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You can apply a permutation to a tuple, in particular to the tuple of generators of the polynomial ring. You can then apply the function to the permuted variables.

Here is an example, from which you could build a function permute_variables(sigma,f) (left as an exercise).

sage: version() # just for reference
'Sage Version 6.3.beta5, Release Date: 2014-07-01'
sage: n = 5
sage: R = PolynomialRing(QQ,n,'x')
sage: R
Multivariate Polynomial Ring in x0, x1, x2, x3, x4 over Rational Field
sage: Sn = SymmetricGroup(n)
sage: f = R.random_element()
sage: f
1/2*x2*x3 + x0 + x3
sage: sigma = Sn.random_element()
sage: sigma
(1,4,5)
sage: R.gens()
(x0, x1, x2, x3, x4)
sage: sigma(R.gens())
(x3, x1, x2, x4, x0)
sage: f(*sigma(R.gens()))
1/2*x2*x4 + x3 + x4


You can apply a permutation to a tuple, in particular to the tuple of generators of the polynomial ring. You can then apply the function to the permuted variables.

Here is an example, from which you could build a function permute_variables(sigma,f) (left as an exercise).

sage: version() # just for reference
'Sage Version 6.3.beta5, Release Date: 2014-07-01'
sage: n = 5
sage: R = PolynomialRing(QQ,n,'x')
sage: R
Multivariate Polynomial Ring in x0, x1, x2, x3, x4 over Rational Field
sage: Sn = SymmetricGroup(n)
sage: f = R.random_element()
sage: f
1/2*x2*x3 + x0 + x3
sage: sigma = Sn.random_element()
sage: sigma
(1,4,5)
sage: R.gens()
(x0, x1, x2, x3, x4)
sage: sigma(R.gens())
(x3, x1, x2, x4, x0)
sage: f(*sigma(R.gens()))
1/2*x2*x4 + x3 + x4


Another example where you input the polynomial and the permutation yourself:

sage: n = 5
sage: R = PolynomialRing(QQ,n,'x')
sage: R.inject_variables()
Defining x0, x1, x2, x3, x4
sage: Sn = SymmetricGroup(n)
sage: f = x1 + x2
sage: sigma = Sn((1,2,3))
sage: sigma
(1,2,3)
sage: sigma(2)
3
sage: f(*sigma(R.gens()))
x0 + x2


Be careful: permutations are indexed from 1 to 5, but the generators of $R$ are indexed from 0 to 4. In particular the permutation $(1,2,3)$ of $S_5$, when acting on $(x_0, x_1, x_2, x_3, x_4)$, acts as the cyclic permutation $(x_0, x_1, x_2)$.

You can apply a permutation to a tuple, in particular to the tuple of generators of the polynomial ring. You can then apply the function to the permuted variables.

Here is an example, from which you could build a function permute_variables(sigma,f) (left as an exercise).

sage: version() # just for reference
'Sage Version 6.3.beta5, Release Date: 2014-07-01'
sage: n = 5
sage: R = PolynomialRing(QQ,n,'x')
PolynomialRing(ZZ,n,'x')
sage: R
Multivariate Polynomial Ring in x0, x1, x2, x3, x4 over Rational Field
Integer Ring
sage: Sn = SymmetricGroup(n)
SymmetricGroup(range(n))
sage: f = R.random_element()
sage: f
1/2*x2*x3 + x0 + x3
x1*x2 - x3^2 + x2*x4 + 3*x1 - 1
sage: sigma = Sn.random_element()
sage: sigma
(1,4,5)
(0,2,4)
sage: R.gens()
(x0, x1, x2, x3, x4)
sage: sigma(R.gens())
(x3, (x2, x1, x2, x4, x3, x0)
sage: f(*sigma(R.gens()))
1/2*x2*x4 + x3 + x4
-x3^2 + x0*x4 + x1*x4 + 3*x1 - 1


Another Rather than a random permutation, you probably prefer to input it yourself. For this, you need to define x0 etc. and make them point to the variables of polynomials in R. This is done with R.inject(variables). And to input permutations as products of cycles, use something like Sn('(0,1)(2,3)'). An example where you input the polynomial and the permutation yourself:

sage: n = 5
sage: R = PolynomialRing(QQ,n,'x')
PolynomialRing(ZZ,n,'x')
sage: R.inject_variables()
Defining x0, x1, x2, x3, x4
sage: Sn = SymmetricGroup(n)
SymmetricGroup(range(n))
sage: f = x1 + x2
sage: sigma = Sn((1,2,3))
sage: sigma
(1,2,3)
sage: sigma(2)
3
f(*sigma(R.gens()))
x2 + x3
sage: sigma = Sn('(0,1)(2,3)')
sage: f(*sigma(R.gens()))
x0 + x2
x3


Be careful: permutations If you define the permutation group as SymmetricGroup(5), they are indexed from 1 to 5, but n, so it's tricky to follow what goes on when you let them act on the generators of $R$ $R$, which are indexed from 0 to 4. n-1 (you have to offset your indices by one, then apply the permutation, then undo the offset). In particular the permutation $(1,2,3)$ of $S_5$, when acting would then act on $(x_0, x_1, x_2, x_3, x_4)$, acts as the cyclic permutation $(x_0, x_1, x_2)$.