Polynomial functions on 0 (2λ+1)

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POLYNOMIAL FUNCTIONS ON 0, 2i+l

B arry W illia m WETHERILT

A th e s is submitted fo r the degree o f Doctor o f Philosophy a t

the U n iv e r s ity o f Warwick, November 1981.

Mathematics In s t it u t e

U n iv e r s ity o f Warwick

BRITISH THESES

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T H IS T H ESIS H A S B E E N M IC R O FILM ED EX A C T LY A S R EC EIV ED

T H E B R IT IS H , L IB R A R Y

D O C U M E N T SU P P L Y C EN T R E

Boston Spa, W etherb y W e s t Yorkshire, LS23 7BQ

Acknowledgements

Summary

§0 P re lim in a r ie s and n otatio n

0.1 The general lin e a r group and polynom ial functions 1

0.2 The Schur algebra 6

0.3 Polynom ial fun ctio n s on subgroups o f G 11

0.4 B ilin e ar forms and subgroups o f G 14

§1 C h e v a l l e y 's th e o re m f o r G a n d Q p & tl

1.1 Unipotent and diagonal subgroups o f G 16

1.2 The Big C e ll and C h e v a lle y 's theorem fo r G 18

1.3 The B ig C e ll and C h e v a lle y 's theorem fo r O211+I 26

11 The kernel of - V - k. I G UlK [r]

2.1 P re lim in a rie s 35

2.2 The kernel of t u_{• K[Ug] *} K[Ur J 36

2.3 The kernel of f T •

t k • K[TC] * K lTr ] 39

2.4 The kernel o f K+[G )[d ‘ ' ] * K l r l l d " ' ] 40

2.5 The K-algebra

b k 47

2.6 The kernel of

57

53 Modular Theory

3.1 The K-algebra K 6 Z[Tq]

3.2 Modular Reduction 64

54 The Schur algebras o f Ooa | j(Q )

4.1 The coalgebras Kr [r ] 66

4.2 The KG module e£ 70

4.3 The r-maps 8 ^ . 71

4,4 T raceless tensors 74

4.5 Decomposition o f e£ in c h a r a c t e r is t ic zero 77

4.6 Connection w ith Schur algebras Sr g ( r ) 80

i i Representation theory I

5.1 Representation theory o f G 92

5.2 Weights o f r = ° 2)l+1 103

5.3 Ir r e d u c ib le modules o f Sr g ( r ) 107

i £ Representation theory I I

6.1 The ir r e d u c ib le Kr-module Ar EK 117

6.2 The Qr nodules g 126

6.3 A Z-form o f g 134

6.6 The Kr modules K 154

B ib !iog rap h y 157

Appendix A 159

Appendix B 162

1

Acknowledgements

I would lik e to thank my su p e rviso r P ro fe s s o r J . A . Green fo r his

help and guidance over the past three y e a r s , my w ife H ila r y fo r

h er i n f i n i t e p atien ce and understanding, my parents f o r t h e ir never

ending support and Peta M c A llis te r fo r her expert ty p in g .

I g r a t e fu lly acknowledge the f in a n c ia l support o f the S .R .C . during

Summary.

This th e s is 1s an attem pt to g e n e ralis e to the odd orthogonal group ,

over an I n f i n i t e f i e l d K not o f c h a r a c te r is tic two, the work o f Schur [ S ] , and more re ce n tly Green [ G ], on the general lin e a r group G^ using the

approach o f Weyl [W ] 1n c h a r a c t e r is t ic zero. The sp e cia l fe a tu re here is th a t we t r e a t r K as merely a group o f m atrices defined by the vanishing

o f polynomials in i t s c o e f f ic ie n ts (th e c la s s ic a l view ) ra th e r than a group

generated by elements d erived from an asso ciated L ie a lg e b ra , the approach

used i n i t i a l l y by C h e v alley and adopted by most authors in re ce n t tim es.

A f te r g e n e ra lis in g G reen's [G ] Schur algebra fo r to in §0

we prove in §1 C h e v a lle y 's famous theorem on the 'B ig C e l l ' in GK and

then, by an easy e x te n sio n , prove i t f o r the B ig C e ll in . C h e v a lle y 's

o r ig in a l proof uses re p re se n ta tio n s o f L ie a lg e b ra s , ours re q u ire s nothing but a l i t t l e knowledge o f the coordinate rin g K+[G ] o f a l l 'polyn o m ial' fu n ctio n s on G^ . We d efine K [ r ] , the co ord in ate rin g o f r K , to be

the space o f a l l polynomial fun ctio n s on G^ r e s t r ic t e d to and in s2

give a generating s e t o f the kernel o f the r e s t r ic t io n map

h,k:K+[G ] -*■ K [ r ] . Th is g e n e ralises W ey l's r e s u lt in c h a r a c t e r is t ic zero.

In §3 we use th is r e s u lt to show th a t the fa m ily , o r 'schem e', o f rings K [ r ] (K varying over a l l i n f i n i t e f ie ld s not o f c h a r a c t e r is t ic two) is

'd e fin e d over Z' ; in f a c t K [ r ] is n a t u r a lly isomorphic to K 0 Z[Tq3 , where Z[Tq] is the subring o f Q [r ] spanned by 'monomial' fu n ctio n s.

This enables us to form ulate a 'm odular' re p re se n ta tio n theory fo r r which

connects polynomial re p re se n ta tio n s o f Tq w ith those o f r K .

In §4 we in v e s tig a te the Schur algebras o f Tq fo llo w in g Weyl [W] and

in §5 fin d a complete s e t o f ir r e d u c ib le s fo r each o f them, once again

fo llo w in g the lead o f Weyl. In §6 we attempt to 're d u ce' these modules modulo p to obtain 'W e yl' modules fo r , a task only p a r t i a l ly

0.0

Throughout K w i l l denote an i n f i n i t e f i e l d o f a r b it r a r y c h a r a c te r is tic

unless stated otherw ise and, when no confusion should a r is e , we s h a ll w rite

the tensor product 8K as 8 . Fo r an in te g e r n > 0 , EK(n ) (o r ju s t

Ek ) w ill denote an n-dimensional K-space w ith basis ( e ^ . e ^ , . . . ,e n> .

I f r > 0 is another in te g e r we denote the r- fo ld tensor product

Ek 8 Ek 8 . . . 8 Ek by e£ and d e fin e e£ to be K .

Denote by I ( n , r ) the s e t o f r- tu p le s w ith e n tr ie s from n = i l , 2 , . . . , n )

Then E^ has basis

(e . : = e . 8 e . 8 . . . 8 e .

_l

₁

_l

_{12 1}

_p

: i = ( i , , ! , ... 1r ) e I ( n . r ) } ._{\ c r}

0.1 The general lin e a r group and polynomial functions

We denote the group o f a l l non-singular n*n m atrices g = (9liv) p ven

w ith e n trie s in K by Gn K o r , when no confusion should a r i s e , v a rio u s ly

by Gn , Gk , G(K) and G .

Then G „ acts n a t u r a lly on the l e f t o f E „ (n ) by extending lin e a r l y

n,K ^

to the whole o f E^(n ) the a ctio n

0.1a

g.e = E g e fo r a l l v e n , geG v .

a v _yeri wv u — n,K

Hence G „ acts on E^ (n ) by extending lin e a r ly to the whole o f E£(n)

n, k n ^

( r > 0) the action

g (e . fi e. « . . . a e . ) • g .e . a g .e, a . . .a g .e .

11

12

V

11

'z

V

( f o r a l l e I ( n , r ) » 9 e Gn «) and Gn>|< a cts t r i v i a l l y

on E^(n ) = K .

Thus, the group algebra KG^ K o f Gr k c o n s is tin g o f formal sums

Z X .g (X e K, f i n i t e l y many 4 0 )

9‘ Gr .K 9 9

acts on E^(n) , making i t in to a l e f t KGn ^-module.

0.1c

For any s e t 0 we d e fin e Kn to be the s e t o f fu n ctio n s f :f i K .

Then Kn is a commutative K-algebra w ith operations defined 'p o in tw is e '

e .g . i f f , f ' e K then f f ' e Kn is the fu n ctio n which takes x e n to

f ( x ) f ' ( x ) e K . L e t r be any group, then we can extend each f e Kr

li n e a r l y to the whole o f the group algebra Kr o f r and id e n t if y Kr

w ith HomK( K r,K ) , the K-algebra o f K - lin e ar maps Kr -*• K . G K

Now fo r p ,v e ri d e fin e cyv e K n’ to be the K-l in e a r fu n ctio n

which maps each 9 e Gn k to i t s (u » v )th c o e f f ic ie n t gy v e K .

Denote by K+l Gn K1 the K-subalgebra o f K n,K generated by these

fu n ctio n s . Then K+|Gn K] i s by d e fin it io n the K-algebra o f polynomial

fu n ctio n s on G „ and sin ce K is i n f i n i t e the c „ ( u ,v e n) are

n,K wv —

a lg e b r a ic a lly independent over K . Thus K+[Gn can be regarded as

the K-algebra o f polynomials in n2 indeterm inates c yv (m,v e n) .

