Presenting quantum Schur algebras as quotients of the quantized universal enveloping

algebra of 𝔤𝔩_{2}

Stephen Doty and Anthony Giaquinto

December 1, 2000

Abstract

We obtain a presentation of the quantum Schur algebras S_{v}(2, d) by generators and relations. This presentation is compatible with the usual presentation of the quantized enveloping algebra U = U_{v}(𝔤𝔩_{2}). In the process we find new bases for S_{v}(2, d). We also locate the ℤ[v, v^{−1}]-form of the quantum Schur algebra within the presented algebra and show that it has a basis which is closely related to Lusztig’s basis of the ℤ[v, v^{−1}]-form of U.

1 Introduction

In [DG] we gave a description of the rational Schur algebra S_{ℚ}(2, d) in terms of generators and relations. This description is compatible with the usual presentation of the universal enveloping algebra U(𝔤𝔩_{2}). We also described the integral Schur algebra S_{ℤ}(2, d) as a certain subalgebra of the rational version and an integral basis was exhibited. In this paper, we formulate and prove quantum versions of those results.

Consider the Drinfeld-Jimbo quantized enveloping algebra U corresponding to the Lie algebra 𝔤𝔩_{2}. It has a natural two-dimensional module E, and thus the tensor product E^{⊗d} is also a module for U. Let

ρ_{d} : U → End(E^{⊗d})

be the corresponding representation. Amongst the various equivalent definitions of the quantum Schur algebra S_{v}(2, d) is that it is precisely the image of the homomorphism ρ_{d}. (We shall not need to consider the other definitions in this work.) Since S_{v}(2, d) is a homomorphic image of U, it is natural to ask for an efficient generating set of Ker(ρ_{d}), thereby giving a presentation of S_{v}(2, d).

In what follows, we obtain a precise answer to this question. Recall that U is generated by elements e, f, K_{1}^{±1}, K_{2}^{±1} subject to various well-known relations (see section 3). Now in the representation ρ_{d}, it is easy to see that K_{1}K_{2} = v^{d}, so we may use this relation to eliminate K_{2} (or K_{1}) from the generating set for S_{v}(2, d). Having done this, then our main result is that the only additional relation needed to give the desired presentation of S_{v}(2, d) is the minimal polynomial of K_{1} in End(E^{⊗d}).

Set A = ℤ[v, v^{−1}]. The algebra U contains a certain A-subalgebra U_{A} which may be viewed as a quantum version of the integral form U_{ℤ}(𝔤𝔩_{2}) of the classical enveloping algebra. The algebra U_{A}, originally constructed by Lusztig, is generated by the v-divided powers of e and f along with K_{1}^{±1} and K_{2}^{±1}. It has an A-basis which is a quantum analog of Kostant’s basis for the integral form of the classical enveloping algebra U(𝔤𝔩_{2}). The image of the homomorphism ρ_{d} upon restriction to U_{A} gives us an “integral” Schur algebra S_{A}(2, d), which can be used to define a version over any A-algebra. We show that the integral Schur algebra has a basis which is closely related to Lusztig’s basis of U_{A}.

Although the results of this paper are quantum versions of those appearing in [DG], the techniques are somewhat different, since some of the arguments given in that paper did not quantize directly. In particular, the treatment here of the degree zero part (generated by the images of K_{1}^{±1} and K_{2}^{±1}) of S_{v}(2, d) is totally different. Specifically, we exhibit an idempotent basis of this subalgebra, whereas in [DG] a PBW-type basis of the degree zero part was used. The idempotent basis is more amenable to computations and is precisely the kind of basis needed to handle the general Schur algebras S(n, d) and their quantizations. Another difference between the results of this paper and the results of [DG] is that here we do not obtain analogues of the “restricted PBW-basis”, although we believe such results should be true in the quantum case.

Finally, it is clear that analogous results hold in general, for any n and d, although one cannot expect to obtain such precise reduction formulas in the general case as those given here and in [DG]. The authors expect to treat the general case in a later paper.

2 Statement of results

The main results of this paper are contained in the following theorems. Let A = ℤ[v, v^{−1}] with fraction field ℚ(v). Each of the first three theorems gives a presentation of the quantum Schur algebra S_{v}(2, d) in terms of generators and relations. The first result gives a presentation which is similar to that of U_{v}(𝔰𝔩_{2}).

2.1 Theorem Over ℚ(v), the quantum Schur algebra S_{v}(2, d) is isomorphic to the algebra generated by e, f, K^{±1} subject to the relations:

KK^{−1} = K^{−1}K = 1, (a) KeK^{−1} = v^{2}e, KfK^{−1} = v^{−2}f, (b)

ef − fe = \frac{K − K^{−1}}{v − v^{−1}} , (c)

(K − v^{d})(K − v^{d−2}) ⋯ (K − v^{−d+2})(K − v^{−d}) = 0. (d)

The next two results give presentations of S_{v}(2, d) which are similar to that of U_{v}(𝔤𝔩_{2}).

2.2 Theorem Over ℚ(v), the quantum Schur algebra S_{v}(2, d) is isomorphic to the algebra generated by e, f, K_{1}^{±1} subject to the relations:

K_{1}K_{1}^{−1} = K_{1}^{−1}K_{1} = 1, (a) K_{1}eK_{1}^{−1} = ve, K_{1}fK_{1}^{−1} = v^{−1}f (b)

ef − fe = \frac{v^{−d}K_{1}^{2} − v^{d}K_{1}^{−2}}{v − v^{−1}} , (c)

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

By a change of variable (K_{2} = v^{d}K_{1}^{−1}) we obtain another equivalent presentation of S_{v}(2, d).

2.3 Theorem Over ℚ(v), the quantum Schur algebra S_{v}(2, d) is isomorphic to the algebra generated by e, f, K_{2}^{±1} subject to the relations:

K_{2}K_{2}^{−1} = K_{2}^{−1}K_{2} = 1, (a) K_{2}eK_{2}^{−1} = v^{−1}e, K_{2}fK_{2}^{−1} = vf (b)

ef − fe = \frac{v^{d}K_{2}^{−2} − v^{−d}K_{2}^{2}}{v − v^{−1}} , (c)

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

For indeterminates X, X^{−1} satisfying XX^{−1} = X^{−1}X = 1 and any t ∈ ℕ we formally set

[\begin{matrix}X \\ t \end{matrix}] = ∏_{s=1}^{t} \frac{Xv^{−s+1} − X^{−1}v^{s−1}}{v^{s} − v^{−s}} .

This expression will make sense if X is replaced by any invertible element of a ℚ(v)-algebra. The next result describes the A-form S_{A}(2, d) in terms of the above generators, and gives an A-basis for the algebra. In this we can take S_{v}(2, d) to be given by either presentation 2.2 or 2.3, but we always assume that K_{1} and K_{2} are related by the condition K_{1}K_{2} = v^{d}.

