有限群の指標環の構造について

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有限群の指標環の構造について

山内, 憲一

https://doi.org/10.11501/3110975

出版情報：Kyushu University, 1995, 博士（数理学）, 論文博士

On the Structure of the Character Ring of a

Finite Group

Contents

Ir1troduction ^{. . . . . . .. . . . . . . . . . . .. .. . . . . .. . . . . . . . . .. .. . . . . . . . . . . .. . . . . . . . . . .. . . . .1}

Chapter

1.

On the units of a finite order in a character ring ^{. .. . . . . . .. . . . . .}

.

^{. .}

1

§ 1.

Introduction

.

^.

...

^{. . . . . . . .. . .}

.

^{. . .}

.

^{.. . . . . . . . . . .. . . . . .}

.

^.

.

^{. . .. . . . .}

.

^{. . . . .}

1

§2.

^A study of units of finite order

... 3

Chapter

2.

^A unit group in a character ring of an alternating group ^{. . . .}

.

^.

.

^.

.. 5

§ 1.

Introduction ^{. .. .. .}

.

^{. . .. . . . . . . . . .. .}

..

^{. . . . . .}

.

^{. .}

. .

^._.^{. . . . . . . . .. . . . . . . . .}

. 5

§2.

Preliminaries ^.

..

^.

..

^.

...

^..

.

^{. . . . . . ..}_.^{. .}_.

..

^{. . . . .. .}

.

^{. . . . . . .}

.

^{. .}

.

^{. .}

.

_.^.

.

^{. .}_.

. 7

§3.

Construction of unit elements

.

^{. . . . . . . .. . . . .}

.

^.

.. . .

^.

..

^{.. . . . .}

....

^{. .}

.

^.

. 1 2 §4.

_{Units in}

R(An) (

n �

5) ... 17

§5.

Some examples ... 2

5

Chapter

3.

_On isomorphisms of a Brauer character ring onto another ^{. . . . . . .}

.

²⁹

§ 1.

Introduction ...

2

⁹

§2.

Preliminaries ^{. . .}

.

^{. .}

.

^{. . . . .}

. .

^{. . . . .}

.

^.

.

^{. .}

.

^{. . . . . . . . .. .}

.

^.

.

^.

.

^.

... . ..

^.

... 3 1 §3.

Main theorems ^.

.

. .

.

. . .

. .. . .

.

.

. . . .

.

.

. .

. .

..

..^.

.

.

. . ....

.

. . ...

.

.

.

.

.

.

. . .

36

Chapter

4.

On automorphisms of a character ring ^.

..

^{. .}

.

^{. .. . . .}

.

^..

. .

^.

.

^{. . . . . . . .}

40

§1.

Introduction ^{. . . .}

.

^{. . .. .}

...

.^..

.

^{. . .}

.

^{. . .. . . . . .}

.

^{. . .}

.

^..

.

^{. .}

.

^.

.

.

..

^{. .}

.

^{. . ..}

.

^.

40

§2.

Proof of Theorem

1.2 ... 4 2

References

...

^{. . . .. . . .}

..

^{. . . ..}

.

^.

.

^.

...

^{. . . .. . . . . .. . . . .}

.

^{. .}

.

^.

...

^.

..

^.

..

^.

...

^{. . .}

.

^.

4 5

Introduction

Throughout this paper

G,

Z, Q and C denote a finite group, the ring of rational integers, the rational field and the field of co1nplex nun1bers respectively. For a finite set

X,

we denote the number of ele1nents in

X

by

lXI.

Let

l11(G)

⁼

{x1(the p1incipal chaTaciel), ...

^,

x

^h

}

be a full set of irreducible complex characters of

G.

Let

R(G)

be the set of generalized characters of

G.

That

IS,

ai

^{E Z}

( i

^{= 1, .}

.

^.

, h) }

Then

R( G)

forms a commutative associative ring with the identity element

x1

under addition and 1nultiplication of characters of

G

and

R( G)

is called the character ring of

G.

In representation theory of a finite group

G

over C,

R( G)

is a fundamental ring and in modular representations and integral representations of finite groups, repre­

sentation rings and Grothendieck rings are treated respectively in place of character rings.([5

]

,

[12),[13), [17],[18],[19],[23],[24],[28), [29])

It seems that a typical representatives of the theorems concerning the structure of the character ring of a finite group is Brauer's induction theorem. Here we state Brauer's induction theorem in terms of character rings. Let £ be the set of elementary subgroups of

G

and

R1 (G)

be the sub ring of

R( G)

generated by the linear characters. ForHan arbitrary subgroup of

G,

induction of characters gives

rise to a Z-homomorphism ind:

R(H)---+ R(G).

Then we ma.y state Brauer's induction theorem in the following fonn.

Brauer's induction theore1n

The

Z-

homom.orphism

ⁱⁿ

d: EB

EEE

R1 (E)

----t

R( G) defined by 2:::: 'ljJ

---+

2:::: 'lj;G is surjective; where 'ljJG is the induced character of

'ljJ.

In connection with the above theorem, in

[1]

B.Banaschewski studied the maxi1nal ideals of

11.( G)

and obtained a theoren1 analogous to Brauer's induction theorern.

There are n1any papers concerning induction theorerns.

([1],[3],[4],[10},[25],

_... etc

)

There are a few papers concerning the units in a character ring. In

[20)

A.I.Saksonov treated the units of finite order in a character ring when he studied the isomorphisn1s of a. character ring onto another. However as long as \Ve know, it seems that there is no paper concerning the units in a character ring which are not of finite order. In this paper we will treat the units in

R(An)

which are not of finite order where

An

is an alternating group on n syrnbols.

Concerning the isomorphisms of a character ring onto another, the theorem that if

R(G)

^�

R(H)

for two finite groups

G,H,

then

G

and

H

have the same character table was proved by D.R.\i\leidrnan in

[27).

