Gelfand Pairs, Representation Theory of the Symmetric Group, and the Theory of Spherical Functions

This thesis gives an introduction to the study of Gelfand pairs and their applications. We begin with a brief introduction to the notion of a Gelfand pair and then move to some of the foundational results concerning Gelfand pairs. Next, we explore specific examples of Gelfand pairs, developing tools of independent interest as we progress. We find that consideration of a specific example of a Gelfand pair and of the tools used in our study naturally leads us to a discussion of the representation theory of the symmetric group. We then conclude our study by developing the theory of spherical functions on groups, which gives us a glimpse of the relevance of Gelfand pairs to areas of mathematics outside of representation theory.