Fo r an in te g e r r > 0 we denote by Kr [Gn the K-subspace of

K+[G^ K] spanned by monomials in the cyv ( u ,v e n) which are o f degree

r , (thus K °[G n K] » K m where TL is the constant fu n ctio n

g 1 (geGn ) . Thus the dimension o f Kr [Gn>((] as a K-space 1s

("V'1» ■

For r > 0 , l e t G ( r ) be the symmetric group on r = { l , 2 , . . . , r > .

Then G (r) acts on the r ig h t o f I ( n , r ) by

0. Id

< V * 2 ... 1r> -' ‘ ... 1i.(r)>

( fo r a l l ( i i »12* — »1 r ) e U n . r ) , it e G ( r ) ) .

Hence G (r) acts on I ( n , r ) * I ( n , r ) : i f i , j e I ( n , r ) and n e G (r )

then (i,j)T T = ( iTTtJTT) .

For i , j e I ( n , r ) we w r ite i ^ j i f i and j are in the same

G (r ) o r b it o f I ( n , r ) , th a t is i f i = Jtt fo r some n e G ( r ) . S im ila r ly

f o r h,k e I ( n , r ) we w r ite ( i , h ) ^ ( j , k ) i f i = jn and h = kir fo r

some ir e G (r ) . Denote by T ( n ,r ) a s e t o f re p re s e n ta tiv es o f the G (r )

o rb its in I ( n , r ) x I ( n , r ) .

Now, given i , j e I ( n , r ) we w rite c ^ fo r the element

CV l C ,2 V ' CV r 1" KrtG">Kl '

0.1e

Kr [ G l

( I . J ) e T ( n . r )

s in ce c le a r ly c ^ runs over a l l monomials o f degree r in the n

generators cyv ( y , v e ji) o f K+[G^] » as (1 *J ) runs over T ( n ,r ) .

Hence

2

O . l f

2 | T (n , r )| . dimK Kr [Gn] . ( " " " ' I •

L e t V be a f i n i t e dimensional l e f t KG^-module w ith basis

( v , , v , , . . . ,vm ) . _{1 2 m} Then

O .lg

g-vb = £ Pab(g )* a fo r a11 be- ’ scGn aem

G

where the pflb e K (a.bem) , ( c a lle d 'c o e f f i c i e n t fu n ctio n s ' o f V ). G

Denote by c f ( V ) the K-subspace o f K n spanned by the pflb (a.bem) .

I t is easy to show th a t th is d e fin it io n is independent o f the choice of

b a s is o f V . We c a l l i t the c o e f f ic ie n t space o f V .

We denote by M ^G^) the category o f f i n i t e dimensional l e f t KGn-modules

V such th a t c f ( V ) c. K+[Gn] . Then every V in H^(Gn) gives r is e to a

f i n i t e dimensional rep resen tation Pv :Gn GL(V) o f Gn c a lle d a polynomial

re p re s e n ta tio n . This extends li n e a r l y to a re p re se n ta tio n (a ls o denoted by

5

-O .lh Example

The module E1^ is in the category MK( ° n) f o r a l l r >_ 0 , sin ce

by (0 .1 a ,b ) we have equations

g .e . - E c . ^ g j e ,

V

J c l ( n . r ) , geG K .

j i e l ( n . r ) 1j 1 n,K

Hence c f (E ^ ) = K-span { c ^ : i , j e l ( n , r ) > .

L e t M^(G ) denote the subcategory o f MK(Gn) co n s is tin g o f those

modules V e MK( Gn) such th a t c f ( V) c. Kr [Gn] . These are the polynomial

modules which affo rd rep resen tation s whose c o e f f ic ie n ts are homogeneous of

degree r in the c^v (g .v c ji) . C le a r ly e£ is in M^(Gn ) .

O . l i Theorem

L e t V e MK(Gn) , then

y . i * v<r > r>0

where each is a KG -submodule o f V w ith c M?3(G ) .

n K n

Proof

The proof given by Schur [S ,p .5 J f o r K = t , the complex numbers,

works f o r any i n f i n i t e f i e l d .

i t is enough to study homogeneous re p resen tation s ie . those in M£(Gn )

fo r a l l r >, 0 .

Remark

K+[Gn] can be regarded as the coordinate rin g o f the a f f in e semigroup

M „ o f a l l nxn m atrices w ith e n trie s in K . This means we can consider n»K

polynomial rep resentations o f Gn K as r a tio n a l re p resen tation s o f Mn K

and v ic e versa .

F i n a l l y , we w rite

M G„ ] Kr [ G l

the space o f polynomials in the c ^ (w .ven) o f degree <_ r .

0.2 The Schur algebra

L e t C be a K-coalgebra with c o m u ltip lic a tio n A:C -*■ CfiC and co u n it

e:C K , (see [Sw] fo r d e fin it io n o f coalgebra and re la te d term in o logy).

0.2a Examples

(1 ) We can consider K as a one dimensional K-coalgebra with

a( 1k) = 1K fi 1K and e ( 1K) = 1^ .

(2 ) The K-algebra K+[G] has a K-coalgebra s tr u c tu r e . C o m u ltip lica tio n

is d efin e d by extending lin e a r ly and m u lt ip lic a t iv e ly to the whole o f K+[G J:

7

-A (c ) - l c fl c ( y , v c n) .

pv pen pp pv

This a ris e s from the equations

V , < S - 9 ' ) *

^

„i9>cpv( i '> (u ' v c

S>

where g ,g ' e G^ K .

The co u n it e :K +[G] + K is defined by extending e (U ) - 1K and

0.2c

1 y = v ( y , v e rO e (c ) = {

“ v 0 y t v

This is e v a lu a tio n a t the id e n t it y o f Gn k •

(3 ) S in ce a (K r IG ] ) c Kr [GJ « Kr [G] and

4(Kr lG )) s Kr [G] a Kr [G] , both Kr [GJ and Kr [G] are K-subcoalgebras

o f K+[G] .

0.2d

Me can make C* = HomK(C ,K ) in to an a s s o c ia tiv e K-algebra w ith u n it

e by d e fin in g m u lt ip lic a tio n as fo llo w s:

i f f , f ' e C* then f f ' - ( f ■ f ' )A:C ♦ K (where K 8K is id e n tif ie d w ith K ).

For any r > 0 we c a l l the dual space s£(Gn) o f Kr [Gn) a Schur

algebra fo llo w in g Green [ G . p . l l ] . Then by (0 .2 d ) i f r > 0 , i , j c I ( n , r )

and e sJ^(G^) we have

♦•♦'(c,,) * i

♦(c

1

k)^‘(ckj) .

1J k e l ( n . r ) J

We denote by Ci j the element o f s£(Gn) defined by

0.2e

1 ( 1 . J ) * (h ,k ) (h .k e I ( n , r ) ) . C4 i (c hk) = t

i j hk 0 ( i , j ) jt- (h .k )

C le a r ly i-. - ch k i f f ( i , j ) n. (h,k> and th e re fore s£(Gn) has basis

i j h,K ^

: ( i . j ) c T ( n , r ) } . For any r > 0 d e fin e e:KGn - W to be

the unique K - lin e a r map such th a t fo r g e G ^ K , e (g ) is e va lu a tio n a t

9 . I . e .

e (g )( c ) = c (g ) fo r a l l c e Kr [Gn] .

We w r ite eg f o r c (g ) ; e ( l G) is the co u n it o f Kr [Gn ) .

L e t mod(s£(Gn) ) denote the category o f f i n i t e dimensional l e f t

s£(Gn) modules.

0 .2 f Proposi tio n

For any r >_ 0 the K - lin e a r map e:KGn s £ (Gn ) an epimorphism

9

-and mod(s£(Gn) ) using the ru le

g.v = e . v _g fo r a l l g e G „,veV_n,K

where V c MJ^(Gn) o r mod(s£(Gn) ) .

Proof

The case K = C is due to Schur [ S ] , fo r the general case see Green

[G, 12.4].

0.2g Remark

I f V e m£(G ) has basis ( V j. V g , . . . , vm> such th a t fo r b e m ,

9 e G„ . K

9 'v b ' £m pa b (9 )v a aem

then the a ctio n o f s£(Gn ) on V is g ive n by

4>. v. = Z ♦(Pab^v a f o r a11 be- ’ 4’eSK^Gn^ ’

a cm

Define Sr K(Gn) to be the extern al d ir e c t sum

0.2h

sk

''<G> ■ ■ * S°K(Gn>

s r K( Gn) has basis

0.21

(Ci j : ( i , j ) e Tr (n ) : » T ( n ,r ) u T(n,r-1 ) u . x » T (n ,0 )} .

I f | = e a . . c . . e S „( G ) we denote by the component

(1.J)eTr (n) r ’K n

of f in S^(Gn ) ie .

