Schur--Weyl duality is a fundamental paradigm in representation theory relating representations of the general linear group to those of the symmetric group. We will discuss the classical case of Schur--Weyl duality and an analog for the unipotent upper triangular matrix group over a finite field. Then we will define supercharacter theory and how a particular supercharacter theory for the unipotent upper triangulars, within the context of this Schur--Weyl duality analog, lead to a class of modules called beach modules and a collection of projection-like maps in the centralizer algebra.
A Day at the Beach