so the above equality in that case simplifies to X_{α}1_{λ} = v^{λ_{j}−1}K_{j}^{−1}1_{λ+α}X_{α}.

Now by Proposition 7.6(a) K_{j}^{−1} acts on 1_{λ+α} as v^{−(λ_{j}−1)}. Thus we obtain the equality in the first part of the proposition in the case λ_{j} > 0.

If λ_{j} = 0 then one sees easily by Proposition 7.4(c) that the righthand-side vanishes. The first part of the proposition is proved. The proof of the other part is similar.

Corollary 7.14. We have in T the equalities X_{α}^{d+1} = 0 for all α ∈ Φ.

Again, the proof is the same as in the classical case.

8. Quantum straightening

Similar to the classical case, we define left and right content χ_{L}, χ_{R} on Kostant monomials by

\begin{matrix}χ_{R}(X_{α}^{(m)}) = m ε_{j}, χ_{L}(X_{α}^{(m)}) = m ε_{i}, \\ χ_{R}([\begin{matrix}K_{i} \\ m \end{matrix}]) = χ_{L}([\begin{matrix}K_{i} \\ m \end{matrix}]) = χ_{R}(K_{i}) = χ_{L}(K_{i}) = 0 \end{matrix} (8.1)

where α = ε_{i} − ε_{j}, and again using the rule χ_{L}(XY) = χ_{L}(X) + χ_{L}(Y) (similarly for χ_{R}) whenever X and Y are Kostant monomials.

The q-analogues of (4.2)–(4.7) hold, with the same argument as in the classical case. In particular, we can work with the description of Y_{+} in which the idempotent appears on the right.

Now we write X_{ij} for the root vector X_{α} when α = ε_{i} − ε_{j} and write K_{α} = K_{ij} = K_{i}K_{j}^{−1} (1 ≤ i ≠ j ≤ n). Note that E_{i} = X_{i,i+1}, F_{i} = X_{i+1,i}. We assume that i < j and k < l. By Xi [Xi1, §5.6] we have the following root vector commutation formulas, listed below in formulas (8.2) and (8.3) (some of which appeared already in [Lu2] and [Ro]). Note that our notation differs from Xi’s: he writes E_{i,j−1} (resp., F_{i,j−1}, K_{i,j−1}) where we write X_{ij} (resp., X_{ji}, K_{ij}).

X_{ij}X_{kl} = \begin{cases}X_{kl}X_{ij} & (j < k or k < i < j < l) \\ v^{−1}X_{kl}X_{ij} & (i = k < j < l or i < k < j = l) \\ vX_{il} + vX_{kl}X_{ij} & (j = k) \\ X_{kl}X_{ij} + (v^{−1} − v)X_{il}X_{kj} & (i < k < j < l) \end{cases} (8.2)

which, as Xi (loc. cit.) proves, lead to the following

X_{ij}^{(M)}X_{kl}^{(N)} = X_{kl}^{(N)}X_{ij}^{(M)} (j < k or k < i < j < l) (8.2a)

\begin{matrix}X_{ij}^{(M)}X_{kl}^{(N)} = v^{−MN}X_{kl}^{(N)}X_{ij}^{(M)} \\ (i & = k < j < l or i < k < j = l) \end{matrix} (8.2b)

\begin{matrix}X_{ij}^{(M)}X_{kl}^{(N)} = ∑_{t=0}^{min(M,N)} v^{(M−t)(N−t)+t}X_{kl}^{(N−t)}X_{il}^{(t)}X_{ij}^{(M−t)} \\ (j & = k) \end{matrix} (8.2c)

\begin{aligned}X_{ij}^{(M)}X_{kl}^{(N)} = ∑_{t=0}^{min(M,N)} v^{−t(t−1)/2}(v^{−1} − v)^{t}[t]!X_{kj}^{(t)}X_{kl}^{(N−t)} & X_{ij}^{(M−t)}X_{il}^{(t)} \\ & (i < k < j < l). \end{aligned} (8.2d)

We note that these formulas lead to others, of the same form as those given above, except for signs and (in some cases) a scalar factor of some integral power of v. The new formulas are obtained from the ones listed above by solving for the term X_{kl}^{(N)}X_{ij}^{(M)} and then interchanging (i, j) and (k, l).

