Ring of Symmetric Functions - The Ring of Symmetric Functions - Defining Individual Symmetric Functions

Defining Individual Symmetric Functions

It should be noted that the name "symmetric function" for elements of Λ_R 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 Λ_n) are no longer polynomials: they are formal infinite sums of monomials. We have therefore reverted to the older terminology of symmetric functions.

(here Λ_n 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 ρ_n (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 ρ_i for i < n to decrease the number of indeterminates, and φ_i 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 m_α, determined by monomial Xα (where α = (α₁,α₂,…) is a sequence of natural numbers); m_α 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 m_α(X₁,…,X_n) for any n large enough to have the monomial Xα. The distinct monomial symmetric functions are parametrized by the integer partitions (each m_α has a unique representative monomial Xλ with the parts λ_i in weakly decreasing order). Since any symmetric function containing any of the monomials of some m_α 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 Λ_R as R-module.

• The elementary symmetric functions e_k, for any natural number k; one has e_k = m_α 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 e_k(X₁,…,X_n) for any n ≥ k.
• The power sum symmetric functions p_k, for any positive integer k; one has p_k = m_(k), the monomial symmetric function for the monomial X₁k. This symmetric function corresponds to the power sum symmetric polynomial p_k(X₁,…,X_n) = X₁k+…+X_nk for any n ≥ 1.
• The complete homogeneous symmetric functions h_k, for any natural number k; h_k is the sum of all monomial symmetric functions m_α 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 h_k(X₁,…,X_n) for any n ≥ k.
• The Schur functions s_λ for any partition λ, which corresponds to the Schur polynomial s_λ(X₁,…,X_n) 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 ρ_n. 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 s_λ(X₁,…,X_n) turn out to be compatible for varying n, and therefore do define a symmetric function.