In addition in

[20)

A.I.Saksonov proved a theorem analogous to vVeidman's theorem which is a strengthened version of \�/ei­

drnan's result. In this paper we will also treat the san1e problem with respect to Brauer character rings.

This paper is composed of four chapters. In chapter 1 we study the units of finite

order in a character ring. In

[20]

A.I.Saksonov detern1ined the units of finite order in A

R( G)

where A is a ring of algebraic integers in a. finite extension of Q and

AR(G)

is an A-algebra spanned by

x1,

... , Xh· Here we state a theorem which is a generalization of Saksonov 's theorern, and gi \ e a short proof of this theorern which is different from Sa.ksonov's proof.

In chapter

2

we treat the units in

R(An) (

n 2::

5).

If \Ve denote the unit group of

R(An)

by

U(R(An))

and the set

{V;21 V;

is a.

unit in _R(An)}

_by

U2(R(An)),

then we will construct c

(

n

)

units

V;1, ... , 1/-'c(n)

in

R( An)

which are not of finite order and show that

U2(R(A�))

�

(�1, ...

,

V; _c

₍

_n

₎

)

_v,rhere

_(V;1, .

_.

. , V;c(n))

is the subgroup of

U(R(An))

generated by

V;1, ... , V;c(n). (

Concerning a nurnber c

( n),

see Definition 2.5 in chapter

2)

Here we would like to throw the main emphasis upon the fact that the units in a character ring previously treated are of finite order.

In chapter 3 we define a. Brauer character ring

BR( G)

and consider isomorphisms of a Brauer character ring onto another. Vle \�rill prove a theorem analogous to the theorerns of D.R.\iVeidrnan and A.I.Saksonov. This theorern is a generalization of the results of D.R.\iVeidman and A.I.Saksonov.

In chapter 4 we study isornorphisn1s of a character ring onto another. \.Ve will determine the form of these isomorphisms. This work is an extension of the result of A.I.Saksonov which is cited above.

The author would like to express his thanks to Professor E.Ba.nnai for his encour­

agement and to Profssors K.Shiratani, H.Tachikawa, S.Koshitani and T.

O

bayashi

for their Yaluable suggestions.

Department of J11athematics Faculty of Education

Chiba University Chiba) 263 Japan 1995

Chapter 1.

ring

§1. Introduction

On the units of finite order in a character

Let n be the exponent of a finite group

G

and let

(

be a prirnitive n-th root of unity. Then ]( ⁼

Q( ( )

C

C

is a splitting field for

G.

In particular, if

G

is a finite abelian group and

A

is the ring of algebraic integers in ](, then any unit of finite order in the group ring

AG

has the form r:g for some g E

G

and son1e unit c E

A.

(See p 263, Theorern 37.4 of

[6])

This result yields an interesting theorem. That is, if

G

and

H

are finite abelian groups such that

ZG

�

ZH,

then

G

�

H.

(See p 264, Theorern 37.7 of

[6]).

\file denote the algebraic closure of Q in

C

by Q and the ring of algebraic integers in

Q

by

Z.

In this chapter, applying the theory of characters of finite groups, we intend to study the units of finite order in the ring Z

R( G)

where

Z R( G)

is the

Z

-algebra of

Z

-linear combinations of irreducible complex characters of a finite group

G.

Afterward we shall show that any unit of finite order in

Z R( G)

has the form EX for sorne linear character x and some unit E in

Z

(See Theorem 2.1

)

and then we shall apply this result to conclude that if

G

and

H

are finite groups such that

R(G)

�

li(H)

as rings, then

Gj D(G)

�

H/ D(H)

where

D(G)

and

D(H)

are commutator subgroups of

G

and

H

respectively.

The theorems concerning the units of finite order in a character ring are stated in

[20]

^and

[30] (

See Theorem 1 of

[20]

and Lemma 6.1 of

[30]),

and Theore1n 2.1 is an extension of these results. Vle also present a short proof of Theoren1 2.1.

§2.

A study of units of finite order

Vve keep the notation in

§1

and in addition use the following notation.

x 1 (

⁼

1c ), ... , Xh-1

and

Xh

denote the irreducible co1nplex characters of

G.

For a E

C,

_a denotes a conjugate complex nun1ber of a and ia

l

an absolute value

of a.

For any ring

B, U1(B)

denotes the set of units of finite order in

B

and

G

the group of linear characters of

G.

For

B,

77 E

ZR( G),

we set

(9, "7)

⁼

1/IGI ^LgEG B(g)TJ(g).

Then we have the following theorem about the units of finite order in

ZR(G).

Theorem 2.1.

U1(ZR(G))

⁼

U1(Z)

^x

G (

a direct product

)

.

Proof. For u ⁼

L:7=1aiXi

E

ZR(G),ai

E

Z,

we set u =

L:7=1ai Xi,

where

Xi

denotes a conjugate character of

Xi (i

⁼

1, ... ,

h

) .

Suppose that

u

E

U1(ZR(G)).

Then

u(g)

^{is a} root of unity for all

g

E

G.

^Hence

lu(g) 12

⁼

u(g )u(g)

⁼

1.

^Therefore

we have

uu

⁼

1c.

From this equation, it follows that

2:7=1 lail2

⁼

(u, u)

⁼

(uu, 1c)

⁼

1 ... (2.1)

For any a E

G(Q/Q),

we set

ua

⁼

2:�1 aixi·

Since

xi

is also an irreducible character of

G,

^{we have}

ua

^E

UJ(ZR(G)).

By the equation of

(2.1),

we have

2::?=1 laf 12

⁼

1

^{for all e7 E}

G( Q /Q).

Hence for each i,

laf I

:::;

1

for all e7 E

G( Q / Q).

Therefore

ai

is either 0 or a root of unity. (i ⁼

1,

^{. .}.