2.4 Theorem The integral Schur algebra S_{A}(2, d) is isomorphic to the A-subalgebra of S_{v}(2, d) generated by

e^{(m)} := \frac{e^{m}}{[m]!}, f^{(m)} := \frac{f^{m}}{[m]!} (m ∈ ℕ), K_{1}^{±1}.

The preceding statement is true when K_{1} is replaced by K_{2}. Moreover, an A-basis for S_{A}(2, d) is the set consisting of all

e^{(a)} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] f^{(c)}

such that the natural numbers a, b_{1}, b_{2}, c are constrained by the conditions a + b_{1} + c ≤ d, b_{1} + b_{2} = d. Another such basis consists of all

f^{(a)} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] e^{(c)}

such that the natural numbers a, b_{1}, b_{2}, c satisfy the constraints a + b_{2} + c ≤ d, b_{1} + b_{2} = d.

The following reduction formulas make it possible, in principle, to express the product of two basis elements as an A-linear combination of basis elements.

2.5 Theorem In S_{A}(2, d) we have the following reduction formulas for all s ≥ 1 and all a, b_{1}, b_{2}, c ∈ ℕ with b_{1} + b_{2} = d:

\begin{matrix}e^{(a)} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] f^{(c)} = \\ ∑_{k=s}^{min(a,c)}(−1)^{k−s} & [\begin{matrix}k − 1 \\ s − 1 \end{matrix}] [\begin{matrix}b_{1} + k \\ k \end{matrix}] e^{(a−k)} [\begin{matrix}K_{1} \\ b_{1} + k \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} − k \end{matrix}] f^{(c−k)} \end{matrix} (a)

\begin{matrix}f^{(a)} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] e^{(c)} = \\ ∑_{k=s}^{min(a,c)}(−1)^{k−s} & [\begin{matrix}k − 1 \\ s − 1 \end{matrix}] [\begin{matrix}b_{2} + k \\ k \end{matrix}] f^{(a−k)} [\begin{matrix}K_{1} \\ b_{1} − k \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} + k \end{matrix}] e^{(c−k)} \end{matrix} (b)

where s = a + b_{1} + c − d in (a) and s = a + b_{2} + c − d in (b).

The next theorem, the quantum analogue of [DG, Thm. 2.4], provides yet another kind of basis for S_{A}(2, d). In this case we will deduce it as a direct consequence of Theorem 2.4, while in [DG] the analogue of the idempotent basis in Theorem 2.4 was not needed. We remark that the analogues of the integral idempotent bases in Theorem 2.4 above do hold in the classical situation.

2.6 Theorem The set

\{e^{(a)} [\begin{matrix}K_{1} \\ b \end{matrix}] f^{(c)} | a, b, c ∈ ℕ, a + b + c ≤ d\}

is an A-basis for S_{A}(2, d). Another such basis is given by the set

\{f^{(a)} [\begin{matrix}K_{2} \\ b \end{matrix}] e^{(c)} | a, b, c ∈ ℕ, a + b + c ≤ d\} .

3 Quantized enveloping algebras

3.1 The Drinfeld-Jimbo quantized enveloping algebra U = U_{v}(𝔤𝔩_{2}) is defined to be the ℚ(v)-algebra with generators e, f, K_{1}, K_{1}^{−1}, K_{2}, K_{2}^{−1} and relations

K_{1}K_{2} = K_{2}K_{1}, (a) K_{i}K_{i}^{−1} = K_{i}^{−1}K_{i} = 1 (i = 1, 2), (b) K_{1}eK_{1}^{−1} = ve, K_{1}fK_{1}^{−1} = v^{−1}f, (c) K_{2}eK_{2}^{−1} = v^{−1}e, K_{2}fK_{1}^{−1} = vf, (d)

ef − fe = \frac{K_{1}K_{2}^{−1} − K_{1}^{−1}K_{2}}{v − v^{−1}} . (e)

Let U^{+} (respectively U^{−}) be the subalgebra generated by e (respectively f) and let U^{0} be the subalgebra generated by K_{1}^{±1}, K_{2}^{±1}. There are ℚ(v)-vector space isomorphisms

U ≅ U^{+} ⊗ U^{0} ⊗ U^{−} ≅ U^{−} ⊗ U^{0} ⊗ U^{+}

and moreover it is well-known that the algebra U has PBW-type bases {e^{a}K_{1}^{b_{1}}K_{2}^{b_{2}}f^{c}} and {f^{a}K_{1}^{b_{1}}K_{2}^{b_{2}}e^{c}} where a, c ∈ ℕ and b_{1}, b_{2} ∈ ℤ.

In the algebra U, set K = K_{1}K_{2}^{−1}, and define U_{v}(𝔰𝔩_{2}) to be the subalgebra of U generated by e, f, and K^{±1}. The familiar relations that these elements satisfy are easily deducible from (a) - (e).

3.2 For r, s ∈ ℤ define

[r] = \frac{v^{r} − v^{−r}}{v − v^{−1}} (a)

[\begin{matrix}r \\ s \end{matrix}] = \frac{[r][r − 1] ⋯ [r − s + 1]}{[1][2] ⋯ [s]}. (b)

These satisfy the well-known identities

[r + s] = v^{−s}[r] + v^{r}[s] (c)

[\begin{matrix}r + 1 \\ s \end{matrix}] = v^{−s} [\begin{matrix}r \\ s \end{matrix}] + v^{r−s+1} [\begin{matrix}r \\ s − 1 \end{matrix}] . (d)

For every m, t ∈ ℕ, c ∈ ℤ and any element X an ℚ(v)-algebra define [m]! = [m][m − 1] ⋯ [1] (e)

X^{(m)} = \frac{X^{m}}{[m]!} (f)

(g) [\begin{matrix}X; c \\ t \end{matrix}] = \begin{cases}∏_{i=1}^{t} \frac{Xv^{c−i+1} − X^{−1}v^{−c+i−1}}{v^{i} − v^{−i}} if t ≠ 0 \\ 1 if t = 0. \end{cases} (X invertible)

We note that [\begin{matrix}X \\ t \end{matrix}], as defined in section 2, coincides with the element [\begin{matrix}X; 0 \\ t \end{matrix}].

