Chapitre 3: groupes de matrices et représentations

G g ∈G
(G×G→G
(g1, g2)→g1.g2
(g1.g2).g3=g1.(g2.g3),∀g1, g2, g3∈G
e∈G
g.e =e.g =g, ∀g∈G
g∈G g−1∈G
g.g−1=g−1.g =e
G
g1.g2=g2.g1,∀g1, g2∈G
Z={. . . −2,−1,0,1,2, . . .}+
0R,+
R\{0} ×
1
Sn
Sn:= {σ n }
Snn!
σ2◦σ1∈Sn
Sn
Snn≥3
S3

D=t12t23, L =t23t12
t12t23 6=t23t12 e=I t23.t23 =I
(t23)−1=t23
B.A
I t12 t23 t13 D L
I I t12 t23 t13 D L
t12 t12 I D L t23 t13
t23 t23 L I D t13 t12
t13 t13 D L I t12 t23
D D t13 t12 t23 L I
L L t23 t13 t12 I D
6 3×3P3
S3P3S3
1,2,3
I=
1 0 0
0 1 0
0 0 1 
,T12 = 0 1 0
1 0 0
0 0 1 
,D= 0 0 1
1 0 0
0 1 0 
,
T23 =
1 0 0
0 0 1
0 1 0 
,T13 = 0 0 1
0 1 0
1 0 0 
,L= 0 1 0
0 0 1
1 0 0 

T12T23 =Dp∈S3
P∈ P3
φ:p∈S3→P∈ P3
Pj
i= 1 j=p(i)Pj i=δj,p(i)φ
φ(p2p1) = φ(p2)φ(p1)
P1=φ(p1)P2=φ(p2)P3=φ(p2p1) (P2P1)k
i=Pl(P2)k l(P1)l i=
Plδk,p2(l)δl,p1(i)=δk,p2(p1(i)) = (P3)k
i
G G0
φ:G→G0
φ(g2g1) = φ(g2)φ(g1),∀g1, g2∈G
φ:G→G0G0∼
=G0
U(1) := {z∈C,|z|= 1}
Cz∈U(1) z=eiθ θ∈R
eiθ1.eiθ2=ei(θ1+θ2)eiθ−1=ei(−θ)U(1)
U(1)
θ∈RU(1)
NRN
N
G N
G×G→G g ∈G→g−1∈G
U(1) θ∈R
θ1, θ2→(θ1+θ2)C∞
θ→(−θ)C∞U(1)

L2(R)
GLn(R)
E n
GL (E) := {A:E→E, }
e=IE
(e1, . . . , en)E A ∈GL (E)
ADet (A)6= 0
A◦BAB
GL (E)
GL (n, R) := {A, n ×n(det (A)6= 0)}
φe:A∈GL (E)→A∈
GL (n, R)GL (E)GL (n, R)
f φf
AA0=φf(A)P∈
GL (n, R)
∀A,A0=φf(A) = P−1AP
P
A0=P−1AP P =Pj
i
fi=PjPj
iej
GL (n, C) := {A, n ×n(det (A)6= 0)}