It was clear from the earlier articles that n (number of variables ) plays a minimal role in the combinatorics of the symmetric polynomials. Hence, removing the parameter n turns out to be quite convenient; the process gives us the formal ring of symmetric functions.
Concrete Definition of Ring Λ
Consider the set of all monomials , where is a vector of non-negative integers such that only finitely many terms are non-zero. Thus and are monomials but is not. Now for , consider the additive group of all sums
Though each has only finitely many non-zero terms, the collection of all ‘s may have infinitely many non-zero terms. E.g. is a perfectly valid element of
Now let be the subgroup of all such that for any permutation of Note that each is a finite free abelian group with basis elements given by:
Let: ; this gives us a homogeneous graded ring, called the formal ring of symmetric functions. E.g. in this ring, we have:
For each n, the map taking gives a graded ring homomorphism
Abstract Definition of Λ
Here is an alternate definition: let be the map (graded ring homomorphism) taking Thus we get a sequence:
Now is defined to be the inverse limit of this sequence, which gives us a graded ring and a set of graded maps one for each n.
Note that if the maps are all isomorphisms; hence is also an isomorphism.
Symmetric Polynomials in Λ
We define the following in , also called the elementary symmetric polynomial, complete symmetric polynomial and the power sum symmetric polynomial.
For a partition , we also define etc, as before. Finally, the monomial symmetric polynomial is the sum of all over all and such that when sorted in decreasing order, becomes Note that
When projected via , the above polynomials becomes their counterpart in For example
Rewriting Earlier Results
We thus have the following:
where Indeed, the above relations hold in for any n and d; when the isomorphism shows that the same relations hold in Similarly, the identities from earlier can be copied over verbatim by projecting to for
Duality in Λ
Recall that we have an involution for each n; this map does not commute with the projection For an easy example, say n=2, the map takes However, upon projecting to , the element while
This can be overcome by taking Since is graded, we thus have the following:
The vertical lines are isomorphisms and the diagram commutes.
Definition. For each , we define the involution to be that induced from for any
Note that in Λ, we have:
for all since these hold in for any