q-Schur algebras and Quantized Enveloping Algebras [Ph.D. thesis]

131 Pages·1996·0.7 MB·English

by  Richard M. Green

Contents
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
x1: Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1 The Quantized Enveloping Algebra . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Schur Algebras and q-Schur Algebras . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Coxeter Systems and their Properties . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
x2: A New Setting for the q-Schur Algebra . . . . . . . . . . . . . . . . . 24
2.1 The classical Schur algebra as a subspace of tensor matrix space . . . . . . . . . . 24
2.2 The v-Schur algebra as a subspace of tensor matrix space . . . . . . . . . . . . . 28
2.3 On certain subalgebras of the q-Schur algebra . . . . . . . . . . . . . . . . . . . 44
x3: A Straightening Formula for Quantized Codeterminants . . . . . . . . . 47
3.0 Conventions and Review of x2 . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1 Hecke algebras and codeterminants of dominant shape . . . . . . . . . . . . . . 48
3.2 The Quantized Straightening Formula . . . . . . . . . . . . . . . . . . . . . . 55
x4: q-Schur Algebras as Quotients of Quantized Enveloping Algebras . . . . . 63
4.0 Conventions and Review of x3 . . . . . . . . . . . . . . . . . . . . . . . . . 63
(cid:0) 0 +
4.1 The restriction of (cid:18) to U , U and U . . . . . . . . . . . . . . . . . . . . . 64
4.2 Codeterminants, explicit surjectivity and ker ((cid:18)) . . . . . . . . . . . . . . . . . 73
1
x5: q-Weyl modules and q-codeterminants . . . . . . . . . . . . . . . . . 78
5.1 Quantized Left Weyl Modules . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2 Quantized Right Weyl Modules . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3 q-Codeterminants and q-Weyl modules . . . . . . . . . . . . . . . . . . . . . . 90
5.4 Remarkson the case n<r . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
x6: Cellular inverse limits of q-Schur algebras . . . . . . . . . . . . . . . 96
6.1 The strong straightening result . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2 Cellular algebras and quasi-hereditary algebras . . . . . . . . . . . . . . . . . . 99
6.3 Epimorphismsbetween v-Schur algebras . . . . . . . . . . . . . . . . . . . . 105
6.4 Cellular inverse limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
2
Acknowledgements
Above all, I would like to thank my supervisor, Professor R.W. Carter, for introducing me to
suchaninterestingarea,forteachingmetowritegoodmathematics,andformakingsomanyhelpful
commentsand corrections to earlier drafts of this thesis.
I alsothank the Mathematics Institute for providingan excellent workingatmosphere,without
which I would not have been able to make so much progress so quickly. I would like to thank all
myfriends and colleagues there, especiallyJohn Cockerton and Robert Marsh, for manyinteresting
discussions.
I am grateful to Professor S. Donkin, Professor R.C. King and Professor G.I. Lehrer for their
interest in my research and for their helpful and encouraging comments during the preparation of
this thesis.
Finally, I would like to thank the Engineering and Physical Sciences Research Council for
funding my research.
Declaration
The material in Chapter 1 is expository. The material in Chapters 2, 3, 4, 5 and 6 is to the
best of my knowledge original,except where otherwise indicated.
3
Summary
The main aimof this thesis is to investigate the relationship between the quantized enveloping
algebra U(gln) (corresponding to the Lie algebra gln) and the q-Schur algebra, Sq(n;r). It was
shown in [BLM] that there is a surjective algebra homomorphism
(cid:0)1
(cid:18)r :U(gln)!Z[v;v ](cid:10)Sq(n;r);
2
where q =v .
x1 is devoted to background material.
In x2, we show explicitly how to embed the q-Schur algebra into the r-th tensor power of a
suitablen(cid:2)n matrixring. Thisgives a product rulefor the q-Schur algebra withsimilarproperties
toSchur'sproductrulefortheunquantizedSchuralgebra. Acorollaryofthisisthatwecandescribe,
in x2.3, a certain family of subalgebras of the q-Schur algebra.
In x3, we use the product ruleof x2 to prove a q-analogue of Woodcock's straightening formula
for codeterminants. Thisgives a basis of \standard quantized codeterminants"for Sq(n;r) which is
heavily used in chapters 4, 5 and 6.
