Polynomial invariants of finite groups over fields of prime characteristics

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Discrete Math. Appl., Vol. 9, No. 4, pp. 343-354 (1999)

©

VSP 1999.

Polynomial invariants of finite groups over fields

of

prime characteristics*

S. A. STEPANOV

Abstract - Let R be a commutative ring with the unit element 1, and let G

=

Sn be the symmetric group of degree n 2'. 1. Let A~n denote the subalgebra of invariants of the polynomial algebra Arnn = R[x11 , ... ,x1n; ... ;xm1, ... ,Xmn] with respect to G. A classical result of Noether [6] implies that if

every non-zero integer is invertible in R, then A~n is generated by polarized elementary symmetric polynomials. As was recently shown by D. Richman, this result remains true under the condition that n! is invertible in R. The purpose of this paper is to give a short proof of Richman's result based on the use of Waring's formula and closely related t0 Noether's original proof.

The research was supported by Bilkent University, 06533 Bilkent, Ankara, Turkey.

1. INTRODUCTION

Let m,n be positive integers, R be a commutative ring with the unit element 1, and let

Arnn= R[xn, ... ,Xrni; ... ;x1n, · · · ,Xrnn]

be the algebra of polynomials in mn variables Xij over R. The symmetric group

G

=

Sn operates on the algebra Arnn as a group of R-automorphisms by the rule

gxij

=

xi,g(j), gEG.

Denote by A~n the subalgebra of invariants of the algebra Arnn with respect to the group G and define polarized elementary symmetric polynomials u,1, ... ,rm E A~n in

n vector variables (x11, ... ,xrn1 ), ... ,(x1n, ... ,Xrnn) by means of the formal identity

n

IT(l+x1jZJ+ ... +XrnjZrn)=l+

L

U,1, .. ,,rmZ1'1 ••• zrn'm. (1)

j=I l:s;r1+ ... +rm:s;n

If R is Noetherian, it follows from the Hilbert-Noether finiteness theorem [4, 6] that A~n is a finitely generated commutative R-algebra and Arnn is finitely generated as a module over A~. Moreover, if every integer is invertible in R, the invariants u,1, ... ,rm form a complete system of generators of A~n over R (see [l], p. 9; [2],

p. 62; [14], p. 37). In other words, every element u of the algebra A~n may be • UDC 519.4. Originally published in Diskretnaya Matematika (1999) 11, No. 3, 3-14 (in Russian). Received May 25, 1999. Translated by the author.

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344 S. A. Stepanov

written as a polynomial in Uri, ... ,,.111 , 1 ~ r1

+ ...

+rm ~ n, with coefficients in R. The

above system of generating invariants contains (m;n) - 1 elements connected with each other by different algebraic relations (see [3], p. 68, and [12]). This result was recently generalized by D. Richman [8] as follows.

Theorem 1. Assume that G

=

S11 and n! is invertible in R. Then A~n is

gener-ated as an R-algebra by the polarized elementary symmetric polynomials u,1, ... ,rm•

1 ~ r1 + ... +rm~ n, of degree at most n.

In particular, if R is a field of a prime characteristic p

>

n, then n! is invertible in R, and we arrive at the following result.

Corollary 1. Let R be a field and G

=

Sn.

If

the characteristic of R is zero or p

>

n, then A~n is generated as an R-algebra by the polarized elementary symmetric polynomials u,1 , ... ,rm• 1 ~ r1

+ ... +

r111 ~ n.

In this paper we give a short and simple proof of Theorem 1 based on polariz-ation of the classical Waring formula and closely related to one of two Noether's original proofs in the case where R is a field of characteristic 0. Several examples presented in the final section of the paper show that the restriction on R stated in Theorem 1 cannot be removed.

More generally, let A

=

R[x1, ... ,x111 ] be a finitely generated commutative R-algebra, G be a finite group of the R-algebra automorphisms of A, and let AG be the subalgebra of invariants of G. If z 1 , ... ,

Zm

are commuting indeterminates, define

F(z1, ... , z,,,)

=

f1

(1

+

't(x1

)z1

+

-r(x2)z2

+ ... +

-r(xm)Zm).

tEG

If every non-zero integer is invertible in R, it follows from the Noether theorem that AG is generated as an R-algebra by the coefficients of F(z1, ... ,zm). The result of Theorem 1 and the standard arguments based on the use of the Reynolds operator and theNoethermap(see [6], [ 10], p. 63, [14], p. 275) lead to the following theorem. Theorem 2.

