On vector invariants of the symmetric group

On vector invariants of the symmetric group*

S. A. STEPANOV

Abstract-The purpose of this paper is to give a proof of the results announced by the author [7] in 1982 on the algebraic independence over a field k of any non-degenerate system of mn distinct basis invariants in the ring k[x11 , ... ,x1,,; ••• ;x,,,1, ••• ,x.u,l with respect to the symmetric group G = S,,. The result of this paper

can be extended to the case of an arbitrary finite group.

The work was partially supported by the Russian Foundation for Basic Research, Grant 94--01--01206-a.

1. INTRODUCTION

Let m, n be positive integers, k be a field of characteristic O and let

be the algebra of polynomials in mn indeterminates xu,

i

=

1, ... , m, j

=

1, ... , n. The symmetric gro.up G = Sn operates on the algebra Amn as a group of k-automorphisms by the rule 8Xu

=

x;,gU>• g e G. Denote by A~n the subalgebra of invariants of the algebra Amn with respect to the group G and define elementary symmetric polynomials

u,, ...

rm e A~n Of Vectors (X11, ... , X1n), ... , (XmJ, ... , Xmn) by means of the formal identity

n

II (

I

+

X1jZt

+ , , , +

XmjZm)

=

1

+

L

u,,, .. ,.,mz~',, .z;.

(])

j=I lsr,+ ... +rmSn

The polynomials

u,, ... , .•

form a basis of the algebra A~n (see [5], [I,

A. IV,

p. 62], [2, p. 9], and [8, p. 37]). In other words, each element F of the algebra A~n may be represented as a polynomial in

u,, ...

,m

with coefficients from k. That basis contains

('";;n) - 1 elements connected with one another by different algebraic relations.

On the other hand by the Noether normalization theorem (see [5] and [2, p. 17]), there exist mn algebraically independent basis invariants u1, ... , Umn among

u,, ...

,m

such thatA~n is finitely generated over k[u1, ... , Umn1 as a module. This means that the transcendence degree of A~n over k is equal to mn (see also [3, p. 681). Since any invariant u e A~n is algebraically dependent on u1, ... ,Umn, the set {u1, ... ,Umn} can be also considered as a basis of another type in the algebra A~n· The theorem of Noether provides only the

• UDC 519.4. Originally published in Diskretnaya Matematika (1996) 8, No. 2 (in Russian). Translated by the author.

136 S. A. Stepanov

existence of such a basis. The purpose of this paper is to give an effective version of the Noether result.

We say that a system of distinct basis invariants u,1 , .... , .. is non-degenerate if for every

positive integer µ

s

m and any integral sequence (i1, ••• , iµ), 1 S i1

< ... <

iµ S m, it

contains at most µn elements u,, ... r,., with the condition that for every i E

{i1, ••• ,

iµ}

there exists at least oner; ~ I and for any i E

{i

1, ••• , iµ} all the corresponding r; ¥e

zeros. Note that any subsystem of a non-degenerate system is also non-degenerate. Theorem 1. Any s S mn elements u,1, ••• ,r,. of a non-degenerate system of distinct

basis invariants are algebraically independent over the field k.

In addition, this paper contains a generalization of the well-known Waring's formulae (see [l, A. IV, p. 99])

.i=l /1+2/2+ ... +nl.=a

where u1, ••• , Un are the elementary symmetric polynomials of the vector (x1, ••• , Xn), c(l ... l)

= (-

l)/2+2/3+ ... +(n-l)I. <1(/1

+ ... +

ln - l)!

] , ' n l I l I '

I···· n·

(2)

and the sum on the right-hand side is over all non-negative integers 11, ••• , ln satisfying

the condition 11 + 2/2

+ ... +

nln

=

<1, to the case of any number m ~ 1 of vectors

(X11, ..• ,Xin), ... ,(XmJ, 0 0 0,Xmn).

