Defining Individual Symmetric Functions
It should be noted that the name "symmetric function" for elements of ΛR is a misnomer: in neither construction the elements are 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. We have therefore reverted to the older terminology of symmetric functions.
(here Λn denotes the ring of symmetric polynomials in n indeterminates), and also in (Stanley, 1999).
To define a symmetric function one must either indicate directly a power series as in the first construction, or give a symmetric polynomial in n indeterminates for every natural number n in a way compatible with the second construction. An expression in an unspecified number of indeterminates may do both, for instance
can be taken as the definition of an elementary symmetric function if the number of indeterminates is infinite, or as the definition of an elementary symmetric polynomial in any finite number of indeterminates. Symmetric polynomials for the same symmetric function should be compatible with the morphisms 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. (An example of a family of symmetric polynomials that fails both conditions is ; the family fails only the second condition.) Any symmetric polynomial in n indeterminates can be used to construct a compatible family of symmetric polynomials, using the morphisms ρi for i < n to decrease the number of indeterminates, and φi for i ≥ n to increase the number of indeterminates (which amounts to adding all monomials in new indeterminates obtained by symmetry from monomials already present).
The following are fundamental examples of symmetric functions.
- The monomial symmetric functions mα, determined by monomial Xα (where α = (α1,α2,…) is a sequence of natural numbers); mα is the sum of all monomials obtained by symmetry from Xα. For a formal definition, consider such sequences to be infinite by appending zeroes (which does not alter the monomial), and define the relation "~" between such sequences that expresses that one is a permutation of the other; then
- This symmetric function corresponds to the monomial symmetric polynomial mα(X1,…,Xn) for any n large enough to have the monomial Xα. The distinct monomial symmetric functions are parametrized by the integer partitions (each mα has a unique representative monomial Xλ with the parts λi in weakly decreasing order). Since any symmetric function containing any of the monomials of some mα must contain all of them with the same coefficient, each symmetric function can be written as an R-linear combination of monomial symmetric functions, and the distinct monomial symmetric functions form a basis of ΛR as R-module.
- The elementary symmetric functions ek, for any natural number k; one has ek = mα where . As a power series, this is the sum of all distinct products of k distinct indeterminates. This symmetric function corresponds to the elementary symmetric polynomial ek(X1,…,Xn) for any n ≥ k.
- The power sum symmetric functions pk, for any positive integer k; one has pk = m(k), the monomial symmetric function for the monomial X1k. This symmetric function corresponds to the power sum symmetric polynomial pk(X1,…,Xn) = X1k+…+Xnk for any n ≥ 1.
- The complete homogeneous symmetric functions hk, for any natural number k; hk is the sum of all monomial symmetric functions mα where α is a partition of k. As a power series, this is the sum of all monomials of degree k, which is what motivates its name. This symmetric function corresponds to the complete homogeneous symmetric polynomial hk(X1,…,Xn) for any n ≥ k.
- The Schur functions sλ for any partition λ, which corresponds to the Schur polynomial sλ(X1,…,Xn) for any n large enough to have the monomial Xλ.
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 "discriminant" is another example of an expression giving a symmetric polynomial for all n, but not defining any symmetric function. The expressions defining Schur polynomials as a quotient of alternating polynomials are somewhat similar to that for the discriminant, but the polynomials sλ(X1,…,Xn) turn out to be compatible for varying n, and therefore do define a symmetric function.
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