If IGI!

is invertible in R, then AG is generated as an R-algebra by the coefficients of

F(z1, ... ,z

111 ). In other words, AG is generated over R by the

invariant polynomials in xi, ... ,x111 of degree at most

IGI.

This result provides us with an efficient algorithm to compute a complete sys-tem of generating polynomial invariants under the condition that

IGI

! is invertible in R. There is another constructive proof of Theorem 1 based on different arguments also ascending to Noether (see [9] and [10], p. 29). The upper bound on the degrees of a set of generating polynomials for the algebra of invariants given by Theorem 2 is known as Noether's bound (see also [9], [10], p. 28, and [11]). In the final section of the paper, we show that the conditions of Theorem 2 cannot be removed. In par-ticular, it will be shown that Noether's bound is false if Risa field of characteristic

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Polynomial invariants of finite groups 345

2 and G

=

Sz. For other results and problems in the theory of polynomial invariants over fields of prime characteristic see [9] and [ 11].

2. GENERATING INVARIANTS OF THE SYMMETRIC GROUP

Let G

=

Sn be the symmetric group of degree n

2'.

1 that operates on the R-algebra Amn

=

R[x11, ... ,Xm1; ... ;x1n, ... ,Xmn] as a group of R-automorphisms, A~ be the subalgebra of invariants of G in Arnn, and Ur1 , ... ,rm, I ::; r1

+ ... +

rm ::; n, be the

polarized elementary symmetric polynomials in A~n·

Let Va1 , ••. ,crm be an invariant polynomial in A~n of the form

If m

=

I, then the well-known Waring formula (see [13], p. 13 and [2], p. 99]) gives

n

~ a ~ ( ) SJ S

Va= LJXJ

=

Li C SJ,··· ,Sn U1 ... Un", (2)

}= I s1 +2s2+ ... +n.1·.=a

where Cs_{1 , •••},en are integers of the form

( ) - ( - )s2+2s3+ ... +(n-l)s.cr(s1+ ... +sn-I)!

C SJ, ... , Sn - I I ·

SJ! .. . Sn.

The following result can be considered as a generalization of the Waring formula to the case where m

>

I (see also [12]).

Proposition 1. Let CJ1, ... , CJm be non-negative integers, Va1, ••• ,am

=

LJ=I

xf) ...

x:j

be the polynomial in A~n of degree cr

=

cr1 + ... + crm, and let u,1, ••• ,rm• 1 ::; r1 + ... +rm ::; n, be the polarized elementary symmetric

polynomi-als of vectors (x11, ... ,XmJ ), 1 ::; j ::; n. For non-negative integers s1, ... , Sn and S1v, ... , Smy satisfying the conditions

SJ +2s2 + ... +nsn = cr, SJy+ ... +smY = Vsy,

1::;

V::; n,

let S I Sv

L

Y·

IT

a w. _{S[v,···,smv}.

=

_~ _I _~ _I u vt r1,, ... ,rmt• R vyJ .... vysv. 't=l

where the sum is over the set R of all non-negative integers r1 't, ... , r msv and

CTvJ, ••• , CJYsv such that

rµJCJyJ + ... +rmsvCJYsv =sµv,

CJyJ + · · · + (JYsv

=

Sy,

(4)

Then

n

L

c(s1,,,. , Sn)

L

TI

Vs1vjl"' ,Smv;, (3) s1+2s2+ ... +nsn=O' S i=l

where the inner sum is over the set S of all non-negative integers Sµvi, · · · , Sµv.

sat-isfying the relations

sµv1

+ ... +

sµv.

