This is rather simpler: you should be using polynomials and not the
symbolic ring:
sage: R.<x1,x2,x3,x4> = QQ[]
sage: f = x1*x2+x3*x4
sage: list({f(y) for y in permutations(R.gens())})
[x1*x2 + x3*x4, x2*x3 + x1*x4, x1*x3 + x2*x4]
Someone has been implementing a module for classical invariant theory
in Sage, but I am not sure what its current status is.
On 12 February 2013 08:15, Jori Mantysalo <[email protected]> wrote:
> Let's examine f(x1,x2,x3,x4)=x1*x2+x3*x4.
>
> It is D4-invariant. I was able to calculate that it has 3 possible outcomes
> out of 4!=24 arrangements of arguments:
>
> var('x1, x2, x3, x4')
> lista = [x1, x2, x3, x4]
> def f(y):
> return lista[y[0]]*lista[y[1]]+lista[y[2]]*lista[y[3]]
> list(set(map(f, arrangements([0,1,2,3], 4))))
>
> --> [x1*x4 + x2*x3, x1*x3 + x2*x4, x1*x2 + x3*x4]
>
> However, this only works because Sage happens to arrange monomials. How to
> compare polynomials? This does not work:
> (x1*x2+x3*x4)==(x2*x1+x3*x4)
> So how to tell Sage that I am comparing polynomials, and not for example
> matrices?
>
> And a harder question: how to show that f really is D4-invariant, not just
> G-invariant where |G|=4!/3? Or, in general, calculate "invariantness" of
> multivariate polynomial?
>
> --
> Jori Mäntysalo
>
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