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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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