symmetric polynomials in terms of the elementary symmetric polynomials, and the latter gives a set of equations that allows doing the inverse. This implicitly shows that any symmetric polynomial can be expressed in terms of the hk(X1, …, Xn) with 1 ≤ k ≤ n: one first expresses the symmetric polynomial in terms of the elementary symmetric polynomials, and then expresses those in terms of the mentioned complete homogeneous ones. Schur polynomials Another class of symmetric polynomials is that of the Schur polynomials, which are of fundamental importance in the applications of symmetric polynomials to representation theory. They are however not as easy to describe as the other kinds of special symmetric polynomials; see the main article for details. Symmetric polynomials in algebra Symmetric polynomials are important to linear algebra, representation theory, and Galois theory. They are also important in combinatorics, where they are mostly studied through the ring of symmetric functions, which avoids having to carry around a fixed number of variables all the time. Alternating polynomials Analogous to symmetric polynomials are alternating polynomials: polynomials that, rather than being invariant under permutation of the entries, change according to the sign of the permutation. These are all products of the Vandermonde polynomial and a symmetric polynomial, and form a quadratic extension of the ring of symmetric polynomials: the Vandermonde polynomial is a square root of the discriminant. See also Symmetric function Newton's identities Stanley symmetric function Muirhead's inequality References Macdonald, I.G. (1979), Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. Oxford: Clarendon Press. I.G. Macdonald (1995), Symmetric Functions and Hall Polynomials, second ed. Oxford: Clarendon Press. (paperback, 1998). Richard P. Stanley (1999), Enumerative Combinatorics, Vol. 2. Cambridge: Cambridge University Press. Polynomials