exercices supplémentaires

U⊂C1G π :G→
GL(V)G V
h·,·i
V
(ζ|ξ) = 1
|G|X
g∈Ghπ(g)ζ, π(g)ξi.
U(V)V
V
π(g)∈U(V)g∈G
g∈G π(g) (ei)
V π(g)
G1α1,··· , αnπ
α1⊕ ··· ⊕ αn
1
G
1
ˆ
G= Mor(G, U)α:G→Uˆ G
G
ˆ
G
ˆ
GMor(G, C∗)
φ:G7→ Mor( ˆ G, U)φ(g)(α) = α(g)
φ
λ G CV=CG
ρ(g)(f) = (h7→ f(hg))
ρ g
f∈CGρ(g)f
ˆ
G
ˆ
G
V φ
ˆ
G
G V
Card ˆ G= Card G φ
ˆ
G
G=Z/nZˆ G'Z/nZ
G'ˆ G G
φ G
Gˆ G

SU(2)
U⊂C1SU (2) ∈M2(C)
1Vn⊂C[X, Y ]
n
(XkYn−k)kVnVn
πnSU (2) Vn
πn(g)(P) = P(aX +cY, bX +dY )g=a b
c d .
πn
z∈Udz=z0
0 ¯zP=Pn
0akXkYn−kVn
WP= Vect{πn(dz)(P)|z∈U} ⊂ Vnπn(dz)z
zn−2iπn(dz)(P)i∈ {0, . . . , n}
aiXiYn−i∈WPi
W⊂Vnπn
W XiYn−i
g∈SU (2) Xnπn(g)(XiYn−i)
h∈SU (2) πn(h)(Xn)
W=Vnπn
m n ≥1φ:Vm−1⊗Vn−1→Vm⊗VnP⊗Q7→
(X⊗Y−Y⊗X)(P⊗Q)ψ:Vn⊗Vm→Vn+mP⊗Q7→ P Q
X⊗Y−Y⊗X V1⊗V1φ ψ
SU (2)
0→Vm−1⊗Vn−1→Vm⊗Vn→Vn+m→0φ
ψKer ψ= Im φ
πn⊗πm'π|n−m|⊕π|n−m|+2 ⊕ ··· ⊕ πn+m
SU (2)
πn
E1=C2En=E⊗n
1σnSnEn
σn(τ)(ζ1⊗···⊗ζn) = ζτ−1(1)⊗···⊗ζτ−1(n).
Fn⊂Enσn
S=1
n!Pτ∈Snσn(τ)Fn
(e1, e2)C2
fk=S(e1⊗···⊗e1⊗e2⊗···⊗e2),
e1k(fk)Fn
ρ1SU (2) C2=E1
ρn(g) = ρ1(g)⊗···⊗ρ1(g)∈GL(En).
ρn(g)σn(τ) = σn(τ)ρn(g)g∈SU (2) τ∈Sn
Fnρn
ρnFnπn

q
C
q∈C∗(n)q= 1 + q+··· +qn−1n∈N∗(0)q= 0
q q q
(0)!q= 1,(n)!q= (n)q(n−1)q···(1)q,k
lq
=(k)!q
(l)!q(k−l)!q
.
q0≤l≤k
q k l q
n
q1≤l≤k
k−1
l−1q
+qlk−1
lq
=k
lq
.
q n q
q
a b ba =qab q
k∈N∗
(a+b)k=
k
X
l=0 k
lq
albk−l.
k=n q n
n≥2n q Tn(q)
Tn(q) = hx, g |gn= 1, xn= 0, xg =qgxialg.
(ei)n−1
i=0 Cnσ τ
E=Cn⊗Cn
σ(ei⊗ej) = ei+1⊗ej, σ(en−1⊗ej) = e1⊗ej,
τ(ei⊗ej) = qiei⊗ej+1, τ(ei⊗en−1) = 0.
φ:Tn(q)→L(E)φ(g) = σ
φ(x) = τ σkτl(e0⊗e0) 0 ≤k, l ≤n−1
(gkxl)0≤k,l≤n−1Tn(q)
Tn(q)⊗Tn(q)Tn(q)
G=g⊗g, X = 1⊗x+x⊗g, ˜g=g−1,˜x=−xg−1.
∆ : Tn(q)→Tn(q)⊗Tn(q)
∆(g) = G∆(x) = X
:Tn(q)→C(g) = 1 (x)=0
(∆⊗id)∆ = (id⊗∆)∆ (⊗id)∆ = (id⊗)∆ = id
S:Tn(q)→Tn(q)op S(g) = ˜g
S(x) = ˜x m(id⊗S)∆(a) = m(S⊗id)∆(a) = (a)1 a∈Tn(q)
a=g a =x
a=bc b c
S
φ Tn(q)
φ(φ⊗id)∆(gkxl) 1
Tn(q)
C C
p2pC[Z/p2Z]C[Z/pZ×Z/pZ]Tp(q)

