Le théorème de Pólya

G X G X
·:G×X→X1·x=x x ∈X g ·(h·x) =
(gh)·x x ∈X g, h ∈G
GS(X)X
G X
SO3(R)R3
G X g ∈G
x∈X x g g ·x=x x ∈X
Stab(x)g∈G x
g∈GFix(g)g
SO3(R)R3
SO2(R)
G X x G
G·x={g·x|g∈G}
X X
G X/G

R3SO3(R)
R3/SO3(R)R+
G X
x|G|
|Stab x|
G G ·x
Stab(x)
G X Y f :X→Y
f(g·x) = f(x)
g x ¯ f:X/G →Y¯ f(G·x) =
f(x)f¯ f
f=k·k2:R3→R
SO3(R)¯ f:R3/SO3(R)→R
G X g ∈GhgiG g
X x ∈X gn·x n ∈ZX
g X hgi
G X
X
G
X

G
X E Qf:X/G →E
X
ω∈X/G
f(ω) = 1
|G|X
g∈GX
ξ∈Fix g
f(ξ)
(g, ξ)∈G×Xg·ξ=ξf(ξ)
X
(g,ξ)∈G×X
g·ξ=ξf(ξ) = X
g∈GX
ξ∈X
g·ξ=ξf(ξ) = X
g∈GX
ξ∈Fix g
f(ξ)
X
(g,ξ)∈G×X
g·ξ=ξf(ξ) = X
ξ∈XX
g∈G
g·ξ=ξf(ξ)
=X
ξ∈X
|Stab ξ|f(ξ)
=X
ξ∈X
|G|
|G·ξ|f(G·ξ)f
E
|G|

=X
ω∈X/G
|ω||G|
|ω|f(ω)
=|G|X
ω∈X/G
f(ω)
X n
G K k
x1, . . . , xk
g∈G
gZ[t1, . . . , tn]ζg=Qω∈X/hgit|ω|ζg=
tα1
1· · · tαn
nαii X
hgig
G ZG
X=1
|G|Pg∈Gζg
G X G ZG=
ZG
X
G
X
ρ
2π
5G ρ
ρ
hρi
X
5ζρ=t2
1t2 5
Sn[[1, n]]
G1G2
X1X2G1×G2
X1tX2(g1, g2)·x1=g1·x1x1∈X1(g1, g2)·x2=g2·x2
x2∈X2ζ(g1,g2)=ζg1ζg2(g1, g2)∈G1×G2
ZG1×G2
X1tX2=ZG1 X1ZG2 X2
G X N
X G
G/N X Z(G/N )
X=ZG X
S G S(X)ZS
X=ZG X
X k
K X γ :X−→ KΓ

k X X K
ΓS(K)
G g ·γ=γ◦g−1
X G
G G
X G
K={n, r, b}γ
X
σ∈S(K)n b r σ ◦γ
ρ2π
5
ρ·γ=γ◦ρ−1
γ∈Γ ind γ=Qξ∈Xγ(ξ)∈Z[K] ind γ=xc1
1· · · xck
k
ciX xi∈Kind
ΓG
ind : Γ/G −→ Z[x1, . . . , xk]AΓ
Γ/G A UA=
Pγ∈aind γ∈Z[K]
ind γ γ UA
A
AΓ/G G
γ
ind γ=b9n2r
n=x1r=x2b=x3ind γ=x2
1x2x9 3
UΓ/G
xiK σ ∈S(K)σΓ/G Γ
G g ∈G γ ∈Γg·(σ◦γ) = σ◦γ◦g−1=
σ◦(g·γ)
σ·UΓ/G =σ·X
ω∈Γ/G
ind ω
=X
ω∈Γ/G Y
ξ∈X
σ(ω(ξ))