Harmonic and Symmetric Polynomials

Abstract

We revert to a more algebraic topic. Courses in linear algebra or more general algebra at the undergraduate level provide foundations, but most often do not have the time to point to more sophisticated ways of putting the material together. The present topic illustrates concepts from linear algebra in a context which up to now has been reserved for much more advanced students. There is no difficulty in any given item, but just as the space of positive definite matrices provided a fertile ground to expand material from advanced calculus, the ring of polynomials and two important subsets, the harmonic polynomials and the symmetric polynomials, provide a fertile ground to illustrate concepts from linear algebra, not to speak of the calculus involved with partial derivatives. When dealing with polynomials, however, the analysis side of things is mostly illusory.