In [Lu], Lusztig investigates many relations which hold in quantized enveloping algebras. Those which we will need are contained in the following Lemma. 3.3 Lemma For any c, n ∈ ℤ and m, t, t′ ∈ ℕ the following identities hold in U: K_{1}^{n} e K_{1}^{−n} = v^{n} e, K_{2}^{n} e K_{2}^{−n} = v^{−n} e, (a) K_{1}^{n} f K_{1}^{−n} = v^{−n} f, K_{2}^{n} f K_{2}^{−n} = v^{n} f, (b)

[\begin{matrix}K_{1}; c \\ t \end{matrix}] e = e [\begin{matrix}K_{1}; c + 1 \\ t \end{matrix}] , [\begin{matrix}K_{1}; c \\ t \end{matrix}] f = f [\begin{matrix}K_{1}; c − 1 \\ t \end{matrix}] , (c)

[\begin{matrix}K_{2}; c \\ t \end{matrix}] e = e [\begin{matrix}K_{2}; c − 1 \\ t \end{matrix}] , [\begin{matrix}K_{2}; c \\ t \end{matrix}] f = f [\begin{matrix}K_{2}; c + 1 \\ t \end{matrix}] , (d)

f^{(m)} e = e f^{(m)} − [\begin{matrix}K_{1}K_{2}^{−1}; m − 1 \\ 1 \end{matrix}] f^{(m−1)}, (e)

f e^{(m)} = e^{(m)} f − e^{(m−1)} [\begin{matrix}K_{1}K_{2}^{−1}; m − 1 \\ 1 \end{matrix}] , (f)

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

[\begin{matrix}K_{i} \\ t \end{matrix}] [\begin{matrix}K_{i}; −t \\ t′ \end{matrix}] = [\begin{matrix}t + t′ \\ t \end{matrix}] [\begin{matrix}K_{i} \\ t + t′ \end{matrix}] , (h)

[\begin{matrix}K_{i}; c \\ t \end{matrix}] = ∑_{j=0}^{t} v^{c(t−j)} [\begin{matrix}c \\ j \end{matrix}] K_{i}^{−j} [\begin{matrix}K_{i} \\ t − j \end{matrix}] (c ≥ 0). (i)

Proof. All of these identities are special cases of those appearing on pages 269-270 of [Lu]. □

4 The algebra B_{d}

In this section we define a homomorphic image of U which will turn out to be isomorphic to the quantum Schur algebra S_{v}(2, d).

4.1 Let d be a fixed nonnegative integer and define B_{d} to be the ℚ(v)- algebra generated by e, f, K_{1}^{±1}, K_{2}^{±1} subject to relations 3.1(a)-(e), along with the additional relations

K_{1}K_{2} = v^{d} (a)

(K_{1} − 1)(K_{1} − v) ⋯ (K_{1} − v^{d}) = 0 (b)

Note that relation (b) can be replaced by the equivalent relation

(K_{1}^{−1} − 1)(K_{1}^{−1} − v^{−1}) ⋯ (K_{1}^{−1} − v^{−d}) = 0 (c)

and in the presence of relation (a) it can also be replaced by either

(K_{2} − 1)(K_{2} − v) ⋯ (K_{2} − v^{d}) = 0, or (d)

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

The defining relations of B_{d} are invariant if e and f are interchanged along with K_{1} and K_{2}. These interchanges therefore induce an automorphism of B_{d}. We shall often make use of this property, which we will call symmetry, in the sequel.

Let B_{d}^{0} be the subalgebra of B_{d} generated by K_{1}^{±1}, K_{2}^{±1}. It follows from relations 4.1(a)-(b) that dim(B_{d}^{0}) = d + 1.

4.2 Lemma In the algebra B_{d}^{0} we have

[\begin{matrix}K_{1} \\ d + 1 \end{matrix}] = 0 = [\begin{matrix}K_{2} \\ d + 1 \end{matrix}] , [\begin{matrix}K_{1}^{−1} \\ d + 1 \end{matrix}] = 0 = [\begin{matrix}K_{2}^{−1} \\ d + 1 \end{matrix}] .

Proof. Using definition 3.2(g) we have that

[\begin{matrix}K_{1} \\ d + 1 \end{matrix}] = ∏_{i=1}^{d+1} \frac{K_{1}v^{1−i} − K_{1}^{−1}v^{i−1}}{v^{i} − v^{−i}}

= ∏_{i=1}^{d+1} \frac{(K_{1}^{−1}v^{1−i})(K_{1}^{2} − v^{2(i−1)})}{v^{i} − v^{−i}}

= ∏_{i=1}^{d+1} \frac{(K_{1}^{−1}v^{1−i})(K_{1} − v^{(i−1)})(K_{1} + v^{(i−1)})}{v^{i} − v^{−i}}

= 0 by relation 4.1(b).

The other equalities follow in a similar manner using 4.1(c)-(e). □ More generally we have

4.3 Lemma In the algebra B_{d}^{0} we have

[\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] = 0

whenever b_{1} + b_{2} = d + 1.

Proof.

[\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] = ∏_{i=1}^{b_{1}} \frac{K_{1}v^{1−i} − K_{1}^{−1}v^{i−1}}{v^{i} − v^{−i}} ∏_{i=1}^{b_{2}} \frac{K_{2}v^{1−i} − K_{2}^{−1}v^{i−1}}{v^{i} − v^{−i}}

= ∏_{i=1}^{b_{1}} \frac{K_{1}v^{1−i} − K_{1}^{−1}v^{i−1}}{v^{i} − v^{−i}} ∏_{i=1}^{b_{2}} \frac{K_{1}^{−1}v^{d+1−i} − K_{2}v^{−d+i−1}}{v^{i} − v^{−i}}

= (−1)^{b_{2}} [\begin{matrix}d + 1 \\ b_{2} \end{matrix}] [\begin{matrix}K_{1} \\ d + 1 \end{matrix}]

= 0 by 4.1(b).

□

An immediate consequence of the preceding lemma is that

[\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] = 0

whenever b_{1} + b_{2} ≥ d + 1. When b_{1} + b_{2} ≤ d, these elements are non-zero, but we shall see that the most important case is b_{1} + b_{2} = d.

4.4 Lemma Suppose that b_{1} + b_{2} = d and t ∈ ℕ. Then for i = 1, 2 the following identities hold in B_{d}^{0}:

K_{i} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] = v^{b_{i}} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] (a)

[\begin{matrix}K_{i}; c \\ t \end{matrix}] [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] = [\begin{matrix}b_{i} + c \\ t \end{matrix}] [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] (b)

Proof. The second equality of the lemma follows immediately from the first using definition 3.2(g), and so we only need to prove (a). Consider the case i = 1. We have

(K_{1} − v^{b_{1}}) [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}]

= (K_{1} − v^{b_{1}}) ∏_{i=1}^{b_{1}} \frac{K_{1}v^{1−i} − K_{1}^{−1}v^{i−1}}{v^{i} − v^{−i}} ∏_{i=1}^{b_{2}} \frac{K_{2}v^{1−i} − K_{2}^{−1}v^{i−1}}{v^{i} − v^{−i}}

= (−1)^{b_{2}}(K_{1} − v^{b_{1}}) ∏_{\begin{matrix}i=1 \\ i≠b_{1}+1 \end{matrix}}^{d+1} \frac{K_{1}v^{1−i} − K_{1}^{−1}v^{i−1}}{v^{i} − v^{−i}}

= (−1)^{b_{2}} (K_{1} − v^{b_{1}}) ∏_{\begin{matrix}i=1 \\ i≠b_{1}+1 \end{matrix}}^{d+1} \frac{v^{1−i}K_{1}^{−1}(K_{1}^{2} − v^{2(i−1)})}{v^{i} − v^{−i}}

= 0 by 4.1(b).

