Geometric Methods in Representation Theory

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

This research is at the interface of geometry and representation theory. Geometry is the mathematical study of shapes and properties of space, and representation theory uses algebraic techniques to study symmetries. Several research projects in geometry and representation theory that enhance the unity of mathematics by connecting algebraic and geometric ideas will be completed. Among them is a project to use a novel representation-theoretic approach to solve a long-standing conjecture about counting the number of Latin squares. In more detail, the proposed projects are summarized as follows: The first project is to answer of a well-known question of Kostant to describe the decomposition of the tensor product of two irreducible representations of a semisimple Lie algebra with the same highest weight; the second project is to extend the celebrated Parthasarathy-Ranga Rao-Varadarajan theorem to a new setting; the third project is to find an analog for more general groups of the classical fact about the general linear group that the cup product structure coefficients in the cohomology of Grassmannians coincide with the Littlewood-Richardson coefficients for tensor products of representations; and the fourth project is to solve a combinatorial conjecture of Alon and Tarsi counting the number of Latin squares of a certain type.