su3-09
TRANSCRIPT
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o
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o
o
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(12)
(123)
(123),(132) (12)(34)
(12),(123), (1234) (12)(34)
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Z3
S3
A4
S4
A5
A4
S4
A5 {1, 2, 3, 4, 5}
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Z3
S3
A4
S4
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A5
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3o
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V
C
V (V) GL(V)
V
V
a GL(V) V a1
G
G G G(s, t) st
G Gs s1
(rs)t= r(st) r, s t G
eG es= se = s sG
ss1 =s1s= e sG
GL(V)
GL(V)
V
G
|G|
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G
G V
: GGL(V).
sG
sGL(V)
s :VV.
s t G
st =st.
V
G
V
V = C GL(V) =
GL(C) = C G : G C s = 1 sG
G
G := Sn
n : Sn C
s =
1,
s
1, s
Sn
G := Z3 ={0, 1, 2} : Z3 C
(k) := k
= e2i/3
1 Z3
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G
g :=|G|
G
V g {es}sG V G t G t : V V t(es) = ets
: G GL(V)
G
G
V n := (V)
GL(n) n
GL(n) :={(aij)nn| (aij)= 0}.
V
n
GL(V)
GL(n)
V
[T] T : V V
GL(V) GL(n)T [T]
G
: GGL(n)
sG
s =
a11(s) a12(s) a1n(s)a21(s) a22(s) a2n(s)
an1(s) an2(s) ann(s)
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aij :G C 1i, jn
aij(st) =nk=1
aik(s)akj(t).
S3
S3
(12) (123)
: S3 GL(2) (12)
(123)
(12)=
1 0
0 1
(123)=
0
0 1
,
= 2i/3 : S3
GL(2)
S3 2
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(12) (123)
: G GL(V)
G V
: G V V(s, v) s(v)
.
: G VV
G
V G
V sG
(s, ) :VV
: G GL(V)s (s, )
G V
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: GGL(V)
G
W
V
s(W)W
s
G
s W
s|W :WW
W
|W : G GL(W)s s|W
.
W
V
: GGL(V)
G
V
W
V W V
G
G
V
: G V V(s, v) s(v)
.
W
V
(s, W)W
sG W G
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: G GL(V)
G
wV w=sG
es s(w) =w
s G W V
w
|W : G GL(W) G
G
G
: GGL(V) : GGL(W)
G V W
sG
s s : V W V W(v, w) (s(v), s(w))
VW
VW
: G GL(V W)s s s
VW G (V) =m (W) =n
: GGL(m) : GGL(n),
: GGL(m + n)
( )s = s 00 s .
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V1, . . . , V n G
V1 Vn
V
W
G
VW =V + W
: GGL(V) G W V P :VW sG
s
P
1s :V
V
V
sP
1s
G
sG
sP 1s
V
V
V
G
W V
P :VW V W P0 :VV
P0 := 1
|G| sGsP
1s .
W
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P
P|W =
W P W W
G P0 W P0|W = W
P20 =P0
P20 = 1
|G|sG
sP01s
= 1
|G| sG s 1
|G|tG tP 1t 1s
= 1
|G|2s,tG
stP 1st
= 1
|G|2s,tGr=st
rP 1r
= 1
|G|2rG
|G|rP 1r =P0
V G W
V
W
V W V =W W
P :VW P0 : V W
W:=
(P0) V =W W
sP01s =P0
sG
vW
P0(v) = 0
P0s(v) =sP0(v) = 0
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sG
vW
s(v)W
s G W V
V
W V V = W W V 0 V
V = 0 V
V
V
G
V V
V 0 V
G
G
V
G
V G
V
(V) =
1 V (V)2 V
V
W V
W
W
V
(W) < (V) (W) < (V)
V =W W
W
W
V
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: G GL(V) (V) > 1
s = V sG V =W1 W
(V)
Wi 1
V
V
Wi
V
: G GL(V) : G GL(W) G
: VW
s =s
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sG
1s s =
V W
s sV W
s G
X G X
V
{ex}xX X sG s :VV s(ex) =esx
sG
s GL(V)
: G
GL(V)
s s G
X
: G H :HGL(V) H
: GGL(V)
G
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V
H
: GGL(V)
G
V W G
V W
V W G T : V W
T
V
T
W
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V
n
V T : V
V
T
[T] = (aij)nn
T
(T) :=nk=1
akk .