We also note that the formulas one obtains in this way, together with the ones already listed, exhaust the possibilities for a commutation of two root vectors labelled by positive roots. Indeed, given two finite intervals [i, j], [k, l] one of the following mutually exclusive possibilities must apply: 1) the intervals are disjoint; 2) one interval is properly included in the other without shared endpoints; 3) one interval is properly included in the other with the two intervals sharing a common endpoint; 4) the intervals coincide; 5) the intervals meet at a single point; 6) the intervals properly overlap. Looking at the formulas listed above, we see that case (a) covers possibilities 1) and 2), case (b) covers possibility 3), case (c) is possibility 5), and case (d) is possibility 6). We do not need a formula for possibility 4).

There are five cases listed in [Xi1, §5.6 (d0)] and only four cases listed above, but the second of the five cases in (loc. cit.) is superfluous, as we have just proved.

There are similar commutation formulas for products of negative root vectors. One way to get them is by applying the isomorphism Ω : U → U^{opp} of [Lu2, 1.2], defined by ΩE_{i} = F_{i}, ΩF_{i} = E_{i}, ΩK_{i} = K_{i}^{−1}, Ωv = v^{−1}, to the positive root vector commutation formulas discussed above.

Xi (loc. cit.) also gives the following, which express the commutation between positive and negative root vectors. Still under the assumption i < j and k < l we have

X_{ij}X_{lk} = \begin{cases}X_{lk}X_{ij} & (j ≤ k or k < i < j < l) \\ X_{lk}X_{ij} + v^{−1}K_{kj}^{−1}X_{ik} & (i < k < j = l) \\ X_{lk}X_{ij} − X_{lj}K_{ij}^{−1} & (i = k < j < l) \\ X_{lk}X_{ij} + [\begin{matrix}K_{ij}; 0 \\ 1 \end{matrix}] & (i = k, j = l) \\ X_{lk}X_{ij} + v^{−1}(v − v^{−1})X_{lj}K_{kj}^{−1}X_{ik} & (i < k < j < l) \end{cases} (8.3)

which, as Xi proves, lead to the following

X_{ij}^{(M)}X_{lk}^{(N)} = X_{lk}^{(N)}X_{ij}^{(M)} (j ≤ k or k < i < j < l) (8.3a)

\begin{matrix}X_{ij}^{(M)}X_{lk}^{(N)} = ∑_{t=0}^{min(M,N)} v^{t(N−t−1)}X_{jk}^{(N−t)}K_{kj}^{−t}X_{ij}^{(M−t)}X_{ik}^{(t)} \\ (i < k & < j = l) \end{matrix} (8.3b)

\begin{matrix}X_{ij}^{(M)}X_{lk}^{(N)} = ∑_{t=0}^{min(M,N)}(−1)^{t}v^{t(M−t)}X_{lj}^{(t)}X_{lk}^{(N−t)}K_{ij}^{−t}X_{ij}^{(M−t)} \\ (i = k < j < l) \end{matrix} (8.3c)

X_{ij}^{(M)}X_{ji}^{(N)} = ∑_{t=0}^{min(M,N)} X_{ji}^{(N−t)} [\begin{matrix}K_{ij}; 2t − M − N \\ t \end{matrix}] X_{ij}^{(M−t)} (8.3d)

X_{ij}^{(M)}X_{lk}^{(N)} = ∑_{t=0}^{min(M,N)} ξ(v, t)X_{lk}^{(N−t)}X_{lj}^{(t)}K_{kj}^{−t}X_{ij}^{(M−t)}X_{ik}^{(t)}
where ξ(v, t) = v^{−t(2N+t−1)/2}(v − v^{−1})^{t}[t]! (i < k < j < l).

(8.3e)

As before, these formulas lead to others, of a similar form as those given above. The new formulas in this case are obtained by first applying Ω to get a formula in U^{opp}, then switching order of factors to get a formula in U, and finally interchanging (i, j) with (k, l).

The formulas one obtains in this way, together with the ones already listed, exhaust the possibilities for a commutation involving a positive root vector followed by negative root vector. For the possibilities 1)

– 6) listed earlier for two finite intervals [i, j], [k, l], we see that case (a) covers possibilities 1), 5), and 2), cases (b), (c) cover possibility 3), case (d) is possibility 4), and case (e) is possibility 6).

There are similar commutation formulas for the case of a negative root vector followed by a positive root vector. They are obtained from the ones already described, by simply solving for the term X_{lk}^{(N)}X_{ij}^{(M)}. The new formulas will be of the same form, except for signs.

