Among other things, this ring plays an important role in the representation theory of the symmetric group.
The ring of symmetric functions can be given a coproduct and a bilinear form making it into a positive selfadjoint graded Hopf algebra that is both commutative and cocommutative.
More formally, there is an action by ring automorphisms of the symmetric group Sn on the polynomial ring in n indeterminates, where a permutation acts on a polynomial by simultaneously substituting each of the indeterminates for another according to the permutation used.
For instance the Newton's identity for the third power sum polynomial p3 leads to where the
denote elementary symmetric polynomials; this formula is valid for all natural numbers n, and the only notable dependency on it is that ek(X1,...,Xn) = 0 whenever n < k. One would like to write this as an identity that does not depend on n at all, and this can be done in the ring of symmetric functions.
The easiest (though somewhat heavy) construction starts with the ring of formal power series
over R in infinitely (countably) many indeterminates; the elements of this power series ring are formal infinite sums of terms, each of which consists of a coefficient from R multiplied by a monomial, where each monomial is a product of finitely many finite powers of indeterminates.
One defines ΛR as its subring consisting of those power series S that satisfy Note that because of the second condition, power series are used here only to allow infinitely many terms of a fixed degree, rather than to sum terms of all possible degrees.
For every k ≥ 0, the element ek ∈ ΛR is defined as the formal sum of all products of k distinct indeterminates, which is clearly homogeneous of degree k. Another construction of ΛR takes somewhat longer to describe, but better indicates the relationship with the rings R[X1,...,Xn]Sn of symmetric polynomials in n indeterminates.
This means that the restriction of ρn to elements of degree at most n is a bijective linear map, and ρn(ek(X1,...,Xn+1)) = ek(X1,...,Xn) for all k ≤ n. The inverse of this restriction can be extended uniquely to a ring homomorphism φn from R[X1,...,Xn]Sn to R[X1,...,Xn+1]Sn+1, as follows for instance from the fundamental theorem of symmetric polynomials.
That construction only uses the surjective morphisms ρn without mentioning the injective morphisms φn: it constructs the homogeneous components of ΛR separately, and equips their direct sum with a ring structure using the ρn.
It is also observed that the result can be described as an inverse limit in the category of graded rings.
That description however somewhat obscures an important property typical for a direct limit of injective morphisms, namely that every individual element (symmetric function) is already faithfully represented in some object used in the limit construction, here a ring R[X1,...,Xd]Sd.
The name "symmetric function" for elements of ΛR is a misnomer: in neither construction are the elements functions, and in fact, unlike symmetric polynomials, no function of independent variables can be associated to such elements (for instance e1 would be the sum of all infinitely many variables, which is not defined unless restrictions are imposed on the variables).
However the name is traditional and well established; it can be found both in (Macdonald, 1979), which says (footnote on p. 12) The elements of Λ (unlike those of Λn) are no longer polynomials: they are formal infinite sums of monomials.
(here Λn denotes the ring of symmetric polynomials in n indeterminates), and also in (Stanley, 1999).
Symmetric polynomials for the same symmetric function should be compatible with the homomorphisms ρn (decreasing the number of indeterminates is obtained by setting some of them to zero, so that the coefficients of any monomial in the remaining indeterminates is unchanged), and their degree should remain bounded.
There is no power sum symmetric function p0: although it is possible (and in some contexts natural) to define
as a symmetric polynomial in n variables, these values are not compatible with the morphisms ρn.
The second definition of the ring of symmetric functions implies the following fundamental principle: This is because one can always reduce the number of variables by substituting zero for some variables, and one can increase the number of variables by applying the homomorphisms φn; the definition of those homomorphisms assures that φn(P(X1,...,Xn)) = P(X1,...,Xn+1) (and similarly for Q) whenever n ≥ d. See a proof of Newton's identities for an effective application of this principle.
The ring of symmetric functions is a convenient tool for writing identities between symmetric polynomials that are independent of the number of indeterminates: in ΛR there is no such number, yet by the above principle any identity in ΛR automatically gives identities the rings of symmetric polynomials over R in any number of indeterminates.
Some fundamental identities are which shows a symmetry between elementary and complete homogeneous symmetric functions; these relations are explained under complete homogeneous symmetric polynomial.
the Newton identities, which also have a variant for complete homogeneous symmetric functions: Important properties of ΛR include the following.
It immediately implies some other properties: This final point applies in particular to the family (hi)i>0 of complete homogeneous symmetric functions.
of rational numbers, it applies also to the family (pi)i>0 of power sum symmetric functions.
The fact that the complete homogeneous symmetric functions form a set of free polynomial generators of ΛR already shows the existence of an automorphism ω sending the elementary symmetric functions to the complete homogeneous ones, as mentioned in property 3.
The fact that ω is an involution of ΛR follows from the symmetry between elementary and complete homogeneous symmetric functions expressed by the first set of relations given above.
Contrary to the relations mentioned earlier, which are internal to ΛR, these expressions involve operations taking place in R[[X1,X2,...;t]] but outside its subring ΛR[[t]], so they are meaningful only if symmetric functions are viewed as formal power series in indeterminates Xi.
explains the symmetry between elementary and complete homogeneous symmetric functions.
These expressions are sometimes written as which amounts to the same, but requires that R contain the rational numbers, so that the logarithm of power series with constant term 1 is defined (by