In x4, we use the theory of quantized codeterminants developed in x3 to describe preimages
under the homomorphism(cid:18)r and the kernel of (cid:18)r.
In x5, we use the resultsof x3 and x4 to linkthe representation theories of U(gln) and Sq(n;r).
We also obtain a simpli(cid:12)ed proof of Dipper and James' \semistandard basis theorem" for q-Weyl
modules of q-Schur algebras.
In x6, we show how to make the set of q-Schur algebras Sq(n;r) (for a (cid:12)xed n) into an inverse
system. Weprovethattheresultinginverselimit,Sbv(n),isacellularalgebrawhichiscloselyrelated
to the quantized enveloping algebra U(sln) and Lusztig's algebra U_.
4
1. Preliminaries
Inthischapter,weintroducethenecessarybackgroundmaterialfortherestofthisthesis. Most
of this material is expository. In x1.1, we introduce the quantized enveloping algebra, and in x1.2,
we introducethe Schuralgebras andq-Schur algebras. In x1.3, westate somewellknown properties
of Coxeter systems,which we use to analyse symmetricgroups and their associated Hecke algebras.
Finally, x1.4 is devoted to combinatoric de(cid:12)nitions.
The symbols q and v always represent indeterminates unless stated otherwise, and are related
2 (cid:0)1
via v =q. Throughout the thesis, we denote Z[v;v ], the ring of Laurent polynomialsover Z,by
A.
x1.1 The Quantized Enveloping Algebra
1.1.1 De(cid:12)nition of the Quantized Enveloping Algebra
We now de(cid:12)ne the algebra U(gln) over Q(v) as in [D4]. It is generated by elements
(cid:0)1
Ei;Fi;Kj;Kj ;
(where 1(cid:20)i(cid:20)n(cid:0)1 and 1(cid:20)j (cid:20)n) subject to the following relations:
KiKj =KjKi; (1)
(cid:0)1
KiKi =1; (2)
+
(cid:15) (i;j)
KiEj =v EjKi; (3)
(cid:15)(cid:0)(i;j)
KiFj =v FjKi; (4)
(cid:0)1 (cid:0)1
KiKi+1(cid:0)Ki Ki+1
EiFj (cid:0)FjEi =(cid:14)ij v(cid:0)v(cid:0)1 ; (5)
EiEj =EjEi if ji(cid:0)jj>1; (6)
FiFj =FjFi if ji(cid:0)jj>1; (7)
2 (cid:0)1 2
EiEj (cid:0)(v+v )EiEjEi+EjEi =0 if ji(cid:0)jj=1; (8)
2 (cid:0)1 2
FjFi(cid:0)(v+v )FjFiFj +FiFj =0 if ji(cid:0)jj=1: (9)
Here,
(
1 if j =i;
+
(cid:15) (i;j):= (cid:0)1 if j =i(cid:0)1;
0 otherwise;
5
and
(
1 if j =i(cid:0)1;
(cid:0)
(cid:15) (i;j):= (cid:0)1 if j =i;
0 otherwise.
The algebra is also equipped with two coassociative comultiplications. One, (cid:1) : U ! U (cid:10)U,
has the following e(cid:11)ect on the generators:
(cid:0)1
(cid:1)(Ei)=1(cid:10)Ei+Ei(cid:10)KiKi+1;
(cid:0)1
(cid:1)(Fi)=Ki Ki+1(cid:10)Fi+Fi(cid:10)1;
(cid:1)(Ki)=Ki(cid:10)Ki;
(cid:0)1 (cid:0)1 (cid:0)1
(cid:1)(Ki )=Ki (cid:10)Ki :
0
The other map, (cid:1) :U !U (cid:10)U, is de(cid:12)ned by
0 (cid:0)1
(cid:1)(Ei)=KiKi+1(cid:10)Ei+Ei(cid:10)1;
0 (cid:0)1
(cid:1)(Fi)=1(cid:10)Fi+Fi(cid:10)Ki Ki+1;
0
(cid:1)(Ki)=Ki(cid:10)Ki;
0 (cid:0)1 (cid:0)1 (cid:0)1
(cid:1)(Ki )=Ki (cid:10)Ki :
0
We will not be using (cid:1) in this thesis, but we include it for completeness because many authors
prefer it to (cid:1).