Thus we see that the formulas listed in (8.2) and (8.3), together with formulas easily derivable from them, give the commutation in U between q-divided powers of any two root vectors.

The most important feature of these commutation formulas, for our purposes, is that every product of the form X_{α}^{(r)}X_{β}^{(s)} may be expressed as a scalar multiple (by an element of A) of the product in the opposite order, modulo an A-linear combination of monomials of degree and (right or left) content which is no greater that that of the original product. This differs from the classical case, where the extra terms in any such commutation always have strictly lower degree and content.

A central result of [Lu2] is a q-analogue of Kostant’s basis for U_{ℤ}. The description of this basis uses the “box ordering” on Φ^{+} defined as follows: if α = ε_{i} − ε_{j} and β = ε_{r} − ε_{s}, then α ≻ β if either s > j or s = j and r > i. For B ∈ ℕ^{n−1} and δ ∈ {0, 1}^{n−1} set

K_{δ,B} = K_{α_{1}}^{δ_{1}} ⋯ K_{α_{n−1}}^{δ_{n−1}} [\begin{matrix}K_{α_{1}}; 0 \\ B_{1} \end{matrix}] ⋯ [\begin{matrix}K_{α_{n−1}}; 0 \\ B_{n−1} \end{matrix}] .

Then Lusztig [Lu2, Thm. 4.5] proves that the set of all elements of the form

F_{A}K_{δ,B}E_{C} (A, C ∈ ℕ^{Φ^{+}}, B ∈ ℕ^{n−1}, δ ∈ {0, 1}^{n−1}) (8.4)

is an A-basis of U_{A}(𝔰𝔩_{n}), provided the products in E_{C} are taken in the box order on Φ^{+} and the products in F_{A} are taken in the reverse box order. In [Xi2, Theorem 2.4] it is shown that the box order is not necessary as one may take any ordering on Φ^{+} when forming F_{A} and E_{C}. By applying the involution ω (see [Lu3, 3.1.3]) which interchanges the E_{i} and F_{i} we obtain another such basis, consisting of all elements of the form

E_{A}K_{δ,B}F_{C} (A, C ∈ ℕ^{Φ^{+}}, B ∈ ℕ^{n−1}, δ ∈ {0, 1}^{n−1}) (8.5)

We note that elements [\begin{matrix}K_{α}; c \\ t \end{matrix}] appear in some of Xi’s commutation

formulas. In the following argument we use the fact that any [\begin{matrix}K_{α}; c \\ t \end{matrix}]

(α ∈ Φ, c ∈ ℤ, t ∈ ℕ) belongs to U_{A}^{0} and thus belongs to T_{A}^{0}. This was proved by Lusztig [Lu1, 4.5] in case α is simple. Lusztig’s argument extends immediately to general α ∈ Φ since one has a version of [Lu1, (4.3.1)] for any α.

We shall also need the relation

X_{γ}^{(a)}X_{γ}^{(b)} = [\begin{matrix}a + b \\ a \end{matrix}] X_{γ}^{(a+b)} (γ ∈ Φ). (8.6)

which follows immediately from the definitions.

Proposition 8.7. The sets Y_{+} and Y_{−} span the algebra T_{A}.

Proof. We just prove the statement for Y_{+}. As in the classical case, the algebra T_{A} is spanned by the set of all “special” monomials of the form

M = X_{γ_{1}}^{(ψ_{1})} ⋯ X_{γ_{m}}^{(ψ_{m})} 1_{μ} (8.8)

for m ∈ ℕ, γ ∈ Φ^{m}, ψ ∈ ℕ^{m}. By (8.6) we may assume that γ_{i} ≠ γ_{i+1} for all i ≤ m − 1. Set χ = χ_{R}(M) and define the deviation of M by δ = ∑_{χ_{i}>μ_{i}}(χ_{i} − μ_{i}). We claim that the set of all special monomials of deviation 0 lie within the A-span of Y_{+}. However, the proof of this claim is different in the present case. By (8.5) we can express the product X_{γ_{1}}^{(ψ_{1})} ⋯ X_{γ_{m}}^{(ψ_{m})} (which lies in the subalgebra U(𝔰𝔩_{n})) in terms of an A-linear combination of terms of the form E_{A}K_{δ,B}F_{C}. In this expression, each of the terms has right content not exceeding that of M, because commutation does not increase degree or right content. We express each K_{δ,B} in terms of an A-linear combination of idempotents 1_{λ}, and use Proposition 7.9 to commute the idempotents all the way to the right. Upon right multiplication of the resulting expression by 1_{μ} we obtain an equality of the form

M = ∑_{A,C} a_{A,C} E_{A}F_{C}1_{μ} (a_{A,C} ∈ A)

in which the right content of each term on the right-hand-side is no greater than that of M. It follows that each term in the right-hand-side of the above equality has deviation 0, and thus lies in Y_{+}. This proves the claim.