This proves identity (a) for i = 1. The case i = 2 follows from symmetry. □

The following result establishes the structure of the algebra B_{d}^{0}; it will later be crucial in determining the structure of the entire algebra B_{d}.

4.5 Theorem In the algebra B_{d}^{0}, the set

\{[\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] | b_{1} + b_{2} = d\}

is a basis of mutually orthogonal idempotents whose sum is the identity.

Proof. By Lemma 4.4 we have that

([\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}])^{2} = [\begin{matrix}b_{1} \\ b_{1} \end{matrix}] [\begin{matrix}b_{2} \\ b_{2} \end{matrix}] [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] = [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}]

and so each [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] is an idempotent. Now suppose that b_{1} + b_{2} =

b′_{1} + b′_{2} = d and b_{1} ≠ b′_{1}. Then either b_{1} + b′_{2} ≥ d + 1 or b′_{1} + b_{2} ≥ d + 1 and so orthogonality follows by Lemma 4.3. Thus in the algebra B_{d}^{0} (which has dimension d + 1), we have a set of d + 1 distinct mutually orthogonal idempotents. It follows that these must form a basis and that their sum is the identity. □

These idempotents have pleasant commutation relations with the elements e and f, given in the next lemma.

4.6 Lemma Suppose that b_{1} + b_{2} = d and a ∈ ℕ. Then in the algebra B_{d}

[\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] e^{a} = \begin{cases}e^{a} [\begin{matrix}K_{1} \\ b_{1} − a \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} + a \end{matrix}] if b_{1} ≥ a \\ 0 if b_{1} < a \end{cases} (a)

e^{a} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] = \begin{cases}[\begin{matrix}K_{1} \\ b_{1} + a \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} − a \end{matrix}] e^{a} if b_{2} ≤ a \\ 0 if b_{2} > a \end{cases} (b)

f^{a} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] = \begin{cases}[\begin{matrix}K_{1} \\ b_{1} − a \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} + a \end{matrix}] f^{a} if b_{1} ≥ a \\ 0 if b_{1} < a. \end{cases} (c)

[\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] f^{a} = \begin{cases}f^{a} [\begin{matrix}K_{1} \\ b_{1} + a \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} − a \end{matrix}] if b_{2} ≤ a \\ 0 if b_{2} > a. \end{cases} (d)

Proof. Each of the relations is similar to prove, so we will only verify the first one. By 3.3(c) and 3.3(d) we have

[\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] e^{a} = e^{a} [\begin{matrix}K_{1}; a \\ b_{1} \end{matrix}] [\begin{matrix}K_{2}; −a \\ b_{2} \end{matrix}]

and thus

[\begin{matrix}b_{2} + a \\ a \end{matrix}]^{−1} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] e^{a} [\begin{matrix}K_{2} \\ a \end{matrix}] = [\begin{matrix}b_{2} + a \\ a \end{matrix}]^{−1} e^{a} [\begin{matrix}K_{1}; a \\ b_{1} \end{matrix}] [\begin{matrix}K_{2}; −a \\ b_{2} \end{matrix}] [\begin{matrix}K_{2} \\ a \end{matrix}] .

Now using 3.3(d) and 3.3(h), this equality becomes

[\begin{matrix}b_{2} + a \\ a \end{matrix}]^{−1} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] [\begin{matrix}K_{2}; a \\ a \end{matrix}] e^{a} = [\begin{matrix}b_{2} + a \\ a \end{matrix}]^{−1} e^{a} [\begin{matrix}K_{1}; a \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} + a \end{matrix}] .

Transforming this further using 3.3(i) and 4.4(b) yields

[\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] e^{a} = [\begin{matrix}b_{2} + a \\ a \end{matrix}]^{−1} e^{a} (∑_{j=0}^{b_{1}} v^{a(b_{1}−j)} [\begin{matrix}a \\ j \end{matrix}] K_{1}^{−j} [\begin{matrix}K_{1} \\ b_{1} − j \end{matrix}]) [\begin{matrix}K_{2} \\ b_{2} + a \end{matrix}] .

Since [\begin{matrix}K_{1} \\ b_{1} − j \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} + a \end{matrix}] = 0 whenever j < a and [\begin{matrix}a \\ j \end{matrix}] = 0 whenever j > a, the only possible non-zero term on the right side of the equality corresponds to the case j = a. Consequently, [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] e^{a} = 0 whenever b_{1} < a and if b_{1} ≥ a then

[\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] e^{a} = v^{a(b_{1}−a)}K_{1}^{−a} [\begin{matrix}K_{1} \\ b_{1} − a \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} + a \end{matrix}] .

The remaining part of claim (a) of the lemma now follows using 4.4(a). □

4.7 Remark An immediate consequence of the preceding lemma is that

e^{a} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] = 0 = [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] f^{a} whenever a + b_{1} ≥ d + 1

and

f^{a} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] = 0 = [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] e^{a} whenever a + b_{2} ≥ d + 1.

In particular, this implies that both e and f are nilpotent of index d + 1. The results obtained thus far show that the sets

\{e^{(a)} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] f^{(c)} | a, c ≤ d, b_{1} + b_{2} = d\}

and

\{f^{(a)} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] f^{(c)} | a, c ≤ d, b_{1} + b_{2} = d\}

are spanning sets for the algebra B_{d}. We will henceforth refer to elements of these spanning sets simply as monomials. These sets, however, are not bases. To establish this fact, we need some terminology. Define the fake

degree of a monomial e^{(a)} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] f^{(c)} (respectively f^{(a)} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] e^{(c)})

to be a + b_{1} + c (respectively a + b_{2} + c) and, in both cases, define its height to be a + c.

4.8 Theorem Suppose that b_{1} + b_{2} = d. Then in the algebra B_{d} all mono-

mials of the form e^{(a)} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] f^{(c)} (respectively f^{(a)} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] e^{(c)}) of

fake degree d + 1 are expressible as ℚ(v)-linear combinations of monomials of the same form of strictly smaller fake degree and height.

Proof. By symmetry it is enough to prove the claim for monomials of the

form e^{(a)} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] f^{(c)}. We use induction on height. The base case here

is height one since there are no height zero monomials of fake degree d + 1. Consider a monomial of fake degree d + 1 and height s ≥ 1. Suppose that a ≥ c. Then a ≥ 1 and by Lemma 4.6 we have

e^{(a−1)} [\begin{matrix}K_{1} \\ b_{1} + 1 \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} − 1 \end{matrix}] e f^{(c)} = [a]e^{(a)} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] f^{(c)}

and so the claim will follow once we show that e^{(a−1)} [\begin{matrix}K_{1} \\ b_{1} + 1 \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} − 1 \end{matrix}] e f^{(c)}

can be expressed in the desired form. But, by 3.3(e) and 4.4(b) we have

e^{(a−1)} [\begin{matrix}K_{1} \\ b_{1} + 1 \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} − 1 \end{matrix}] e f^{(c)}