<-( i. J ) e l<-( p .n ) au ' u

We s h a ll sometimes w rite 4> as the r+ l-tuple .

9 e Gn K » * r ^ is e v a lu a tio n a t 9

0 .2 j P ro p o sitio n

For any r .> 0 er :KGn s r K( Gn) is an e PimorPh'ism o f K-algebras.

Proof

The case r = 0 is immediate . C le a r ly

£r (KGn) c S p, K(Gn) ■ Suppose Sr>K(Gn) ,ji er (KGn) , then there e x is ts a

non zero element c e Kr [Gn] such th a t er ( g ) ( c ) = c (g ) = 0 f o r a l l

g c Gn K . By the ir r e le v a n c e o f a lg e b raic in e q u a lit ie s , lW ,p.4] we

- 11

0 . 3 Polynomial fu n ctio n s on subgroups o f G .

L e t r be any subgroup o f G^ . Elements o f K+1G] can be regarded,

by r e s t r ic t io n to r , as fu n ctio n s on r o r lin e a r fun ctio n s on the group

algeb ra Kr . We d e fin e K [r ] c. Kr to be the s e t o f these r e s tr ic te d

fu n c tio n s . Then K [ r ] is a K-subalgebra o f Kr and in h e r its a K-coalgebra

s tru c tu re from K+[G] . The r e s t r i c t i o n map \ : K +[G] -► K [r ] Is a s u r je c tiv e

morphism o f K-algebras and K-coalgebras. The subspace ’r K(K r. [G ]) in h e r its

a K-coalgebra s tr u c tu r e from Kr [G] , denote i t by [ r ] and fo r

1 . j e I ( n , r ) w r ite c j j f o r '^K( c i ^ ) . Denote by Sr>K( r ) the dual o f

Kp[ r ] . By (0.2d) we can g iv e Sr K( r ) a K-algebra s tr u c tu r e . The su r­

j e c t i v e morphism o f K-coalgebras

V . K : Kr [G1^ Kr [ r l

c 1j * C1J

induces an in je c t iv e morphism o f K-algebras

0.3a

S r,K<r > ’ Sr.K<G) •

We s h a ll id e n t if y Sr K( r ) w ith i t s image 1n Sr>K(G) .

We c a ll Sr K( p) a ' Schur alg e b ra' of r .

0.3b Lenina

Fo r any subgroup r o f G and non-negative in te g e r r we have

where Kr is considered as a K-subalgebra o f KG .

Proof

Le t 4> e Sr k(G) , then <J> e Sf K( r ) i f f the kernel o f Kr [G] -*• K

contains the kernel o f f K , (th e n 4> can be considered as a fu n ctio n

on

Kr [rl) .

Le t 4> = £r (a ) (aeK r) then c e ker v r K im p lies that c ( a ) = er ( a ) ( c ) = 0 and hence e p(K r ) Ç Sr>K( r ) •

Now, suppose Sr K( r ) <£ er (K r ) . Then there e x is ts some non-zero

c e Kr [ r ] w ith c = v r K( c *) f o r some c ' e Kr [G] such that er ( a ) ( c ' ) = c ' ( a ) = 0 fo r a l l a e Kr i e . c ' e k e r f ^ so th a t

V r K^c ' ) = c = 0 » a c o n tra d ic tio n .

We d efin e e j: K r -*■ Sr K( r ) to be the r e s t r ic t io n o f er :KG -*• Sr K (G)

to Kr .

L e t V be a f i n i t e dimensional l e f t Kr-module with basis t v i * v2 ** * **vm^*

then fo r b e m

9■vb * E P Î b ( 9 )v a (9 £ r ) aem

where pîjb e Kr (a ,b e m) .

As in §0.1 we d efine the c o e f f i c i e n t space c f ( V) to be the K-span

o f the p£b (a ,b e m) and as befo re i t is independent o f the ch o ice of

b asis.

13

-V such th a t c f ( -V ) c Kr [ r ] . As an example a module in Mr K(G) is

the d ir e c t sum o f modules from m£(G ) , m£ ^ ( G ) , ... M^(G) by (0 .1 1 ).

We have the analogue o f ( 0 . 2 f ) .

0.3c Pro p o s itio n

L e t r be a subgroup o f G . Then the category o f f i n i t e dimensional

l e f t Sr K( r ) modules and Mr K( r ) are e q u iv a le n t using the ru le

a .v = e j ( a ) . v (a e K r, v e V)

where V is in e ith e r o f the c a te g o r ie s .

Proof

The proof follow s th a t o f ( 0 . 2 f ) given in [G, S2.4J and uses (0 .3 b ).

0.3d Remark

S in c e , f o r V e MK( r ) we have V e Mr K( r ) fo r some r > 0 , to study

the Kr modules in MK( r ) i t is enough by (0 .3 c ) to work w ith the f i n i t e

dimensional a s s o c ia tiv e K-algebras Sr K( r ) ( r > 0) . A ll th is a p p lie s to

r = G o f co u rse, although by ( O . l i ) and (0 .2 f ) we can confine our atte n tio n

to s£(G) (p > 0) .

In the case c h a r a c te r is tic K equals zero, th is was the technique used

by Schur [S ] on G and Weyl [W] on the other c la s s ic a l groups (see below ).

Green [G] has extended S ch u r's work to an a r b it r a r y i n f i n i t e f i e l d and we

0.4 B ilin e a r forms and subgroups o f G

L e t B:E^ * E^ -*■ K be a b i l i n e a r form defined by a matrix

B = (B (e ,e ) ) _ . Then the s e t — ' ' p ’ v " u , v e j i

0.4a

r B = igeGn K | B (g e .g e ') = B ( e , e ') V e .e 'c E ^ }

i s a subgroup o f ^ and i f we denote by g*" the transpose o f the

m atrix g , then

0.4b

( i i ) The m atrix B^ = ^ y ie ld s the even orthogonal group, 02)l(K ) •

( i i i ) The m atrix B = J 2ä+1 y ie ld s the odd orthogonal group, 02i + l ^ * r B = (9 tG n,K I gt- 9 = ’

0.4c Examples

Fo r an in te g e r i. > 0 , l e t

J, 0

(ixfc)-m atrix.

i

0

0 J

1 y ie ld s the sym plectic group, Sp2a(K ) ( i ) The m atrix JÎ

15

-Henceforth r „ K (n = 2*+ l) , or when no co nfu sion should a r is e r K ,

r (K ) or r , w i l l denote the odd orthogonal group 02ji+1(K ) '

0.4d Remark

Many o f the techniques used on 02a+i ( K ) ™ subse(luent sections can

also be a p p lie d , with some m o d ific a tio n s , to the sym plectic and even

orthogonal groups. These s im ila r it ie s (and d i s s i m i l a r i t i e s ) are o u tlin ed

§1. C h e v a lle y 's theorem fo r G and (^fc+l

1.1 Uni potent and diagonal subgroups o f G

We d e fin e the fo llo w in g subgroups o f G :

UG: = (u n i potent upper tria n g u la r m a tr ic e s ), a uni potent group

Ug: = iu n ip o te n t lower tria n g u la r m a tr ic e s ), a u nipotent group

Tg: * (non s in g u la r diagonal m a tr ic e s ), a commutative group

WG: = {perm utation m a tr ic e s ), isomorphic to G (n) .

1.1a

I t is easy to see th a t J Ug J n - U3 , where J n is the m atrix

n n

defin ed in (0 .4 c ).

We can now s ta te the fo llo w in g w e ll known theorem.

1.1b Theorem (B ru h a t, C h e valley)

We have G = U^.w.Tg.Ug (a d i s jo i n t u n ion ).

w e

W

g

Proof

Stein b erg [ S t , p .36] proves (by hand) th a t G = L J Ug.w.Tg.Ug .

wcWG

S in ce J (= 0n) is a permutation m atrix we can rep lace Ug by UgJ ,

a ls o JG = G and th e re fo re

G - U (OUgJ).w. T U - U U^.

w

.T

g

.Ug .

17

-1.1c

I t 1s c l e a r th a t K[UG] and K[U^] are polynomial rin gs over K

f r e e l y generated by the ( 2) coordinate fu n ctio n s (c ^ v : 1 < u < v < n)

and i c l : 1 £ v < u <_ n l r e s p e c t iv e ly , where c^v and cjjv denote the

image o f c yv e K+[G] 1n K[UG] and K[U^1 r e s p e c t iv e ly . A lso K[TGJ

is a polynomial rin g over K f r e e ly generated by { c j y : yen} , where

c uu ^ma9e c u v e ^[TG] . C le a r ly K[TG] has a basis

o f monomials:

( « { , ' ’ ‘ c22> 2

where V X 2... xn c Z>?