Theorem 2. Let (1, <11, •• ,, C1m be fixed, and let Li, ... , ln; r1µ, ... , rmµ; Siv, ... , Smv

bear-bitrary non-negative integers satisfying the relations

<11

+

<12

+ ... +

<1n

=

Li

+

212

+ ... +

nln

=

C1, r1µ

+

r2µ

+ ... +

rmµ

=

V, I S µ S Iv,

lsvsn,·

and let I, v.,·,v ... .\',,,v

=

LIT

u,Iµ, .... ,,,,,,, µ.:.1

where the sum is overall r 11 , ••• , r11,; ••• ; rmi, ... , rm,, such that r;1 + ... +ru,

=

S;v, 1 Si Sm. Then

n I I n

'°"

_{L..., X 1}a

_{j ...}

_X~i a

₌

<11 • • • • _I <1m ·

'°"

_{L..., c(/1, ... , ln)}

'°"

_L...,

IT

_{V,,·,,,. ....}

_,m,•

.i=I (1. v=I

where the internal sum is over all s11 , ..• , s1n; ... ; Smi, . .. , Smn such that S;i + ...

+

S;n =

<1;, 1 S i S m, and the external sum is over all non-negative integers Li, ... , ln, satisfying the condition 11

+

212

+ ... +

nln

=

<1.

These results allow us to pass from mn indeterminates x11 , ••• ,X1n; ... ;Xm1, ... ,Xmn to

mn new indeterminates u,, ... ,,,,, which can be chosen among the basis invariants u,, ... ,,,, in an arbitrary way. Note also that the results can be extended easily to the case of positive characteristic p > c(m, n).

2. NOTATION AND LEMMAS

Let CT= (CT11 , •.• , CT1n; ... ; CTmi, ... , CTmn) be a binary sequence of length mn and}1, ... ,}µ be

the numbers of all those subsequences CT= (CT11, ... , CTmJ), 1 :SJ :Sn, each of which has at least one non-zero component. Puts;, 1 :S i $ m, equal to the number of j, 1 :S j :S n, such that CTu = 1, and define the weight w( CT) of the sequence CT as

m n

w(CT)

=LL

CTu.

i=I )=I

For any subset

{Ii, ... ,/.,}

of the set {1,2, ... ,n} we denote by

u~'.:·:):,>

the elementary symmetric polynomials of the vectors X; = (xu)1,,1s,,,Jit{L, ... L,}, 1 Sis m, and put

uu, ...

u

= { 1 if (r1, ... ,rm)= (0, ... , 0),

,, ... ,,,. 0 if r;

<

0 at least for one i = 1, 2, ... , m.

Lemma 1. Let CT= (CT11, ... , CT1n; ... ; CTm1, ... , CTmn) be a binary sequence of length mn and let w(CT} =· v.

(i) If at least one subsequence CTj

= (

CTIJ, ... , CTmJ), 1 S j S n, contains at least two components CT;,J

=

1 and CT;v

=

1, then

(ii) If every subsequence CTj

=

(CT11, ... , CTmJ), 1 SJ Sn, contains at most one component CTu = 1, then

aw<rr>u

---''-···-··'-"'--- = UU1, .. '.J,.,> ..

:I CJ11 :\ 0'1n :\ 0'1111 : ) ~ r, -s1, ... ,r111-·lm

ax11 .. . ax1n .. . aXml .. . a.Ai,',';

Proof The first statement trivially follows from relation ( 1 ). To prove the second

statement, we use induction on the weight w(CT) Sn of CT. For w(CT) = 0 the statement is

obvious. We assume that the statement is true for w(CT)

=

v, 0

s

v < n, and prove its validity for w( CT) = v

+

1 S n.

Let CT

= (

CT11, •.• , CT1n; ... ; CTm1, ... , CTmn) be a binary sequence with the above property and let {)1, ... ,Jv}, w(CT), {s1, ... ,sm} be the corresponding parameters defined by this sequence. If w( CT)

= v,

by the induction assumption and by definition of ii),·-.:_·_~' ... . , ... -.,·,,, we

138

have

S. A. Stepanov

aw<u>u

'' , ... ,,,,, U1 , ... Jv>

ax<J11 ') O'!n ') a,,,, ') (J, = u,,-.\'(, ... ,r,,,-,\;,,, 11 .. • axln . .. axml .. . ax,,:.;;;

n

II (

I

+

X ·Z

+

+

X ·Z ) - "'u(j,, ... j,) Zr,-.,, zr..,-s.., IJ I ··· mJ m - ,L...,, r1-,\· 1, ••• ,r,,,-s,,, I ··· m , i=I jt{i, .... ,j,} (3)

(4)

where the sum is over all r1, ... ,rm, such that O ~ (r1 ,-s1)+ ... +(rm - sm) ~ n-:- v.

