Table Of ContentPresenting quantum Schur algebras as
1
0
0 quotients of the quantized universal enveloping
2
n algebra of gl
a 2
J
5
Stephen Doty and Anthony Giaquinto
]
T
R December 1, 2000
.
h
t
a
m Abstract
[
We obtain a presentation of the quantum Schur algebras Sv(2,d)
2
by generators and relations. This presentation is compatible with the
v
4 usual presentation of the quantized enveloping algebra U = U (gl ).
v 2
6
In the process we find new bases for S (2,d). We also locate the
1 v
1 Z[v,v−1]-form of the quantum Schur algebra within the presented al-
1 gebra and show that it has a basis which is closely related to Lusztig’s
0 basis of the Z[v,v−1]-form of U.
0
/
h
t
a 1 Introduction
m
:
v In [DG] we gave a description of the rational Schur algebra S (2,d) in terms
Q
i
X of generators and relations. This description is compatible with the usual
r presentation of the universal enveloping algebra U(gl ). We also described
a 2
the integral Schur algebra S (2,d) as a certain subalgebra of the rational
Z
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 correspond-
ing to the Lie algebra gl . It has a natural two-dimensional module E, and
2
thus the tensor product E⊗d is also a module for U. Let
ρ : U → End(E⊗d)
d
1
be the corresponding representation. Amongst the various equivalent defini-
tions of the quantum Schur algebra S (2,d) is that it is precisely the image of
v
the homomorphism ρ . (We shall not need to consider the other definitions
d
in this work.) Since S (2,d) is a homomorphic image of U, it is natural to
v
ask for an efficient generating set of Ker(ρ ), thereby giving a presentation
d
of S (2,d).
v
In what follows, we obtain a precise answer to this question. Recall that
U is generated by elements e, f, K ±1, K ±1 subject to various well-known
1 2
relations (see section 3). Now in the representation ρ , it is easy to see that
d
K K = vd, so we may use this relation to eliminate K (or K ) from the
1 2 2 1
generating set for S (2,d). Having done this, then our main result is that the
v
only additional relation needed to give the desired presentation of S (2,d) is
v
the minimal polynomial of K in End(E⊗d).
1
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 (gl ) of
Z 2
the classical enveloping algebra. The algebra U , originally constructed by
A
Lusztig, is generated by the v-divided powers of e and f along with K ±1
1
and K ±1. It has an A-basis which is a quantum analog of Kostant’s basis
2
for the integral form of the classical enveloping algebra U(gl ). The image of
2
the homomorphism ρ upon restriction to U gives us an “integral” Schur
d A
algebra S (2,d), which can be used to define a version over any A-algebra.
A
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 ap-
pearing 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 and
1
K±1) of S (2,d) is totally different. Specifically, we exhibit an idempotent
2 v
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
2
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 (2,d) in terms of generators
v
and relations. The first result gives a presentation which is similar to that of
U (sl ).
v 2
2.1 Theorem Over Q(v), the quantum Schur algebra S (2,d) is isomor-
v
phic to the algebra generated by e, f, K±1 subject to the relations:
(a) KK−1 = K−1K = 1,
(b) KeK−1 = v2e, KfK−1 = v−2f,
K −K−1
(c) ef −fe = ,
v −v−1
(d) (K −vd)(K −vd−2)···(K −v−d+2)(K −v−d) = 0.
The next two results give presentations of S (2,d) which are similar to
v
that of U (gl ).
v 2
2.2 Theorem Over Q(v), the quantum Schur algebra S (2,d) is isomor-
v
phic to the algebra generated by e, f, K ±1 subject to the relations:
1
(a) K K −1 = K −1K = 1,
1 1 1 1
(b) K eK −1 = ve, K fK −1 = v−1f
1 1 1 1
v−dK 2 −vdK −2
(c) ef −fe = 1 1 ,
v −v−1
(d) (K −1)(K −v)(K −v2)···(K −vd) = 0.