0
The maps (cid:1) and (cid:1) are also algebra homomorphisms. This means that if M is a U-module,
(cid:10)r
we can make M into a U-module via
r(cid:0)1
u:(m1(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)mr)=(cid:1) (u)(m1(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)mr):
r(cid:0)1
This is well-de(cid:12)ned because (cid:1) is coassociative. Note that (cid:1) is an algebra homomorphismU !
(cid:10)r r(cid:0)1
U . It may be checked by a simple inductive argument that the e(cid:11)ect of (cid:1) on the generators
(cid:0)1
Ei;Fi;Ki;Ki is as follows:
r(cid:0)1 (cid:0)1
(cid:1) (Ei)=(1(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)1(cid:10)Ei)+(1(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)1(cid:10)Ei(cid:10)KiKi+1)+(cid:1)(cid:1)(cid:1)
(cid:0)1 (cid:0)1
(cid:1)(cid:1)(cid:1)+(Ei(cid:10)KiKi+1(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)KiKi+1);
r(cid:0)1 (cid:0)1 (cid:0)1 (cid:0)1 (cid:0)1
(cid:1) (Fi)=(Ki Ki+1(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)Ki Ki+1(cid:10)Fi)+(Ki Ki+1(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)Ki Ki+1(cid:10)Fi(cid:10)1)+(cid:1)(cid:1)(cid:1)
(cid:1)(cid:1)(cid:1)+(Fi(cid:10)1(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)1);
r(cid:0)1
(cid:1) (Ki)=Ki(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)Ki;
r(cid:0)1 (cid:0)1 (cid:0)1 (cid:0)1
(cid:1) (Ki )=Ki (cid:10)(cid:1)(cid:1)(cid:1)(cid:10)Ki :
We introduce certain elements of Q(v), as follows.
6
We willde(cid:12)ne the quantum integer [a], where a is a nonnegative integer, to be
a (cid:0)a
v (cid:0)v
:
v(cid:0)v(cid:0)1
We also de(cid:12)ne quantized factorials by
Ya
[a]!:= [k];
k=1
and quantized binomial coe(cid:14)cients by
h i
a [a]!
:= :
b [b]![a(cid:0)b]!
Note that when v is specialised to 1, these become ordinary integers, factorials and binomial
coe(cid:14)cients, respectively.
(c)
If X is an element of U and c is a nonnegative integer, then the divided power X is de(cid:12)ned
to be
c
X
:
[c]!
Sometimes,itisconvenienttoworkwithanintegralformofU(gln),whichisdenotedbyUA(gln),
or (when the context is clear) by U. This is an A-algebra which is generated by the elements of
U(gln) given by
(c)
Ei (1(cid:20)i<n; c2N) (10)
(c)
Fi (1(cid:20)i<n; c2N) (11)
Kj (1(cid:20)j (cid:20)n) (12)
(cid:20) (cid:21)
Kj;0
(1(cid:20)j (cid:20)n; t2N): (13)
t
Here,
(cid:20)Ki;c(cid:21) Yt Kivc(cid:0)s+1(cid:0)Ki(cid:0)1v(cid:0)c+s(cid:0)1
:= :
t vs(cid:0)v(cid:0)s
s=1
(cid:0)
TheA-algebraUA isgenerated by theelementsin(11) subject torelationsofform(7) and (9).
0
The A-algebra UA is generated by the elements in (12) and (13), subject to relations of form
(1) and (2).
+
TheA-algebraUA isgeneratedbytheelementsin(10),subjecttorelationsofform(6)and(8).
It is known (see [L3, x3.2]) that U (cid:24)=U(cid:0)(cid:10)U0(cid:10)U+ as A-modules.
A good reference for the general theory of quantized enveloping algebras is [L3].
7
1.1.2 Root systems and root vectors
We now state without proof some properties of root systems. The general theory of these can
be found in any good text on Lie algebras.