Denote by iv+I one of the numbers} E { I, 2, ... ,

n}

for which Oj

=

(a1j, ... , O'mj)

=

(0, ... , O),

and apply the operator dldxij,., to both sides of identity (4). As a result we obtain the relation n Z;

II

(I

+

X1jZI

+ .. , +

XmjZm) j=l jt {i, , ... j,.,}

=

"'

(-d-u(j,, ...

j..,) ) zr,-s, xr..,-s,,, ~

ax..

r1-,\'1, ... ,r,,,-s,,, l ··· m · OS(r1 -,\'1 )+ ... +(,,,, -.\'111)Sn-v Uv+l

Now we use the identity

n Z;

II

(I

+

X1jZ1

+ .. , +

XmjZm) j=l jt {i, , ... j,.,}

=

Z· ( " ' U(i,, ... j,.,> z r,-s, z r..,-.,·..,) 1 ,L...,, r1 -.,· 1, ••• ,r,,,-s,,, I • • • m Os(r1 -s,)+ ... +(r111 - . \ '111)Sn-v-l

= U(h , ... Jv+I) z'I -s1 r111-s111

r1 -s1, ... ,r;-,,·i-l, .... rm-,,·,,, I • • .zm ,

~(r1 -,,·1 )+ ... +(rm -s,,,)Sn-v

and then compare it with the previous relation. As a result we get =

uv, ....

j,.,>

,, -,\'1, ... ,r;-s;-l, ... ,r,,,-,,.,,,,

which implies, in view of (3), that

(i,, ... j,.,)

= u,,-s1, ... ,r;-s;-l, .... r,,,-s,,.·

Jxfl' .. .

Jxf~· ... dxu,., ... dx~'1' ... dx~~·

This proves the lemma.

Denote by Rmn the set of all sequences r

=

(r1, .. . , rm) of non-negative integers

r1, ... , rm, satisfying the condition I ~ r1

+ ...

+rm ~ n, and consider a subset ~ of the set Rmn of cardinality s. Let us index the elements of the set~ by the numbers I, 2, ... , s,

setting r = r(k), 1 $; k $; s, and for a given set

'JI£,

=

{Ci, ,j, ), ... ,

Ci.,,f,.)}

of distinct integral

pairs (i1,j1) , where I $; i1 $; m, 1 $; j1 $; n, 1 $; l $; s, let us introduce into the consideration

the determinant

II

OUr(k)

II

!ls= det - - .

oxu (iJ)e 'Jo(_,. J,;Jcss

Let T = ('i11 , ... , T1n; ... ; Tmi, ... , Tmn) be an integral sequence of length mn with

non-negative components Tu and let

m n

w(T)=

LL

Tu

i=I j=I

be the weight of this sequence.

For each integral pair (i,j), I $;

i

$;

m,

I $; j $;

n,

we consider the set of all sequences

k ij = (k(I) u , ... , u k(Tij)) wit mutua y 1stmct components . h II d" . k(I) u , ... , k(T;j) ij e {I 2 , , ... , s , } an put d ku

=

0 for Tu

=

0. Set <Ju(k) = {

~

"fk {k(I) k(Tij)} I E ij ' ... , ij ' "fk {k(I) k(Tij)} I 11: ij ' ... , ij ' <J(k) = ( <J1 I (k), · · ., <Jin(k); ... ; O"mJ (k), ... , <1mn(k)), m n w(cr(k))

=LL

<Jjj(k) i=I j=I

and denote by !l.,.(k11 , ... ,k1n; ... ;km1, ••• ,kmn) the determinant which is obtained from !l., if we replace the elements

ou,(k)

oxu

of the kth column by the elements 0w(a(k))+I U,(k)

(i,j) E 9{,,

ox""(k) ox"'•<*> ox""''(k) o""'"m(k>;)x .. '

II · • · In • • · ml • · • "mn IJ

(i' j) E

'JI£,.

