An algebraic approach to harmonic polynomials on S3

Abstract

In this thesis we are going to study harmonic polynomials on spheres, with the particular attention to the 3-sphere S³. As a Lie group, the algebraic structure of S³ ≃ SU(2) enables us to study harmonic polynomials from an algebraic point of view. A purely algebraic approach has several advantages, such as the possibilities to extend to more general matrix groups and work over general fields such as finite fields, and importantly to rational numbers. This algebraic approach allows us to construct an explicit basis for the space of harmonic polynomials on S³ and describe them from a geometrical point of view. Finally, we give an explicit algebraic formula for zonal harmonic polynomials on S³ which can be identified as characters of Lie group S³ ≃ SU(2).