,h).

That is, it follows tha.t

u

⁼

EiXi

for sorne i, where Ei is a root of unity. Since

1Xi(1)1

⁼

lci-1u(1)1

⁼

1, Xi

1nust be a linear character of

G.

This

completes the proof. Q.E.D.

As a consequence of Theorem

2.1,

we can easily obtain the following corollary.

Corollary 2.2.

U1(R(G))

⁼

{±1}

^x

G

(a direct product)

Theore1n 2.3

If R(G) � R(H) as rings for two finite groups G, H; then we have G/D(G) � H/D(H).

Proof. Since

R(G) � R(H),

we see that

UJ(R(G)) � UJR(H)).

By Corollary 2.2, we have

{±1}

^x

G � {±1}

^x

H.

By the fundan1ental theoren1 of finite abelian groups,we obtain

G � H.

Hence we have

G/D(G) � G � H � HjD(H)

This completes the proof. Q.E.D.

Chapter 2. A unit group 1n a character r1ng of an alternating group

§1.

Introduction

In what follows,

G

denotes a finite group, Z the ring of rational integers: Q the field of rational numbers, C the field of complex numbers. In addition \Ve fix _the following notation.

R(G):=

the character ring of

G U(R(G)):=

the unit group of

R(G)

U1(R(G)):=

the subgroup of

U(R(G))

which consists of units of finite order in

R(G)

Sn, An:=

a symmetric group and an alternating group on n symbols respectively for a natural number n

In section 2, we will prove that

U ( R( G))

is finitely generated. Hence a factor group

U(R(G))fUJ(R(G))

is a free abelian group of finite rank.

In the same section , we also state the results in

§6,

_{Ch.VI in}

[2]

which play a funda1nenta.l role in this chapter, and define a non-negative rational integer c

(

_n

)

_for

a natural number n. This number is very in1portant. That is, we will construct c

(

_n

)

units

'l/;1, ... ,'fc(n)

in

R(An)

in section 3 and show that

U2(R(An)) S: ('l/;1, ... ,'l/Jc(n))

in section 4 _where

U2(R(An)) = {'l/;2 J _'lj;

E

U(R(An))}

and

('l/;1, ... ,'l/Jc(n))

is the subgroup of

U(R(An))

generated by

'l/;1, ... , 'l/Jc(n)·

As a direct consequence of this

result, we obtain that c

(

n

)

= the rank of

U(R(An))jU1(R(An)).

In section 5, as an application of the above results, \Ve state son1e exarnples such that {±1} X

(1/;1, . .. , 1/;c(n))

⁼

U(R(An))

by finding generators of

U(R(An))

concretely.

§2.

Prelin1inaries

\i\e first show that

U(R(G))

is finitely generated.

Theore1n 2.1.

For a finite group G, U(R(G)) is finitely generated.

Proof. Let ( be a prinutive IGI-th root of unity, and let J( = Q(() be the

smallest subfield of C containing Q and (

.

Let us denote the ring of algebraic integers in I< _by

A.

Let C1, ... , Ch be a full set of conjugacy classes in

G

and let

cl = 1' _...

, Ch

be the representatives of c]' _.

.

.

, ch

respectively. Let "J be an element of

U(R(G)).

Then there exists u' E

R( G)

such that

uu' ⁼ x1

(

the principal character)

Hence u.

(

ci

)

^· u'(ci

)

= 1

(i

= 1,

..

.

, h).

^If xis an irreducible con1plex character of

G,

then x(ci) E A

(i

= 1,

... , h).

Therefore u(ci

)

^E

A,

u'(ci

)

^E

A (i

= 1,

... , h).

That is, u(ci) and ·u'(ci

)

are units in

A (i

= 1, .

.

.

, h).

vVe denote a unit group of

A

by

U(A).

row we define a 1napping ¢ fron1

U(R(G))

to a direct product of

h

copies of

U(A);

¢: U(R(G))

³ u ^�

(

u

(

c1

)

,

... ,u(ch))

^E

U(A)

x ^{· · ·} x

U(A) (h

copies)

Then it is clear that ¢ is a homomorphism and injective. Since

A

is the ring of algebraic integers in J(,

U(A)

is finitely generated by Dirichlet's Theorem. Therefore

U(A)

x · · · x

U(A)

is an abelian group which is finitely generated. As

U(R(G))

is

isornorphic to a subgroup of

U(A)

x

· · ·

x

U(A), U(R(G))

is finitely generated. The

theorem is proved.

Q.E.D.

There are three irreducible con1plex characters of

A3.

\Ve denote then1 by

x1,

x2,

x3.

Each

Xi

is a linear character and

Xi(x)

E

Q(-/=3)

for x E

A3.

Hence for any

'ljJ

_E

R(A3),'tf;(x)

E

Q(-/=3)

for

x

E

A3.

Since

U(Q(-/=3))

=

{±1,±p,±p2}

where

p ⁼⁼

( -1

+

-/=3)/2,

by the proof of Theorem 2.1, we can see that any unit in

R(A3)

is of finite order. Therefore we have

U(R(A3))

=

U1(R(A3))

=

{±x1, ±x2, ±x3}

by Corollary 2.2 in §2, Ch.l.

A4

has four irreducible cmnplex characters

X1, X2,

X3, X4 such that Xr

(

1) ⁼ X2 (1) ⁼

x3(1)

= 1 and

x4 (

1

)

⁼ 3. For anyx E

A4,Xi(x)

E

Q(-/=3) (i

= 1,2,3,4).Analogously

we have

U(R(A4))

⁼

U1(R(A4))

⁼

{±x1, ±x2, ±x3}.

For a natural nurnber n

� 5, An

is a simple group. And so

An

⁼

D(An)

(the comn1utator subgroup of

An)·

Hence

An

has only one linear character Xr (i.e. the principal character).By Corollary 2.2 in §2, Ch.1, we have

UJ(R(An))

⁼

{±x1}.

