The theory of symmetric functions, which generalizes some ideas that came up in the previous discussion of Polya theory, can be motivated by thinking about polynomial functions of the roots of a monic polynomial . Problems on high school competitions often ask for the sum of the squares or the cubes of the roots of a given polynomial, and those not familiar with symmetric functions will often be surprised that these quantities end up being integers even though the roots themselves aren’t integers. The key is that the sums being asked for are always symmetric in the roots, i.e. invariant under an arbitrary permutation. The coefficients of a monic polynomial are the elementary symmetric polynomials of its roots, which we are given are integers. It follows that any symmetric polynomial that can be written using integer coefficients in terms of the elementary symmetric polynomials must in fact be an integer, and as we will see, every symmetric polynomial with integer coefficients has this property.
These ideas lead naturally to the use of symmetric polynomials in Galois theory, but combinatorialists are interested in symmetric functions for a very different reason involving the representation theory of . This post is a brief overview of the basic results of symmetric function theory relevant to combinatorialists. I’m mostly following Sagan, who recommends Macdonald as a more comprehensive reference.
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