Denote by A (G) the s e t o f n-tuples w ith e n t r ie s from Z>Q and fo r

\ = (Xr X2, . . . , X n ) e A(G) by xA K the fu n c tio n o f TG given by (1 -Id)

i e .

l . l e

xX,K^d i a 9 { t r t 2 * ' ' ” t n ^ = *1 fc2 t n »

(where d i a g ( t 1 , t 2, . . . , t n) e TG is the m atrix w ith t 1#t 2... t R on the

diagonal from l e f t to r i g h t ) .

J G We c a l l elements o f A(G) ‘w e ig h ts' and f o r XeA(G) xx>K e K is

the 'c h a r a c te r o f weight X '. C le a r ly (x x>K : X e A (G )} is a basis o f

1.2 The Big C e ll and C h e v a lle y 's theorem fo r G

Define the B ig C e ll fig in G to be the s e t o f elements

1.2a

(u 't u : u~eUg , t e Tg , u e Ug)

Now, fo r g e Gn denote by g ° the m atrix obtained from g by d e le tin g

the n**1 row and column. I f f e K+[Gn] is a polynomial in the cyv

(1 £ p,\> £ n-1) we can consider f as an element o f K+ ] by using

the obvious in c lu s io n K+[Gr _^J d K+[Gn] , ( r e c a ll K+[Gn_^] is fre e on

the indeterm inates { c yv : 1 £ p , v £ n - U and K+tGn] on indeterm inates

{ c yy : 1 £ m,v £ n J) . Then, f ( g ) = f ( g ° ) , remembering that i f g°

is s in g u la r we can s t i l l defin e f ( g ° ) (Remark, p .6 ). I t is easy to prove

th at:

1.2b

For u"eUg , w e Wg , t e Tg and u e Ug then (u "w tu )° =

n n n n

u 0w °t °u 0 . A lso fig = fig

n n-1

We now prove the fo llo w in g important r e s u lt.

1,2c P ro p o sitio n

For s e n d efin e D$ e K+[G$] <£ K+[Gn] to be the determinant o f the

m atrix (c ) _ and d „ : = n D„ .

' wv'p.ves^ n Scn s

Then fi~ =

[image:27.340.22.333.23.442.2]

- 19

Proof

We prove th is by in d u ctio n on n . C le a r ly the r e s u l t holds i f n = 1

Assume the r e s u l t is true fo r n-1 , so th a t

fig

= {geG^ ^ |dn

.j

(g ) + 0} n-1

Consider u tu e U~ Tg Ug , then by (1 .2 b ) u ° t ° u ° e fig so th a t

n n n n-1

dn(u "tu ) = Dn (u "tu )d ni (u 't u )

■ Dn( t ) d n_ 1<u-°t0u0 )

is non zero, (co n sid e rin g K+[Gn_ j ] C K+[Gn] ) .

To show these are the only elements of Gn s a t is f y in g th is we must

use (1.1b) which s ta te s th a t Gn is the d is jo in t union

O UG .w.Tq .Ug

weWg n n n

where Wr is the group o f permutation m a trice s.

Notice th a t fir is the term w ith w = 1 e Wfi

Case ( i )

I f 1 t weWg has a 1 in the ( n ,n ) th p la ce , then 1 / w ° e Wg

n n-1

and th e re fore by in d u ctio n dn_ j(u 0w °t °u 0 ) = 0 so th a t dn_ j( u wtu) = 0

by (1.2 b ) and hence dn(u wtu) = 0 .

Case ( i i )

[image:28.343.28.317.25.464.2]

so th a t Dn_ j ( w ° ) = 0 . I t now fo llo w s th a t (u w t u )° = u~0w °t °u 0 is

s in g u la r so th a t Dn-1 (u ’ wtu) = 0 and th e re fore dn (u "w tu ) = 0 .

1.2d Remark

Before proceeding fu rth e r we make some general remarks concerning

rin g s o f fr a c t io n s and r e fe r the reader to [A.M.S3] fo r more d e t a ils .

L e t R be any commutative rin g w ith id e n t it y and S be a m u ltip li-

c a t i v e l y closed subset con tain in g the id e n t it y . We d e fin e an equivalence

r e la t io n ~ on R*S as fo llo w s:

( r , s ) -v (r-|,s.|) i f f ( r s 1 - r-jS)s' = 0 fo r some s ' e S .

We denote by r/s the equivalence c la s s o f ( r , s ) and g iv e the set

R [S _1] o f a l l these cla sse s a comnutative rin g s tru c tu re :

th is is not in j e c t i v e , but i f every element o f S is not a zero d iv is o r

then r / l R = r j/ 1 R i f f ( r - r ^ s - 0 f o r some s e S , i e . r - r ] - 0 .

Thus R can be id e n t if ie d w ith a subring o f R IS * 1] in th is case and w<

w r ite r f o r r / l R .

stru c tu re in the obvious manner. I f S i s generated by a s in g le element

Hence, = (geGn : dn(g) ^ 0} •

n

ss

There is a rin g homomorphism R -*■ R [S given by r -*■ r / l R . In general

I f R is a K-algebra then R [S -1] in h e r its from R a K-algebra

21

-1,2e Example

Consider the a s s o c ia tiv e K-algebra K+[Gn] [ d n^ ]. Sin ce K+l® nl

is an in te g ra l domain ( i t is a fre e polynomial r in g ) i t can be considered

as a K-subalgebra o f [Gn] [d "1 ] . Also every element o f K+[GnJ [t*” ' ]

is o f the form q/d" fo r some q t K+(GnJ and non 2er0 tn teger m ,

(though not u n iq u e ly ).

When no confusion should a r is e we w rite ii f o r fig and d f o r

dn-1.2f

We defin e * :K +[G ][d _1] - Kn as fo llo w s:

l e t q e K+[G] and m e Z>Q , then

4>(q/dm) : g q(g)/dm(g ) f o r a l l g e n .

We must show * is w ell d efined. m.

Suppose th a t q/dm = q ^ d f o r some q , q 1 e K+[G] and m,m1 e Z>Q

Then qd”1' = q ^ " 1 i f f q lg ld ” 1 (g ) = q, (g )dra(g) f o r a l l g t i i f f

q (g )d m,(g ) ■ q1(g)d m(g ) fo r a l l g e l l (s in c e , i f g ( a then

m m.

dm(g ) * d ' (g ) = 0) i f f q (g )/d m(g ) « q1 (g)/d (g ) fo r a l l g e t i ie .

*(q/dm) = * (q 1/d"'1 ) . Thus, * is w e ll defined and we have also shown i t is

in je c t i v e . C le a r ly i t is a K-algebra morphism, so henceforth we id e n tify

I^ G Jld ’ 1] w ith i t s image under * in K° .

1.2g

[image:30.344.23.330.25.404.2]

procedure to show th a t the K-algebra A = K[Ug] 8 K[Tg] Cl K[Ug] can be

uG*TG*UG

id e n tif ie d w ith a K-subalgebra o f K v ia the map which takes

f^ 8 f 2 8 f^ e A to the fu n ctio n mapping ( u " , t , u ) to f j ( u J f g i t J f j f u )

For a proof th a t th is is a w e ll defined in je c t iv e K-algebra morphism see

fo r example [H, §2.4]. We id e n tif y A w ith i t s image in

U^»t0«Ug

Now, w r ite 6(6^) f o r the element (18dy81) e A , where dy eK[Tg]

is the r e s t r i c t i o n o f d e K+[G] to Tg ( in f a c t dy = X ( n>n_ ^ ,1 ),K ^ *

UG*TG*^G

Then, sin ce we consider A as a K-subalgebra of K , i t i s easy

to show th a t A[6~*] can be id e n tif ie d w ith a K-subalgebra of

UpxTpxUp _ _I

K v ia the map which takes q/6 e A[6 ] to the fu n ctio n mapping

( u " , t , u ) to q (u " ,t ,u )/ 6 m( u " , t , u ) . Henceforth we id e n t if y A[6 with

ur xTGxUr

i t s image in K . A ls o , sin ce A is an in te g ra l domain ( i t is

isomorphic to a free polynomial r in g ) we can id e n tify i t w ith a subalgebra

o f A ( S '’ ] .

1,2h

D efine a map * = » Tg < U j » nG by *n ( u ', t , u ) - u 'tu .

n n n n

C le a r ly t h is is s u r je c t iv e . Suppose th a t ♦n(u’ , t , u ) = ♦n( ui » t j * ui ) »

then u "tu = u^tjU-| • Hence (u ^ )” ^ u "t = t^u^u ^ and sin ce the L .H .S .

is a lower tr ia n g u la r m atrix and the R .H .S . is an upper tr ia n g u la r m atrix

we see th a t t = t j , u = u^ and u = u-j . Thus <i>n is a ls o i n je c t i v e .

23

-1,2 i Theorem (C h e v a lle y )

The map <|>n :Ug x Tg x Ug ■+ nG induces an isomorphism o f K-algebras

n n n n

V M Gnl l d n 'j * ] • X[Tg ] a K[Ug ) ) [ * ; ' )

n n n

such th a t $n( l/ d n ) = l /6n .