Applying to !l., the well-known differentiation rule of determinants with functional ele-ments and noting that

for all k = 1, ... , s and for all (i,j), I $; i $;

m,

I $; j $;

n,

such that <Ju > I, we obtain the

following result.

Lemma 2. For a given integral sequence T = (T11 , ... , T1n; ... ; TmJ, ... , Tmn) with non-negative components,

140 S. A. Stepan"v

For a given sequence

5\,(

=

{r(k) E Rm,. J IS k S

s},

let us consider the corresponding system

U.,

=

{ur<k> J r(k) E 5\.(, IS k S

s}

of vector invariants u,ck> and denote by

1.,

=

II

OUr(k)

II

OXij ISiSm, I ~ . ISks.,

the Jacobian of the system

U,, ..

Lemma 3. For a given non-degenerate system

U.,·

=

{ur<k> J r(k) E 5\,(, Is ks

s},

of distinct vector invariants u,, it is possible to find an integral sequence 1' = (1'11, ... ,1'1n; •.. ;1'm1, ... ,1'mn), IS 1';j S s, and a set

'Xi,=

{(i1,j1), •.• ,(i.,,J,)} of distinct

integral pairs (i1,j1), 1 S i1 Sm, 1 Sj1 Sn, 1

s

ls s, such that

aw<r>

II

~

ou

II

- I I I I

r r t det (") M - 1'11 ..•. 1'Jn .... 1'mJ ... . 1'mn. det A.,,

OX1'i', . • OX1~ .. .

cJx,;;·/ ...

ax::in;;

OXij

'j~~~"

(5)

where A., is a binary s

x

s matrix of the rank s, which may be reduced by elementary row

operations to the unit s x s matrix

0 I.,=

0

Proof We prove the lemma by induction on

s,

1 S

s

S mn. Let u, = u,, ... , ... be an arbitrary elementary symmetric polynomial in m vector variables

(X1 I, ... , X1,.), .•. , (XmJ' •• • , Xmn)

and let w(r) = r1 + ... +rm be the weight of the sequencer= (r1, ••• , rm).

By Lemma I we have

aw<r>u,

- - - =

_{'.} ... '.} '.} ... ::i} u<l, .... .1.> _{o ... o}

=

I, aX1 I·· .UXJr1 aX2.r,+I · • .aX2.r,+r2 • • .uXm.r1+ ... +r..,_1+1 • • .uXm,r,+ ... +r..,

We assume that the assertion holds for s - 1 ~ I and prove its validity for s ~ 2. Let

Ms

= {r(k) E Rmn

I

1 ::; k::; s} be an ordered set of cardinality s, consisting of elements

r = (ri, ... , rm) E Rmn, let Us= { ur<k>

I

r(k) E

Ms}

be the corresponding non-degenerate

system of distinct vector invariants u,<k>• 1 ::; k ::; s, and let

II

dur(k)

II

ls=

-dXij 1:Sism. 1,ai:Sn, 1:Sk:Ss

be the Jacobian of the system U,,. We assume that the set

Ms

is ordered in such a way that w(r(l)) ::; w(r(2)) ::; ... ::; w(r(s)). The case where all elements of the system

Af,.

have weight 1 is trivial, hence without loss of generality we can assume that at least one sequence r(k)

=

(r1 (k), ... , r mCk)), I ::; k ::; s, has weight w(r(k)) ~ 2. Thus we can assume that v

=

w(r(s)) ~ 2.