T :VV V T
T
T :VV S:VV
(T S) =
(ST)
: G GL(V)
G
V
V
V :G C
V(s) := (s).
V
V
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z =a +bi
z=a bi
V n
(1) =n
(s1) =(s)
s
G
(tst1) =(s) s, tG
(1) = V V
n
(1) = (
V) =n
sG 1, . . . , n s G k Z sk =e ks =sk =e = V |i|k = 1 i= 1, . . . , n ii = 1
(s) = (s) =
ni=1
i = (1s ) = (s1) =(s
1).
(tst1) = (tst1) = (
1t ts) = (s) =(s),
G G
G G G(t, s) tst1
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sG
[s] ={tst1|tG}
s
s1, s2G
[s1] [s2]= [s1] = [s2]
: G C
1
G
=
: S3 (2) S3
[e] ={e}
[(12)] ={(12), (13), (23)}[(123)] ={(123), (132)}.
(e) = 2
((12)) =
0 1
1 0
= 0
((123)) =
0
0 1
=1
= e2i/3
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G
G
V1 V2 G
T :V1V2
V1 T
V2 T
(T) V1
V1 V1
(T) = 0
(T) =V1 T
(T) = 0
T
(T)
V2
V2 V2 (T) = 0
(T) =V2 T
(T) =V2
T
V
G
T :VV
C
T = V
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V
T T:=T V T v V T
T(v) =T(v) v= 0.
T
T= 0
T = V
G
G 1
: G
GL(V)
G sG G
st =st =ts =ts,
tG
s =s V
sG
s
sG
1 V
s
s
G
V (V) = 1
1 :GGL(V1) 2 :GGL(V2)
T :V1V2 T0 :
V1V2 T0 = 1|G|
sG
(2s)1T 1s.
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T0
tG
(2t )1T01t =
sG
(2t )1(2s)
1T 1s1t =sG
(2st)1T 1st =T0
T01s =
2sT0 T0
T
V1 V2 G
T :V1V2
T0 := 1
|G| sG(2
s
)1T 1s
.
1 2 T0 = 0
V1=V2 1 =2 T0= V1 =
(T)
(V1)
T0
1
T0 = 0 2
T0 = V1 (T0) = (V1)
(T0) =sG
((2s)1T 1s) = (T).
(V1) = (T).
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V
V
, : V V C(v, w) v, w
: GGL(V) G V
,
V
s(u), s(v)=u, v
sG u, vV
V s
: G GL(V) G
,
V
V
, 0
V
, : V V C
u, v= sG
s(u), s(v)0.
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,
tG
t(u), t(v)=sGst(u), st(v)0
=rGr(u), r(v)0
=u, v,
: G V
,
sG
sts(u), v=s(u), s(v)=u, v
u, vV
sts(u)
u, v
v V
v := sts(u)u
sts(u) =u
uV
s
t =s1 sG V n
s = (aji(s))nn aij(s) = aij(s1)
sG
G
G
C[G]
C[G] :={f :G C}.
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V
G
C[G] f, g C[G]
f, g= 1|G|sG
f(s)g(s).
C[G]
1 : G GL(V1) 2 : G GL(V2) G n1 n2
V1 V2
1
2
is =
ai11(s) ai1ni(s)
aini1(s) a
inini
i= 1, 2
T :V1V2
T
T =
t11 t1n1
tn21 tn2n1
.
V1 V2 G
n1 n2 T :V1V2
1
2
T
V1 V2
1|G|sG
a2kl(s1)a1ji(s) = 0
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i,j,k
l
1 2
1
|G|sG
a2kl(s1)a1ji(s) =
1
n1ikjl .
