S4 est un groupe de pavage (ou presque) - Agreg

S4
3S4
2 2
4
S4'< t1, tj2, rO,2π/3, sOx > / < t2, t2j2>
tj2
t1
A= 2Z+ 2j2Z
B= 1 + A
C=j2+A
D= 1 + j2+A
S={A, B, C, D}G=< t1, tj2, r, s > s
r O 2π/3G S
g(s)∈S s ∈S

t1(A) = 1 + A=B
t1(B) = 2 + A=A
t1(C) = 1 + j2+A=D
t1(D) = 2 + j2+A=j2+A=C
t1(AB)(CD)
r1 + j+j2= 0
r(A) = jA = 2jZ+ 2j3Z= (−2−2j2)Z+ 2Z= 2Z+ 2j2Z=A
r(B) = jB =j+jA =j+A=−1−j2+A=D
r(C) = jC =j3+A=B
r(D) = jD =j+j3+A=−j2+A=C
r(BDC)
s(A)=2Z+ 2 ¯ j2Z= 2Z+ 2jZ= 2Z+ 2(1 + j)Z= 2Z−2j2Z= 2Z+ 2j2Z=A
s(B) = 1 + ¯ A= 1 + A=B
s(C) = ¯ j2+A=j+A=−j+A= 1 + j2+A=D
s(D) = 1 + ¯ j2+A= 1 + j+A=−j2+A=j2+A=C
s(CD)
G S g ∈G E ∈S
ρ(g)(E) = g(E)
ht2, t2j2i ⊂ Ker(ρ)
g∈Ker(ρ)g(0) ∈A t2t2j2g
0g∈G+g g 0g(1) ∈B
g|g(1)|= 1 g(1) = 1 + 2a+ 2j2b b = 0
1 = |1 + 2a+ 2j2b| ≥ 2 sin(2π/3)|b| ≥ √3|b| |1 + 2a|= 1 a= 0
a=−1g(1) = 1 g(1) = −1g(1) = −1g π
G LG+={~g, g ∈G+}=h~rig∈G+
g r s
~g −−−→ srs−1srs−10
1r−1~g ∈ h~ri |L+
G|= 3
π2g(1) = 1 g=Id
g∈G−sg ∈G+B sg Ker(ρ)s B
g sg(1) = 1 −1−1
π G sg =Id s ∈Ker(ρ)
g=Id Ker(ρ) = ht2, t2j2iG/Ker(ρ)
S4(AC)(BD) (CD) (BDC)
tj2s r (AC)(BD)(CD) = (ACBD)
(BDC)(CD)(BCD)=(BC)Snn
¯g g ∈G
< t2, t2j2> e R2S4

ρ(¯g) = Mate(~g)
(),(AB),(ABC),(ABCD) (AB)(CD) (CD)
(ABC) 2π/3 (ABCD) (AB)(CD)
() (CD) (BDC) (ACBD) (AB)(CD)
22+ 6 ×02+ 8 ×(−1)2+ 6 ×02+ 3 ×22= 4 + 8 + 12 = 24

< t1, tj2, rO,2π/3, sOx >
< t1, tj2, rO,2π/6>
πKer(ρ)
Ker(ρ) = ht2, t2j2, rπi
•
•
•
•
•
•
•
•
•

•
•
•
•