Associated with the Lie algebra sln, or (in our case) gln, is a certain collection of vectors in
(n(cid:0)1)-dimensional Euclidean space known as a root system of type An(cid:0)1. It is well-known that
this root system contains an independent subset (the fundamental roots) f(cid:11)1;:::;(cid:11)n(cid:0)1g such that
any other root is of form
(cid:11)=(cid:11)i+(cid:11)i+1+(cid:1)(cid:1)(cid:1)+(cid:11)j (1(cid:20)i(cid:20)j <n)
or of form
(cid:11)=(cid:0)(cid:11)i(cid:0)(cid:11)i+1(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:11)j (1(cid:20)i(cid:20)j <n):
In the (cid:12)rst case, the root (cid:11) is called positive, and in the second case, the root (cid:11) is called negative.
+ (cid:0)
Denote these two sets of roots by (cid:8) and (cid:8) , respectively.
We willalso write (cid:11)(i;j+1) to denote the positive root (cid:11)i+(cid:11)i+1+(cid:1)(cid:1)(cid:1)+(cid:11)j.
We de(cid:12)ne the height, h((cid:11)), of (cid:11)=(cid:11)(i;j) to be j(cid:0)i.
Following [L2, x2.2], we de(cid:12)ne the function g((cid:11)(i;j)) =j(cid:0)1. (The function g (cid:12)nds the index
of the highest fundamental root occurring with nonzero coe(cid:14)cient in its argument.)
+ +
The bilinear map ( ; ):(cid:8) (cid:2)(cid:8) !Zis de(cid:12)ned to satisfy
(
2 if i=j;
((cid:11)i;(cid:11)j):= (cid:0)1 if ji(cid:0)jj=1;
0 otherwise.
+
Associated to each positive root (cid:11) in gln, we de(cid:12)ne an element E(cid:11) in U and an element F(cid:11)
(cid:0)
in U .
Let (cid:11) = (cid:11)i +(cid:11)i+1 +(cid:1)(cid:1)(cid:1)+(cid:11)j be a (typical) positive root in type An(cid:0)1, where the (cid:11)i are, as
usual,the fundamentalroots. If i=j, we de(cid:12)ne E(cid:11) :=Ei and F(cid:11) :=Fi. If i6=j, welet (cid:13) =(cid:11)(cid:0)(cid:11)j
and (cid:12) =(cid:11)(cid:0)(cid:11)i and de(cid:12)ne, by induction on j(cid:0)i,
(cid:0)1
E(cid:11) :=E(cid:13)Ej (cid:0)v EjE(cid:13)
and
(cid:0)1
F(cid:11) :=F(cid:12)Fi(cid:0)v FiF(cid:12):
We also order the elements E(cid:11) and the elements F(cid:11) as follows.
TheelementE(cid:11)(i;j) precedes(orappearstotheleftof)theelementE(cid:11)(k;l) ifi>kor(i=kand
j >l). We denote by (cid:12)N;:::;(cid:12)2;(cid:12)1 the sequence of positiveroots correspondingto thissequence of
root vectors.
8
TheelementF(cid:11)(i;j) precedes(orappearsto theleftof) theelementF(cid:11)(k;l) ifj <l or(j =l and
i<k). We denote by (cid:13)N;:::;(cid:13)2;(cid:13)1 the sequence of positiveroots correspondingto this sequence of
root vectors.
Example
Let n=4. In this case, the positive roots are
(cid:11)1; (cid:11)1+(cid:11)2; (cid:11)1+(cid:11)2+(cid:11)3; (cid:11)2; (cid:11)2+(cid:11)3; (cid:11)3:
The ordering on the elements E(cid:11) corresponds to the ordering
(cid:11)(3;4)<(cid:11)(2;4)<(cid:11)(2;3)<(cid:11)(1;4)<(cid:11)(1;3)<(cid:11)(1;2)
on the positive roots, and the ordering on the elements F(cid:11) corresponds to the ordering
(cid:11)(1;2)<(cid:11)(1;3)<(cid:11)(2;3)<(cid:11)(1;4)<(cid:11)(2;4)<(cid:11)(3;4)
on the positive roots.
1.1.3 Basis theorems for Quantized Enveloping Algebras
In this section, we work with the integral form U = UA. The aim is to de(cid:12)ne a Poincar(cid:19)e-
Birkho(cid:11)-Witt type basis for U by using the elements E(cid:11) and F(cid:11) which were introduced in x1.1.2.