T0
T0 := 1
|G
| sG(2s)
1T 1.
T0
t0ki T0
t0ki = 1
|G|j,l
sG
a2kl(s1)a1ji(s)
tlj
t0ki
1 tlj 1 l n2 1 j n1 V1 V2
t0ki = 0
i k V1
V2 t
0ki =ki =
1n1
l,jljtlj
t0ki = 1n1l,j
kiljtlj .
tlj
G
G
, = 1
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V W
G V, W= 0
: G GL(V) G
n
s = (aij(s))nn (s) =i aii(s) , =
ni,j=1
aii, ajj =ni,j=1
1
|G|sG
aii(s)ajj (s).
V
aii(s) =aii(s1)
, =n
i,j=11
|G
| sGaii(s)ajj (s
1).
, =ni,j=1
iijjn
=ni,j=1
ijn
= 1.
V W
V , W= 0.
G
C[G]
G
G
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V
G
G
V G
V
V =W1 Wn.
W G
Wi W ,
Wi W
i Wi
= 1+ + n.
, =1, + + n, .
i,
= 1 Wi W
i, = 0 Wi W
W
V
,
V
W
G
V =W V W
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V =W V, i=W, i
i G
Wi G
i V W
W1, . . . , W n G
1, . . . , n V
V =Wm1
1 Wmnn
mi
V
mi =, i i= 1, . . . , n
, =n
i=1
m2i =n
i=1
, i2.
V G
V , = 1
,
= 1
ni=1, i2 = 1 i
V = i
i V
G
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G
G
G g :=|G| V g
{es
}sG
G G
: G GL(V)t t
t :VV t(es) =ets
(t) =
|G|, t= e
0,
t=e
t= e e = V
(e) =
(
V) =|G|. t= e ts= s s G
t {es}sG
(t) = (t) = 0
V
G
V
G
(V)
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V , V
, V= 1|G|sG
(s)V(s1) =V(e) = (V).
V1, . . . , V k
G
1, . . . , k
n1, . . . , nk s
G
{e
}
ki=1
nii(s) = 0.
ki=1 n
2i =|G|
(s) =ki=1
nii(s)
sG
s=e
s = e i n
2i =|G|
C[G]
G
: G GL(V)
G
s
1
sG
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f : G C G
sG
[s]
[s] :={tst1|tG}.
f
G
f(r) =f(s) r[s]
f(tst1) =f(s)
s, tG
: GGL(V) G f :G C f :VV
f :=sG
f(s)s.
f
tG
1t ft =sG
f(s)1t st =sG
f(s)t1st =u=t1st
f(tut1)u.
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f
f(u) =f(tut1)
1t ft =uG
f(u)u=f.
: G GL(V)
G
f
G f :VV
f :=sG f(s)
s.
f = V = |G|
(V)
f,
f
f
f = V
= 1
(V) (f) =
1
(V)
sG
f(s)(s) = 1
(V)f, ,
G
C[G]
C[G] G
[G]
G
[G] :={f :G C|f }.
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[G] C[G]
G G
[G]
: GGL(V)
f G
f =sG
f(s)s
f f V
C[G]
G
[G]
V
S={v1, . . . , vn} V w V
w, vi = 0 i = 1, . . . , n
w /
(S)
(S)
V
S
:={1, . . . , n}
G
[G]
[G]
[G]
f
[G]
f, i= 0
i = 1, . . . , n
f = 0
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: GGL(V)
f =sG
f(s)s
f = 0 Vi
G i
f = Vi
= |G|
(Vi)
f, i.
= 0
f = 0
f= 0
G
: GGL(V)
e1
f(e1) =sG
f(s)s(e1) =sG
f(s)es.
f = 0 f(s) = 0 sG
G
G G
G
c1, . . . , ck
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G
f : G C
f ci
i = 1, . . . , k i
i(s) =
1,
sci
0, s /ci
i= 1, . . . , k
f=ki=1sci
f(s)i.
