70 VIII. W I T T INVARIANT S
P R O O F . Assum e tha t n i s even , s o tha t n = 2m . (Th e cas e wher e n i s od d
is similar. ) Le t a b e a n elemen t o f lnv(Et
ni
W). A s i n 24.10 an d 25.5 , le t H
be th e produc t o f m copie s o f Z/2Z , viewe d a s th e subgrou p o f S
n
generate d b y
the transposition s (12), (34) , .. . A n elemen t a o f H 1 {k^H) i s (cf . 27.14) a n m -
tuple ( a i , . . . , a
m
) an d define s a multiquadrati c algebr a E
a
= Yli E
ai
r E
OLi
th e
quadratic algebr a k[X}/(X 2 c^). Th e ma p a i— a(Ea) i s a Wit t invarian t o f
iJ 1 (*,iJ) , henc e (b y Th . 27.14) ca n b e writte n uniquel y a s a linea r combinatio n
Y ^ c j -a/(a) = V ^ ci-{aj) wit h c / i n W(fco) .
By usin g th e actio n o f S
m
o n G , on e see s (a s i n th e proo f o f Th . 17.3) tha t cj
depends onl y o n th e numbe r o f element s d o f / , s o tha t w e ma y rewrit e th e abov e
sum a s
m
^2cd-bd(a) withb
d
(a)= ^ (a/) .
d=0 \I\=d
On th e othe r hand , th e trac e for m q of th e algebr a E
a
i s give n b y
( 2 ) - g = ( l , . . . , l ) 0 ( a
1
, . . . , a
m
) .
One deduce s fro m thi s tha t th e exterior power s X d(q) fo r d = 0 , . . . , m ar e related t o
the bdipt) b y a triangula r invertibl e matri x wit h coefficient s i n W(ko), wher e thes e
coefficients depen d onl y o n m. Fo r example ,
X°(q) = b
0
(a)
A1(^) = (2)-6i(a ) + m-(2)-6
0
(a)
X2(q) = 62(a ) + m-2 -b
x
(a) + ! ^ L z i l
M a )
This show s tha t ther e exist s a uniqu e choic e o f element s c'
d
G W(ko) suc h tha t
a(E) =
YJc'd-\d{qE)
when E i s multiquadratic , henc e fo r ever y etal e algebr a E b y Th . 29.1.
Note tha t i n 29.2 , i run s throug h th e integer s m = [n/2] . Thi s implie s
in particula r tha t th e A 2(^£;), i m , ca n b e writte n a s linea r combination s o f 1,
Xx{qE)j •, Am (g#). Thi s ca n als o b e deduce d fro m th e followin g mor e precis e fact .
Define a quadrati c for m uj
n
b y
^
=
f ( 2 , 2 , . . . , 2 ) (rankm ) i f n = 2 m
1 ' j ^ n I (1, 2 , . . . , 2 ) (ran k ra + 1) i f n = 2 m + 1.
T H E O R E M 29.4 . X i(qE - v
n
) = 0 for all i m.
(Here qE ujn i s viewe d a s a n elemen t o f WGr(k), s o tha t X l(qE ujn) make s
sense.)
P R O O F . B y Theore m 29.1, it i s sufficien t t o prov e thi s whe n th e etal e algebr a
E i s multiquadratic . Bu t i n tha t cas e qE i s of th e for m
qE = u
n
© Q,
where Q i s a quadrati c for m o f ran k m . I t i s the n obviou s tha t X
l
Q = 0 fo r al l
i m, an d th e resul t follows .
Previous Page Next Page