By the induction assumption we can find an integral sequence

I ( I I • • I I ) T

=

T1J>·••,T1n•···•Tm1'···,Tmn • andan(s- l)x(s- l)matrix

,

II

dur(k)

II

Hs-1

= - -

,

dxu (ij)e!i\l.:_,. J:Sk,;.,-J N.";_1 = {(i1,}1), ... , (i.,.-1.J.,-1)},

(6)

consisting of the first s - 1 columns and s - 1 rows of the Jacobian l.n such that the determinant ~:_ 1

=

det

H.'.-

i of the matrix

H.'.-

i satisfies the relation

aw<r')~/

s-1 - 1 _{I ' I} I _I I _Id tA'

_r. ___ r_' _ _ r' _ _ _ r'_ - T11 ····Tin···· Tm!···· Tmn · e J-1 •

dx1\' •• • dx1~ •• • Jx,;;'/ ... Jx,;:~

(7)

where A:_1 is a non-singular binary (s - 1) x (s - 1) matrix, which may be reduced by elementary row operations to the unit matrix /.,_ 1.

Now we fix the set

'JIC_

1 =

{Ci

1 ,} 1 ), ... , (i,,._

i.J.,_

1)} and the sequence T' = (T;J> ... , T;n; ... ; T~J> ... , T~n) with the above mentioned properties, and show that there

exists an integral pair ((,.,}.,), 1 ::;

i., ::;

m, 1 ::;

J., ::;

n, and an integral sequence

T = (T11, ... ,Tin; ... ; Tm1, ... , Tmn), ] ::; Tu ::; S, such that equality (5) is valid for the determinant

A - d t

II

du,<kl

II

L.J., - e

. dX·

IJ (ij)e '}{_,, l:Sk,;.,

Next we describe in detail the induction process necessary to construct the required pair

Ci.,,J.,)

and the required sequence T = (T1i, . . . , T1n; ... ; Tm 1, ... , Tmn). Let

142 S. A. Stepanov

be a binary sequence with parameters

v = w(CT) = w(r(s)) = r1(s)

+ ... +

rm(s), (si, ... Sm)= (r1(s), ... , rm(s)),

which has the property that any its subsequences of the form CJ'p = (CJ'ip, .•. , Cl'mp), I ~ p ~ n,

contains at most one component CT;p = 1. Let p1,

···,Pv

be the numbers of all those

sequences CJ'p

=

(CT1p, ... , Cl'mp), I ~ p ~

n,

which contain exactly one component CJ';p

=

I

and assume thatf,. e

{pi, ... , p.,}.

Then by Lemma I we have

Jw<a>u

_ _ _ _ _ _ r<_.,_·J _ _ _ - U(p1, .•• ,p.J -

1

"I a11 "I U1n "I Um1 "\.,a - 0, ... ,0 - •

ax11 .. • ax1n .• . aXmi •. . a.11,,,:;: (8)

Denote by m = (m11, ••. , m,n; •.. ; Wm1, ••• , Wmn) the binary sequence which is obtained from

CJ'= (CJ'11 , ... , Cl'1n: .•• ; Cl'mi, ••• , Cl'mn) if we replace a subsequence Cl'j• = (0, ... ,0, 1,0, ... ,0)

beginning with

i'

zeros by the zero sequence (0, ... , 0). There are

n!

L,.

=

-. r,(s)!. .. rm(s)!(n - r,(s) - ... - rm(s))!

different sequences CJ'= (CT11 , •.. , CT1n; ••. ; Cl'mi, ••• , Cl'mn), satisfying relation (8), and these sequences generate

m I

'°'

n.

M,

=

L.,

· i=I r,(s)!. .. (r;(s) - 1)!. .. rm(s)!(n - r1(s) - ... - rm(s) + I)! binary sequences a= (a11 , ..• , a1n; ... ; a,,,1, ... , <Xmn), with components

which corresponds to all possible sequences CJ' = ( CT11 , ... , Cl'1n; ••• ; Cl'm1, ... , Cl'mn). The

sequences

a

are not necessarily distinct, but for any given i, 1 ~ i ~

m,

they do contain exactly

"f,= (r,(s)+ ... :rm(s)-1)

=

(v:

1)

distinct sequences of the form

a

= (0, ... , O; ... : ail, ... , <X;n; ... : 0, ... , 0), where each se-quence a;

=

(a;1, ••• , a;n) has exactly r1 (s)+ ... +rm(s)-1 zeros and n-r,(s)- ... -rm(s)+ I

units. This provides a lot of essentially different candidates for the last column of the matrix A., which correspond to various integral sequences 7:

= (

7:11, ... ,

-r,n; ... ;

'l:m1, .•. , 'l:mn) of the form

(9)

We will use this fact later on, and now we establish the validity of relation (5). Namely, we prove by induction on s that for any given integral pair (i.,,f,.)

e:

91C_

1, I ~

i.,

Sm, 1 S

J.

