NIL-THETAFUNCTION S 3
Thus w e hav e reduce d th e isomorphis m proble m fo r th e genera l bilinea r
form t o th e cas e o f alternatin g forms . W e ma y no w us e th e fundamenta l fac t
about alternatin g form s tha t state s tha t ther e exist s a nonsingula r linea r
transformation
C : V - V
such tha t
fo
K
o\
CAC = I -
jr
l 0 0
l /
V o o o y
where I i s th e r X r identit y matri x an d zero s denot e zer o matrice s o f
the appropriat e sizes . (Se e [8] . ) I n othe r words , th e kerne l o f a n alternatin g
bilinear for m i s the only invarian t o f th e form . Fo r mos t o f th e res t o f thes e
notes, w e ma y assum e tha t al l A hav e bee n writte n i n th e abov e form . W e
will no w sho w tha t th e abov e discussio n easil y implie s tha t i f z ( ) denote s
the cente r o f th e grou p i n th e bracket ,
dimz(N(A)) = l+dim{kerne l o f A } .
For le t w z(N(A)) ; the n fo r (v , 0) N(A )
(w,0)(v,0) = (w+v,A(w,v) )
(v,0)(w,0) = (w+v,A(v,w)) .
Hence A(w,v ) = A(v,w) = A(v,w) = 0 fo r al l v V an d w i s i n th e kerne l
of A .
Conversely, le t w b e i n th e kerne l o f A . Sinc e
0 = A(w , v ) , v V ,
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