CAPÍTULO II: MARCO TEÓRICO-CONCEPTUAL
2.8. INFORME COSO
2.8.3. COSO III
I t follow s th a t there i s a non s in g u la r b ilin e a r form
6.6c
< ■ ) : VI , K " °X , K * K
given by d e fin in g , f o r x e K , y e e£ (C (x) } :
( x *®x,K<y) ) : ■ AXB<x , y >
where K : e£ (C (x) ) (6.5 m ).
6.6d Remark
When K « Q by (6. 4h)
kergBX • kerge .(C (x )>
and th e re fo re v j g as defined in (5.39) co in cid e s w ith our d e fin itio n h e re , sin ce i f x c Vx>gn k« rg6 the" x ■ y< C (x )> fo r some y c E j . B u t x.(C (x > ) ■ y { C ( x » . | C ( x ) | • |C (x )|y (C (x > ) ■ |C (x )|x . so that x = |C (X )| * ' x(C(x)> c ke rge(C(X)> and Vx g rs kerg 6 = Vx g n kergSX • The re ve rse in c lu s io n fo llo w s from the f a c t th a t
dimg(v x g fi kerQe *) - dinig Dj g (s in c e (6.6b) is non s in g u la r ) and d1ll\ )(Vx , g f' k e r Q6) ■ dtmQ g (s in c e ( , ) i s non s in g u la r ).
6.6e Remark
156 -
w ith modules fo r r we would lik e to prove the r e s u lt corresponding to (4.6 d ) i e .
Conjecture
Le t r ^ 2 , then <j>e Sr K( r ) i f and o n ly i f co n d ition s ( i ) and ( i i ) o f (4 .6 d ) hold to geth er w ith :
«">
♦ C b ,+ -
*
and
«*>
♦ C b
'
♦
BIBLIOGRAPHY
[AM] M.F. A tiy a h , I.G . Macdonald; "In tr o d u c tio n to commutative a lg e b ra ", Reading, M ass., Addison-Wesley, 1969.
[ B ] A. Borel ; P ro p e rtie s and re p re se n ta tio n s o f C h evalley groups, L e ctu re notes in Maths. No. 131, S p r in g e r , B e r lin (1971) 1-55.
[C L ] R.W. C a r t e r , G. L u s z tig ; On the modular re p re se n ta tio n s o f the general l i n e a r groups, Math. Z e i t . 136 (1974) 193-242.
[D ] J . Desarmenien; appendix to "T h éo rie com binatoire des in v a ria n ts c la s s iq u e s ", G-C. Ro ta, S e rie s de Math, pures e t a p p l. l/S-01, U n iv e r s ité Lo u is- Paste u r, S trasb o u rg, 1977.
[G ] J . A . Green; "Polynom ial re p re se n ta tio n s o f GLn " , Lectu re notes in maths. No. 830, S p rin g e r, B e r l i n , 1980.
[ G ' ] J . A . Green; L o c a lly f i n i t e re p re s e n ta tio n s , Jo urnal o f A lgeb ra, 41 (1976) 137-171.
[H ] J . E . Humphreys; 'L in e a r a lg e b ra ic g r o u p s ', Graduate texts in maths. No. 21, Sp rin g e r, New York, 1975.
[L T ] G. L a n c a s te r, J . Towber; Rep resen tatio n fu n ctors and f la g algebras f o r the c la s s ic a l groups I I , P r e p r in t , De Paul U n iv e r s ity , Chicago.
[M ] T. M a r tin s ; Ph.D. T h e s is , Warwick U n iv e r s it y , Coventry 1981.
[ R ] J . J . Rotman; "Notes on homological a lg e b r a " , Van Nostrand, New York, 1970.
- 158
[ S ] I . Schur; Über ein e Klasse von M atrize n , d ie sich e in e r gegebenen M atrix zuordnen lassen (1 9 01), in I . S c h u r, "Gesammelte Abtrandlungen I " , S p r in g e r , B e r lin (1973) 1-70.