(
[G]) = k
G
G
G
G G
G
s G
[s]
#[s]
G
G
G
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[s] [0] [1] [2]
#[s] 1 1 11 1 1 11 1
2
2 1 2
Z3
n1, . . . , nk G
i n
2i =|G|
1, . . . , k n1, . . . , nk
G
i nii(s) =
0 sG s=e
: G H
G
H
1
Z3
1
Z3
3 3
3
Z3
=e2i/3 Z3
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[s] [e] [(12)] [(123)]
#[s] 1 3 21 1 1 1 1 1 13 2 0 1
S3
S3
S3 1
S3 3
n
12 + 12 + n2 =|S3|= 6 n= 2
2
S3
S3
A4
A4
K :={e, (12)(34), (13)(24), (14)(23)} A4,
A4/K= Z3
A4 Z3 : Z3GL(V)
Z3
A4 A4
K
GL(V)
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[s] [e] [(123)] [(132)] [(12)(34)]
#[s] 1 4 4 31 1 1 1 12 1
2 13 1
2 14
A4
A4 s t
A4 (s) = (t)
((s)) =((t))
A4
Z3
4
4
S4
A4 K =
{e, (12)(34), (13)(24), (14)(23)}
S4
S4/K=S3
S3 S4
S4 S4K
GL(V).
A4
S3
4
5
S4
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[s] [e] [(12)] [(123)] [(1234)] [(12)(34)]
#[s] 1 6 8 6 31 1 1 1 1 1 1 1 1 1 13 2 0 1 0 24 5
S4
[s] [e] [(123)] [(12)(34)] [(12345)] [(13245)]#[s] 1 20 15 12 12
1 1 1 1 1 1
I1 3 0 1 1+5
21
5
2
I2 3 0 1 15
21+5
2
V 4 1 0 1 1 5
1 1 0 0
A5
A5
A5
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A4, S4 A5
X G
X
V
{ex}xX X s G
s :V V s(ex) =esx :G GL(V)
s s
X
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G
X
G
X : G GL(V) X V s G
s
V(s) = #{xX|sx= x}.
:={ex}xX V
X
xX
s(ex) =esx (aij) s
k
xX
akk =
1, sx= x
0, sx
=x
.
V(s) = (s) = (aij) = #{xX|sx= x},
A4
T
e, (123), (132)
(12)(34)
A4
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(123), (132) (12)(34)
[s] [e] [(123)] [(132)] [(12)(34)]
T 4 1 1 0
A4
T
A4
4 T
T T, T
T
T, 1= 1
1
T
T = 1W
W
A4 3
T =1+ W,
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(12), (123),(1234)
(12)(34)
W =T 1 W
W, W= 1 W
W =4 A4
1+ 4
S4
C
e, (12), (123), (1234) (12)(34)
C
C
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[s] [e] [(12)] [(123)] [(1234)] [(12)(34)]
C 4 2 1 0 0
S4
C
C, 1 = 1 C
C= 1
W
W
S4
3
C=1+ W,
W
W, W= 1 W
4
5 W W
A5 60
5
[e], [(123)], [(12)(34)], [(12345)]
[(13245)]
A5
A5
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e
(123)
1 := 23
1/2 3/2 03/2 1/2 0
0 0 1
;
(12)(34)
1 0 00 1 00 0 1
;
(12345)
1 := 25
1 1 0
1 1 0
0 0 1
;
(13245) 2 := 45
2 2 0
2 2 0
0 0 1
;
I1
A5
1
2
e, (123) (12)(34)
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[s] [e] [(123)] [(12)(34)] [(12345)] [(13245)]
5 2 1 0 0
A5 {1, 2, 3, 4, 5}
I2
I1, I1 =I2, I2 = 1
A5
A5 {1, 2, 3, 4, 5}
, 1= 1
1
V 1 V V :=1 V, V= 1 V
A5 1, I1, I2 V
12+32+32+42 = 35
5
A5
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