Re: algebraic topology question

> Let s be the cyclic permutation 1->2->3-> ... ->n->1.  Then s acts on
> the polynomial ring R = C[x_1,...,x_n] by shifting the indices.  What
> you are looking for is a generating set f_1,...,f_N for the subring S
> of R consisting of the s-invariant polynomials.  Once you have that,
> the map p -> (f_1(p),...,f_N(p)) will have the property you want.
>
> Now the action on R is linear, and it is straightforward to analyse
> the action on the linear part R1 of R.  The characteristic polynomial
> of s on R1 is t^n - 1, so there are n distinct complex eigenvalues of
> the form a^i where a is a primitive n-th root of unity.
> Diagonalizing s on R1 amounts to finding a different basis y_1,...y_n
> consisting of linear combinations of the x_i, and such that s(y_i) =
> a^i*y_i for i=1,...,n.  For example, we may take y_n=sum_i x_i.
>
> Now R = C[y_1,...,y_n], and in this representation the invariant ring
> is clearly generated by monomials in the y_i's, in fact those
> monomials y_1^b_1* ... * y_n^b_n such that sum_i i*b_i is divisible
> by n.

Thanks for your response. I will have to forward this to someone in my math
department to explain me in simple terms.

What kind of changes do I need to make it also invariant on mirror
permutation, for example:

(a,b,c,d) and (d,c,b,a)

and all the cyclic permutations of both?

Thanks once again. I would appreciate if you could put it into the terms of
"dummy's guide to algebraic topology" as I am very new to the field of
algebra.