Proof

1 * UG *TG *^G

Given f e K+[Gn] [ d ‘ ' ] we d e fin e *n( f ) e K n to be the

com position fo<j>n » then <f>n( f ) ( u , t , u ) = f (u tu ) .

We must f i r s t show th a t the image o f <)>n l i e s in

A - l « '1) ■ ]

a

K[TC]

a

K[UG ] ) [ « ■ ']

.

n n n

Now, sin ce 4>n is c le a r ly a K-algebra map we need only consider f = c _{yv '} (y .v c n ) and f = l/ d „ .

— ' n

( i ) I f f * c yv (u .v e n ) then ♦ „ ( % „ ) ( u \ t , u ) « c y v (u ‘ tu )

= E cU (u ” )c T ( t ) c U (u ) . Hence,

pen up

'

p p'

'

p v'

♦*(c ) = E c U

a

CT

a

CU e A i« ’ 1] . rr p v ' pen pp pp pv n ' n

(1 1 ) I f f * l/ d n then ♦ * (l/d n) ( u ' , t , u ) - l/ d n( u 't u ) - 1 / d „(t) .

Hence ♦ *(l/d fl) . 1/1 I d j I I ■ l / s n .

* * Now <i>n is s u r je c t iv e , so th a t 4>n is in je c t iv e sin ce i f <|>n( f ) * 0

then <|>*(f ) (u ~ , t , u ) = f(u ~ tu ) ■ 0 fo r a l l u tu e iig i e . f = 0 .

n n

To prove the theorem i t remains to show th a t $n is s u r je c t iv e .

We proceed by in d u ctio n on n . I f n = 1 then G^ = Tg^ and

K [G jj = [ d ^ ] is c e r t a in ly isomorphic v ia 4^ to (lflK [T g j f t l ) [6^ ] Suppose th a t the theorem is true fo r n-1 . Now, as remarked above, we

can co n sid er K+[Gn_^J as a K-subalgebra o f K+[GnJ • We then have an

In c lu s io n i : K+[Gn_ , ) [ d ^ , l * K t [Gn) [ d ‘ ’ l such th a t i ( c uv) ■ c pv

(u ,v e n-1) and 1 (l/ d -j) = Dn/dn • C le a r ly we can a ls o consider

K[Ug ] , K[Tg ] and K[Ug ] as K-subalgebras o f K[U^ ] , K[Tg ]

n- 1 n- 1 n- 1 n n

and K[Ug ] r e s p e c t iv e ly . Thus we o b tain an in c lu s io n

n _

i ■

V l 1'«-!1 * V

O

such that J(C • 1 *

‘ c“u * 1 ® ’ •

JO »

cjv a l) -

1

* cjv a l , JO a l a e“v) - i a i i c “( (u.vcnO)

and J ( l / « n. , ) » (1

a

X ( l i , ..., ) « l ) / « n ■ Now, i t is easy to see th a t

0 - J ** ., : K. l G„ - l l l d n -l) * AnISn 'l • B* , " duct,on C l 1s an 1s°

-morphism and i t then follow s th a t c|Jv a 1 81 (y ,v e n - l) , H 1

(X c A(Gn) : xn * 0) and 1 fi 1 fi c^v (u ,v c n - l) are a l l in the image of

<t>* . F u rth e r, s in ce 4>ni ( l / d n_-,) = (1 a X(1, 1 , . . . , 1 )® 1^ 6n then

1 « x (0 ,0 ,..0 ,1 )® ' - ( ’ • ‘ (1 .1 ...,1 )» ' ) 2 ‘ n-2/ s n is 1n the * “ * of ‘ n

Hence every 1 B xx ® 1 (X e A(Gn) ) is in th is image.

I t remains to show th a t y 8 1 a 1 (ven) and 1 a 1 l c ^ n (uen) *

are in the image o f <t>n . Now

E C

p e]l

i f

lp

a

c

u

25

-but c“ p = 0 when p > 1 and = 1 . Thus, we can re w rite

M e , ) _{n in} as 1 8 c l , _{i i} 8c "_in and sin ce there e x is ts a X c A(G ) such _n th a t

( 1 a xA » l / « n ) ( u ' , t , u ) - c ^ i t) - 1

we see th a t 1 a 1 a c^n is in the image o f 4>n . S im ila r ly

and sin ce c^ S a c^

there e x is t a X e A(G )

8 ‘ Ì , 8 ‘ f n + 18« ¡ 1 8 c2n

is now shown to be in the image of

such th a t

and

[ l i x , < v « n ) ( u ' . t , u ) - 4 2( t ) - ’

we see th a t 1 8 1 8 Cgn is also in the image.

Proceeding in th is way we can show th a t 1 8 1 8 c pn is in the image of

<j>* fo r a l l y e n_ and by a s im ila r method c|{v 8 1 8 1 (ve n ) are also in the image.

This completes the p roof.

1,2 j C o ro lla ry to the proof o f ( 1 .2 i)

L e t K = Q and d e fin e Z+[Gq] to be the subring o f Q+[Gq] Z-spanned

by monomials in the co ordinate functions c yv (u .v e n ) and Z[UG^ ] ,

r e s t r i c t i o n maps. Then

♦ * ( V G„1> ■ <z l% Q )l 8 ZITG(Q)i 8 ZWoiQ)))“ ' ' ) •

Proof

L e t K = Q in ( 1 . 2 i ) and observe th a t the proof works over Z .

1,2k Remark

The proof o f (1 .2 1 ) d if f e r s from C h e v a lle y 's o r ig in a l proof in so

f a r as our approach uses p re cis e knowledge o f the two K-algebras involved

ra th e r than the a ctio n s o f generating elements o f the group on c e r ta in

modules. Of course i t is not s u rp ris in g th a t we can do th is f o r the

r e l a t i v e l y s tra ig h tfo rw a rd group G , but we s h a ll a ls o use th is approach

on the odd orthogonal group in the next s e c tio n . These proofs u n derline our

approach, th a t is we a re d e fin in g our groups as the zeros o f a s e t o f p oly­

nomials ra th e r than groups generated by c e r ta in m atrices.

1.3 The Big C e ll and C h e v a lle y 's theorem f o r 02A+1

Henceforth unless s ta te d otherw ise we s h a ll assume th a t the c h a r a c t e r is t ic

o f K i s not 2 . The reason behind th is r e s t r ic t io n is explained in Remark

(1 .3 h ).

We defin e the fo llo w in g subgroups o f the odd orthogonal group r = r n (K )

(n-2 1+1, t>0) :

- 27

and the subset fir : = fig^r , (the Big C e l l ) .

1,3a Lemma

The subset fip is equal to

( i ) tg e r|d r (g ) / 0} where dr e K [r ] is the r e s t r ic t io n o f

d e K+[G] to T ,

( i i ) u ;T r ur .

Proof

S in ce fig = (g e G |d (g ) ^ 0} (1 .2 c) then ( i ) is immediate. To prove

( i i ) we introduce an automorphism o:G G defined by o (g ) = J ( g t ) ^ J .

I t is not hard to show th a t o preserves Ug • tg and Ug and th a t fo r geG , o (g ) = g i f and only i f ger (0 .4 b ). Thus, r is the s e t of fixed

points o f a . Now, fig = Ug TgUg so suppose th a t g = u "tu e f ig a r then

o (g ) = o ( u " ) o ( t ) o ( u ) = g . But we have shown th a t every element o f fig

has a unique expression as a product o f elements from Ug , Tg and Ug

(1 .2 h ). Hence o(u ‘ ) = u" , o ( t ) = t and o (u ) = u so th a t u~ e u ’ ,

t e Tp and u e Uf . Thus fip c Tp Uf and the reverse in clu s io n is

o b viou s.

We in v e s tig a te the elements o f Up :

L e t * be the (n*n) m atrix:

A D B A,C are upper tr ia n g u la r m atrices

0 1 E B i s a *XJI matrix

0 0 C D i s a 1* 1 ma t r i x

[image:36.343.26.326.22.452.2]

Then * e

ur i f and o n ly i f n * « J n ie .

A t 0 0 0 0 J £ A D B 0 0

D l l 0 0 1 0 0 1 E * 0 1 0

Bl Efc Ct J t 0 0 o o r>

₀

0

Expanding th is we ob tain the fo llo w in g r e la tio n s :

1,3b

( R l ) A ^ C ■ J ,

(R2) E * D ^ C - 0

(R3) Bl J t C + El E + c ‘ j t B * 0

(R4) Cl J t A . J ,

(R5) C ^ D . E1 - 0 .