Now we proceed by induction on deviation, with the above claim forming the base step in the induction. Let M be a special monomial of the form (8.8) of deviation δ = δ(M) > 0. Set χ = χ_{R}(M). Since χ ⪯̸ μ, there is an index j for which χ_{j} > μ_{j}. Fixing this index j, we call β ∈ Φ bad if β = ε_{i} − ε_{j} for some i ≠ j and call β good otherwise. We extend this terminology to the factors X_{γ}^{(ψ_{γ})} of M. We define the defect D of the monomial M by the equality D(M) = ∑ ψ_{α} where the sum is over the set {X_{α}^{(ψ_{α})}} of good factors in M appearing to the right of some bad factor. The defect of M is 0 if and only if all the bad factors in M appear as far to the right as possible. As in the classical case we have by Proposition 7.9 that M = 0 whenever D(M) = 0.

Now suppose that D(M) > 0. Then there exists at least one good factor X_{α}^{(a)} appearing to the right of some bad factor. We may assume that X_{α}^{(a)} is the leftmost such good factor in M. It has one or more bad factors immediately to its left. If there is just one bad factor X_{β}^{(b)} to the left of X_{α}^{(a)} we apply relations (8.2), (8.3) to the product X_{β}^{(b)}X_{α}^{(a)}. In all cases except (8.2d), (8.3e) the argument is similar to the classical case, and we obtain a multiple (by an element of A) of a Kostant monomial of the same deviation as M but of strictly lower defect, modulo an A-linear combination of monomials of strictly lower deviation. In cases (8.2d), (8.3e) (as written) we obtain an A-linear combination of monomials of the same defect and deviation as M. But in relation (8.2d) the factors X_{ij}^{(M−t)}, X_{il}^{(t)} commute (up to a power of v), and so do the factors X_{kj}^{(t)}, X_{kl}^{(N−t)}. After interchanging those pairs of factors, the factors X_{kj}^{(t)}, X_{il}^{(t)} will be adjacent, and they commute. Thus we see that the right-hand-side of formula (8.2d) can be rewritten in such a way that the bad factors X_{kj}^{(t)}, X_{ij}^{(M−t)} appear on the right, and thus all the monomials we obtain after applying (8.2d) to M are of strictly lower defect than that of M. In case (8.3e) similar commutation applies to obtain the same result: here X_{ij}^{(M−t)}, X_{ik}^{(t)} commute up to a power of v, then X_{ik}^{(t)}, X_{lj}^{(t)} commute by case (8.3a) since k − 1 < j. (One needs also in this case to express the factor K_{kj}^{−t} as an A-linear combination of idempotents, and then commute those all the way to the right.)

Now suppose there is more than one bad factor to the left of X_{α}^{(a)}. Let X_{β}^{(b)} be the rightmost such bad factor. We again apply relations (8.2), (8.3) to the product X_{β}^{(b)}X_{α}^{(a)} and repeat the argument given above. The result in the cases (8.2d), (8.3e) is an A-linear combination of Kostant monomials of the same defect and deviation as that of M, but with fewer bad factors to the left of the good factor X_{α}^{(a)}. By induction on the number of such factors, we are done.

This proves that M lies in the A-span of Y_{+}, and hence that Y_{+} spans T_{A}. The argument for Y_{−} is similar.