Lemma 1.1
+ + + + +
De(cid:12)ne : U ! U by (Ei) := En(cid:0)i. Then extends naturally to an A-algebra
+ + +
isomorphism :U !U .
(cid:0) (cid:0) (cid:0) (cid:0) (cid:0)
De(cid:12)ne : U ! U by (Fi) := Fn(cid:0)i. Then extends naturally to an A-algebra
(cid:0) (cid:0) (cid:0)
isomorphism :U !U .
Proof
+
The map is self-inverse, and preserves the relations (6) and (8).
(cid:0)
The map is self-inverse, and preserves the relations (7) and (9).
Lemma 1.2
(cid:0) (cid:0) + (cid:0)
There is an A-algebra isomorphism! :U !U given by ! (Fi)=Ei.
+ + (cid:0) +
There is an A-algebra isomorphism! :U !U given by ! (Ei)=Fi.
(cid:0) +
Proof Since ! and ! are mutual inverses, and they are clearly surjective, it su(cid:14)ces to check
that each one preserves the relations. This is immediate from the nature of the relations (6), (7),
(8) and (9).
9
Lemma 1.3
+ (cid:0)
(i) (! (F(cid:11)(i;j)))=E(cid:11)(n+1(cid:0)j;n+1(cid:0)i).
+ (cid:0)
(ii) F(cid:11) precedes F(cid:12) in the ordering on the elements F(cid:13) if and only if (! (F(cid:11))) precedes
+ (cid:0)
(! (F(cid:12))) in the ordering on the E(cid:13).
Proof
We (cid:12)rst prove (i), using induction on h = h((cid:11)). The case h = 1 follows from the de(cid:12)nition of
(cid:0)
! .
(cid:0)1
For the general case, F(cid:11)(i;j) = F(cid:11)(i+1;j)F(cid:11)(i;i+1) (cid:0) v F(cid:11)(i;i+1)F(cid:11)(i+1;j), by de(cid:12)nition. By
induction, we have
+ (cid:0) (cid:0)1
(! (F(cid:11)(i;j)))=E(cid:11)(n(cid:0)j+1;n(cid:0)i)E(cid:11)(n(cid:0)i;n(cid:0)i+1)(cid:0)v E(cid:11)(n(cid:0)i;n(cid:0)i+1)E(cid:11)(n(cid:0)i;n(cid:0)j+1);
+ (cid:0)
because ! is an algebra isomorphism. The result now follows from the de(cid:12)ntion of
E(cid:11)(n(cid:0)j+1;n(cid:0)i+1):
The proof of (ii) is immediatefrom the claimof (i) and the de(cid:12)nitions of the two orders.
De(cid:12)nition
De(cid:12)ne V(cid:0) to be the A-algebra given by generators fFb(cid:11)(c) :(cid:11)2(cid:8)+;c2Z(cid:21)0g (where F(cid:11)(0) =1)
and relations
(cid:20) (cid:21)
Fb(cid:11)(c)Fb(cid:11)(b) = c+b Fb(cid:11)(c+b); (14)
c
Fb(cid:11)(ci)Fb(cid:11)(b) =Fb(cid:11)(b)Fb(cid:11)(ci) if ((cid:11);(cid:11)i)=0 and i<g((cid:11)); (15)
X
Fb(cid:11)(b)Fb(cid:11)(c0) = v(cid:0)j(cid:0)(c(cid:0)j)(b(cid:0)j)Fb(cid:11)(c0(cid:0)j)Fb(cid:11)(j+)(cid:11)0Fb(cid:11)(b(cid:0)j); (16)
j(cid:21)0;j(cid:20)c;j(cid:20)b
vcbFb(cid:11)(c0)Fb(cid:11)(b+)(cid:11)0 =Fb(cid:11)(b+)(cid:11)0Fb(cid:11)(c0); (17)
vcbFb(cid:11)(b+)(cid:11)0Fb(cid:11)(c) =Fb(cid:11)(c)Fb(cid:11)(b+)(cid:11)0: (18)
0
The relations (16), (17) and (18) are each subject to the restrictions that ((cid:11);(cid:11)) = (cid:0)1 and
0 0 0
either ((cid:11) =(cid:11)i and i<g((cid:11))) or (h((cid:11))=h((cid:11))+1 and g((cid:11))=g((cid:11))).
10