Représentations des groupes finis II

G G
ρ:G−→ (V)
VCn V
n
ρ:G−→ (V)G
g∈G ρ(g)V
Og∈G ρ(g)V g
k ρ(g)k=Vpρ(g)|Xk−1∈C[X]
pρ(g)ρ(g)M
W⊂V ρ
ρ(g)(W)⊂W
g∈G
ρ|W:G−→ (W)
g−→ ρ(g)|W
ρ
ρ0:G−→ (V0)
G u ρ ρ0
u:V−→ V0
g∈G
u◦ρ(g) = ρ0(g)◦u
u ρ ρ0
(u) (u)ρ ρ0
u u ρ ρ0
ρ ρ0ρ ρ0

ρ:G−→ (V)
VCρ=χV
χ
χ:G−→ C?
G
Gd G G (G)
Sn
G|X|
X x −→ exCX|X|
ρX:G−→ (CX)
g∈G ρX(g)CX
ρX(g)(ex) = egx x∈X
G
G ρG:G−→ (CG)
SnX={1,· · · , n}
SnCn(ei)1≤i≤n
ρ:Sn−→ (Cn)
σ∈Sn1≤i≤n ρ(σ)ei=eσ(i)
V=C
n
X
i=1
eiW={z= (zi)1≤i≤n/
n
X
i=1
zi= 0}
ρ
ρ|V:Sn−→ (V)
n−1
ρ|W:Sn−→ (W)
Sn

ρ:G−→ (V)G
W V W 0
Op∈ LK(V)W
P=1
(G)X
g∈G
ρ(g)◦p◦ρ(g)−1∈ LK(V)
P W (P)⊂W x ∈W
g∈G ρ(g)◦p◦ρ(g)−1(x) = ρ(g)◦ρ(g)−1(x) = x P (x) = x
P◦P(x) = P(x)x∈V
P ρ h ∈G
ρ(h)◦P◦ρ(h)−1=1
(G)X
g∈G
ρ(hg)◦p◦ρ(hg)−1=P
G W 0= (P)
W ρ M
ρ:G−→ (V)V
{0}V
ρ:G−→ (V)ρ0:G−→ (V0)G
G
ρ⊕ρ0−→ (V⊕V0)
g−→ ρ(g)⊕ρ0(g)
ρ:G−→ (V)
Oρ W V
W0ρ=ρ|W⊕ρ|W0
VO
ρ:G−→ (V)ρ0:G−→ (V0)
G u ρ ρ0
Ou ρ ρ0(u)V
(u) = {0}(u)V0(u) = V0
uM
ρ:G−→ (V)u
ρ
Ou ρ λ u v =λ−v
ρ v = 0 u=λM

ρ:G−→ (V)G ρ
χρ:G−→ C
g−→ (ρ(g))
ρ:G−→ (V)χρ(1) ρ
ρ:G−→ (V)χρ(g−1) = χρ(g)g∈G
ρ:G−→ (V)χρ(g)g∈G
ρ:G−→ (V)χρ(hgh−1) = χρ(g)g, h ∈G
χρG
ρ:G−→ (V)ρ0:G−→ (V0)
χρ=χρ0
ρ:G−→ (V)ρ0:G−→ (V0)
χρ⊕ρ0=χρ+χρ0
Oχρ(1) = ( V) = (V)
ρ(g)V λi1≤i≤n
χ(g−1) = (ρ(g−1)) =
n
X
i=1
λ−1
i
=
n
X
i=1
λi=
n
X
i=1
λi= (ρ(g)) = χ(g)
u V v (vuv−1) = (u)
(ρ⊕ρ0)(g) = ρ(g) 0
0ρ0(g)M
χ:G−→ C?
χGρG:G−→ (CG)
χG(g) = (0g6=e
(G)g=e
Og6=e χG(g) (eg)g∈GCG0
M

χXρX:G−→ (CX)
G X
χX(g) = ({x∈X/gx =x})
OM χX(g)XCXx∈X
Mx,x =δgx,x M
Sn
Sn
ρ|W:Sn−→ (W)
1≤i≤n−1i=ei−ei+1 (i)1≤i≤n−1Cn
W={z= (zi)1≤i≤n/
n
X
i=1
zi= 0}
c c = (1,· · · , n)
ρ(c)(1) = 2
ρ(c)(n−2) = n−1
ρ(c)(n−1) = −(1+· · · +n−1)
ρ(c) (i)1≤i≤n−1W
CP=
0 0 · · · 0−1
1 0 · · · 0−1
0
0 0 −1
0 1 −1
P=Xn−1+· · · +X+ 1 = Xn−1
X−1χρ(c)=pρ(c)=P
A pAA χA= (X−A)
A A
P χA=pA=P
P ek2π i
n1≤k≤n−1
W
W=
n−1
X
k=1
Wek2π i
n(ρ(c))
ζ v =
n−1
P
i=1
vii∈Wζ(ρ(c))
ρ(c)(v) = ζ v