I t is c le a r th a t ( R l ) is e q u iva le n t to (R4) and (R2) to (R 5 ). Thus

( R l ) , (R2) and (R3) are necessary and s u f f ic ie n t co n d itio n s fo r * to belong

to r . Now, i f we choose A a r b i t r a r i l y (upper tr ia n g u la r u n ip oten t) we

then determine C by ( R l ) . F u r th e r , i f we choose D a r b i t r a r i l y we determine

E by (R 2 ). We must now check th a t (R3) does not in te r f e r e w ith our freedom

to choose A and D a r b i t r a r i l y and see to what ex te n t we are fr e e to choose

B :

L e t F ■ < V u . v . £ = " V * " d " " ( \ v > U , v « i ' E ‘ E • The" (" 3)

can be r e w ritte n as:

29

-g iv in -g f yv + f yp = -hyv ( p . v e l) .

Hence, each (ue£) is determined by A and D and i f \

f (u<v) a r b i t r a r i l y , A and D determine each f (u>\>) , JJV ' '

is no r e la t io n between A and D .

t- 1 t

Now, s in ce B = (by v ) y sttl ■ J t Cfc F we can choose the by ,

a r b i t r a r i l y , the r e s t being determined by A and D v ia C and

We can now prove:

1,3c P ro p o s itio n

The fo llo w in g s e t o f fu n ctio n s on Up f r e e ly generate

K-algebra K[Ur ) :

1.3d

<‘ !!v : (w .v ) < R ,}

where c y ^ e K[Ur ] is the r e s t r ic t io n o f cyv e K+[G] to Up

= { ( y , v ) e nxn : 1 < _ v < t , u+1 £ v < 2t+ l-u )

We can rep resent these elements on a schematic m atrix (*• =

1 * * 1 * ■ * *

1 * ¡ * ! *

1 '• * •

i(r = element o f (1.3 d )

‘ choose

but there

(u+v<i)

F .

the

and

Proof

The fu n ctio n s (1 .3 d ) a l l H e in K[Ur J , sin ce 1t 1s c e r ta in ly

generated by the s e t o f functions

1.3e

Ur

{c : 1 < y < v < n ) .

yv - —

The c a lc u la tio n s above on ♦ show th a t i f a r b it r a r y values in K

are assigned to each o f the fun ctio n s in (1.3d) then there is a uniquely

defined element o f Up on which they take these v a lu e s . This shows th a t

the fu n ctio n s (1.3 d ) are a lg e b r a ic a lly independent and th a t every fu n ctio n

in (1 .3 e ) is a polynomial over K in them. This completes the proof.

We can also prove in a s im ila r way:

1,3 f P ro p o s itio n

The fo llo w in g s e t o f *2 fu n ctio n s on Up f r e e ly generate the

K-algebra K[Up] :

K 3 a

( u .v ) e R;>

where c ^ e K[U~J is the r e s t r ic t io n o f cyv e K+[G] to Up and

R* = t ( y , v ) e n^n : 1 <_v < l , v+1 < y ± 2t+ l-v)

1,3h Remark

31

y ie ld s 2f = -h (pen) . I f the c h a r a c t e r is t ic o f K i s 2 th is

J mm p p ' —

im plies t h a t a l l o f the e n tr ie s in E are zero. Then (R2) im plies th a t

D is determined by A . On the other hand the e n tr ie s {b^v : p+v <_ * } can

be chosen a r b i t r a r i l y ( t more than b e fo re ). A ll th is means th a t K[Ur J

2 2

is no lon g er f r e e ly generated by the 9. fu n ctio n s (1 .3 d ) but by the i

functions

Ur

{ c y^ : l<y<£, p+l<y<t, t+2<_v<2t+2-pl .

I t is to a vo id th is com p lication th a t we assume the c h a r a c te r is tic o f K is

not 2.

1.3i Remark

U_ * U

I f K = Q , observe th a t each c ^ u . v e n ) li e s in Z [cy v : (p .vJe R ^ ] ,

where Z is the subring o f Q generated by Z and J . The need to

ad join J to Z a ris e s from the r e la t io n 2fyji = - h ^ (yen ) , ( c f . (1 .3 h )).

We now turn our a tte n tio n to Tr . I t is r e a d ily seen th a t a diagonal

m atrix d i a g { t ] , t2>. . . , t n> l i e s in r i f and o n ly i f t ^ t ^ ^ = 1 (yen ) . Thus t-j , t 2, . . . , t A can be chosen a r b i t r a r i l y non zero and t| +1 = 1 . I t follow s t h a t K[Tr ] has a basis o f monomials

U i

Tp a , T Ciy Ip o a

(c l j ) (c 2,2 ^ ^c t+l,» + l^

1 p

Denote by A ( r ) the s e t o f *.+1-tuples a = (c^ ,<»2, . . . ,a Jl+1) such

t h a t a.j .cig,. . . e Z and e {0 ,1 } . As with Tg we can define

the ch a ra cte r K o f r g iv e n by ( 1 . 3 j ) . Thus

1.3k

xo.K<d

1

*

9

(tl

>t2

.... ‘I-1-*!’

where c = ±1 .

. t , '> ■ t / t .“ l “ 2

We c a l l elements o f A ( r ) 'w e ig h ts ' and x£ K (a e A (r )) a 'ch a ra cte r

o f weight a ' . C le a r ly K[Tp] has a basis (x£ K - a e A (r ) } .

Follow in g ( 1 .2 f ) and (1 .2 g ) we can id e n t if y the K-algebras K l r H d " 1] fir

and Ap * K[Up] 8 K[Tr J 8 K[Ur ] w ith K-subalgebras o f K and

U’ xT »U

K r e s p e c t iv e ly . N otice however th a t K [r] is no longer an

in te g r a l domain. A ls o , the co u n te rp art o f 6, 6r : = (1 8 dy 8 1) e Af i s in v e r t ib le s in c e the r e s t r i c t i o n dy o f d e K+[G] to Tr is the

c h a ra cte r xr (n _ 1 > n . 3... 0) which has In v e rs e x [ , . „ >3_ n ...0) i " \ ■

Thus Ar [ s ' ’ ] = Ar .

We a ls o have the analogue o f (1 .2 h ):

1 .31

The map ♦I>:U" * Tr * Up - o r d efined by ♦r (u ’ , t , u ) - u’ tu is a

b i je c t io n .

33

-1,3m Theorem (C h e v a lle y )

The map 4>r : U~ * Tf * Up ftp Induces an isomorphism o f K-algebras:

♦ * : K I r ]

[dj:'j

* Ktuf]

a

K[Tr ]

a

Klu,.] .

Proof

i * ur*Tr*ur

G iven f e K [r] [ d” 1 ] we defin e ♦r ( f ) e K to be the composition

fo<|>r , then -|>*(f ) ( u ~ , t , u ) = f jj “ tu ) .

As in the proof o f ( 1 . 2 i ) <J>* is an in je c t iv e K-algebra morphism

in to Ar . We have o n ly to show th a t <t-r is s u r je c t iv e .

Take any element a e Aj, , then th ere e x is ts an element

a e ( K[U^J a K[Tg] fi K[Ug] ) [6_1] such th a t a r e s tr ic te d to U* » Tf * Up is p r e c is e ly a , ( o f course a is not uniquely determined by a ) . By

(1 .2 1 ) th ere e x is ts q e K +[G] and m e Z>Q such th a t ♦ (q/dm) = a ie .

q (u " t u )

dm(u~tu)

a ( u " , t , u )

fo r a l l (u , t , u ) e UG * tg * UG .

This eq uation holds in p a r tic u la r f o r a l l (u , t , u ) e Uf * Tf x Uf and

i t fo llo w s th a t

• a

where q e K [r] i s the r e s t r ic t io n o f q e K+[G] to r .

1,3n C o ro lla ry to the p roof o f (1.3m) and (1 .2 J ) .

Le t K ■ Q and d e fin e

Z[rp)

. zCu7 (q)3 • ZCTr(Q) 1 and ZCUr(Q)-1

to be the images o f Z+[Gp ] under the re s p e c tiv e r e s t r ic t io n maps.

Then

♦ • (Z LI-Q Kd ^ l)

• zcu;(Q)] »

Z [T r ( Q ) ]

a z w r(p)j •

Proof

35

-§2. The kernel o f ^ : K ^ G I -» K [ r ]

2.1 P r e lim in a r ie s

In t h is ch apter we s h a ll fin d a generating s e t fo r the kernel o f the

r e s t r ic t io n map »K:K+COI -> M r ] as an id e al o f K+[G ] . However, our

f i r s t move s h a ll be to compute the kernel o f the r e s t r ic t io n map

2.1a

¿ ¡ f - ^ i M V a m t g i a m u g i * Mu"r i a K iTr j a M U r i

where _*K -U’ xTvU , i f _*K a v j a yJJ and

is r e s t r ic t io n M - V - K[U’ ] ,

’ ¡t is r e s t r ic t io n m t g i K [Tr l and

is r e s t r ic t io n m ug : - K[Up] .