9. Proof of Theorems 2.1 and 2.3.

Since S is the image of the representation U → End(V^{⊗d}), it follows from Lemma 7.1 that the images of the K_{i} in the quantum Schur algebra S satisfy the defining relations for the algebra T, so the surjection U → S factors through the algebra T. In particular, this gives a surjection T → S. Thus, as in the classical case, to complete the proof of Theorem 2.1 it is enough to produce a spanning set in T of cardinality equal to the dimension of S. We know from Proposition 8.7 that Y_{+} and Y_{−} span the algebra T, and it is clear that these sets have cardinality equal to the dimension of S, since they are in bijective correspondence with the set Y_{+}. This proves that T ≃ S and Theorem 2.1 follows. It also follows that the sets Y_{+} and Y_{−} are ℚ(v)-bases for S. Hence these sets are linearly independent over ℚ(v), and thus also over A. Thus they are A-bases for T_{A}. By [Du], the restriction to U_{A} of the map U → S gives a surjection U_{A} → S_{A}. It follows that the restriction map T_{A} → S_{A} is an isomorphism (of A-algebras), and that the sets Y_{+}, Y_{−} are A-bases for the q-Schur algebra S_{A}. Moreover, by Proposition 7.4, K_{i} and K_{i}^{−1} lie in the subalgebra of T_{A} generated by

the [\begin{matrix}K_{i} \\ b \end{matrix}]. This proves Theorem 2.3.

Remark 9.1. We conjecture that for any fixed i_{0} (1 ≤ i_{0} ≤ n) either of the sets

{E_{A}K_{B}F_{C}}, {F_{A}K_{B}E_{C}}

(B ∈ ℕ_{i_{0}}^{n}, A, C ∈ ℕ^{Φ^{+}}, |A| + |B| + |C| ≤ d) is an A-basis of S_{A}(n, d). These are a truncated form of Lusztig’s analogue for U_{A} of Kostant’s basis for U_{ℤ}. The conjecture is true when n = 2; see [DG2].

10. Proof of Theorem 2.4

To prove Theorem 2.4, one sets K_{j} = ∑_{λ} v^{λ_{j}}1_{λ}, K_{j}^{−1} = ∑_{λ} v^{−λ_{j}}1_{λ} and verifies that these elements, along with the E_{i}, F_{i}, satisfy the relations (Q1)–(Q7). The argument is similar to that given in the proof of Theorem 1.4. The details are left to the reader.

11. Applications

In this section we apply our main results to study some subalgebras of S. In what follows, we will focus entirely on the quantum case as the corresponding results for the classical case are essentially the same.

Recall the triangular decomposition (7.3), which we now write in the form S_{A} = S_{A}^{+}S_{A}^{0}S_{A}^{−} in light of the identification S_{A} ≅ T_{A}. We consider the plus part S_{A}^{+}. We give a new proof for the following result of R.M. Green [RG2, Prop. 2.3].

Proposition 11.1. Let E_{A} ∈ S_{A}^{+} and suppose the products are taken in the box order. Then E_{A} = 0 if |A| > d. Similarly if F_{C} ∈ S_{A}^{−} and the products are taken in the reverse box order, then F_{C} = 0 if |C| > d.

Proof. Let E_{A} ∈ S_{A}^{+} be given in the box order with |A| > d. For each j = 2, …, n set

E_{A_{j}} = ∏_{i=1}^{j−1} X_{j−i,j}^{(A_{j−i,j})},

where we write X_{j−i,j} (resp., A_{j−i,j}) short for X_{ε_{j−i}−ε_{j}} (resp., A_{ε_{j−i}−ε_{j}}). Then E_{A} = E_{A_{n}} ⋯ E_{A_{2}}. According to Proposition 7.9, if E_{A_{j}}1_{λ} ≠ 0, then E_{A_{j}}1_{λ} = 1_{μ}E_{A_{j}} where μ_{j} = λ_{j} − ∑_{r=1}^{j−1} A_{j−r,j} and μ_{s} = λ_{s} if s > j. Therefore for E_{A}1_{λ} ≠ 0 it is necessary that λ_{j} ≥ ∑_{r=1}^{j−1} A_{j−r,j} for all j = 2, …, n, and thus

∑_{j=2}^{n} λ_{j} ≥ ∑_{j=2}^{n} ∑_{r=1}^{j−1} A_{j−r,j} = |A|.

This inequality cannot be satisfied since Σ_{j=2}^{n}λ_{j} ≤ d and |A| > d. Therefore E_{A}1_{λ} = 0 for all λ ∈ Λ(n, d). This forces E_{A} = 0, since 1 = ∑ 1_{λ}. This completes the proof for the claim about the plus part. The proof for the other claim is similar.

Proposition 11.2. Fix an order on Φ^{+}. The set of E_{A} (resp., F_{A}) such that |A| ≤ d, with products of factors taken in the designated order, is an A-basis of S_{A}^{+} (resp., S_{A}^{−}).