1. Sn, and any given integral sequence f

=

(t11, ... ,fin; ... ; 't'm1, ... , 't'mn) of form (9) relation (5) holds with a unique binary

s

x

s

matrix A.,. The matrix A., can be constructed as follows. Expanding the determinant A., according to the last column, we obtain

( -1 )·''+'

aur(.,)

A •.

u,J,s-1·

(iJ)e '>l,, (iJ)=(i1J1) By the induction assumption we have

ax;j

with a unique binary (s - 1) x (s - 1) matrix A;J,.,-i, and according to the well-known Leibniz formula we find that

ax?i' .. .

axr~ .. .

ax::;-/ ... ax;::~

T11 Tum

(f ) ('[ )

aw(v)+] U . aw(t)-W(V)-1 f:l .. .

=

'°' (

-1)·1_·+1

'°' ... '°'

11 . . . _{mn r(.,)} IJ,s-1

~ ~ ~ V V

ax

axv11

a

Vn,n

a

T11-V11

a

1i,in-V,nn

(iJ)e 'JI(_,, (iJ)=(i1,i,) v11=0 v..,.=O II mn ij 11 · · · Xmn X11 · · · Xmn

=

L (-

Ir' (

't']]) ... ('t'mn) <011 !. .. COmn !

i:

I!. .. i:n ! detA;J,.,-1 (iJ)e 'JI(_,, (iJ)=(itJt) <011 COmn

=

f1 I!.·· f1n !. · · 't'm1 !. · · 't'mn ! detA.,.

Further we show that for any integral pair (i,,,J.,) 11:

'11[.'_

1' I

s

i.,

s

m,

1

s ).

1.

s

n,

it is possible to choose an integral sequence f

= (

t 11 , ... , tin; ... ; 't'mi, ... , 't'mn) of form (9) satisfying the condition that the last column of the corresponding matrix A., in equality

(5) differs from all its other. columns. For any given /, I

s

l

s

s,

we denote by

n

1 the number of positions (i1,j), I

s

j

s

n,

where the last column of the matrix

H,.

=

II

aur(k) II

.

ax ..

IJ (i,j)e 'JI(_,, ISks.1·

has exactly

n

1 non-zero entries, and set p., = n 1

+ ... +

n.,.

If P., = 0, then the last column of the matrix A.,· consists only of zero elements, hence it differs from all other columns

of A.,. Now we assume that

p.,

~ 1.

It

follows from the previously mentioned arguments

that for any l

=

1, ... ,s such that r;(s) ~ 1 with i

=

i1 there is exactly

N.,

=

C~

_{1) distinct} binary sequences a;

=

(a;1, ... , a;n), containing v - 1 zeros and n - v + 1 units. If we remove all entries of the vector a; situated at any n - n1 positions, then we obtain exactly

Vo-I ( )

'°'

n,

q,=

~

a=v11-I -(n-n1) CJ

distinct binary sequences of length

n

1, where v0

=

min(v,n -

v

+ 2) and {:)

=

0 for

144 S. A. Stepanov

Assume that n1 ~ 1 for some l

=

1, 2, ... , s and consider the n1 rows of the matrix A.,· indexedbyintegralpairs(i,,j) e 9{,.withgiveni1• Letk1, .. ,,kµ, 1 ~k1

< ... <

kµ

<

s,be

the numbers of all those columns of A., which intersect the considered rows by non-zero binary vectors of length n1, say vkp ... , vk". It follows from the structure of the matrix

A.,

that all remaining entries of the corresponding columns, situated outside of the given

n

1 positions, are zeros. Since the matrix A~_ 1 is non-singular and the system U., is

non-degenerate, we find that µ ~ .;\.1

=

min(n,,

n -

1) and, moreover, that the vectors vkp .... vk" are distinct. Thus we have q1 different possibilities to choose the last column of the matrix A., with the condition that all their entries situated outside of the given n1 positions are zeros, and only µ ~ .;\.1 different possibilities for all other columns with the same property. Since q1

> .;\.