We have:

ke r a ki tg ] a K[UG] +

J a K[UG] ♦ K[U^ 1 a K[TG 1 a ker v

u

_{K •}

Hence i t is enough to consider the kern els of

s e p a r a te ly , th is s h a ll be done in (2 .2 ) and (2 .3 ). We s h a ll then use

the isomorphisms given by C h e v a lle y 's theorem (1 .2 1 , 1.3m) to achieve our

aim. B e fo re we commence, a d e fin it io n :

2.1c D e fin itio n

For any subgroup G o f G and a ,B c n we d e fin e the follow in g

elements o f K[G] :

where f o r ven v : = n+l-v and 6 _{a , B}-?

0 a/B

N o tice th a t i f C an (n*n) m a trix , then

2. le

2.2 The kernel o f 4^ : KCU^l -» KfUp]

2.2a Pro p o s itio n

37

-: K[Ug ] - K[Ur ]

is generated as an id e al o f KCUg] by the s e t o f elements

a.ßefi

and a ls o by the elements

a )

a.ßefl , ß > a }

Proof

Consider the m atrix CU = ( c U )_{' p v 'y .v e jl}

By ( 2 . Id ) FU„ (a ,S e n ) is the ( a , e ) th c o e f f ic ie n t o f the m atrix

ctp —

(CU) t J CU - J . L e t F be the id e a l o f K[Ur ] generated by a l l these

' ' n n o

elem ents. Then, sin ce CU is an upper tria n g u la r uni potent m a trix , i t is

not d i f f i c u l t to see th a t F is a ls o generated by the s e t o f elements (2 .2 b ),

(th ese are the e n tr ie s o f (CU) t J nCU - J R below the diagonal which runs from

the top r i g h t hand to bottom l e f t hand c o rn e rs ). Now, sin ce f o r any g e Up ,

q t j g - J is a zero m a trix i t is c le a r th a t F is contained in the kernel

3 nJ n

o f tjJrKCUg] - K[Ur 3 .

Denote by c^v (u .v e n ) the image o f c|Jv in K[UG]/F and l e t Z

be the m atrix (c ) . Then Z is an upper tr ia n g u la r uni potent matrix

' jjv'u»veri

and Cl J C - J ; 0 . I t fo llo w s th a t 1f we submit t to the process

n n

described in ( S i . 3) we s h a ll show, as 1n the proof o f ( 1 .3 c ), th a t every

{ c yv : ( u . v ) e R£} , (though these elem ents do not n e c e s s a r ily f r e e ly

generate K[UG] / F ) . I t fo llo w s imm ediately th a t

KCUg] - K[cJ[u : ( u . v ) t « , ] ♦ F . . . . (1 )

But we a ls o have th a t

K[Ug 3 - KDcMv : ( ii , v ) e R£ ] ® ker . . . . (2 )

the sum being d i r e c t sin ce t4'|((CyV' : (v » v ) e f r e e ly generate the K-algebra y]J(K[UG] ) = K[Ur ] , (1 .3 c ).

Now, l e t z e ker , then using (1 ) we have z = x+y with

x e K[c[Jv : (m,v) e R£] and y e F . B u t F c ker vjj , thus

x c Ker K [cjjv : ( u ,v ) e R£] = (0 ) by ( 2 ) , and th e re fo re z e F .

This proves th a t the s e t o f elements (2 .2 b ) generates ke r 4^ . To show th a t the s e t o f elements (2 .2 c ) a ls o generates the kernel

we submit the lower tr ia n g u la r u nipotent m a trix (CU) t to a process s im ila r

to th a t f o r upper tr ia n g u la r unip oten t m a trice s in ( s i . 3 ) , and thereby show

th a t

K[Ug ] - Krc“ jU : ( n .v ) e « ¡1 + H

where H Is the id e al o f K[Ug] generated by the elements ( 2 .2 c ). Now,

i t is easy to see th a t ( v , u ) e R£ i f f ( u .v ) e R£ . Hence

K[Ug 3 ■ K [cjjv : (w ,v ) c R£] + H

- 39

and the r e s t o f the proof fo llo w s as b efo re.

In a s im ila r way we can prove:

2.2d Proposi tio n

The kernel o f the r e s t r i c t i o n map

: KCUg] - K[U~]

is generated as an id e al o f K[U g ] by the s e t o f elements

2.2e

{F U : a , Sen , B > a )

ap —

and also by the s e t o f elements

2.2f

{H^„ : a,Ben , 6 < a )

aB

2.3 The kernel o f 4^ : K[TC] -» K [Tr ]

We have shown in (1 .1 c) t h a t K [Tg] is f r e e l y generated as a K-algebra

by the elements { c ^ : pen} > and in ( 1 .3 j) th a t KfTj,] is the K-algebra

K(Xr X2... Xr _At+1

where X

V

I t follow s th a t the kernel o f yJ is th e re fo re generated as an

id e a l o f KCTq] by the elements

2.3b

CT c l- - li/rT n (wen)

uu u p k l i g j

B u t HT- = FT- = cT C7- - L r y ! (Me£) •

p p pu p p u p k l,g j

Hence, we have

2.3c Pro p ositio n

The kernel o f the r e s t r ic t io n map

»K : « V * « V

is generated as an id e a l o f K[Tg] by the s e t o f elements

{H^— = F^— : aen , 0 > a }

aa act — ~

Now, by (2 .1 b ) p rop o sition s (2 .2 a ,d ) and (2 .3 c) give gen eratin g elements

o f the kernel o f as an id e a l o f K[Uq] fi K[Tg] 8 K[Uq]

2.4 The kernel o f : K ,[ G][d~^ 1 - K m t d' 1 T

We prove a general lemma concerning rin g s o f f r a c t io n s .

2.4a Lemma

L e t R and Rj be commutative rin g s w ith id e n t it y and f : R -*■ Rj be

an epimorphism o f r in g s . L e t S be a m u lt ip lic a t iv e ly closed subset o f R

- 41

0 - (ker f > [S' 1 ] - R [ S " ’ ] T* R ^ f l S ) ' 1) - 0

i s e xact, where f o r r e R , s e S

T ( r / s ) ■ f ( r ) / f ( s ) e R , [ f ( S ) ' 1J .

P ro o f

We must f i r s t check th a t T : R[S~^] -*• R [ f ( S ) is w e ll d e fin e d .

Suppose th a t r/s = r ' / s ' in R [S , then ( r s ' - r ' s ) s " = 0 in R fo r

some s " e S . Thus ( f ( r ) f ( s * ) - f ( r * ) f ( s ) ) f ( s * *) = 0 in R^ so th a t

f ( r ) / f ( s ) * f ( r ' ) / f ( s ' ) in R , [ f ( S ) ' 1] g iv in g T ( r / s ) - T ( r V s 1) , which

shows th a t T is w e ll defin ed as re q u ire d .

Now, sin ce f is an epimorphism i t fo llo w s th a t T is a ls o . I t

th e re fo re remains to show th a t ker T = (k e r f ) [S ^] = {r/ s : reker f , s e S } .

Suppose th a t 7 ( r / s ) = 0 then f ( r ) / f ( s ) is zero in R -|[f(S) so th a t

th e re is some s ' e S w ith f ( r ) f ( s ' ) = f ( r s ' ) = 0 in R^ . Then r s 'e ker f

and r/s = r s '/ s s ' e (k e r f ) [ S ' ^ ] as re q u ire d .

We can now co n s tru ct the fo llo w in g coirm utative diagram:

_U"*T*U

(ke r ij* >,T><U) [ a' 1 ] * AK( s " ’ ] _{Ar ( K ) [<r}

u * t i

♦k

wn * r ( K )

- k+[G ][d -1]

K

^ k [ r ] [ d ‘ ' ]

+ + +

where AK « K[U£] ft K[Tg] fi K[Ug]

Ar (| () - K[ Up] i K[Tp] Q K[Up]

J K = kernel o f : K+[G] K [r ]

and d e K+[G] , 6 e A^ are the elements described in (1 .2 c ,g ) and d „ e K [r ] , 6r e Ar ,„x t h e ir co u n terp arts with re sp ect to r (K ) . R e c a ll

r i 1 1* ;

Ar ( K ) l * r ' * Ar ( K ) ■ ( 1 ' 3, p ' 3 2 )'

io

U’xT*U

The f i r s t and second h o riz o n tal rows are exact by (2 .4 a ), i'K

and being the n atu ral maps induced by 4^ *T*U and r e s p e c t iv e ly .

The upper row o f v e r t i c a l isomorphisms a r is e from C h e v a lle y 's theorem (1 .2 1 )

and (1.3m) and the bottom row o f v e r t i c a l maps are a l l monomorphisms s in c e K+[G] is an in te g ra l domain (1 .2 e) and:

2.4c Lenina

The element dr e K [ r ] is not a zero d iv is o r , th e re fo re K [r] can be

id e n t if ie d w ith a K-subalgebra o f K ^ H d ” 1] , (1 .2 d ).

Proof

See appendix A.

2.4d D e fin it io n

L e t I K be the id e a l o f K+[G] generated, as an id e a l, by the s e t of

<FG_a

6 a,8 e n)

43

-D efine BK to be the K-algebra K+[ G ] / I K , then we have a s . e . s .

