Symmetric Polynomials
Introduction
Symmetric polynomials are a fundamental concept in the field of algebraic combinatorics and invariant theory. They are polynomials in several variables that remain unchanged under any permutation of the variables. This property of invariance under permutation makes them a useful tool in various branches of mathematics, including algebraic geometry, representation theory, and number theory.
Definition
A polynomial in variables x1, x2, ..., xn is called symmetric if it is invariant under all permutations of the variables. In other words, if we interchange any two variables in the polynomial, the value of the polynomial remains the same. Formally, a polynomial P in n variables is symmetric if for any permutation σ of the symmetric group S_n, we have P(x1, x2, ..., xn) = P(xσ(1), xσ(2), ..., xσ(n)).
Fundamental Theorem of Symmetric Polynomials
The Fundamental Theorem of Symmetric Polynomials states that every symmetric polynomial can be expressed as a polynomial in the elementary symmetric polynomials. This theorem is a cornerstone in the theory of symmetric polynomials and has profound implications in various mathematical disciplines.
Elementary Symmetric Polynomials
Elementary symmetric polynomials are a specific type of symmetric polynomial that play a crucial role in the theory. For n variables x1, x2, ..., xn, the k-th elementary symmetric polynomial is the sum of the products of all distinct k-tuples of the variables.
Power Sums and Newton's Identities
Power sums are another important class of symmetric polynomials, defined as the sum of the variables raised to a certain power. Newton's identities provide a relationship between power sums and elementary symmetric polynomials, and are a key tool in the study of symmetric polynomials.
Symmetric Functions
Symmetric polynomials can be generalized to symmetric functions, which are functions in infinitely many variables that are invariant under permutations of the variables. Symmetric functions have a rich structure and are a central object of study in algebraic combinatorics.
Applications
Symmetric polynomials and functions have wide-ranging applications in mathematics. They appear in the solution of the Galois problem of expressing roots of polynomials in terms of coefficients, in the representation theory of the symmetric group and general linear group, and in the geometry of symmetric spaces, among others.