Nilpotent Orbits, Representation Theory, and Combinatorics

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

The principal investigator will study several questions in representation theory, algebraic geometry, and combinatorics which are related to nilpotent orbits in Lie algebras. More specifically, the investigator will continue his work on ideals in the poset of positive roots and their connection to representation theory and the theory of cluster algebras. He will also continue to study the Lusztig-Vogan bijection and to seek out its explicit computation. He will try to complete the determination of which nilpotent orbits in the exceptional Lie algebras have normal closure and prove results (in all Lie algebras) about the structure of functions on covers of nilpotent orbits that rely on similar techniques. Representation theory is a branch of modern algebra that is concerned with understanding symmetries. A central idea is that complicated algebraic or geometric structures can be represented by a certain set of matrices (arrays of numbers), which are easier to understand than the original structure. For example, the symmetries of the square can be represented (in one possible way) by a certain set of eight two-by-two matrices. This view of representation theory has many applications in chemistry and physics. The investigator's work will contribute to understanding the representation theory of Lie algebras and Lie groups, which are essential to understanding the symmetry of objects which arise in mathematics.