Ring of Symmetric Functions - Symmetric Polynomials

Symmetric Polynomials

The study of symmetric functions is based on that of symmetric polynomials. In a polynomial ring in some finite set of indeterminates, there is an action by ring automorphisms of the symmetric group on (the indices of) the indeterminates (simultaneously substituting each of them for another according to the permutation used). The invariants for this action form the subring of symmetric polynomials. If the indeterminates are X1,…,Xn, then examples of such symmetric polynomials are

and

A somewhat more complicated example is X13X2X3 +X1X23X3 +X1X2X33 +X13X2X4 +X1X23X4 +X1X2X43 +… where the summation goes on to include all products of the third power of some variable and two other variables. There are many specific kinds of symmetric polynomials, such as elementary symmetric polynomials, power sum symmetric polynomials, monomial symmetric polynomials, complete homogeneous symmetric polynomials, and Schur polynomials.

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