Fron1 now on, we may assun1e n

� 5,

when we consider about

U(R(An)),

and we use a notation

"U(R(An))/{±1}"

in place of

:'U(R(An))/UJ(R(An))"

for simplicity, by identifying

{±1}

with

{±xi}·

Now we state the irreducible cornplex characters of an alternating group

An.

The irreducible characters of the syrmnetric groups which are not self-associated, are also irreducible characters of the alternating groups.

Every self-associated character of the syrmnetric group

Sn

is the sum of two ir-

reducible characters of the alternating group An- These two irreducible characters of An take exactly half the values of the character of Sn� except for the conjugacy class for which the value of the character of Sn is ±1. This conjugacy class splits into two for An, and it is for these conjugacy classes alone that the two irreducible characters of An differ, the characteristic values in the two conjugacy classes being interchanged for the second character. Again we repeat these circumstances explic- itly. (See p222 of

[2])

Let

[m1�

^{· · ·}

� 1nr L m1

+ ^{· · ·} +

mr =

n be a self-associated frame.

In the following way� we can assign to

[n1-1, ... , 7nr]

a conjugacy class of Sn with cycles of odd lengths

q1

>

q2

> ^{· · ·} >

qk, q1

+

q2

+ ^{· · ·} +

qk =

n; let

q1

be the length of the "

hook �' consisting of the first row and the first column;

q1 =

2

m1 - 1.

If this hook is deleted, another self-associated frarne ren1ains, from which we detennine

q2

in the same way;

q2 =

2(

m2 - 1) -

1

= 21n2

^- 3. VVe continue thus until there is nothing left.

Here we use the following notation;

(q1, q2, ... , qk):=

a conjugacy class of Sn with cycles of lengths

q1

>

q2

> ^{· · ·} >

qk, q1

+

q2

+ ^{· · ·} +

qk =

n.

Then the following two theorerns, which play a fundarnental role, are well known (See p 222-223 of

[2]).

Theorem

2.2.

The character of a self-associated representation of Sn which corresponds to a self-associated frame

[m1, ... , 1nr], 1n1

+ ^{· · ·} +

mr =

n is

( -1)�(n-k)

⁼

( -1)�(p-I)

zn the conJugacy class

( qi, q2, ... , qk)

which is assigned to

[rni:

^···:

1nr]

where p

qiq2

^{• · ·}

qk;.

in all other conjugacy classes it is an even number.

Theorein 2.3. (Frobenius 's theore1n) Let

X

be a self-associated character of

Sn

which corresponds to a self-associated frame

[mi, ... , mr], mi

^{+ · · · +}

mr

^{= n.} Then we have

( 1)

If we consider

X

as a characteT of

An, X

is the sum of two irreduciblt- char-acters

(2)

^If

(q1, q2, ... , qk)

is a COJugacy class which is assigned to

[1ni, ... , mrL

then

( qi, q2, ... , qk)

spZ.its into two conjugacy classes C', C" of

An.

The ¹ a lues of

XI and

x2

are

A±

y/pX

2

in the two classes C', C") where A =

( -1) �(n-k)

⁼

( -1) �(p-I)

and p ⁼

qi q2

^{· · ·}

qk.

The values of

XI

and

X2

are equal in all other conjugacy classes of

An; XI

⁼

Let r =

[mi, ... , mr], mi

^{+ · · · +}

mr

^{= n} be a self-associated frame. Then we assign to r a conjugacy class C ⁼

(qi, q2, ... , qk)

of

Sn

with cycles of odd lengths

set p = q1q2 · · · qk· In addition,we assu1ne that p ⁼

1(mod.4)

and pis not the square of a number (i.e.

yP �

Q). Then we state the following two definitions.

Definition 2.4. In the above situation we call r a self-associated fran1e of real type and v.,re also say that

(f, C,

^p

)

is a triple of a self-associated fra1ne of real type r.

Definition 2.5. For a natural nu1nber ⁿ we define a non-negative integer

c( n)

as follows

c(

ⁿ

)

^{: =} the number of self-associated frames of real type such that

Example. Vle co1npute

c(15).

There are three self-associated fra1nes;

[8, 1, ... , 1], [5,4,3,2,1),(4,4,4,3].

_Vile can assign to

(8,1, ... ,1],[5,4,3,2,1L[4,4,4,3]

conju- gacy classes of sl5

(15),(9,5,1),(7,5,3)

respectively. And conjugacy classes

(15), (9, 5, 1), (7� 5, 3)

determine odd numbers

15,9

^x

5

^x

1

=

45,7

^x

5

^x

3

=

105

respectively.

45

⁼

1(mod.4), 105

⁼

1(mod.4).

Therefore we have

c(15)

= 2.

In

§4

we will show that the rank of

U(R(An))/{±1}

=

c(n).

§3.

Construction of unit elen1ents

In this section we construct unit ele1nents of

R(An)

which are not of finite order.

Let r =

[ ml' ... 'mr], 7TI1

^{+ ... +}

mr

= n be a self-associated fran1e of real type and let

(q1,q2, ... ,qk)

be a cojugacy class of

Sn

which is assigned to

[m1, ... ,mr]·

We set

p

⁼

q1 q2

· · ·

qk.

Then

p

=

1 ( m o d

.4

)

and

p

is not the square of a. nun1ber.

Hence

Q (yip)

is the real quadratic field. Here we state severa.l lenuna.ta in the above situation.

Leinina 3.1. A

conjugacy class (q1: q2, ... , qk) of Sn consists of ISni/P ele·ments.

Proof. Since

(q1, q2, ... qk)

is a cojugacy class with cycles of lengths

q1

>

q2

>

· · · >

qk, ql

⁺

q2

^{+ · · · +}

qk

⁼ n, then it consists of

n!