Proof. First consider the box order on Φ^{+}. From Lusztig [Lu2, §4.7] we know that the set of all E_{A} (products taken in the box order) is an A-basis of U^{+}. Thus, by Proposition 11.1, the set Γ consisting of those E_{A} satisfying the condition |A| ≤ d is an A-spanning set of S_{A}^{+}. The cardinality of this spanning set is equal to dim_{ℚ(v)} S^{+}, so it follows that it forms a basis (over ℚ(v)) of S^{+}. Hence the elements in the set are linearly independent over ℚ(v), and thus also over A. Hence they form an A-basis of S_{A}^{+}.

Now we form products E_{A} with respect to an arbitrary (but fixed) order on Φ^{+}. By [Xi2, Theorem 2.4] the set of such products spans S_{A}^{+}. But the commutation formulas (8.2) do not increase degree; hence when we express an element of Γ as an A-linear combination of E_{A}’s (in the given fixed order on factors) the degree cannot increase. It follows that the set of E_{A} (|A| ≤ d) must span S_{A}^{+} (over A). It follows that this set is also an A-basis of S_{A}^{+}.

This proves the statement for the plus part. The proof for the minus part is similar.

Next we consider the Borel Schur algebras S_{A}^{⩾0} = S_{A}^{0}S_{A}^{+} and S_{A}^{⩽0} = S_{A}^{0}S_{A}^{−}.

Proposition 11.3. Fix an order on Φ^{+}. With products taken in the specified order, the set {E_{A}1_{λ} | χ(E_{A}) ⪯ λ} (resp., {1_{λ}F_{C} | χ(F_{C}) ⪯ λ}) is an A-basis of S_{A}^{⩾0} (resp., S_{A}^{⩽0}).

Proof. From the preceding result and the decomposition S_{A}^{⩾0} = S_{A}^{+}S_{A}^{0} we see that the set of all E_{A}1_{λ} spans S_{A}^{⩾0}. We can argue as in the proof of Proposition 8.7 that with the restriction χ(E_{A}) ⪯ λ the set still spans. Clearly this set is linearly independent over A since it is a subset of an A-basis of S_{A}. This proves the first statement. The proof of the other case is similar.

Remark 11.4. We conjecture that the set of all elements of the form E_{A}H_{B} (A ∈ ℕ^{Φ^{+}}, B ∈ ℕ_{i_{0}}^{n}) satisfying |A| + |B| ≤ d is an A-basis of S_{A}^{⩾0}, with a similar statement applying to S_{A}^{⩽0}.

Now we consider an application to Hecke algebras. Suppose that n ≥ d. Let ω = (1^{d}). Then the subalgebra 1_{ω}S(n, d)1_{ω} is isomorphic with the Hecke algebra H = H(Σ_{d}). If E_{A}1_{λ}F_{C} (χ(E_{A}F_{C}) ⪯ λ) is any basis element of S then by Propositions 7.4 and 7.9 we see that 1_{ω}E_{A}1_{λ}F_{C}1_{ω} = 0 unless

λ + ∑_{α∈Φ^{+}} A(α)α = ω = λ + ∑_{α∈Φ^{+}} C(α)α, (11.5)

in which case 1_{ω}E_{A}1_{λ}F_{C}1_{ω} = E_{A}1_{λ}F_{C} = 1_{ω}E_{A}F_{C}1_{ω} = 1_{ω}E_{A}F_{C} = E_{A}F_{C}1_{ω}. Therefore the nonzero elements E_{A}1_{λ}F_{C} of Y_{+} satisfying condition (11.5) comprise an A-basis of H. There is a similar basis for H as a subset of the basis Y_{−}.

Taking d = n, we can see that H is generated by the elements t_{i} = 1_{ω}E_{i}F_{i}1_{ω} (1 ≤ i ≤ n − 1). One can check directly from relations (Q1)–(Q7) and the propositions in §7 that these generators satisfy the following relations:

t_{i}^{2} = [2] t_{i} (H1) t_{i}t_{j} = t_{j}t_{i} (|i − j| > 1) (H2) t_{i}t_{i+1}t_{i} − t_{i+1}t_{i}t_{i+1} = t_{i} − t_{i+1}. (H3)

Setting e_{i} = t_{i}/[2] and putting q = v^{2} establishes the equivalence of the above presentation with the presentation in terms of generators e_{i} given in Wenzl [We, §2].