= e^{(a−1)} [\begin{matrix}K_{1} \\ b_{1} + 1 \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} − 1 \end{matrix}] (f^{(c)}e + [\begin{matrix}K_{1}K_{2}^{−1}; c − 1 \\ 1 \end{matrix}] f^{(c−1)})

= e^{(a−1)} [\begin{matrix}K_{1} \\ b_{1} + 1 \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} − 1 \end{matrix}] f^{(c)}e

+ [b_{1} − b_{2} + c − 1]e^{(a−1)} [\begin{matrix}K_{1} \\ b_{1} + 1 \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} − 1 \end{matrix}] f^{(c−1)}

where the second term on the right hand side is zero if c = 0. However, in general, the second term is a monomial of fake degree d and height s − 2, which is of the desired form. To analyze the first term, we need to use the fact that if M is a monomial of fake degree d′ and height s′, then Me is an ℚ(v)- linear combination of monomials of fake degree at most d′ and height at most s′ + 1 (this claim can be verified using 3.3(e) and 4.4(b)). Now by induction,

e^{(a−1)} [\begin{matrix}K_{1} \\ b_{1} + 1 \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} − 1 \end{matrix}] f^{(c)} is expressible as an ℚ(v)-linear combination of

monomials of fake degree at most d and height at most s − 2. Thus, by our

claim, e^{(a−1)} [\begin{matrix}K_{1} \\ b_{1} + 1 \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} − 1 \end{matrix}] f^{(c)}e is expressible as a linear combination of

terms of fake degree at most d and height at most s − 1. This completes the proof in the case a ≥ c; the case a ≤ c is similar and is omitted. □

5 Identifications

In this section we show that B_{d} is isomorphic to S_{v}(2, d).

5.1 Let E be an ℚ(v)-module with basis {e_{1}, e_{2}}. There is a canonical representation ρ : U → End_{ℚ(v)}(E) defined by:

e ↦ (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) , f ↦ (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}) , K_{1} ↦ (\begin{matrix}v & 0 \\ 0 & 1 \end{matrix}) , K_{2} ↦ (\begin{matrix}1 & 0 \\ 0 & v \end{matrix}) .

Since U is a bialgebra, we obtain a representation

ρ_{d} : U → End_{ℚ(v)}(E^{⊗d})

in the dth tensor power of E, for every d ∈ ℕ. Specifically, ρ_{d} = ρ^{d} ∘ Δ^{d−1}, where Δ^{d−1} : U → U^{⊗d} is the iterated comultiplication map. As stated earlier, S_{v}(2, d) is the image of the homomorphism ρ_{d}.

5.2 Lemma Write \overline{X} for the image of X ∈ U under the representation ρ_{d} : U → End_{ℚ(v)}(E^{⊗d}). Then we have the identities

\overline{K_{1}}\overline{K_{2}} = v^{d} (a)

(\overline{K_{1}} − 1)(\overline{K_{1}} − v) ⋯ (\overline{K_{1}} − v^{d}) = 0 (b)

Proof. We have \overline{K_{1}}\overline{K_{2}} = ρ_{d}(K_{1}K_{2}) = ρ_{1}(K_{1}K_{2})^{⊗d} since Δ(K_{i}) = K_{i}⊗K_{i}. But by 5.1,

ρ_{1}(K_{1}K_{2}) = (\begin{matrix}v & 0 \\ 0 & 1 \end{matrix}) (\begin{matrix}1 & 0 \\ 0 & v \end{matrix}) = v (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) = v1_{V}

and so ρ_{1}(K_{1}K_{2})^{⊗d} = v^{d}1_{V^{⊗d}}, which proves claim (a). The case d = 1 for

claim (b) follows immediately since ρ_{1}(K_{1}) = (\begin{matrix}v & 0 \\ 0 & 1 \end{matrix}). Now ρ_{d}(K_{1}) = K_{1}^{⊗d} is

a diagonal matrix whose entries are 1, v, …, v^{d} (not counting multiplicities). The claim follows. □

Henceforth we shall omit the bar, writing the image of X simply as X instead of \overline{X}.

5.3 Theorem The Schur algebra S_{v}(2, d) is isomorphic as a ℚ(v)-algebra

to B_{d}. Moreover, the set of all monomials e^{(a)} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] f^{(c)} (a + b_{1} + c ≤

d) is a basis over ℚ(v). Similarly, another such basis consists of all the

monomials f^{(a)} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] e^{(c)} (a + b_{2} + c ≤ d).

Proof. By the preceding lemma we see that the surjection ρ_{d} : U → S_{v}(2, d) factors through B_{d}, giving the commutative diagram

\begin{matrix}U & ⟶ & S_{v}(2, d) \\ ↓ & & ↑ \\ B_{d} & & B_{d} \end{matrix} (a)

in which all arrows are surjections. By Theorem 4.8, the set of all monomials

of the form e^{(a)} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] f^{(c)} satisfying a + b_{1} + c ≤ d spans the algebra

B_{d}. This spanning set is in one-to-one correspondence with the set of all monomials in 4 commuting variables of total degree d (set one of the variables equal to 1 to get the correspondence). It is well known that this number is the dimension of S_{v}(2, d). Hence the map B_{d} → S_{v}(2, d) must be an isomorphism and the spanning set is a basis. A similar argument establishes that the set

of all f^{(a)} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] e^{(c)} satisfying a + b_{2} + c ≤ d is also a basis of B_{d}. This

completes the proof. □

5.4 Theorems 2.2 and 2.3 follow immediately from the previous result by using the relation K_{1}K_{2} = v^{d} to remove either K_{1} or K_{2} from the generating set. However, the results of the preceding theorem are not sufficient to establish Theorem 2.4 since we only know that the coefficients in the reduction formulas are elements of ℚ(v), not necessarily of A. In the next section we indeed show that these coefficients all lie in A. The process by which we prove this does not rely on Theorem 5.3 and so the forthcoming results give alternative proofs of Theorems 2.2 and 2.3 as well as a proof of 2.4.

We now prove Theorem 2.1. In the algebra S_{v}(2, d), set K = v^{−d}K_{1}^{2}. Using Lemma 4.4 and Theorem 4.5 we obtain the equality

K_{1}^{2} = v^{d}K = ∑_{b_{1}+b_{2}=d} v^{2b_{1}} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}] = ( ∑_{b_{1}+b_{2}=d} v^{b_{1}} [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}])^{2} .

Thus, e, f, and K^{±1} comprise another generating set for S_{v}(2, d). Relations (a)-(c) of Theorem 2.1 are easily seen to be equivalent to the corresponding parts of Theorem 2.2. Relation (d) of these theorems are also equivalent. To see this, note that

∏_{i=0}^{d}(K − v^{d−2i}) = v^{−d^{2}−d} ∏_{i=0}^{d}(K_{1}^{2} − v^{2(d−i)})

= v^{−d^{2}−d} ∏_{i=0}^{d}(K_{1} − v^{(d−i)})(K_{1} + v^{(d−i)}).

But by Theorem 2.2 this expression is equal to zero. This completes the proof of Theorem 2.1.