1 1 1 1
By a change of variable (K = vdK−1) we obtain another equivalent
2 1
presentation of S (2,d).
v
3
2.3 Theorem Over Q(v), the quantum Schur algebra S (2,d) is isomor-
v
phic to the algebra generated by e, f, K ±1 subject to the relations:
2
(a) K K −1 = K −1K = 1,
2 2 2 2
(b) K eK −1 = v−1e, K fK −1 = vf
2 2 2 2
vdK −2 −v−dK 2
(c) ef −fe = 2 2 ,
v −v−1
(d) (K −1)(K −v)(K −v2)···(K −vd) = 0.
2 2 2 2
For indeterminates X,X−1 satisfying XX−1 = X−1X = 1 and any t ∈ N we
formally set
X t Xv−s+1−X−1vs−1
= .
t vs −v−s
(cid:20) (cid:21) s=1
Y
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 (2,d) in terms
A
of the above generators, and gives an A-basis for the algebra. In this we
can take S (2,d) to be given by either presentation 2.2 or 2.3, but we always
v
assume that K and K are related by the condition K K = vd.
1 2 1 2
2.4 Theorem The integral Schur algebra S (2,d) is isomorphic to the
A
A-subalgebra of S (2,d) generated by
v
em fm
e(m) := , f(m) := (m ∈ N), K ±1.
1
[m]! [m]!
The preceding statement is true when K is replaced by K . Moreover, an
1 2
A-basis for S (2,d) is the set consisting of all
A
K K
e(a) 1 2 f(c)
b b
1 2
(cid:20) (cid:21)(cid:20) (cid:21)
such that the natural numbers a,b ,b ,c are constrained by the conditions
1 2
a+b +c ≤ d, b +b = d. Another such basis consists of all
1 1 2
K K
f(a) 1 2 e(c)
b b
1 2
(cid:20) (cid:21)(cid:20) (cid:21)
such that the natural numbers a,b ,b ,c satisfy the constraints a+b +c ≤
1 2 2
d, b +b = d.
1 2
4
The following reduction formulas make it possible, in principle, to ex-
press the product of two basis elements as an A-linear combination of basis
elements.
2.5 Theorem In S (2,d) we have the following reduction formulas for
A
all s ≥ 1 and all a,b ,b ,c ∈ N with b +b = d:
1 2 1 2
K K
e(a) 1 2 f(c) =
b b
1 2
(cid:20) (cid:21)(cid:20) (cid:21)
(a)
min(a,c)
k −1 b +k K K
(−1)k−s 1 e(a−k) 1 2 f(c−k)
s−1 k b +k b −k
1 2
k=s (cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21)
X
K K
f(a) 1 2 e(c) =
b b
1 2
(cid:20) (cid:21)(cid:20) (cid:21)
(b)
min(a,c)
k −1 b +k K K
(−1)k−s 2 f(a−k) 1 2 e(c−k)
s−1 k b −k b +k
1 2
k=s (cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21)
X
where s = a+b +c−d in (a) and s = a+b +c−d in (b).
1 2
The next theorem, the quantum analogue of [DG, Thm. 2.4], provides yet
another kind of basis for S (2,d). In this case we will deduce it as a direct
A
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
K
e(a) 1 f(c) | a,b,c ∈ N, a+b+c ≤ d
b
(cid:26) (cid:20) (cid:21) (cid:27)
is an A-basis for S (2,d). Another such basis is given by the set
A
K
f(a) 2 e(c) | a,b,c ∈ N, a+b+c ≤ d .
b
(cid:26) (cid:20) (cid:21) (cid:27)
5
3 Quantized enveloping algebras
3.1 The Drinfeld-Jimbo quantized enveloping algebra U = U (gl ) is de-
v 2
fined to be the Q(v)-algebra with generators e, f, K , K−1, K , K−1 and
1 1 2 2
relations
(a) K K = K K ,
1 2 2 1
(b) K K −1 = K −1K = 1 (i = 1,2),
i i i i
(c) K eK−1 = ve, K fK−1 = v−1f,
1 1 1 1
(d) K eK−1 = v−1e, K fK−1 = vf,
2 2 2 1
K K−1 −K−1K
(e) ef −fe = 1 2 1 2.