ζ∈V ζ 6= 0
π(g)ζ6= 0 π(g)hπ(g)ζ, π(g)ζi>0g(ζ|ζ)>0
h G g G gh
G π(h)
(π(h)ζ|π(h)ξ) = X
g∈Ghπ(gh)ζ, π(gh)ξi= (ζ|ξ).
π(g)
π(g)π(h) = π(gh) = π(hg) = π(h)π(g)G
π(g)
n g G Xn−1
π(g)
(ei)i g
αi(g)π(g)(ei) = αi(g)eiπ(g) 1
g G π(g) = id αi:G→C∗
αi(g)αi(h)ei=αi(g)π(h)(ei) = π(h)(αi(g)ei) = π(h)π(g)(ei) = π(hg)(ei) = αi(hg)ei,
αi(hg) = αi(g)αi(h) = αi(h)αi(g)
C∗'GL(C)αiGC
α=α1⊕···⊕αnGCnu∈L(Cn, V )
u(λ1, . . . , λn) = λ1e1+···+λnen
α π
u◦α(g)(λ1, . . . , λn) = u(α1(g)λ1, . . . , αn(g)λn) = α1(g)λ1e1+··· +αn(g)λnen
=π(g)(λ1e1) + ··· +π(g)(λnen) = π(g)◦u(λ1, . . . , λn).
GU U
Mor(G, U) 1
Mor(G, U)f g ∈Mor(G, U)
(fg)(hk) = f(hk)g(hk) = f(h)f(k)g(h)g(k) = f(h)g(h)f(k)g(k)=(fg)(h)(fg)(k)
(fg)(1) = f(1)g(1) = 1 fg ∈Mor(G, U)f−1= (h7→ f(h)−1)∈Mor(G, U)
f∈Mor(G, C∗)c= (1|1)
|f(g)|2= (f(g)1|f(g)1) = c g f(e)=1 c= 1 f
UG g ∈G
n∈N∗gn=e f(g)n= 1 f(g)∈U
α∈Mor(G, U)g h ∈GMor( ˆ G, U)
φ
φ(gh)(α) = α(gh) = α(g)α(h) = φ(g)(α)φ(h)(α)=(φ(g)φ(h))(α).
φ(e)(α) = α(e) = 1 1 Mor( ˆ G, U)
ρ(e) = id
ρ(gh)(f) = (k7→ f(kgh)) = ρ(g)(k7→ f(kh)) = ρ(g)ρ(h)(f).
f∈Mor(G, U)ρ(g)(f) = (h7→ f(hg)) = (h7→ f(h)f(g)) = f(g)f f
ρ(g)f g
α(g)f(hg) = α(g)f(h)h α
GC∗Mor(G, U)h=e
f=f(e)α
V
ρ(g)ˆ Gˆ G

V g ∈G φ(g)=1
α(g)=1=α(e)α∈ˆ G f(g) = f(e)f∈V
g=e φ
α∈ˆ G∆(α)(g, h) = α(gh) = α(g)α(h)=(α⊗α)(g, h)V⊗V
Gˆ G V
ˆ
G V Card ˆ G= dim V=
Card G φ
Z/nZψ:ˆ G→C∗α7→ α(¯ 1)
Un
n α(¯ k) = α(¯ 1)kk α(¯ 1)n=α(¯n) =
α(¯ 0) = 1 α(¯ 1) ∈Unα ω ∈Unω ψ
α(¯ k) = ωkk
SU(2)
(XkYn−k)Vnn+ 1
g∈SU (2) πn(g)
πn(g)(P+λQ)=(P+λQ)(aX +cY, bX +dY )
=P(aX +cY, bX +dY ) + λQ(aX +cY, bX +dY ) = πn(g)(P) + λπn(g)(Q).
πn(e) = id πnGL(Vn)
πn(g)πn(g0)(P) = πn(g)P(a0X+c0Y, b0X+d0Y)
=P(a0(aX +cY ) + c0(bX +dY ), b0(aX +cY ) + d0(bX +dY ))
=P((aa0+bc0)X+ (ca0+dc0)Y, (ab0+bd0)X+ (cb0+dd0)Y),
aa0+bc0ca0+dc0ab0+bd0cb0+dd0gg0
πnGL(2) Vn
πn(dz)πn(dz0)(P) = πn(dzdz0)(P) = πn(dzz0)(P)z0∈UWp
πn(dz)z∈U
zn−2iπn(dz)(P) = Pn
k=0akz2(k−i)XkYn−k.
z pq zpq p 6= 0 q≥2
Pz∈Upq zp=pPz∈Uqz= 0.
N= 2 ppcm{2(k−i)|0≤k≤n, k 6=i}zUN
WPk6=i
NaiXiYn−i∈WpaiXiYn−i∈Wp
W P =Pn
0aiXiYn−i∈W i ai6= 0
XiYn−i∈WPW P ∈W WP⊂W
g=h=1
√21 1
−1 1 .
πn(g)(XiYn−i)=2−n/2(X−Y)i(X+Y)n−iπn(h)(Xn)=2−n/2(X−Y)n
πn(g)(XiYn−i)W
W Xn∈W πn(h)(Xn)
XkYn−kW W =Vn