TD1. Représentations linéaires

e
WC#G{eh, h ∈
G}G:G→GL(W)
G(g)(eh) = egh .
GG G
H W PGegH
W1G:G→C×, g 7→ 1
V G v {g(v), g ∈G}
V V G
ABC S3
S={A, B, C}
(A B) 3 (A B C)
ρS3CV
B={eA, eB, eC}
ρ(σ)(eS) = eσ(S),
σ∈S3S∈ S
Bρ(A B)ρ(A B C)
(A B C)(A B) (A B)(A B C)
Bρ(A C)ρ(B C)
BρS3
G V =CG
G
V
C
VVV ρ :G→VV
ρ(g)(α)(h) = α(g−1h).
ρ G
ρGV H W
σ:G→VV
σ(g)(α)(h) = α(hg).
σ G

σeσeσ(g)(α)(h) = α(gh)
ρ σ
U V W G
(U⊕V)⊕W U ⊕(V⊕W)C
(U⊕V)⊗W(U⊗W)⊕(V⊗W)
G V, W G
L(V, W )CV W C
ρ G L(V, W )
ρ(g)(α)(v) = gα(g−1v),
g∈G v ∈V gv ρV(g)(v)ρV
G V W ρ G L(V, W )
V W
G U G UGu∈U
gu =u g ∈G
UGU
L(V, W )G