=

crµ, SJv;

+ ... +

Smv;

=

isi, 1 ~ µ ~ m, 1 ~ i ~ n. Proof In (2) we set Xj

=

XJjZI

+.,,

+XmjZm, 1 ~ j ~ m. Since we have n cr! L(XJjZI + ... +XmjZmf

=

L

₁ ₁ ·-1 CJ + +CJ =CJ c:JJ • • · • c:Jm • ] - I ... m

On the other hand,

V

L

n

(XJj,ZI

+" · +

Xmj,Zm)

=

L

Ur1 , ... ,rmZ~1 " ·

z~,

I::s;}J < ... <Jv:::,:;ns=l r1 + ... +rm=V

and hence, in view of (2),

n

L

(x1jZI

+,,,

+xmJZmf

j=l

As a result we find that cr! cr1+ .. tcrm=CJ CJ1 ! • • • CJm!

( ~ ~ x l j ... xmj CJ1 O'm) O'J z1 .. -Zm O'm

J=l

L

c(s1, .. ,,sn)f:r(

L

Ur1, .. ,,rmZ~1 ...

z~)sv

s1+s[i+ ... +nsn=O' V=l r1+ ... +rm=V

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Polynomial invariants of finite groups 347 Note that ~ S1v ...Smv L_; Ws1v,,,. ,Smvzl ''';cm ' s1v+ ... +smv=Vsv where S ! Sv w - ""' v·

ITuavr

S}y, ... ,smv - L.J (J I (J I ri,, ... ,rm,' R vi···· Vsv · -r=I

and the set R is defined in the statement of the proposition, therefore we find that

""' (

~

cr1 Om) cr1 Om L.J _

f:

x 11 ... xmj z1 ... Zm

a1 + ... +am-a J-1

L (

L

c(s1, ... ,sn)LITWs1v;, ... ,smv;)zf1 ... z~m,

a1+ ... +am=a s1+2s2+ .. .+ns.=a S i=I

where the set S is defined in the statement of the theorem. Thus, we arrive at the relation

~ a1 Om_<JJ! ... <Jm! ""' ( )""'Iln

L_;Xi}"'Xmj- <J! L.J CSJ, .. ,,Sn L.J Wslv;, ... ,Smv;>

}=I s1+2s2+ ... +nsn=a S i=I

which proves the theorem.

If cr

=

<J1

+ ... +

<Jm :Sn+ l, then <JJ ! · · · <Jm ! V _{cr1, ... ,crm-}- _cr! n

I

c(s1, ... ,crm)IlvSiv;,···,mv; s1+2s2+ ... +nan=a i=I

involves only the polarized elementary symmetric polynomials Ur1 , ... ,rm, 1 :S r1

+

... +

rm

:S

n. Moreover, the coefficients of v01 , ... ,am are rational numbers whose

denominators are not divisible by any prime p

>

cr. As a consequence of this obser-vations we get the following result.

Corollary 2. lfn! is invertible in Rand cr

=

cr1

+ ...

+crm :Sn+ 1, then

is a polynomial over R in the polarized elementary symmetric polynomials Uri, ... ,rm,

1 ~ r1 + ... +rm :Sn, of degree at most n.

Now we show that any invariant in A~n can be represented as a polynomial over

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Proposition 2. Let f be a monomial in Arnn and v=

I.

u.

uE{,(f)i,EG}

Then v is a polynomial over R in the invariants

where cr1, ... , cr"' are non-negative integers such that O :,S _cr1

+ ... +

crm ::; degf. Proof We write f in the form f

=

!1 ...

fn, where each fJ is a monomial in

R[x11, ... ,XmJl· We set

d(J) = max (degf1) l:SJ:Sn

and prove the assertion by induction on 8(!)

=

deg/ - d(f). Suppose first that 8(/)

=

0. Then f

=

!J

=

xr) ...

x~j for some j E

{1,2, ...

,n} and (ar, ... ,a111 ),

0 :'S a1

+ ... +am

'.'S

degf, therefore

/l

_ "'"' _"'"' a1 am

V - L.J U - L.JXIJ ... xmj·

uE{,(f)l,EG} J=l

Suppose now that 8(!)

> 0 and let j E

{1,2, ...