Note that (with q = v^{2}) the elements T_{i} = v^{2}−vt_{i} satisfy the relations

T_{i}^{2} = (q − 1)T_{i} + q (H1′)

(H2′)  T_{i}T_{j} = T_{j}T_{i} (|i − j| > 1)
(H3′)  T_{i}T_{i+1}T_{i} = T_{i+1}T_{i}T_{i+1}

which is the “usual” presentation of H.

Note also that by (Q2) and Proposition 7.6(a) we have 1_{ω}E_{i}F_{i}1_{ω} = 1_{ω}F_{i}E_{i}1_{ω}, so the alternative ordering of the basis elements of the Schur algebra does not yield another presentation of H.

Example 11.6. The easiest way to write down the basis elements in the basis Y_{+} (Y_{−} is similar) uses the alternate description in the qanalogue of (4.7), with the idempotent on the right. Once one has the basis elements, it is a simple matter to rewrite them with the idempotent anywhere one pleases using 7.9 repeatedly.

Given n, d and λ ∈ Λ(n, d) set Y_{+}(λ) = {E_{A}F_{C}1_{λ} | χ_{R}(E_{A}F_{C}) ⪯ λ}, so that Y_{+} = ⋃_{λ} Y_{+}(λ) (disjointly). The partition pieces Y_{+}(λ) are obtainable as follows. One must choose orders on the factors in E_{A}F_{C} (the two parts can be ordered independently). Once that is done, then for each j = 1 …, n one writes out the set of monomials in variables X_{ij} (i ≠ j) of total degree not exceeding λ_{j}. (As before, we write X_{ij} short for X_{α} with α = ε_{i} − ε_{j}.) Then one takes the ordered Cartesian product of these sets over j, respecting the given order, with the factor 1_{λ} at the right. (From this it is easy to write a formula for |Y_{+}(λ)| as a product of binomial coefficients.)

Note that it suffices to describe Y_{+}(λ) just for dominant λ ∈ Λ(n, d), since the sets indexed by non-dominant λ can be obtained from the dominant one in its orbit by applying the appropriate permutation to the indices (and then reordering the product to conform to the specified orders on factors of E_{A}, F_{C}, if necessary). We list below the elements E_{A}F_{C} such that E_{A}F_{C}1_{λ} ∈ Y_{+}(λ), for dominant λ. The elements corresponding with basis elements of the Hecke algebra are underlined.

For S_{A}(2, 2) the elements in question are

1_{(2,0)} : {1, X_{21}, X_{21}^{(2)}}
1_{(1,1)} : \{\underline{1}, X_{12}, X_{21}, \underline{X_{12}X_{21}}\}.

Thus dim S(2, 2) = 2 · 3 + 4 = 10. (There are two sets in the (2, 0) orbit, each of cardinality 3.)

For S_{A}(3, 3) we fix the order (12) ≺ (13) ≺ (23) ≺ (21) ≺ (31) ≺ (32). Then the sets are determined by the elements

1_{(3,0,0)} : {1, X_{21}, X_{31}, X_{21}^{(2)}, X_{31}^{(2)}, X_{21}X_{31}, X_{21}^{(3)}, X_{31}^{(3)}, X_{21}^{(2)}X_{31}, X_{21}X_{31}^{(2)}}
1_{(2,1,0)} : {1, X_{12}, X_{21}, X_{31}, X_{32}, X_{12}X_{21}, X_{12}X_{31}, X_{21}^{(2)}, X_{31}^{(2)}, X_{21}X_{31},
X_{21}X_{32}, X_{31}X_{32}, X_{12}X_{21}^{(2)}, X_{12}X_{31}^{(2)}, X_{12}X_{21}X_{31}, X_{21}^{(2)}X_{32},
X_{31}^{(2)}X_{32}, X_{21}X_{31}X_{32}}
1_{(1,1,1)} : \{\underline{1}, X_{12}, X_{13}, X_{23}, X_{21}, X_{31}, X_{32}, X_{12}X_{13}, X_{12}X_{23}, \underline{X_{12}X_{21}},

X_{12}X_{31}, X_{13}X_{21}, \underline{X_{13}X_{31}}, X_{13}X_{32}, X_{23}X_{21}, X_{23}X_{31}, \underline{X_{23}X_{32}},

X_{21}X_{32}, X_{31}X_{32}, X_{12}X_{13}X_{21}, X_{12}X_{13}X_{31}, X_{12}X_{23}X_{21},

\underline{X_{12}X_{23}X_{31}}, \underline{X_{13}X_{21}X_{32}}, X_{13}X_{31}X_{32}, X_{23}X_{21}X_{32}, X_{23}X_{31}X_{32}\}.

