Représentations des groupes finis I : le cas abélien

A A
χ:A−→ C?
x∈A χ(x)n n A
e e A χ(x)−1=χ(x)
|χ(x)|= 1
b
A A C?
G χ1, χ2∈b A
χ1χ2:A−→ C?
x−→ χ1(x)χ2(x)
A n a
a:b A−→ Un
χ−→ χ(a)
UnC?n
b
A n
Oχ∈b A χ(a)∈Una n a A
a
aζ∈Unx∈A x =akk
n ζn= 1 ζkx χ(x) = ζkχ∈b A
χ(a) = ζM
f:A−→ B
b
f:b B−→ b A
χ−→ χ◦f
d
A=b
Ag:B−→ C g ◦f=b
f◦bg
fb f

B A
j:B−→ A
b
j:b A−→ b B
χ−→ χ◦j=χ|B
O[A:B] [A:B]=1
[A:B]≥2η∈b B x ∈A\B{k∈Z/xk∈B}
Z{0}o(x)
{k∈Z/xk∈B}=Zr
η(xr) = ζrζ∈C?η η0B0
B x t ∈Zy∈B
η0(xty) = ζtη(y)
η0xty=xt0y0
xt−t0=y0y−1∈B t −t0=kr
ζt0η(y0) = ζt−krη(yxkr)
=ζtη(x−kr)η(yxkr)
=ζtη(y)
η0B0η
[A:B0]<[A:B]x∈B0\B
χ A η0χ η M
A x ∈A\ {e}χ∈b A
χ(x)6= 1
O< x > k =o(x)
a:\ <x> −→ Uk
η−→ η(x)
η∈\ < x > η(x)6= 1 χ∈b A A
η χ(x) = η(x)6= 1 M
A e x e
B A A =B.<x>
Ox e
a:\ <x> −→ Ue
η−→ η(x)
Ue=< ζ > C?e
η∈\ < x > η(x) = ζ\ <x> Ue
η χ ∈b A χ(A)⊂Uee
A
χ η χ \ <x> Ue
χ(A) = Ue
B= (χ)y∈B∩< x > χ(y) = η(y)=1 y= 1 B∩<x>={1}
z∈A χ(z)∈Uext∈< x > χ(z) = χ(xt)
y=zx−t∈B z =xtyM

A B
νA,B :b A×b B−→ \ A×B
(α, β)−→ α⊗β
α∈b A β ∈b B α ⊗β A ×B(α⊗β)((x, y)) = α(x)β(y)
(x, y)∈A×B
Oχ∈\ A×B χA(x) = χ(x, 1B)x∈A χB(y) = χ(1A, y)
y∈B χA∈b A χB∈b B χ =χA⊗χB(α⊗β)A=α
(α⊗β)B=βM
A
d1,· · · , dr≥2d1| · · · ,|dr
A'Z/d1Z× · · · × Z/drZ
d1· · · drA drA
OA e A x ∈A
(x) = e A
A=B. < x > B
B'Z/d1Z× · · · × Z/dr−1Z
d1| · · · ,|dr−1dr−1B dr=e dr−1|dr
B dr−1
A'Z/d1Z× · · · × Z/dr−1Z×Z/drZ
M
A A b A
OA n b A n A
A x e A
A'B×<x> b A'b B×\ <x>
<x> \ <x>'<x> b B'BM
A
ιA:A−→ bb A
x−→ (χ→χ(x))
Ox∈A\ {e}χ∈b A χ(x)6= 1
iotaA
Ab Ab Abb A
ιAM
FA(A)A
FA
(ϕ, ψ) = 1
(A)X
g∈A
ϕ(g)ψ(g)

b
AFA
Oχ∈b A y ∈A Sχ(y) = P
x∈A
χ(xy)
Sχ(y) = Sχ(e) = χ(y)Sχ(e)
Sχ(e) = (0χ= 1
(A)
ϕ, ψ ∈b A
Sϕψ−1(e) = (0ϕ=ψ
(A)
b
A ψ−1=ψ(b
A) = (FA)
M
A A
ρ:A−→ (E)
ECn E
n
ρ:A−→ (E)n A
(vi)1≤i≤nE(χi)1≤i≤nχi∈b A1≤i≤n
x∈A
ρ(x)vi=χi(x)vi
OA={ρ(x)/x ∈A}E
x∈A ρ(x)
ρ(x)k=Epρ(x)|Xk−1∈C[X]pρ(x)
ρ(x)
(vi)1≤i≤nE ρ(x)
x∈A
ρ(x)vi=χi(x)vix∈A1≤i≤n
χi(x)∈C?ρ(x)E
χi∈b A1≤i≤n x, y ∈A
ρ(xy)vi=χi(xy)vi
=ρ(x)ρ(y)vi
=ρ(x)χi(y)vi
=χi(y)ρ(x)vi
=χi(y)χi(x)vi
M

A x, y ∈A m =o(x)n=o(y)
xy o(xy) = mn
O(xy)mn = (xm)n(yn)m= 1 o(xy)mn
(xy)h= 1 xh=y−h∈<x>∩< y > m n
<x>∩< y >={1}m n h mn h xy mn
M
A x, y ∈A m =o(x)n=o(y)
z∈< x, y >⊂A o(z) = (m, n)
Om=pa1
1· · · par
rpar+1
r+1 · · · par+s
r+sn=pb1
1· · · pbr
rpbr+1
r+1 · · · pbr+s
r+sai≥bi1≤i≤r
ar+j< br+j1≤j≤s
m0=pa1
1· · · par
rn0=pbr+1
r+1 · · · pbr+s
r+sxm/m0m0yn/n0n0
z=xm/m0yn/n0m0n0= (m, n)M
A e
A A e A
A
Ox∈A o(x) = e y ∈A o(y)
z∈< x, y > o(z) = (e, o(y)) e o(z) = e
o(y)eM
u∈ LK(E)K
pu∈K[X]u χu= (XE−u)∈K[X]
u
λ∈K
Eλ(u) = (λE−u)
λ∈K
Eλ(u)6={0}
λ χu
λ pu
Oχu(λ) = (λE−u)χu(λ)=0 K
λE−u E Eλ(u)6={0}1) ⇔2)
λ∈K puX−λ q ∈K[X]
pu=q(X)(X−λ) + pu(λ)
pu(u) = 0
q(u)◦(u−λE) + pu(λ)E= 0
χu(λ) = 0 x∈Eλ(u)\ {0}
q(u)(u(x)−λ x
|{z }
=0
) + pu(λ)x= 0