6 The integral reduction

In this section we prove Theorems 2.4, 2.5, and 2.6. For typeographic convenience, we will throughout this section use the abbreviation K_{b_{1},b_{2}} in place

of the more cumbersome notation [\begin{matrix}K_{1} \\ b_{1} \end{matrix}] [\begin{matrix}K_{2} \\ b_{2} \end{matrix}].

6.1 Theorem In the algebra B_{d} we have the equality

e^{(a)}K_{b_{1},b_{2}}f^{(c)} = ∑_{k=1}^{min(a,c)}(−1)^{k−1} [\begin{matrix}b_{1} + k \\ k \end{matrix}] e^{(a−k)}K_{b_{1}+k,b_{2}−k}f^{(c−k)}

for all a, b_{1}, b_{2}, c ∈ ℕ satisfying a + b_{1} + c = d + 1 and b_{1} + b_{2} = d. Similarly we have the equality

f^{(a)}K_{b_{1},b_{2}}e^{(c)} = ∑_{k=1}^{min(a,c)}(−1)^{k−1} [\begin{matrix}b_{2} + k \\ k \end{matrix}] f^{(a−k)}K_{b_{1}−k,b_{2}+k}e^{(c−k)}

for all a, b_{1}, b_{2}, c ∈ ℕ satisfying a + b_{2} + c = d + 1 and b_{1} + b_{2} = d.

Proof. By symmetry it suffices to prove the first reduction formula only. Suppose first that c = 0. Then the claim follows from the remark following Lemma 4.6. If c > 0 we will prove the desired result by induction on a. The base case here is a = 0, which also follows from the remark mentioned above. The identity to be proved can be rewritten in the form

0 = ∑_{k=0}^{min(a,c)}(−1)^{k} [\begin{matrix}b_{1} + k \\ k \end{matrix}] e^{(a−k)}K_{b_{1}+k,b_{2}−k}f^{(c−k)}. (a)

Suppose this equation is satisfied for some fixed quadruple a, b_{1}, b_{2}, c satisfying the conditions a + b_{1} + c = d + 1, b_{1} + b_{2} = d and c ≥ 0. From this we will derive the result for the case a + 1, b_{1} − 1, b_{2} + 1, c. The idea is to multiply (a) on the right by e and commute it all the way to the left. Using Lemma 3.3(e) we obtain

0 = ∑_{k=0}^{min(a,c)}(−1)^{k} [\begin{matrix}b_{1} + k \\ k \end{matrix}] e^{(a−k)}K_{b_{1}+k,b_{2}−k}f^{(c−k)}e

= ∑_{k=0}^{min(a,c)}(−1)^{k} [\begin{matrix}b_{1} + k \\ k \end{matrix}] e^{(a−k)}K_{b_{1}+k,b_{2}−k}

× (ef^{(c−k)} − [\begin{matrix}K_{1}K_{2}^{−1}; c − k − 1 \\ 1 \end{matrix}] f^{(c−k−1)}) .

Then by Lemmas 4.4 and 4.6 this equation becomes

0 = ∑_{k=0}^{min(a,c)}(−1)^{k} [\begin{matrix}b_{1} + k \\ k \end{matrix}] e^{(a−k)}eK_{b_{1}+k−1,b_{2}−k+1}f^{(c−k)}

− ∑_{k=0}^{min(a,c)}(−1)^{k} [\begin{matrix}b_{1} + k \\ k \end{matrix}] [b_{1} − b_{2} + k + c − 1]e^{(a−k)}K_{b_{1}+k,b_{2}−k}f^{(c−k−1)}

which is equivalent to the identity

0 = ∑_{k=0}^{min(a,c)}(−1)^{k} [\begin{matrix}b_{1} + k \\ k \end{matrix}] [a − k + 1]e^{(a−k+1)}K_{b_{1}+k−1,b_{2}−k+1}f^{(c−k)}

+ ∑_{k=1}^{1+min(a,c)}(−1)^{k−1} [\begin{matrix}b_{1} + k − 1 \\ k − 1 \end{matrix}] [a − b_{1} − k + 1]

× e^{(a−k+1)}K_{b_{1}+k−1,b_{2}−k+1}f^{(c−k)}.

(The equality b_{1} − b_{2} + k + c − 1 = −(a − b_{1} − k) follows from the hypotheses a + b_{1} + c = d + 1 and b_{1} + b_{2} = d.) Writing m for min(a, c) we obtain

\begin{matrix}0 & = ∑_{k=1}^{m} Re^{(a−k+1)}K_{b_{1}+k−1,b_{2}−k+1}f^{(c−k)} \\ +[a + 1] & e^{(a+1)}K_{b_{1},b_{2}}f^{(c)} + (−1)^{m}[a − b_{1} − m]e^{(a−m)}K_{b_{1}+m,b_{2}−m}f^{(c−m−1)} \end{matrix} (b)

where

R = [\begin{matrix}b_{1} + k \\ k \end{matrix}] [a − k + 1] − [\begin{matrix}b_{1} + k − 1 \\ k − 1 \end{matrix}] [a − b_{1} − k + 1]

= [a + 1] [\begin{matrix}b_{1} + k − 1 \\ k \end{matrix}] .

Now the last term in (b) is zero if m = c. Otherwise m = a < c and the term takes the form

(−1)^{a} [\begin{matrix}b_{1} + a \\ a \end{matrix}][−b_{1}]K_{b_{1}+a,b_{2}−a}f^{(c−a−1)}

= (−1)^{a+1}[a + 1] [\begin{matrix}b_{1} + a \\ a + 1 \end{matrix}] K_{b_{1}+a,b_{2}−a}f^{(c−a−1)}.

This shows that all the terms in equation (b) have a common factor of [a+1]. Putting these terms together and dividing by [a + 1] we obtain the equality

0 = ∑_{k=0}^{M}(−1)^{k} [\begin{matrix}b_{1} + k − 1 \\ k \end{matrix}] e^{(a−k+1)}K_{b_{1}+k−1,b_{2}−k+1}f^{(c−k)}

where M = m = min(a + 1, c) in the case m = c and M = m + 1 = a + 1 = min(a + 1, c) otherwise. In either case M = min(a + 1, c) and the induction is complete. □

6.2 Theorem Suppose a, b_{1}, b_{2}, c ∈ ℕ with b_{1} + b_{2} = d. Then if s = a + b_{1} + c − d > 0 we have the equality

e^{(a)}K_{b_{1},b_{2}}f^{(c)} = ∑_{k=s}^{min(a,c)}(−1)^{k−s} [\begin{matrix}k − 1 \\ s − 1 \end{matrix}] [\begin{matrix}b_{1} + k \\ k \end{matrix}] e^{(a−k)}K_{b_{1}+k,b_{2}−k}f^{(c−k)}

and if s = a + b_{2} + c − d we have the equality

f^{(a)}K_{b_{1},b_{2}}e^{(c)} = ∑_{k=s}^{min(a,c)}(−1)^{k−s} [\begin{matrix}k − 1 \\ s − 1 \end{matrix}] [\begin{matrix}b_{2} + k \\ k \end{matrix}] f^{(a−k)}K_{b_{1}−k,b_{2}+k}e^{(c−k)}.