,n} satisfy the condition d(J)

=

deg!J

<

degf. Define

v

1 and

vJ,

setting

v· -_{1 -} "'"' _L.J u _' _VJ=I "'"' _L.J _U.I

uE{,(fj)l,EG} u'E{,(J / Jj)laEG}

The induction hypothesis implies that

v

J and

vJ

are polynomials in Va 1 , ... ,am, 0 :::; cr1

+ ... +

crm

:'S

degf. For every p E G, we define Up as the set of all pairs

(u, u')

such that

and note that the map

(u,u')-+

(t(u),t(u'))

is a bijection. Thus,

I

UP

I

=

I

Uid

I

for all p E G. Note also that d ( uu')

2:

d (!) for

all

u

E {

t(!J)

I

t

E G} and

u'

E {

t(f /

!J)

I

t

E G} with equality if and only if

uu'

E

{ t(f)

I

t

E G}. Therefore,

v1v

1 =

1uidl

I,

+

I,

L

u.

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Polynomial invariants of finite gmups 349

By the induction hypothesis, the invariant

VjV1-IVidl

Ii

U=

Ii

u

uE{-r(J)l-rEG} f': degf'=dcgf,d(I')>d(I) 11E{-r(J')l-rEG}

is a polynomial over R in vcr1, ... ,crm, 0 ::; er1

+ ... +

er111

:S

deg

f.

The cardinality of Uid does not exceed the cardinality of

{-r(Jj)

J

-r

E G}, and the last cardinality does not exceed the cardinality of

{xf) ... x~j

J l

:S

j ::;

n },

therefore 1

:S

IVidl

:Sn.

Since n! is invertible in R, we conclude that

Li

u

uE{-r(J)l-rEG}

is a polynomial over R in Vcr1, ... ,crm• 0

:S

er1

+ ... +

er111

:S

degf. This completes the proof.

3. PROOF OF THEOREM 1

Let G

=

Sn be the symmetric group of degree n. Suppose that f is a monomial in Arnn and w E A~n is a polynomial invariant of G. Since

-r(w)

=

w for any 't E G, the polynomials

w

and

-r(

w)

have equal coefficients. This shows that every polynomial invariant of G is an R-linear combination of the invariants

v=

Li

u,

uE{-r(J)l-rEG}

where f varies over the monomials which appear in w.

Let

(i1,

iz, ... ,

iµ)

be a sequence of elements i1, i2 , ... , iµ E { 1, 2, ... ,n }. At first we prove that every invariant wµ of the form

n

Wµ

=

LXi1,j ... X;µ,j

j=I

is a polynomial over R in the polarized elementary symmetric polynomials Uri, ... ,rm,

1 ~ r1

+ ... +rm::; n.

Ifµ::;

n+

1, the assertion follows from Proposition 4. Assume

now that µ

>

n

+

1 and proceed the proof by induction on µ. We set

_ {Xi,,j

Xi,,j

=

Xin+1,jXin+2,j · • .Xiµ+1,j

ifs ::; n,

ifs= n+

1,

for j

=

1,2, ...

,n,

and write

n

Wµ+ I

=

L

Xii ,j · • • Xin ,jXin+ 1,j ·

(8)

Let Arnn= R[x11, ... ,xm1; ... ;x1n, ... ,Xmn], and let A~n be the subalgebra of invari-ants of Arnn· It follows from Corollary 2 that Wµ+I is a polynomial over R in the

polarized elementary symmetric polynomials ur1 , ... ,rm E Arnn, 1 :S r1

+ ...

+rm

'.S

n.

Since every such polynomial ur1 , ... ,rm has the form

a

r1, ... ,rm -

-

a,

uE{-r(/)1-rEG}

for some monomial

J

E Arnn of degree at most n, by Proposition 2 it can be written as a polynomial over R in invariants

n

- _ ~ -01 -Om

Vcr1 , ... ,crm - LJ Xlj • · · Xmj j=I

of degree at most n. Therefore,

n

- _ ~ SI Sj

Vcr1 , ... ,CJm - LJ Xlj · · .Xmj j=I

with 1 :S s1

+ ... +

Sm

'.S

µ. The induction hypothesis implies that every invariant

v

01 , ... ,crm is a polynomial over R in Ur1 , ... ,rm, 1 :S r1

+ ... +

rm

:S

n, so Wµ+ 1 is also a polynomial over R in these polarized elementary symmetric polynomials.

