In his 1901 thesis, Issai Schur discovered a connection between the representation theory of the symmetric group and general linear group.  One way to understand this connection is through a finite dimensional algebra called the Schur algebra.  I will outline this picture and then describe a new algebra, defined in joint work with Tom Braden, which enhances the Schur algebra and provides a new window into the representation theory of symmetric groups.  Finally, I will explain how we came to discover this algebra by studying the geometry of Hilbert schemes of points in the plane and how this fits into my larger program to uncover representation theory in the geometry of symplectic singularities and their resolutions.