Note that for the λ = (2, 1, 0) case we took the ordered Cartesian product of {1, X_{12}, X_{32}} with {1, X_{21}, X_{31}, X_{21}^{(2)}, X_{31}^{(2)}, X_{21}X_{31}} and for the λ = (1, 1, 1) case we computed the ordered Cartesian product of the sets {1, X_{13}, X_{23}}, {1, X_{12}, X_{32}}, and {1, X_{21}, X_{31}}. There are 10 elements in the (3, 0, 0) piece, 18 in the (2, 1, 0) piece, and 27 in the (1, 1, 1) piece. There are 3 pieces in the (3, 0, 0)-orbit, 6 in the (2, 1, 0)- orbit, and 1 in the (1, 1, 1)-orbit. Thus dim S(3, 3) = 3·10+6·18+27 = 165.

References

[BLM] A.A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of GL_{n}, Duke Math. J. 61 (1990), 655–677.

[CL] R.W. Carter and G. Lusztig, On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193–242.

[DJ1] R. Dipper and G.D. James, The q-Schur algebra, Proc. London Math. Soc. 59 (1989), 23–50.

[DJ2] R. Dipper and G.D. James, q-tensor space and q-Weyl modules, Trans. Amer. Math. Soc. 327 (1991), 251–282.

[Do] S. Donkin, On Schur algebras and related algebras III: integral representations, Math. Proc. Cambridge Philos. Soc. 116 (1994), 37–55.

[DG1] S. Doty and A. Giaquinto, Presenting Schur algebras as quotients of the universal enveloping algebra of 𝔤𝔩_{2}, Algebras and Representation Theory, to appear.

[DG2] S. Doty and A. Giaquinto, Presenting quantum Schur algebras as quotients of the quantized enveloping algebra of 𝔤𝔩_{2}, preprint, Loyola University Chicago, December 2000.

[DG3] S. Doty and A. Giaquinto, Generators and relations for Schur algebras, Electronic Research Announc. Amer. Math. Soc. 7 (2001), 54–62.

[Du] Jie Du, A note on quantized Weyl reciprocity at roots of unity, Algebra Colloq. 2 (1995), 363–372.

[Gr] J. A. Green, Polynomial Representations of GL_{n}, (Lecture Notes in Math. 830), Springer-Verlag, New York 1980.

[RG1] R.M. Green, Ph.D. thesis, University of Warwick, 1995.

[RG2] R.M. Green, q-Schur algebras as quotients of quantized enveloping algebras, J. Algebra 185 (1996), 660–687.

[Ji] M. Jimbo, A q-analogue of U(𝔤𝔩(N + 1)), Hecke algebra, and the Yang-Baxter equation, Letters Math. Physics 11 (1986), 247–252.

[Ko] B. Kostant, Groups over ℤ, Proc. Symposia Pure Math. 9 (1966), 90–98.

[Lu1] G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Advances in Math. 70 (1988), 237–249.

[Lu2] G. Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 1, 257–296.

[Lu3] G. Lusztig, Introduction to Quantum Groups, Birkhäuser Boston 1993.

[Ro] M. Rosso, An analogue of P.B.W. theorem and the universal R-matrix for U_{h}sl(N + 1). Comm. Math. Phys. 124 (1989), no. 2, 307–318.

[Sc] I. Schur, Über die rationalen Darstellungen der allgemeinen linearen Gruppe, 1927; reprinted in: I. Schur, Gesammelte Abhandlungen, Vol. III, pp. 68–85, Springer-Verlag, Berlin, 1973.

[We] H. Wenzl, Hecke algebras of type A_{n} and subfactors, Invent. Math. 92 (1988), 349–383.

[Xi1] Nanhua Xi, Root vectors in quantum groups, Comment. Math. Helv. 69 (1994), 612–639.

[Xi2] Nanhua Xi, A commutation formula for root vectors in quantized enveloping algebras, Pacific J. Math. 189 (1999), no. 1, 179–199.

Loyola University Chicago, Chicago, Illinois 60626 U.S.A.

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