1 for any

n

1 ~ 1, we can choose the last column in such a way that it differs from all other columns of the matrix

A.,.

Moreover, if

n

1

>

1, we can choose the last column with the above property as a non-zero vector. Indeed, if v0

=

2, then the set { vk,, .. . , vk"} contains at most one vector of weight I or n1 - 1, and we have

n1 > 1 possibilities to choose the last column of A., _as a vector of the same weight. Now, if v0

>

2, we have q1

> .;\.

1 + 1, and again we can choose the last column of A., as a

non-zero vector.

Now we fix an integral sequence 't'

=

(i-11 , ... , i-1"; ... ; 't'mi, ... , 't'mn) satisfying the just described property and find an integral pair {i,,,i,)

e:

!liC-1'

1 ~

i.,

~

m,

1

~f,

~

n,

which provides the condition that the matrix A., in (5) can be reduced to the unit matrix /.,. Set

and consider the matrix

H" -_{.,-1 -} II dUr(k)

a

11

.

X;j (i j)e ~.:'.... , ' 2S/cs.,

By the induction assumption we can find an integral pair

(i. ..

f,) and an integral sequence

II - ( II II • • II .JI ) 1 < II < h th t th d t . t A II - d t H" 't' - 't'11, ... , 't'1n• .... 't'ml• ... , ''mn , - 't'ij - 't';j, sue a e e erm1nan o.,_] - e .,-1

satisfies the relation

aw(rll) llll

s-1 _ 11 1 11 I II I II I d

A"

t.' " r'' r'' - 't'11 .... 't'ln""'t'm1 .... 't'mn' et .,·-I•

:) II :) tin :l 1111 :l mn

aX1 I .. • ax1n .. . aXml .. . aXmn

(IO)

where A_:'.__1 is a non-singular binary (s - 1) x (s - 1) matrix, which can be reduced by

elementary row operations to the diagonal matrix /.,_ 1.

Now we fix the integral sequence

-r"

=

(-r;'i, ... , -r;~; ... ; <;i, ... , -r;")

and the integral pair (i_,.J,) which satisfy (10), and show that under the above choice of (i.,.J,) and

't' = ('t'11, ... ,

-r

1n; ... ; 't'mi, ... , 't'm11 ) the matrix A.,· in equality (5) possesses all the properties

stated in the lemma. Indeed, it follows from Lemma 2 and relations (8), (9) that :iw(r) A

a o., I I I Id A

- , - - - , - - - t - - : i -

=

't'11 .... 't'1n0 0 . . 't'm1 .... 't'mn· et n

dX1'i' . .. dX1~ ... ox,;;'i' ... ax;;:~ where

is a binary s x s matrix containing (s - I) x (s - I) submatrices

A:-1 = llaull1siJs.,-1• A:'...1 = llaull2s;J,s.1··

Since the matrices A:_

₁

and A�.'...

1

may be transformed by elementary row operations to

the unit matrix /.,-

i,

as a result the matrix A

_s

may be reduced by the similar operations

to the form

I

0

0

C1s

0

I

0 0

A.,=

c.,I 0

0

I

Because all the columns of the matrix A.,· are distinct, all the columns of the matrix A.,

are distinct as well. In that case c

₁

.,c

_.,1

= 0, and hence the matrix A

_.,

is equivalent to the

unit matrix /

_.,

. This proves the lemma.

3.

PROOF OF THEOREM 1

We assume that there is a non-degenerate system consisting of s s;; mn distinct vector

invariants

u,

=

_{u,, ... , ...}

algebraically dependent over

k,

and show that this assumption

leads to a contradiction.

