Devoir à la maison

n≥2k k E
n
E
p≥2r s {1, . . . , p}r6=s λ ∈k T (p)
r,s (λ)
GLp(k) 1 (r, s)λ
GpGLp(k)
T(p)
r,s (λ)µ∈k∗D(p) µ
(1,...,1, µ)Tr,s(λ)
T(p)
r,s (λ)DµD(p) µ
Tr,s(λ)Tr,s(λ0)Tr,s(λ)
M∈GLp(k)p≥2
M Tr,s(λ)M MTr,s(λ)
A B GpM0=AMB m0
11 = 1
C D GpCMD
1 0 . . . 0
0
f
M
0
f
M∈GLp−1(k)
p≥2P∈Gp−1
1 0 . . . 0
0
P
0
Gp
p M GLp(k)RDµS
R S Gpµ k∗
Gp= SLp(k)MGLp(k)RDµ
R Gpµ k∗

L(E)E
N(E)E
F(E)u E im(u)⊕ker(u) = E
T+
n(k)T− n(k) Mn(k)
d∈ {1, . . . , n −1}
g∈L(E)
g∈F(E) rg(g) = d
BE P GLd(k)
MatB(g) = P0
0 0 .
GLd(k)T+
d(k)∪T− d(k)
Q∈GLd(k)Q0
0 0 T+
n(k)∪T− n(k)
M∈T+
n(k)N∈Mn(k)
M=NJn
Jn=
0. . . . . . . . . 0
1
0
0. . . 0 1 0
.
u∈L(E)
v∈L(E)w∈F(E)u=w◦v
{e1, . . . , en}E{e1, . . . , es}ker(u)
v∈L(E)v(ei) = 0 1 ≤i≤s v(ei) = ei−1
i≥s+ 1 vker(v) = ker(u)
Tim(u) im(v)
w∈L(E)u=w◦v w|T= 0
w∈F(E) dim im(u)≤dim im(v) dim ker(u)≤dim ker(v)