Subject: Ravenel's question
From: Tom Goodwillie <tomg@math.brown.edu>
Date: Thu, 7 Jul 2005 13:09:43 -0400
To: Don Davis <dmd1@lehigh.edu>

I have just one thing to add: This is mostly about the limiting 
case as n goes to infinity, which is clearly not what Doug was 
looking at, but I thought I would mention it because it's a clean 
simple statement.

Of course, geometrically Z[x_1,...,x_m] can be thought of as the 
ring of functions on affine m-space A^m, and Doug's ring R, the 
ring of Sigma_n-invariants in the nth tensor power of that ring, 
is then the ring of functions on the nth symmetric power SP^n(A^m).

Any function f on X=A^m determines functions on SP^n(X), for example 
the elementary symmetric polynomials e_j(f). These vanish for j>n. 
It is a well-known fact that in the case when m=1 the ring of 
symmetric functions has the elements {e_j(x_1), j=1 to n} as a polynomial 
basis.

The case of m>1 is a lot more complicated, but here is one nice fact: 
in the (inverse) limit as n-->infinity, the ring is again a polynomial 
ring with a basis that is easy to specify.

Doug was in effect trying to generate the ring of symmetric functions 
in the case of general m by considering e_j(f) where f is a generic 
linear function c_1x_1+...+c_mx_m. His elements are the coefficients 
of e_j(f) with respect to the various monomials in the "dummy" variables 
c_i. 
If those elements did generate the ring, then in particular the elements 
e_j(f) 
where f is a linear polynomial in (x_1,...,x_m) (with integer 
coefficients) 
would generate, which I guess they don't.

However, the elements e_j(f) where f(x_1,...,x_m) is an arbitrary 
polynomial 
do generate. And using formulae like 
e_2(f+g)=e_2(f)+e_1(f)e_1(g)-e_1(fg)+e_2(g) 
one can reduce to the case when f(x_1,...,x_m) is a monomial. And using 
formulae 
like e_2(f2)=e_1(f)2-2e_2(f) one can reduce to the case when the monomial 
is not 
a power of another monomial. These are independent generators; that is, if 
j 
ranges over all positive integers and d=(d_1,...,d_m) ranges over all 
ordered 
m-tuples of nonnegative integers having no common prime factor, and x^d is 
shorthand for x_1^{d_1}...x_m^{d_m}, then the elements e_j(x^d) generate 
the 
ring of invariant functions, and in the limit as n becomes infinite they 
satisfy no relations.

 - Tom Goodwillie