Proof. By symmetry we need only verify the first equality. We proceed by induction on s. The case s = 1 is the content of Theorem 6.1. Let a, b_{1}, b_{2}, c be given such that a + b_{1} + c − d = s + 1 and b_{1} + b_{2} = d. If c < s then a + b_{1} ≥ d + 1 and so by Lemma 4.6 we have

e^{(a)}K_{b_{1},b_{2}}f^{(c)} = 0.

By a similar argument one sees that this also holds if a < c. Hence we may assume that both a and c are ≥ s. It is enough to prove the result for the case a ≥ c ≥ s since the other case is similar.

Thus a ≥ 1 and we have by induction

e^{(a)}K_{b_{1},b_{2}}f^{(c)} = \frac{e}{[a]}e^{(a−1)}K_{b_{1},b_{2}}f^{(c)}

= \frac{e}{[a]} ∑_{k=s}^{min(a−1,c)}(−1)^{k−s} [\begin{matrix}k − 1 \\ s − 1 \end{matrix}] [\begin{matrix}b_{1} + k \\ k \end{matrix}] e^{(a−1−k)}K_{b_{1}+k,b_{2}−k}f^{(c−k)}

= \frac{1}{[a]} ∑_{k=s}^{min(a−1,c)}(−1)^{k−s} [\begin{matrix}k − 1 \\ s − 1 \end{matrix}] [\begin{matrix}b_{1} + k \\ k \end{matrix}] [a − k]e^{(a−k)}K_{b_{1}+k,b_{2}−k}f^{(c−k)}

= \frac{[a − s]}{[a]} [\begin{matrix}b_{1} + s \\ s \end{matrix}] e^{(a−s)}K_{b_{1}+s,b_{2}−s}f^{(c−s)}

+ \frac{1}{[a]} ∑_{k=s+1}^{min(a−1,c)}(−1)^{k−s} [\begin{matrix}k − 1 \\ s − 1 \end{matrix}] [\begin{matrix}b_{1} + k \\ k \end{matrix}] [a − k]e^{(a−k)}K_{b_{1}+k,b_{2}−k}f^{(c−k)}.

Now the first term of the last equality can be expanded by Theorem 6.1 since (a − s) + (b_{1} + s) + (c − s) = a + b_{1} + c − s = d + 1. Putting this in and shifting the index of summation we obtain the equalities

e^{(a)}K_{b_{1},b_{2}}f^{(c)}

= \frac{[a − s]}{[a]} [\begin{matrix}b_{1} + s \\ s \end{matrix}] ∑_{k=1}^{c−s}(−1)^{k−1} [\begin{matrix}b_{1} + s + k \\ k \end{matrix}] e^{(a−s−k)}K_{b_{1}+s+k,b_{2}−s+k}f^{(c−s−k)}

− \frac{1}{[a]} ∑_{k=s+1}^{min(a−1,c)}(−1)^{k−(s+1)} [\begin{matrix}k − 1 \\ s − 1 \end{matrix}] [\begin{matrix}b_{1} + k \\ k \end{matrix}] [a − k]e^{(a−k)}K_{b_{1}+k,b_{2}−k}f^{(c−k)}

= \frac{[a − s]}{[a]} [\begin{matrix}b_{1} + s \\ s \end{matrix}] ∑_{k=s+1}^{c}(−1)^{k−(s+1)} [\begin{matrix}b_{1} + k \\ k − s \end{matrix}] e^{(a−k)}K_{b_{1}+k,b_{2}+k}f^{(c−k)}

− \frac{1}{[a]} ∑_{k=s+1}^{min(a−1,c)}(−1)^{k−(s+1)} [\begin{matrix}k − 1 \\ s − 1 \end{matrix}] [\begin{matrix}b_{1} + k \\ k \end{matrix}] [a − k]e^{(a−k)}K_{b_{1}+k,b_{2}−k}f^{(c−k)}

Now the second term in the last equality above can be taken from s + 1 to c since min(a−1, c) is different from c only if a = c, in which case the additional term in the sum will be zero (the factor [a − k] is zero when k = c = a). Putting the two sums together and using the identity

\frac{[a − s]}{[a]} [\begin{matrix}b_{1} + s \\ s \end{matrix}] [\begin{matrix}b_{1} + k \\ k − s \end{matrix}] − \frac{[a − k]}{[a]} [\begin{matrix}k − 1 \\ s − 1 \end{matrix}] [\begin{matrix}b_{1} + k \\ k \end{matrix}] = [\begin{matrix}k − 1 \\ s \end{matrix}] [\begin{matrix}b_{1} + k \\ k \end{matrix}] .

we obtain

e^{(a)}K_{b_{1},b_{2}}f^{(c)} = ∑_{k=s+1}^{c}(−1)^{k−(s+1)} [\begin{matrix}k − 1 \\ s \end{matrix}] [\begin{matrix}b_{1} + k \\ k \end{matrix}] e^{(a−k)}K_{b_{1}+k,b_{2}−k}f^{(c−k)}

and this completes the induction. □

6.3 Theorem 6.2 combines with the isomorphism B_{d} ≅ S_{v}(2, d) to prove Theorem 2.5.

We now prove Theorem 2.4. First recall the representation

ρ_{d} : U → End_{ℚ(v)}(E^{⊗d}) (a)

of section 5.1. Let U_{A} be the subalgebra of U generated by all e^{(m)}, f^{(m)} (m ∈ ℕ) and K_{i}^{±1}. The map in (a) restricts to give a map U_{A} → End_{ℚ(v)}(E^{⊗d}). Let E_{A} be the A-submodule of E spanned by the canonical basis elements e_{1} and e_{2}. It is clear that E_{A} is stable under the action of U_{A} and hence so is E_{A}^{⊗d}. Thus the image of the above map is contain in End_{A}(E_{A}^{⊗d}). Consequently we have a representation

ρ_{d}^{A} : U_{A} → End_{A}(E_{A}^{⊗d}) (b)

and by a result of Du [Du] (see also [Gr]) the algebra S_{A}(2, d) is precisely the image of this representation.

A fundamental result in [Lu] (applied to the 𝔤𝔩_{2} case) is that U_{A} is a free A-module with basis

{e^{(a)}K_{1}^{δ_{1}}K_{2}^{δ_{2}}K_{b_{1},b_{2}}f^{(c)} | a, b_{1}, b_{2}, c ∈ ℕ, δ_{i} ∈ {0, 1}.} (c)

Hence S_{A}(2, d) is spanned over A by the images of the basis elements of (c). By Lemma 4.4 and Theorem 4.5 the elements K_{1}^{δ_{1}}K_{2}^{δ_{2}}K_{b_{1},b_{2}} with δ_{i} ∈ {0, 1} and b_{1}, b_{2} arbitrary are A-linear combinations of those K_{b_{1},b_{2}} with b_{1}+b_{2} = d. Therefore S_{A}(2, d) is spanned by all e^{(a)}K_{b_{1},b_{2}}f^{(c)} with b_{1} + b_{2} = d. But by Theorem 6.2 we see that the set of all such terms satisfying the condition a+b_{1}+c ≤ d is a spanning set for S_{A}(2, d) over A. Being linearly independent over ℚ(v), this set is also linearly independent over A. By symmetry, the set f^{(a)}K_{b_{1},b_{2}}e^{(c)} with b_{1} + b_{2} = d and a + b_{2} + c ≤ d is linearly independent over A. This completes the proof of Theorem 2.4.