ISnl

qlq2 . . . qk p

elements (Seep

31

of

[2]).

The lermna is proved.

Le1nma

3.2. T¥e set p

⁼

f2Po: (po: square-free). Then we have

(i) Po

=

1(mod.4)

Q.E.D.

(ii) If 2(t 1

⁺

uyip), t, u

^{E Z}

is an algebraic integer in Q(yiii;); then t

=

u(mod.2)

(iii) If

E

is a fundamental unit of Q( yiii;); then the units of Q( yiii;) which take the

form of � ^(t

⁺

^uJp),t,u

^{E Z,}

are given by ±En

(n =

0,±1,±2,

...

), where

E

= Ee

for some natural number e.

Proof. It is clear that (i) and (ii) hold. For (iii): for exarnple: see p319 of

[26].

Q.E.D.

Definition 3.3. We call a unit E which appears in Lernn1a 3.2 (iii), a standard unit in

Q(V'P)(= Q(ffo))

for convenience.

Lem1na 3.4

There exists a und of Q( V'P) which takes the fonn of

1

2(a + bylp) +

^1,

a, bE Z,pia (i.e. a divides by p),b-=/=

^0.

and of which the norm over Q is equal to

^1.

Proof. By Lemma 3.2, there exists a unit TJ ⁼

� (t + uy'p), t, u E Z

such that lVTJ

=

¹ where flTJ denotes the nonn of TJ over Q. Hence

t2 - pu2

^{= 4.} Thus

t2 = pu2 +

^4. If we set

a = pu2, b = tu,

then we obtain

1 1

TJ2 ⁼

-(t2 + pu2 +

²

tuV"P) =-(a+ bV'P) +

1

4 2

because a equation

t2 = pu2 +

⁴

=a+

⁴ holds. Thus

2(a + bV'P) +

1 ¹ is the desired

unit of Q(

V'P)

and so the proof is complete.

Q.E.D.

Now we construct a unit of R(An) which is not of finite order.

Let

[m1, ... , mr], m1 +

^{· · ·}

+ mr =

ⁿ be a self-associated fran1e of real type and let

( ql, q2, ... , qk)

be a conjugacy class of Sn which is assigned to

[ m1, ... , mr]; ( q1 =

2rn1

-

1, ^q2

^{= 2m.}

²

^{- 3,} ... ) . Let C', C" be the two conjugacy classes of An into which

square of a nu1nber.

Let

2"(

1

a+ bVft)

+

1, ^{a, b E}

^Z

^{(pJa, b #- 0)}

be the unit of

Q( Vft)

which is stated in

Len11na 3.4. Then we have Theore1n 3.5.

Theore1n 3. 5.

There exists a unit 'ljJ of R(

An)

such that

'l/J(x)

⁼

1 ^for

^x

^E

An, X� C',C".

1 1

'lj;(c')

⁼

2(a + bJP)

+ 1,

¢(c")

⁼

2(a- bV]J) + 1

where c', c" are the representatives of

C', C"

respectively.

Proof. First \Ve note that a. self-associated character B of Sn which corresponds to the self-associated frame

[n�1,

^{. . .}

, mr],

is the sun1 of two irreducible characters

By Theorem 2.3, we may a.ssu1ne that

¢,(c')

⁼

�(1 ^{+ V'ft),} ¢,(c")

⁼

�(1- ^V'ft)

¢2(c')

⁼

�(1

^-

^V'ft), ¢2( c")

⁼

�(1 ^{+ ,fP)}

cP1 ( x)

⁼

^cP 2 (X) E

Z for

X E

An,

X �

C', C".

Let

X ¹ (the principal char acter),

..^. , Xs be all other irreducible characters of An.

Then

Xi(c')

⁼

Xi(c") E

Z

(i

⁼ 1, ^...

,s).

Here we show that the class function

'ljJ

which is stated in this theorem, is actually written as a linear con1bination of

^Xi

and

�j (i ⁼ _1, ..

.

, s;j ⁼ _1, 2) with integral coeefficients.

Now we pay attention to the fact that IC'I = _IC"I = IAni/P (See Len11na 3.1) _and

that

(1/J-xi)(x)=O forxEAn,x�C',C"

(,P- x,)(c') = � ^(a+ b-JP), (,P- )(t)(c") = � ^{(a- b-JP)}

V·.fe denote by (A, f1) the inner product of two class functions A,

11

of An· That is,

1 -

(A,p) ⁼ _IAnl LgEAnA(g)f.l(g)

where f.l(g) is the conjugate complex nun1ber of p(g).

Here

^we

con1pute several inner products as follows

1 ^{--- ---}

(V'- ^{Xl,Xi) =} IAnl {IC'I(1/J- xi)(c')Xi(c') + IC"I(1/J- xi)(c")xi(c")}

1

( a

₊

byip ^a

^-

bJp) ^{( ')}

^a

^{( ')}

Z

= -

+ Xi c = -xi c _E

p

2 2 p

because Xi( c') = Xi( c") _E

_Z

_and a divides by p.

(,P- ^{Xt, ¢,) =} _l � nl {IC'I(,P- Xt)(c')¢t(c') + IC"I(,P- Xt)(c")¢,(c'')}

=�(a+ byip

1

+ Jp ^a- byip _1- Jp) =�(a b ) z

P

2 2 ⁺ 2 2 2p ⁺ p _E

because

^{a =}

b (mod.2),p is an odd number and a divides by p. Analogously we have

( 1/J - ^X

^1,

¢2) ⁼ 2 � ^{(a - bp) E}

^Z.