Let the above system be of the form U

_.,

= {

u,

I

r

e �}, where � is a subset of

the set R

mn

of cardinality

s,

and let T., = {t,

I

re

�} be the corresponding system of

independent variables t, = t,1, ... ,r.,.· From our assumption of algebraic dependence of the

vector invariants

u,, r

e � over the field

k,

it follows that there exists a polynomial

F e k[T..l such that

F(CJ..)

= 0

(11)

identically with respect to x11, ...

,x1n, . . . ,Xmi, ... ,Xm11•

We may assume that Fis a poly­

nomial of minimal degree satisfying equality ( 11 ). If we differentiate identity ( 11) with

respect to x

_u

, 1 s;;

i

s;;

m,

I s;; j s;;

n,

we obtain the system

"'"'

aF(U.,) au,

__

_O,

�

I s;; i s;;

m,

1 s;; j s;;

n,

M

au, ax

_ij

re,,.,

(12)

consisting of mn linear equations, with respect to the s polynomials aF(U.,)lau,,

r e �- Since F is a polynomial of minimal degree satisfying relation (11 ), we

have aF( u.,.)lau,

,f.

0 at least for one sequence r = (r

₁

' ... , r

_m

) E �- Therefore

to get the desired contradiction, it is sufficient to establish the existence of a set

'Jl. = { (i

_i

,}

1), ... ,

U..,J.,)} consisting of

s

integral pairs (i

1

,j

1),

I s;; Is;;

n,

where I s;; i

1

s;;

m,

I s;;J, s;; n, for which the determinant

II

au,

II

t:.., =det -

_ax

u (ij)e� .. re9'1;

is not zero.

The existence of such a determinant t:.., for any s, 1 s;; s s;; mn, is provided by Lemma 3,

and this concludes the proof.

146 S. A. Stepanov

4. PROOF OF THEOREM 2

In (2) we set Xj = x1jZI

+ ... +

XmjZm, I ~) ~ n. We have

<1! (X1jZ1

+ , , , +

XmjZm)"

=

L

I I

a,+ ... +a,.=a 0'1 .. .. O'm, and then

On the other hand

V

L

II<xu,Z1

+ ...

+Xmj,Zm)

=

L

u,, ... , .. z~' ... z;

lsj, < ... <j,Sn .,=I r1+ ... +rm=V

and then, in view of (2),

=

L

c(l1, ... , ln)

II (

L

u,, ... , .. z~' ... z~·) ,.

11+2/z+ ... +nl.=a v=I r,+ ... +r,.=v Hence ( LX1j ... xmj n a, a,,, ) Z1 ... 0'1 zm Oin J=I n ( ) (l l ) _II

'°'

t''•

., ...

C I, · · ., n ~ Vs,v, ... ,smv I • • .zm li+212+ ... +nln=cr v=I .,·1v+, .. +.t111v=Vlv

=

and therefore n 0'1 ! ... O'm ! n

'°'

~ xa' xa'" - - - - -l j " ' mj - <1!

'°'

~ c(l 1, ... , l ) n

'°'

~ II v .,,, ... ,,.,, j=I 11+212+ ... +nl,=a .,11+ ... +.,,.=a;, ISiSm v=I

REFERENCES

I. N. Bourbaki, Elements <!{Mathematics, Algebra II. Springer, Berlin, 1990.

2. D. J. Benson, Polynomial Invariants <!f Finite Groups. Cambridge Univ. Press, Cambridge, 1993. 3. J. A. Dieudonne and J.B. Carrel, Invariant Theory, Old and New. Academic Press, New York, 1971. 4. D. Mumford, Geometric Invariant Theory. Springer, Berlin, 1993.

5. E. Noether. Der Endlichkeitssatz der lnvarianten endlicher Gruppen. Math. Ann. ( 1916) 77, 89-92. 6. T. A. Springer, Invariant Themy. Springer, Berlin, 1977.

7. S. A. Stepanov, Rational exponential sums and L-functions of Arlin. Dokl. Soviet Acad. Sci. ( 1982)

265, 39-42.