UC1G π :G
GL(V)G V
h·,·i
V
(ζ|ξ) = 1
|G|X
gGhπ(g)ζ, π(g)ξi.
U(V)V
V
π(g)U(V)gG
gG π(g) (ei)
V π(g)
G1α1,··· , αnπ
α1⊕ ··· ⊕ αn
1
G
1
ˆ
G= Mor(G, U)α:GUˆ
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
fCGρ(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)
UC1SU (2) M2(C)
1VnC[X, Y ]
n
(XkYnk)kVnVn
πnSU (2) Vn
πn(g)(P) = P(aX +cY, bX +dY )g=a b
c d .
πn
zUdz=z0
0 ¯zP=Pn
0akXkYnkVn
WP= Vect{πn(dz)(P)|zU} ⊂ Vnπn(dz)z
zn2iπn(dz)(P)i∈ {0, . . . , n}
aiXiYniWPi
WVnπn
W XiYni
gSU (2) Xnπn(g)(XiYni)
hSU (2) πn(h)(Xn)
W=Vnπn
m n 1φ:Vm1Vn1VmVnPQ7→
(XYYX)(PQ)ψ:VnVmVn+mPQ7→ P Q
XYYX V1V1φ ψ
SU (2)
0Vm1Vn1VmVnVn+m0φ
ψKer ψ= Im φ
πnπm'π|nm|π|nm|+2 ⊕ ··· ⊕ πn+m
SU (2)
πn
E1=C2En=En
1σnSnEn
σn(τ)(ζ1⊗···⊗ζn) = ζτ1(1)⊗···⊗ζτ1(n).
FnEnσn
S=1
n!PτSnσn(τ)Fn
(e1, e2)C2
fk=S(e1⊗···⊗e1e2⊗···⊗e2),
e1k(fk)Fn
ρ1SU (2) C2=E1
ρn(g) = ρ1(g)⊗···⊗ρ1(g)GL(En).
ρn(g)σn(τ) = σn(τ)ρn(g)gSU (2) τSn
Fnρn
ρnFnπn
q
C
qC(n)q= 1 + q+··· +qn1nN(0)q= 0
q q q
(0)!q= 1,(n)!q= (n)q(n1)q···(1)q,k
lq
=(k)!q
(l)!q(kl)!q
.
q0lk
q k l q
n
q1lk
k1
l1q
+qlk1
lq
=k
lq
.
q n q
q
a b ba =qab q
kN
(a+b)k=
k
X
l=0 k
lq
albkl.
k=n q n
n2n q Tn(q)
Tn(q) = hx, g |gn= 1, xn= 0, xg =qgxialg.
(ei)n1
i=0 Cnσ τ
E=CnCn
σ(eiej) = ei+1ej, σ(en1ej) = e1ej,
τ(eiej) = qieiej+1, τ(eien1) = 0.
φ:Tn(q)L(E)φ(g) = σ
φ(x) = τ σkτl(e0e0) 0 k, l n1
(gkxl)0k,ln1Tn(q)
Tn(q)Tn(q)Tn(q)
G=gg, X = 1x+xg, ˜g=g1,˜x=xg1.
∆ : 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(idS)∆(a) = m(Sid)∆(a) = (a)1 aTn(q)
a=g a =x
a=bc b c
S
φ Tn(q)
φ(φid)∆(gkxl) 1
Tn(q)
C C
p2pC[Z/p2Z]C[Z/pZ×Z/pZ]Tp(q)
ζV 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
gGhπ(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 Xn1
π(g)
(ei)i g
αi(g)π(g)(ei) = αi(g)eiπ(g) 1
g G π(g) = id αi:GC
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···αnGCnuL(Cn, V )
u(λ1, . . . , λn) = λ1e1+···+λnen
α π
uα(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)f1= (h7→ f(h)1)Mor(G, U)
fMor(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
nNgn=e f(g)n= 1 f(g)U
αMor(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).
fMor(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
GCMor(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)αˆ
G f(g) = f(e)fV
g=e φ
αˆ
G∆(α)(g, h) = α(gh) = α(g)α(h)=(αα)(g, h)VV
Gˆ
G V
ˆ
G V Card ˆ
G= dim V=
Card G φ
Z/nZψ:ˆ
GCα7→ 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)
(XkYnk)Vnn+ 1
gSU (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)z0UWp
πn(dz)zU
zn2iπn(dz)(P) = Pn
k=0akz2(ki)XkYnk.
z pq zpq p 6= 0 q2
PzUpq zp=pPzUqz= 0.
N= 2 ppcm{2(ki)|0kn, k 6=i}zUN
WPk6=i
NaiXiYniWpaiXiYniWp
W P =Pn
0aiXiYniW i ai6= 0
XiYniWPW P W WPW
g=h=1
21 1
1 1 .
πn(g)(XiYni)=2n/2(XY)i(X+Y)niπn(h)(Xn)=2n/2(XY)n
πn(g)(XiYni)W
W XnW πn(h)(Xn)
XkYnkW W =Vn
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