Therefore we obtain

a

s ,

a + bp a

-

bp

'if;== X1 + -·L Xi(c )Xi+

₂

cP1 +

₂

rP2

_E

R(An)

Pi=1 P P

Now we denote by 'if;' the class function of

An

which satisfies

'lj;'(x) ==

1 for

x

_E

An, X� C',C"

'lj;'(c') ==�(a- bJP) +

1,

'lj;'(c") ==�(a+ bJP) +

1

2 2

Then we obtain by the same rnethod

I

a

S I

a

-

bp

I

a + bp

I

'ljJ == X 1 +

-

L Xi (

C

)

_{Xi + 2}

(/Jl +

₂

(/)2

E

R( An)·

Pi=l P P

By the proof of Lermna 3.4, we can see that

r;2 == 2(a +

1

bJP) +

1,

flr; ==

1 where

77 is a unit of

Q(ylp).

Since

l'l(r;2)==

(a

₊

bylp

) ( a - bylp )

----=---

+

1

+

1

==

1,

2 2

we have

'lj;'lj;' == Xl·

Therefore 'if; is a unit of

R(An)

which is not of finite order.

This completes the proof of Theorem 3.5. Q.E.D.

§4. Units in R(An) (n ^2:: 5)

Let r =

[

m1, ... , 1nr

]

, n�1 + ^{· · ·} + 1nr = n be a self-associated fran1e of real type and let (f, C,

p)

be a triple of r . Let C', C" be the t\\ o conjugacy classes of An into which C splits and let c', c" be the representatives of C', C" respectively. Let

E =

� ^(t

⁺

u,JP), (t, u

E Z,

tu # 0)

be the standard unit in Q(

Jp).

We denote by

JV(E)

the nonn of

E

over Q. ·Then we have the following theorem.

Theoren1

4.1. In the above situation; we define a class function 'ljJ of An as follows

In case JV(E)

= 1

In case N(E)

= -1

t/J (

X

)

= -1

f

⁰

r

^X E

An,

X

�

C ^1, C ¹¹

Then tf; is a unit in R( An) which is not of finite order.

Proof. In case

JV(E)

= 1, by both Len1ma 3.4 and Theorern 3.5 we can see that

'1/J is a unit in

R(An)

which is not of finite order and so in case

Jl(E)

= -1, we prove that

1/;

is a unit in

R(An)·

_Since

N(E)

=

�(e- ^pu2)

⁼ -1, we have

t2

=

pu2-

4.

Hence we get the following equation

1 1

£2

= -(t

₄ ^{2 +}

pu

^{2 +}

2tuJp) = -(2pu2-

₄ ^{4 +}

2tuJp)

where

a=

^{pu2 and}

b = tu (# 0).

Therefore we have

('1/J

⁺

xi)(x) = 0

^{for X E}

An,

^X

1:- C',C"

( '1/J

^{+ X 1}

) ( c') = 2 (a

1 ⁺

bJp)

(,P

⁺

xl)(c") = � (a- by')J), pia, b"'

⁰

where

x1

is the principal character of

An·

^By the same proof as that of Theore1n 3.5 we can prove that

'1/J

is actually written as a linear combination of irreducible complex characters of

An

\vith integral coefficients and that

1/J

is a unit in R(An) and so we on1it its proof. Thus the proof is cornplete.

Q.E.D.

Let

(f 1, C1, pi), ... , (f c(n), Cc(n), Pc(n))

be the triples of self-associated frarnes of real type and let )q,

... , Ac(n)

be the characters of self-associated representations of

Sn

':vhich correspond to r

1,

^{... , r}

c(n)

respectively. If we consider

Ai

as a character of An, then

Ai

is the sum of two irreducible cmnplex characters

¢�, ¢�'

^of

An; Ai

⁼

¢�

⁺

¢�' ( i =

^1,

.

^.

.

^{, c}

(

ⁿ

))

^{. Let}

CL C?

be the two conjugacy classes of

An

into which ci splits, and let

<, <'

be the representatives of

CL cr

respectively

( i =

^{1' . .}

. , c(

ⁿ

))

^.

'0/e denote by Ei the standard unit in Q

( ffi) ( i

^{= 1,}

.

^{. . , c}

(

ⁿ

))

and we keep these

notations throughout this section. Then we ha\·e

Theoren1 4.2.

In the above situation let t/J be a unit in R(A

ⁿ

)

^u·h.Zch

is not of finite order such that

where E: is the conjugate number of Ei oveT Q and the sign of Et

^is

equal to that

0* _J

E�j;

t •

Then we have 1V(Ef')

^{= 1}

(i

^{= 1,}

..

^.

, c(n)) where JV(Et) denotes the norm of Et

over Q.

Proof. Let

XI (the p1·incipal char acter), ... , Xk

be the irreducible co1nplex char-

acters of An such that

{XI, ... ; Xk}

^U

{ ¢�,¢�'I

^{i = 1, . . .}

, c(n)}

is a full set of irreducible complex characters of

A

^{n- Now} we assun1e that 'ljJ is \Vritten as a linear co1nbination of irreducible co1nplex characters of

A

ⁿ with integral coefficients as follows

If we set

c

(

ⁿ

)

^c

(

ⁿ

) k

'1/J

⁼

� ^ai¢�

⁺

� ^bi¢�'

⁺

� CjXj, ai, bi, Cj

^{E Z}

i=I i=I j=I

c

(

ⁿ

)

^c

(

ⁿ

) k

1/J'

⁼

� ^bi¢�

⁺

� ai</J�'

⁺

� ^CjXj,

i= I i=I j=I

then by Theoren1 2.3, we can see that

1/J'(x)

⁼

1/J(x)

for

x

^E

A

^n;

X� C�;C/ (i

^{= 1, . . .}

,c(n))

\vhere the sign of

E?

is equal to that of

Ef'.

Therefore it follows that

( <j;l/J')

(

x) = ±

1 or (?j;tfJ')(x) is a unit in an ilnaginary quadratic field for ^X E

An:X � c::c?