v −v−1
Let U+ (respectively U−) be the subalgebra generated by e (respectively f)
and let U0 be the subalgebra generated by K±1, K±1. There are Q(v)-vector
1 2
space isomorphisms
U ∼= U+ ⊗U0 ⊗U− ∼= U− ⊗U0 ⊗U+
and moreover it is well-known that the algebra U has PBW-type bases
eaK b1K b2fc and faK b1K b2ec where a,c ∈ N and b ,b ∈ Z.
1 2 1 2 1 2
InthealgebraU,setK = K K−1,anddefineU (sl )tobethesubalgebra
(cid:8) (cid:9) (cid:8) 1 2 (cid:9) v 2
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
vr −v−r
(a) [r] =
v −v−1
r [r][r −1]···[r −s+1]
(b) = .
s [1][2]···[s]
(cid:20) (cid:21)
These satisfy the well-known identities
(c) [r +s] = v−s[r]+vr[s]
r +1 r r
(d) = v−s +vr−s+1 .
s s s−1
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
6
For every m,t ∈ N, c ∈ Z and any element X an Q(v)-algebra define
(e) [m]! = [m][m−1]···[1]
Xm
(f) X(m) =
[m]!
t Xvc−i+1 −X−1v−c+i−1
X;c if t 6= 0
(g) = vi −v−i (X invertible)
t i=1
(cid:20) (cid:21) 1Yif t = 0.
X X;0
Wenote that , asdefined insection 2, coincides with the element .
t t
(cid:20) (cid:21) (cid:20) (cid:21)
In [Lu], Lusztig investigates many relations which hold in quantized en-
veloping 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:
(a) K neK−n = vne, K neK−n = v−ne,
1 1 2 2
(b) K nf K−n = v−nf, K nf K−n = vnf,
1 1 2 2
K ;c K ;c+1 K ;c K ;c−1
(c) 1 e = e 1 , 1 f = f 1 ,
t t t t
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
K ;c K ;c−1 K ;c K ;c+1
(d) 2 e = e 2 , 2 f = f 2 ,
t t t t
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
K K−1;m−1
(e) f(m)e = ef(m) − 1 2 f(m−1),
1
(cid:20) (cid:21)
K K−1;m−1
(f) f e(m) = e(m)f −e(m−1) 1 2 ,
1
(cid:20) (cid:21)
K ;c+1 K ;c K ;c
(g) i = vt+1 i +vt−cK −1 i ,
t+1 t+1 i t
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
K K ;−t t+t′ K
(h) i i = i ,
t t′ t t+t′
(cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21)
t
K ;c c K
(i) i = vc(t−j) K −j i (c ≥ 0).
t j i t−j
(cid:20) (cid:21) j=0 (cid:20) (cid:21) (cid:20) (cid:21)
X
7
Proof. All of these identities are special cases of those appearing on pages
269-270 of [Lu]. (cid:3)
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 (2,d).
v
4.1 Let d be a fixed nonnegative integer and define B to be the Q(v)-
d
algebra generated by e, f, K ±1, K ±1 subject to relations 3.1(a)-(e), along
1 2
with the additional relations
(a) K K = vd
1 2
(b) (K −1)(K −v)···(K −vd) = 0
1 1 1
Note that relation (b) can be replaced by the equivalent relation
(c) (K−1 −1)(K−1 −v−1)···(K−1 −v−d) = 0
1 1 1
and in the presence of relation (a) it can also be replaced by either
(d) (K −1)(K −v)···(K −vd) = 0, or
2 2 2
(e) (K−1 −1)(K−1 −v−1)···(K−1 −v−d) = 0.
2 2 2
Thedefining relationsofB areinvariant ifeandf areinterchanged along
d
with K and K . These interchanges therefore induce an automorphism of
1 2
B . We shall often make use of this property, which we will call symmetry,
d
in the sequel.