To complete the proof, we note now that every element v E A~n can be written, in view of Proposition 2, as a polynomial over R in the invariants wµ.

4. EXAMPLES

Example 1. Let m

=

3 and n

=

2. Let us show that the cubic

which is invariant with respect to G

=

S2, cannot be written as a polynomial in the

invariants u100, uo10, uoo1, u200, u110, u101, uo20, uo11, uoo2 over F2, the prime finite field of characteristic 2. By Proposition 1, we have

2v111

=

2u1oouo10uoo1 - (u1oouo11

+

uo10u101

+

uoo1 u110),

and since the generating elements u1oo, uo10, u001 , u110, u101, uo11 of the algebra Q[x11,x21,x31 ;x12,x22,x32]G are algebraically independent over Q, this decomposi-tion is unique. Hence it follows that v111 cannot be expressed over F2 as a polyno-mial in u100, uo10, uoo1, u200, u110, u101, uo20, uo11, uoo2- Thus, the Noether bound is false in characteristic 2. Therefore, the conditions of Theorem 1 and Theorem 2 cannot be removed. Moreover, we see that u100 , uo10, uoo1, u011, u101, u110 are algebraically dependent over F2.

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Now we show that any polynomial /cr1cr2cr3 of the form

can be expressed as a polynomial in uoo1, uo10, u100, uoo2, uo11, uo20, u101, u110, u200,

and v111 with integer coefficients. At first we show that any polynomial

is a polynomial in u01 , u10 , uo2, u11, u20 over Z. Indeed, we have

XI] +x12

=

UJO, x21 +x22

=

uo1 and

2 2 2 ₂

X11 +x12

=

u10 - u20, x11x21 +x12x22

=

uo1u10-u11, x21 +x22 2 2

=

u01 -2 2

uo2-If er=

cr1

+

cr2

> 2, we can assume without loss of generality that

cr2

2:

2. In that case,

fcr1cr2

=

(x21 +x22)/cr1cr2-I - (xffx~r 1x22 +x~fx~i-lX21)

=

(x21 +x22)/cr1cr2-I -x21x22/cr1,cri-2

=

u01/cr1cr2-I - uo2/cr1cri-2, and we can use the double induction on m and

cr.

Similarly, if cr

=

cr

1

+

0'2

+

0'3

>

3, we can assume without loss of generality that cr3

2:

2. In that case,

, ( ) , ( cr1 cr2 <13-I cr1 cr2 <13-I ) Jcr1cr2cr3

=

X31 +x32 J<J1<J2<J3-I - X11X21X31 X32 +x21X22X32 X31

=

uoo1/cr1,cr2,cr3-I - uoo2/cr1cr2cr3-2,

and the assertion follows with the use of the double induction on m and

cr.

Example 2.

Let m

=

4 and n

=

2. Let us show that the quartic

which is invariant with respect to G

=

S2, cannot be expressed over F2 as a polyno-mial in u,1 ,rin,,4 , 1 ~ r1 + r2 + r3 + r4 ~ 4. Assume, for a contradiction, that

v1111

=

au10oouo10ouoo10uooo1

+

b(

u10oouo10ouoo11 + u1ooouoo10uo101 + u1ooouooo1 uo110)

+

b(

uo1oouooo1 u1010 + uo1oouoo10u1001 + uoo10uooo1u1100)

(10)

with some a,b,c E F2, and observe that

and

UJO()() =X11 +x21, UQ]()() =X21 +x22,

UOQJO = X31 +x32, XQQOI = X41 +x42

UJJOO = X11X22 +x12x12, UJOJO = X]JX32 +x12X31, U]QOI = X11X42 +x12x41,

uo110=x21x32+x22X31, UQJOI =X21X42+x22X41, UQQ]] =X3JX42+x32X4J.