(i

=

1 ^,

... : c (

ⁿ

) )

and that

('lf'lf')(cD = ('lf'lf')(<') = JV(Ef') =

±1 fori= 1, ^{· · ·}

:

^c

(

ⁿ

)

^.

Thus we can conclude that

'l/J'l/J'

is a unit in

R(An)

which is of finite order. Since

U1(R(An)) = {±x1} (

ⁿ

� 5),

we have

'lf'lf' = ±x1.

Since

('l/;'l/J')(l) =

¹ for the identity

ele1nent

¹ of

An�

^we have the equation 'l/J'l/J'

= x1.

This i1nplies that IV

( E/'") =

^{1 for i}

=

1:

.

^.

.

^{, c}

(

ⁿ

)

^. Thus the proof is

co1nplete. Q.E.D.

V1e assume further that

E1, ... , Er

are the standard units such that

fl(E·J =

· · ·

= JV(Er) =

1 and

Er+l: ... , Er+s(= Ec(n))

are the standard units such that N

( Ej)

= -1

(j =

^{r +} 1,

. . .

^{, 1· + s = c}

( n)).

Then, for ea.ch i E

{

1,

...

^r

}

^, we set

and for each

j

^E

{

^{r +} 1,

. ..

^{, 1· + s} =

c(

ⁿ

) }

^, we set

By Theorem 4.1, it follows that

'l/J1, ... ,'lfr+s(= 'lfc(n))

are units in

R(An)

which are

not of finite order� and we fix these units throughout this section. Then \\·e ha\·e

Theoren1 4.3.

For any unit 1/J

ⁱⁿ

R(An) which

is

not of finite order; we can write

Proof. Since

f\l(Ek) = -1

fork E

{r

+

1:

^{. . . , r + s}

=

^c

(

ⁿ

)}

^, by Theoren1 4.2 we have

where the sign of

E � ^jk

is equal to that of

E);2jk.

Hence we have

( 1

_'t'

; ,2 )

^fc' _\k

) = E4jk (�

_{k i "f/}

! .2) (

^c"

_k ) ^{= E-4jk}

k .

On the other hand, for

h

E

{ 1,

^..^.

, r}

we have

. .

1/J(

^c

�) = ±E�\ 1/J(

^c

�) ^{= ±E};2h}

for sorne ih E Z where the sign of

E�h

is equal to that of

E};2h

.Therefore if \Ve set

then we can see that

J-L(x) = 1

for

x

^E

c:

or ^X E

C:' (

i

= 1�

^.

.

^.

, r

^{+ s}

=

^c

(

ⁿ

))

^.

Thus it follows that J-l is a unit in

R(An)

which is of finite order. Since

UJ(R(An)) =

{±xi} (

ⁿ �

5),

"'Ne have J-l

=±XI·

For an identity element

1

of

An,J-L(1) =

¹ holds and so we obtain f-l

= XI·

This implies that

Thus the result follo\vs.

Q.E.D.

Corollary

4.4.

Let '1/J be any unit in R(An)· Then '1/J(x) is a real num.ber fo1· all

X E

An- In particularJ'l/;(x)

⁼ ±1

for

^X E

An,

^X

� c:,cr (i

⁼ 1: "'; c

(

ⁿ

))

^.

Proof.

It is clear that '1/J( x) is a real nurnber for x E

c:

or x E

C;' ( i

⁼ 1,

.

.. ^{, c}

(

ⁿ

))

^.

By Theorem 4.3 we can see that ('1/J2)(x) ⁼ 1 for x E

An,

x

� c:,cr (i

⁼ 1, ^{... , c}

(

ⁿ

))

^.

Thus the result follows.

Q.E.D.

Vve denote the subgroup of

U(R(An))

generated by

r</;1,

...

, ·1/Jc(n)

_by

_('1/J1, ... ,1/Jc(n))

and the set

{

'1/J2

j 'ljJ

E U (

R( An))}

by U2 (

R( Ar.)).

Then the following theoren1 is ^a.

direct consequence of Theorern 4.3.

Len11na 4.6.

The rank of ('1/J1, ...

^, '1/J

c(n

)

)

=

c(n).

Proof. Suppose that '1/J�1 ^{• · •} '1/J;(�)) ⁼

x1 (e1, ... , ec(n)

E Z).Then we have

Hece

ei

= 0

(i

⁼ 1, .

.. , c(n)).

Therefore we obtain the rank of ('1/J1,

... , '1/Jc(n))

= c

(

ⁿ

)

^.

The lernrna is proved.

Q.E.D.

The following result is a direct consequence of Theoren1 4.5 and Le1n1na 4.6.

Theoren1 4.7.

The rank ofU(R(An))/{±1}

^{== c(n}

)

^.

Let r ⁼⁼

[

m1, ... , mr

]

^{, m1 + · · · + 1nr = n} be a self-associated fran1e of real type and let (f ⁱ c ⁱ

p)

be a triple of r. Let C' ⁱ C" be the t\VO conjugacy classes of An into which

C splits and let c', c" be the representatives of C', C" respectively.

Let

1 2(t

+ u

yp) ^(tu -::f. 0)

be the unit in

Q(JP).

Then we have the following theoren1.

Theore1n 4.8.

In the above situation: let 1/J b the un£t in R(An) such that

'l/;(x)

=

±1 for ^x

^E

An, X�

^C',C"

'if;(c')

⁼

�(t+

^u

^y/p)

^,

'if;(c")

⁼

� ^(t-

^u

^Jp)

Then

the following conditions are equivalent.

(i) 7/J is a difference of two irreducible con�plex characters of An.

(ii)

u =

±1.

Proof. \Ve denote by x1 the principal character of

An

and by (A,

11)

the inner product of two class functions A,

11

^of

An.