It remains to prove Theorem 2.6. By Lemma 4.4 and Theorem 4.5 it follows that

[\begin{matrix}K_{1} \\ b \end{matrix}] = [\begin{matrix}K_{1} \\ b \end{matrix}] · 1 = [\begin{matrix}K_{1} \\ b \end{matrix}] ∑_{b_{1}+b_{2}=d} K_{b_{1},b_{2}}

= ∑_{b_{1}+b_{2}=d} [\begin{matrix}b_{1} \\ b \end{matrix}] K_{b_{1},b_{2}}

and similarly we have

[\begin{matrix}K_{2} \\ b \end{matrix}] = ∑_{b_{1}+b_{2}=d} [\begin{matrix}b_{2} \\ b \end{matrix}] K_{b_{1},b_{2}}.

Moreover, the matrix of coefficients in these equations is, with respect to an appropriate ordering of its rows and columns, triangular with 1’s on the main diagonal. So these equations can be inverted over A to obtain formulas expressing each K_{b_{1},b_{2}} (b_{1} + b_{2} = d) as an A-linear combination of the

[\begin{matrix}K_{1} \\ b \end{matrix}] or the [\begin{matrix}K_{2} \\ b \end{matrix}] as b ranges from 0 to d. Thus it follows that the sets

\{1, [\begin{matrix}K_{1} \\ 1 \end{matrix}] , …, [\begin{matrix}K_{1} \\ d \end{matrix}]\} and \{1, [\begin{matrix}K_{2} \\ 1 \end{matrix}] , …, [\begin{matrix}K_{2} \\ d \end{matrix}]\} are spanning sets (over A)

for the algebra S_{A}^{0}(n, d). Since it is already known that the rank of this free A-module is d + 1 and since the ring A is commutative, it follows that the two sets are in fact A-bases for S_{A}^{0}(n, d).

6.4 Lemma Let a, b, c be nonnegative integers satisfying a + b + c > d.

Then the elements e^{(a)} [\begin{matrix}K_{1} \\ b \end{matrix}] f^{(c)} and f^{(a)} [\begin{matrix}K_{2} \\ b \end{matrix}] e^{(c)} of degree a + b + c are

each expressible as A-linear combinations of elements of the same form but of degree not exceeding d.

Proof. From the remarks preceding the lemma we know that [\begin{matrix}K_{1} \\ b \end{matrix}] and

[\begin{matrix}K_{2} \\ b \end{matrix}] are expressible as A-linear combinations of the idempotents K_{b_{1},b_{2}} (b_{1}+

b_{2} = d). Thus the elements e^{(a)} [\begin{matrix}K_{1} \\ b \end{matrix}] f^{(c)} (resp., f^{(a)} [\begin{matrix}K_{2} \\ b \end{matrix}] e^{(c)}) are expressible

as A-linear combinations of elements of the form

e^{(a)}K_{b_{1},b_{2}}f^{(c)} (resp., f^{(a)}K_{b_{1},b_{2}}e^{(c)}) (a) where b_{1} + b_{2} = d and where the fake degree a + b_{1} + c (resp., a + b_{2} + c) is strictly greater than d. By Theorem 2.5 it follows that each element of the form (a) above is expressible as an A-linear combination of elements of the same form

e^{(a′)}K_{u_{1},u_{2}}f^{(c′)} (resp., f^{(a′)}K_{u_{1},u_{2}}e^{(c′)}) (b)

where a′ < a, c′ < c, u_{1} + u_{2} = d, and a′ + u_{1} + c′ ≤ d (resp., a′ + u_{2} + c′ ≤ d). Now by expressing K_{u_{1},u_{2}} as an A-linear combination of elements of

the form [\begin{matrix}K_{1} \\ b′ \end{matrix}], (resp., [\begin{matrix}K_{2} \\ b′ \end{matrix}]) where 0 ≤ b′ ≤ d we obtain (via left and

right multiplication by appropriate elements) formulas expressing each of the elements of the form (b) in terms of A-linear combinations of elements of the form

e^{(a′)} [\begin{matrix}K_{1} \\ b′ \end{matrix}] f^{(c′)} (resp., f^{(a′)} [\begin{matrix}K_{2} \\ b′ \end{matrix}] e^{(c′)}). (c)

If an element of this form satisfies the constraint a′ + b′ + c′ ≤ d then we leave it be, but for those elements which do not satisfy this constraint we repeat the entire process given above, replacing the element by an A-linear combination of elements of the same form, in which for each element the degree of e is strictly smaller, as is the degree of f. After repeating the process finitely many times we obtain the desired result. □

Now we can prove Theorem 2.6. By Theorem 2.4 we know that the set

B = {e^{(a)}K_{b_{1},b_{2}}f^{(c)} | a + b_{1} + c ≤ d, b_{1} + b_{2} = d}

is an A-basis for S_{A}(2, d). Now consider the set

B′ = \{e^{(a)} [\begin{matrix}K_{1} \\ b \end{matrix}] f^{(c)} | a + b + c ≤ d\} .

The sets B and B′ have the same cardinality and the ring A is commutative, thus it will follow that B′ is a basis once we can show that it spans.

We know that S_{A}(2, d) is spanned by elements of the form e^{(a)} [\begin{matrix}K_{1} \\ b \end{matrix}] f^{(c)}

since S_{A} = S_{A}^{+}S_{A}^{0}S_{A}^{−}. By the above lemma we know that each such element not satisfying the constraint a + b + c ≤ d is expressible as an A-linear combination of elements which do satisfy that constraint. It follows that the set B′ is a spanning set for S_{A}(2, d). This proves the first part of Theorem 2.6. The second part of the theorem follows by symmetry.

References

[DG] S. Doty and A. Giaquinto, Presenting Schur algebras as quotients of the universal enveloping algebra of 𝔤𝔩_{2}, preprint, Loyola Univ. Chicago, Sept. 2000.

[Du] J. Du, A note on quantized Weyl reciprocity at roots of unity, Algebra Colloq. 2 (1995), no. 4, 363-372.

[Gr] R. Green, q-Schur algebras as quotients of quantized enveloping algebras, J. Algebra 185 (1996), no. 3, 660-687.

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

Mathematical and Computer Sciences Loyola University Chicago Chicago, Illinois 60626 U.S.A. E-mail: doty@math.luc.edu

tonyg@math.luc.edu