[ S e ] J- P . S e rr e ; "Groupes de Grothendieck des schemas en groupes re d u ctifs d é p lo yé s ", P u b l. I .H . E .S . No.34 (1968) 37-52.
[ S t ] R. S te in b e rg ; "Lectures on C h e v a lle y gro u p s", Yale U n iv e r s ity , New Haven, 1966.
[Sw ] M .E. Sw e ed le r; "Hopf a lg e b r a s ", W.A. Benjam in, New York, 1969.
[W ] H. W eyl; "The c la s s ic a l gro u p s", P rin ce to n U n iv e r s ity Pre s s , 1946.
[Wo] W .J. Wong; Representations o f C h e valley groups in c h a r a c t e r is t ic p , Nagoya Math. Jo urnal 45 (1971) 39-78.
Appendix A
2.4c Lemma
The element dp e KCr] is not a zero d iv is o r .
U n fo rtu n ately we have only been a b le to prove t h is lemma using a lg e b ra ic v a r ie ty th e o ry. This is u n s a tis fa c to r y in so f a r as th is is the o n ly departure from our ra th e r p u r ita n ic a l approach to orthogonal group viewed as ju s t a group o f m atrices.
Pro o f o f (5 .3 c ) ( f o r a lg e b ra ic v a r ie t y theory see f o r example [ H ] ) .
Le t
D
denote the determinant fu n ctio n inK+[G]
andOj,
■^ ( D )
* the r e s t r ic t io n tor
. Then D* -1
inK[r]
, s in c e a l l elements o fr
are o f determ inant ±1
. Denote byr+ (r")
the subset o f elementso f
r
which have determ inant+ 1 ( - 1 ) -
Thenr = r+ u r
andr+
iso f course the s p e c ia l orthogonal group. Both
r+ , r
are closed s u b v a rie tie s o f r and are in f a c t i t s connected components and th e re fo re t h e ir coordinate rin g sK[r+]
andK[r~]
. the r e s t r i c t i o n o f fu n ctio n s inK[r]
tor+
and
r"
r e s p e c t iv e ly , are in te g ra l domains. Now, weClaim
K t r ] * (1 - Dr ) K [ r ] § ( H D r ) K m . . . (A )
Pro o f o f claim
- 160 -
(ch ar K i 2 ) , thus the id e n t it y o f K [ r ] is in (l- D r ) K [ r ] ♦ (l« D r ) K [ r ] and th e re fo re
K [ r ] - (l- D r ) K [ r ] ♦ (l+ D r ) K [ r ] .
We need o n ly show the sum is d ir e c t. Suppose (1-Dr )x = (1+Dp)y (x .y e K C r]) then m u ltip ly in g through by (1-Dr ) we g e t
<1-Dr ) 2x = (1-D2)y hence (l-2 D r tD2)x - 0 <D2 = 1) and then 2(1-Dr )x = 0 so th a t (1-Dr )x = 0 , proving the c la im .
Ue can id e n t if y (l+ D r ) K [ r ] w ith K [r +: as fo llo w s : l e t T* : M r ] •* K [ r * ] be the r e s t r i c t i o n map. C le a r ly
(l- D r ) K [ r 3 C ker»+ sin ce f +(Dr ) - 1 in K [ r * ] . Suppose ( 1+Dp)x c k e r / then (1+Dr )x (g ) * 0 f o r a l l g e r + and i f g e r then (1+Dr )x {g ) ■ 0 a ls o s in ce Dp(g ) . -1 . Hence (1+Dr )x i 0 on r and hence (1+Dr )x - 0 in K t r ] . Thus (1-Dr )K C r] - k e rt+ and we may id e n t if y (l+ 0 r ) K [ r ] w ith K [r +] . S i m i l a r l y , we may id e n tif y K [ r ‘ ] w ith (l- D r ) K [ r l and th is means th a t both are in te g r a l domains.