Let B0 be the subalgebra of B generated by K ±1, K ±1. It follows from
d d 1 2
relations 4.1(a)-(b) that dim(B0) = d+1.
d
4.2 Lemma In the algebra B0 we have
d
K K K−1 K−1
1 = 0 = 2 , 1 = 0 = 2 .
d+1 d+1 d+1 d+1
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
8
Proof. Using definition 3.2(g) we have that
K d+1 K v1−i −K−1vi−1
1 = 1 1
d+1 vi −v−i
(cid:20) (cid:21) i=1
Y
d+1 (K−1v1−i)(K2 −v2(i−1))
= 1 1
vi −v−i
i=1
Y
d+1 (K−1v1−i)(K −v(i−1))(K +v(i−1))
= 1 1 1
vi −v−i
i=1
Y
= 0 by relation 4.1(b).
(cid:3)
The other equalities follow in a similar manner using 4.1(c)-(e).
More generally we have
4.3 Lemma In the algebra B0 we have
d
K K
1 2 = 0
b b
1 2
(cid:20) (cid:21)(cid:20) (cid:21)
whenever b +b = d+1.
1 2
Proof.
K K b1 K v1−i −K−1vi−1 b2 K v1−i −K−1vi−1
1 2 = 1 1 2 2
b b vi −v−i vi −v−i
1 2
(cid:20) (cid:21)(cid:20) (cid:21) i=1 i=1
Y Y
b1 K v1−i −K−1vi−1 b2 K−1vd+1−i −K v−d+i−1
= 1 1 1 2
vi −v−i vi −v−i
i=1 i=1
Y Y
d+1 K
= (−1)b2 1
b d+1
2
(cid:20) (cid:21)(cid:20) (cid:21)
= 0 by 4.1(b).
(cid:3)
An immediate consequence of the preceding lemma is that
K K
1 2 = 0
b b
1 2
(cid:20) (cid:21)(cid:20) (cid:21)
whenever b +b ≥ d+1. When b +b ≤ d, these elements are non-zero, but
1 2 1 2
we shall see that the most important case is b +b = d.
1 2
9
4.4 Lemma Suppose that b + b = d and t ∈ N. Then for i = 1,2 the
1 2
following identities hold in B0:
d
K K K K
(a) K 1 2 = vbi 1 2
i b b b b
1 2 1 2
(cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21)
K ;c K K b +c K K
(b) i 1 2 = i 1 2
t b b t b b
1 2 1 2
(cid:20) (cid:21)(cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21)(cid:20) (cid:21)
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 K
(K −vb1) 1 2
1 b b
1 2
(cid:20) (cid:21)(cid:20) (cid:21)
b1 K v1−i −K−1vi−1 b2 K v1−i −K−1vi−1
= (K −vb1) 1 1 2 2
1 vi −v−i vi −v−i
i=1 i=1
Y Y
d+1 K v1−i −K−1vi−1
= (−1)b2(K −vb1) 1 1
1 vi −v−i
i=1
i6=Yb1+1
d+1 v1−iK−1(K 2 −v2(i−1))
= (−1)b2 (K −vb1) 1 1
1 vi −v−i
i=1
i6=Yb1+1
= 0 by 4.1(b).
This proves identity (a) for i = 1. The case i = 2 follows from symmetry. (cid:3)
The following result establishes the structure of the algebra B0; it will
d
later be crucial in determining the structure of the entire algebra B .
d
4.5 Theorem In the algebra B0, the set
d
K K
1 2 | b +b = d
b b 1 2
1 2
(cid:26)(cid:20) (cid:21)(cid:20) (cid:21) (cid:27)
is a basis of mutually orthogonal idempotents whose sum is the identity.
Proof. By Lemma 4.4 we have that
2
K K b b K K K K
1 2 = 1 2 1 2 = 1 2
b b b b b b b b
1 2 1 2 1 2 1 2
(cid:18)(cid:20) (cid:21)(cid:20) (cid:21)(cid:19) (cid:20) (cid:21)(cid:20) (cid:21)(cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21)
10