Differentiating the both sides of this equality with respect to x11 and taking into account that and au1000 _ l ax11 - ' auo100 -0 ax11 - ' auoo10 - - = 0 ax11 ' au1100

-a--

=X22, XJJ duo110 _ ₀ dX11 - ' au1010

-a--

=x32, X11 duo101

=

₀ dXJJ ' du1001 - a - - =X42, X11 dUQQII -O ax11 - ' we obtain

x21x31X41 = auo1oouoo10uooo1 + b(uo1oouoo11 + uoo10uo101 + uooo1uo110)

+ b(uo10ouooo1x32 + uo1oouoo1ox42 + uoo10uooo1x22)

+ c(uoo11x22 + uo101x32 + uo110x42).

Setting now x21 = x31 = x41 = x12 = x22 = X32 = x42 = 1 in the last equality, we arrive at the relation 1

=

8a

+

24b

+

6c, which is impossible in F2.

Finally, it follows from Proposition 2 that 6v1111 = 6u10oouo10ouoo1ouooo1

- 2(

u10oouo10ouoo11

+

u10oouoo1ouo101

+

u10oouooo1 uo110)

- 2(uo1oouooo1 u1010 + uo10ouoo1ou1001 + uoo1ouooo1 u1100)

+ (u1 J()()U()()]] + UJQJOUQJO] + UJOQ] uo110),

so the polynomials u1100 , u1010 , u1001, uo110, uo101, uoo11 are algebraically dependent over F2 • On the other hand, they are algebraically independent over

Q.

Example 3. The above examples can be generalized as follows. Let m

>

n ~ 2 be an integer and p be a prime divisor of n. Let Fp be a prime finite field of characteristic p

>

0 and

n

VJL..11

=

L

X!j .•. Xmj

(11)

be the homogeneous polynomial of degree m. Let us show that the polyno-mial v11 ... 11 cannot be expressed as a polynomial in the invariants ur1 , ••• ,um, 1 S

r1

+ .. .

rm

Sn,

over Fp.

Assume, for a contradiction, that

n Sv

V]] ... 11

=

L

asi, ... ,sn

L IT IT

Uriav,···,rmav

s1 +2s2+ ... +ns.=m R(s1 , ... ,s.) V= I crv= I

where a 51 , ... ,sn E Fp and the summation in the second sum is over the set R(s1, ... , sn)

of all non-negative integers YJcrv, ... ,rmav• 1 S VS m, such that Y]crv

+ · · · +

rmav

=

V, 1 S CTv S Sv, 1 S VS n,

ricr1

+ ... +

rua.

=

1, 1

S

i

S

m.

We may assume without loss of generality that if k

s

n is the smallest positive integer such that sk ~ 1, then

{1 if crk

=

1, r1ak

=

.

0 If 2

S

CTk

S

Sk.

Differentiating the both sides of the above equality with respect to x 11 and taking into account that

au

r1, ... ,rm

{o

dXJ I

=

U(I,0, ... ,0) O,ri, ... ,rm

if r1

=

0, if r1

=

1,

where u~~!::::

:~~

is the corresponding elementary symmetric polynomial of vectors (x2j, ... ,xmj), 1 S j

Sn,

we obtain

n .

X21 .. . Xml

=

L

as1, ••. ,s.

L

'¥1{! ...

,s.,

(4)

s1 +2s2+ ... +nsn=m j=I

where

where the set R(s1, ... , sn) is defined above. Denote by Wo,r2 , ... ,rm the value of

uo,,i, ..

.,rm at the point (x11, ... ,Xm1; ... ,;x1n, ... ,Xmn)

=

(1, ... '1; ... ; 1. .. '1). Since

m

>

n ~ 2, each binary sequence (0, r2, ... , rm) encountered in the last equality con-tains l non-zero elements for some 1

s

l

S

n. In that case,

Wo,ri, ...

,rm

=

n(n-1) ... (n-l

+

1),

and setting x11

= ... =

XmJ

= ... =

XJn

= ... =

Xmn

=

1 in (4), we arrive at the

relation

1

=

n

L

hs1, ••• ,s.,

s1 +2s2+ ... +nsn=m

(12)

354 S. A. Stepanov

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