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One context in which symmetric polynomial functions occur is in the study of monic univariate polynomials of degree n having n roots in a given field.These n roots determine the polynomial, and when they are considered as independent variables, the coefficients of the polynomial are symmetric polynomial functions of the roots.
The following lists the power sum symmetric polynomials of positive degrees up to n for the first three positive values of . In every case, = is one of the polynomials. The list goes up to degree n because the power sum symmetric polynomials of degrees 1 to n are basic in the sense of the theorem stated below.
The Newton identities also permit expressing the elementary symmetric polynomials in terms of the power sum symmetric polynomials, showing that any symmetric polynomial can also be expressed in the power sums. In fact the first n power sums also form an algebraic basis for the space of symmetric polynomials.
Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric -tensors on a vector space is isomorphic to the space of homogeneous polynomials of degree on . Symmetric functions should not be confused with even and odd functions, which have a different sort of symmetry.
Therefore, the symmetric algebra over V can be viewed as a "coordinate free" polynomial ring over V. The symmetric algebra S(V) can be built as the quotient of the tensor algebra T(V) by the two-sided ideal generated by the elements of the form x ⊗ y − y ⊗ x. All these definitions and properties extend naturally to the case where V is a ...
For any commutative ring A, denote the ring of symmetric polynomials in the variables X 1, ..., X n with coefficients in A by A[X 1, ..., X n] S n. This is a polynomial ring in the n elementary symmetric polynomials e k (X 1, ..., X n) for k = 1, ..., n. This means that every symmetric polynomial P(X 1, ..., X n) ∈ A[X 1, ..., X n] S n has a ...
The complete homogeneous symmetric polynomials are characterized by the following identity of formal power series in t: = (, …,) = = = = = (this is called the generating function, or generating series, for the complete homogeneous symmetric polynomials).
(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 ...