2.4e

where e :K +[G] -*-*■ BK is the n atu ral map.

By app lyin g (2 .4 a) we a ls o have the s . e . s .

2.4f

I K[d_1 J Kt [ O J [ d '’ ] e~ BK[d ‘ ' ]

where dg: = 0(d ) e B^ .

C le a r ly ¡ K c J K = k e r f K : K+[G] -► K [r ] . Hence y d ' 11 C. J K[d_1 ) ,

but in f a c t we have:

2.4q Pro p o s itio n

The id e a ls I ^ d ' 1] and . y d ' 1] of K+[G ][d -1J a re e q u a l.

Proof

I t is enough to show e q u a lity o f the id e a ls _{*K( I K}

[ d _ 1 1 > a " d

4, * ( J Kld-1] ) in AKl«- ’ l .

By (2 .4 b ). ♦ * (J|t[ d " 1 ) ) - (k e r ” T“ U ) l « _1]| and by (2.1b ), (2.2a ,d ) and (2 .3 c ) the fo llo w in g elem ents o f AK d AK[<5 1'] generate i t as an id e a l

o f Ak[ « _1] :

FU 81 a 1 a,0 e n 0 > a

a0

1 S FT- S 1 a e n a > a

act

l

a

l

a

hu „ o.b t n

L e t X = { (a ,@ ) : a,3 e n 6 > « ) and t o t a l l y order i t le x ic o g r a p h ic a lly so th a t ( a ' ,3 ’ ) < ( a , 6 ) i f f o ' < a or a ' = a and 3' < 6 •

Now, fo r ( a ,3) e X d e fin e I( a ,3) to be the id e al o f AK[6-1] generated by the elements

♦ X v » •

w ith ( o ' , S ' ) e S and ( a ' , 3 ') ± ( “ »8)

Claim

For any ( « . 8 ) e

x , I

( a ,3) is the id e al o f AK[6~1] generated by the elements

FU" , 1 1 1 1 , _{a 3} 1 8 FT . . . 8 1 , _{a 3} 1 8 1 8 h“ , ._{a p}

f or a l l ( o \ B ' ) c X , ( o \8' ) < ( a , 9) . (note t lw t F^. 0. * 0 unless o ' = S ' and f

“ ,-,

= 0 - h

“ ,-, ) .

Proof o f Claim

L e t ( a , 3) e X , then

(1)

p,h,keri

.U‘ U ' a T c T 9 r U CU

uhci k " c hhc kk " c huc kB

Now, sin ce ■0 i f w > v , the sum can be taken over a l l h,k e n_ with h <_ a a nd k <_ 3 .

- 45

O U

E

h<o k<8

< C » « M *

CU c u c ha kg

) + 6 — (18 F — 8 1 )

Note th a t sin ce 8 _> a the co n d itio n s h <_ a , k <_ 8 imply 1T> b > a > h . Thus the terms Fdk have (h ,k ) e X and (h ,k ) <. ( a ,8)

in our o rd e rin g . We show th is is p r e c is e ly $K(F® ) .

Expanding each Fdk and F^— we have:

h<_a k<8

u,h,keri ■U c ”

'yh wk • ‘ I / k k » « 6-_h<o £ sh . ï ( ,a c hhc k<8

Now 6^ = 0 unless h = IT , but then a = h and k = 6 sin ce

It > B > a > h . Thus the second sum is 6a g (l ^ cI a Caa S 1 ) and th is shows th a t 4>K( Fa e ) equals ( i i ) .

S i m i l a r l y , i t can be shown t h a t f o r ( a ,8) e X

« " > ■ h£a (c«hc ek 1

k<8

“ Huh t ) * « .5(1 » F i r « ' )

We can now prove the c la im by in d u ctio n on ( o ,6) e X . F ix ( a ,8) e X and denote by (“0»3Q) the elem ent o f X next below ( a ,6) in our ordering. Assume the cla im is tru e f o r I ( a0,60 ) then by the id e n t it ie s ( i i ) and ( i i i ) we have, modulo I ( a0»30 ) »

and

♦*(HG I = H c T t ! . l H U, i l r ( l l F - l l ) .

Kv aB aa BB «6 aB v aa '

F u fi 1 a 1 and 1 a 1 a Hun 1n the case B > a

aB aB

(u s in g the f a c t th a t (1 a Ca0c B8 a 1) ^ c A K [6 ^ and by

Hence the claim is true fo r ( a , B ) . I t remains o n ly to s t a r t the

in d u c tio n . L e t ( a ,B ) = (1 ,1 ) , the minimal element o f X , then

M F n> > n > tt H

U

11

and sin ce (1 a c T lcTl ® E AK[6_1] the Proo f of the claim is complete.

Now, s e t ( a ,B ) = ( n , l ) , the maximal element o f X . Then

* -l * ~1

I ( n , l )

c. <>K(IKld

] ) by d e fin it io n and by the claim I ( n , l ) = <*>K( J K[d ] ) s in c e they are both generated as id e a ls by the same elements o f A ^ C A^[6

S in c e we alread y have I K[ d " ^ ] c . J K[d i t follow s th a t

♦K(iKi d' ' ] ) = ♦ *(JK« r , )>

p ro vin g the theorem.

47

-2.4h C o ro lla ry

There is an isomorphism o f K-algebras

♦ : BK[d'B’]

K[r] [d

;1

]

such t h a t fo r x e K+[G] , m e Z >_ o

* < 8 (x)/<£) = .

Pro of

Using ( 2 .4 b ,f ,g ) we have the fo llo w in g commutative diagram:

I K[ d ' ' ] ->K+[ G l[d ‘ ' ] - i — ■~ B K[d ‘ ' ]

id id : q

* ♦ f a *

J K[d_1] -*• K+[G] [d“ 1] K [ r ] Id “ 1] .

Hence, there must be an isomorphism 4- between BK[ d ^ ] and K [ r ] [dp^]

g ive n by

♦ (0(x)/d£) = f K(x)/d™

fo r a l l x e K+[G] , m e Z>Q

2.5 The K-algebra BK

The aim o f th is se ctio n is to show th a t BK is a reduced a lg e b ra , th a t

is i t s n i lr a d ic a l, which is the id e a l o f BK co n s is tin g o f a l l n ilp o te n t

2.5a Pro p ositio n

The K-algebra bK[ d^' ) is reduced.

Pro o f

By (2.4h) and the id e n t if ic a t io n o f ( S I . 3 ), BK( d " ' ] is isomorphic to

. Op

a su b alye b ra of the alg e b ra o f fu n ctio n s K . S in c e K c e r t a in ly

c o n ta in s no non zero n ilp o te n t elem en ts, the n ilr a d ic a l o f B^[dg ] is

z e ro . I t is th e re fo re a reduced alg e b ra as re q u ire d .

We need to show th a t BK has a Hopf algebra s tr u c tu r e , to t h is end we

p rove:

2.5b Lemma

I. is a c o id e a l o f K [G] so th a t in h e r it s a coalgebra s tru c tu re

from K+[G] .

P ro o f

R e c a ll th a t in (0 .2 a ) we gave K+[G] a coalgebra s tru c tu re w ith

c o m u ltip lic a tio n

a:K +[ü] - K+[G] fi K +[G]

such t h a t a(c ) = E c S c (u .v e n ) and c o u n it e :K [G] -*■ K ,

wv pen pp pv

e v a lu a tio n a t the id e n t it y o f G .

By d e f in it io n , I K is a co id e al i f

a ( i K) c k+(G]

a

I K

♦

I K

a

k +[G]

49

-Now, fo r any a, 6 e n i t is c le a r th a t

f> g> ■ 0 - O V and i t is not hard to show that

:G ) = i _{_ u v} f g a c_{pa \>B} c 0 + ( i » f GJ

P » v e £ ' aB

f i m) = z c c . a hg + (HG„ « 1)

o 6 ; otp B v p v

p »veil aB

S in c e these elements generate I K as an id e al o f K+[G] and

a ( x y ) = A ( x ) A (y ) * e ( xy ) = e ( x ) e ( y ) f o r a l l x .y e K+[G] i t follow s th a t I K is a co id e al. Then BK = K+[ G J / I K can be given a coalgebra s tru c tu re

by d e fin in g c o m u ltip lic a tio n Ag : BK -► BK a BK by

V x* 'k> ■ E ( x ( l ) * *K> 8 <x (2 ) * V

where l ( l | ■ [ l ( | | I l ( I ) ( x c K +[ G] ) (using the notatio n o f [S w ]) .

and c o u n it e B : b k K by

e B (x + I K) = e (x ) (x e K+[ G ] ) .

T h is completes the proo f.

To show that BK is a Hopf algeb ra we now o n ly need a K-algebra

morphism s B -.B^ B^ , c a lle d an 'a n t ip o d e ', s a t is f y in g

2.5c