(i)===? (ii) Since 1/J is a difference of two irreducible cmnplex characters of

An

and

7/J(x) (x

E

An)

is a real number, we have

(l/J2:Xd = ( 1/J 'I/J:Xl ) = (1/J,l/J) ₌ 2

^{. . . . . .. . . . . . . .. . . . . . . . . . . . . .. .}

_(4.1)

On the other hand by Theorem

4.2 _N( � (t + uJP)) = 1

and so we deri,·e 12

pu2 + 4.

Frorn this fonnula we get

( ^{t ± u.JP} ) 2 pu2 ± tuViJ

= +1

2 2

Hence we have

By Lenlllla 3.1 we have

IC'I ₌ IC"I ₌ �IAnl·

Now we calculate an inner product

p (?/J2- XI, XI)·

(o

_lfl

/ .2-

_X ^/ _1,

_X

₁

) =

^_1_

( ^IAnl ( ^{pu2 +tuft} )

^I

^IAnl ( ^{pu2- tuft} )) ^{= 2 (}

^')

⁾

I An I _p 2

^T

_p

^{2 U .}

^{. . . 4.}

^__,

Therefore it follows that

(l}J2,x1) = 1 + u2.

Hence by the fonnula

(4.1)

^{Vle have}

1

⁺

u2 = 2

and so we get

u = ±1.

(ii )=*(i) \Ne assume that

u = ±1.

Then by the formula

( 4.2),

we get (

l/J2 -

Xr,

XI) _{= 1}

and so we have

(l/J2,XI) ₌

(1/J,l/J)

=

^(1/J,l/J)

= 2

Because 1/J is a unit in

R(An),

it follows that 1/J(l)

= ±1

for the identity elernent

1

of

An.

Hence we can see that 1/J is a difference of two irreducible cornplex characters of

An.

This completes the proof of Theore1n

4.8. Q.E.D.

§5.

Son1e exan1ples

Example

1.

U

( R( A1a)).

\Ale will find the generators of U

( R( A10)).

^{First we}

compute ^c

(10)

^. There are two self-associated fra.n1es;

[4:3:2,1]:[5:2,13].

\\·e assign to

[4, 3, 2, 1], [5: 2, 13]

cnjuga.cy classes of

510: (7: 3), (9, 1)

respectively and conjugacy

classes

(7, 3), (9, 1)

detennine odd nurnbers

7

^x

3

=

21

⁼

1(rnod.4): 9

^x

1

⁼

32

respectively. Therefore \Ve have ^c

( 10)

= 1.

Now we set ^E =

�(5

₊

_hi).

_{Then E is} a fundamental unit in Q(

v'21). (At

^the

same tin1e ^E is ^a standard unit in Q(

--/21)

^{and J\1}

(

^t

)

(the non11 of E over Q) is equal to

1.)

Secondly we prove that there is no unit f-l in

R(A1a)

such that

p,(x)

⁼

±1

fl(

^c'

)

^{= ±E:}

�tc'; � �:�� � C',C" }

^{. . . . .. . . .. . . . . . . . . . . . .. . . . . . . . . . .}

^(5.1)

where

C',C"

are the conjugacy classes of

A10

into vvhich the conjugacy class (7:

3)

of

S10

splits: and c', c" are the representatives of

C'� C"

respectively.

Assun1e by way of contradiction that there is a. unit p, in

R( A10)

which satisfies the equations of

( 5.1).

^{Let A} be a. self-associated character of

S10

which corresponds to the frame

[4,3,2,1]

and let

'lj;1, 'lj;2

be the two irreducible complex characters of A10 into which A splits. By Theorem 4.8 we can see that f-l is a. difference of two irreducible cornplex characters of

A10

and so we rnay assurne that f-l =

±( 'lj;1- x)

^for

some irreducible complex character

x

of

A1a.

Now we can easily compute

deg

A = 768. (See p

78

Theorem

3.9

of

[16].)

Hence we have

deg'lj;1

⁼

deg'lj;2

⁼

384.

Since

p,(1)

=

±('lj;l(1) - x(1))

=

±(384- x(1))

=

±1,

it follows that

x (1)

=

383

or

x(1) ⁼ 385. But there is no irreducible cornplex character X of

A10

such that x(1) ⁼ 383 or x(1) ⁼ 385, because

IAlOI

x(1)

10!

� z

^and

IAIOI

⁼ 10!

2 ^X 383 X(1) 2 ^X 385

---

_2x5x7x11 10!

^t;z.

This contradiction implies that there is no unit f1 in

R( A10)

^\V hich satisfies the equations of (5.1).

Let 'ljJ be the class function of

A10

such that

'l/;(x)

⁼ 1 for

x

^E

A10,x �

C',C"

Then by Theorem 4.1, it follo,�rs that 'ljJ is a unit in

R(A10).

Therefore we have

Theorern

4.3.)

U(R(A10))

⁼

{±'1/Ji I

^{i E}

Z}. (

See the proofs of Theorem 4.2 and

Exarnple 2. U

( R( Ap)).

Let

p

be a prirne nun1ber such that

p

⁼ 1

( mod.4)

and

c(p)

⁼ 1. For exarnple 5, ₁₃ and 17 are the prime nurnbers which satisfy these conditions. Then we will find the generators of

U(R(AP)).

Let E be a fundarnental unit of

Q(vlp),

then

fl(c)

⁼ -1.

(

See p 316 Problen1 5 of

[26].)

There is a self-associated frame;

[ ^p ;

^{1, 1?}

)

^{. We} assign to this frame a conjugacy class of SP,

(p).

Then the conjugacy class

(p)

splits into two conjugacy classes C',C"

[ ^p

^{+ 1}

E=!l

of

AP.

Let /\ be a self-associated character of S

p

which corresponds to -

2-, 1 ^{2 •} 'A hen we consider ,\ as a character of

Ap,

by Theorem 2.3 we can see that ,\ is the