Subject: Computing rings of invariants
From: rusin
Date: Jul 24 1998 14:00 (timestamp) [revised  2001/11/23 ]
To: rusin (once again talking to himself)

What is the ring of invariants of the diagonal action of  Sym(3)
polynomial ring of  SIX variables? Certainly includes the invariants
under  Sym(3) x Sym(3), which is  k[ s1, s2, s3; s1', s2', s3']. But
it ought to be a degree-6 extension of this. We propose a candidate
extra generator, and we're right:

//Magma input
//
//K:=GF(31991);  
K:=RationalField();
R<a,b,c,r,s,t,S,T,U,V,X,Y,Z>:=PolynomialRing(K,13,"lex");
 I:=ideal<R|
 a*t+b*s+c*r-V,
 a*b*c-U,
 a*b+b*c+c*a-T,
 a+b+c-S,
 r*s+s*t+t*r-Y,
 r+s+t-X,
 r*s*t-Z
>;
time L:=EliminationIdeal(I,6);
Write("S3_2",(L));
//
quit;


>Output at console:
>
>Magma V2.3-1      Fri Jul 24 1998 13:27:05 on shuksan  [Seed = 462880732]
>Type ? for help.  Type <Ctrl>-D to quit.
>Time: 457.800
>
>Total time: 459.066 seconds

Contents of file  S3_2:

Ideal of Polynomial ring of rank 13 over Rational Field
Lexicographical Order
Variables: a, b, c, r, s, t, S, T, U, V, X, Y, Z
Basis:
[
S^6*Z^2 - 2*S^5*V*Y*Z + S^4*T*X*Y*Z - 9*S^4*T*Z^2 + 2*S^4*V^2*X*Z + 
    S^4*V^2*Y^2 - 2*S^3*T*V*X^2*Z - S^3*T*V*X*Y^2 + 15*S^3*T*V*Y*Z - 
    2*S^3*U*X^3*Z + S^3*U*X^2*Y^2 + 9*S^3*U*X*Y*Z - 4*S^3*U*Y^3 - 
    2*S^3*V^3*X*Y - 2*S^3*V^3*Z + S^2*T^2*X^3*Z - 9*S^2*T^2*X*Y*Z + 
    S^2*T^2*Y^3 + 27*S^2*T^2*Z^2 + 3*S^2*T*V^2*X^2*Y - 9*S^2*T*V^2*X*Z - 
    6*S^2*T*V^2*Y^2 - 2*S^2*U*V*X^3*Y + 9*S^2*U*V*X*Y^2 - 27*S^2*U*V*Y*Z + 
    S^2*V^4*X^2 + 2*S^2*V^4*Y - S*T^2*V*X^3*Y + 9*S*T^2*V*X^2*Z + 
    3*S*T^2*V*X*Y^2 - 27*S*T^2*V*Y*Z + S*T*U*X^4*Y + 9*S*T*U*X^3*Z - 
    9*S*T*U*X^2*Y^2 - 27*S*T*U*X*Y*Z + 18*S*T*U*Y^3 - 2*S*T*V^3*X^3 + 
    5*S*T*V^3*X*Y + 9*S*T*V^3*Z + 2*S*U*V^2*X^4 - 9*S*U*V^2*X^2*Y + 
    27*S*U*V^2*X*Z - 2*S*V^5*X - 4*T^3*X^3*Z + T^3*X^2*Y^2 + 18*T^3*X*Y*Z - 
    4*T^3*Y^3 - 27*T^3*Z^2 + T^2*V^2*X^4 - 6*T^2*V^2*X^2*Y + 9*T^2*V^2*Y^2 -
    2*T*U*V*X^5 + 15*T*U*V*X^3*Y - 27*T*U*V*X^2*Z - 27*T*U*V*X*Y^2 + 
    81*T*U*V*Y*Z + 2*T*V^4*X^2 - 6*T*V^4*Y + U^2*X^6 - 9*U^2*X^4*Y + 
    27*U^2*X^2*Y^2 - 27*U^2*Y^3 - 2*U*V^3*X^3 + 9*U*V^3*X*Y - 27*U*V^3*Z + 
    V^6
]

Maple re-presents this for us:

eq:=

(27*S^2*Z^2*T^2-4*S^3*Y^3*U-9*S^4*Z^2*T+Y^2*T^3*X^2+S^2*Y^3*T^2+S^6*Z^2+X^6*U^2-4*Y^3*T^3-27*Z^2*T^3+S^4*Z*Y*T*X+S^3*Y^2*X^2*U-27*Y^3*U^2+S^2*Z*T^2*X^3+S*Y*T*X^4*U-2*S^3*Z*X^3*U+9*S^3*Z*Y*X*U-9*S^2*Z*Y*T^2*X+9*S*Z*T*X^3*U-9*S*Y^2*T*X^2*U-27*S*Z*Y*T*X*U+18*S*Y^3*T*U-4*Z*T^3*X^3-9*Y*X^4*U^2+27*Y^2*X^2*U^2+18*Z*Y*T^3*X)

+(-(S^2*Y+T*X^2-3*Y*T)*(2*S^3*Z+S*Y*T*X+2*X^3*U-9*S*Z*T-9*Y*X*U+27*Z*U))*V

+(2*S^4*Z*X+S^4*Y^2+3*S^2*Y*T*X^2+2*S*X^4*U+T^2*X^4-9*S^2*Z*T*X-6*S^2*Y^2*T-9*S*Y*X^2*U-6*Y*T^2*X^2+27*S*Z*X*U+9*Y^2*T^2)*V^2

+(-2*S^3*Y*X-2*S*T*X^3-2*S^3*Z+5*S*Y*T*X-2*X^3*U+9*S*Z*T+9*Y*X*U-27*Z*U)*V^3

+(S^2*X^2+2*S^2*Y+2*T*X^2-6*Y*T)*V^4

+(-2*S*X)*V^5

+V^6


So  V  does indeed generate an extension field of degree 6, which must be
the invariant subfield. 

So it appears that a description of the fixed-field in Galois theory:
   R(a,b,c;r,s,t)^Sym(3) = R(S,T,U;X,Y,Z)[V]/(V^6 + l.o.t.)  .

The invariant sub_ring_ is a little different. It's contained in here,
and so every element is a sum of 6  powers of  V, each times a rational
function in STUXYZ. Many such expressions are not even in R[a,b,c,r,s,t]
because of the denominators, but some are. Indeed, the invariant subring
has a presentation as a free module over  R[S,T,U;X,Y,Z]  on  6
generators, but the properly contains the free module on [1, V, V^2, ..., V^5].

One module basis for the fixed ring consists of  1, V, and four more
symmetrized monomials, e.g. V2=sum(x_i*y_i^2) is an (invariant) symmetrization
of  x1 y1^2. Likewise we may symmetrize  x1^2 y1,  x1^2 y1^2,  and
x1^2 x2 y1^2 y2;  these 6 generate the invariant subring as a module
over the ring of elementary symmetric functions.

It is not too hard to carry out a similar procedure for the invariants
under the diagonal action of  Sym(4), although for larger groups
than this, the processing time begins to become prohibitive.