The investigator will examine combinatorial problems related to representations of Lie groups, both finite and infinite-dimensional. The aim is to gain an explicit combinatorial description of the representations, and conversely to use representation theory to answer combinatorial questions. First he will work in Sperner Theory, which examines methods to organize sets and numbers in ``non-interfering'' ways. The investigator will use a new connection with crystal bases of quantum group representations to address combinatorial problems involving Symmetric Chain Decomposition of posets. Next, he aims to apply combinatorics, specifically the Littelmann and Kyoto path models, to control and clarify the algebraic structure of loop group representations. Finally, the investigator will explore generalizations of combinatorial and geometric representation theory to configuration varieties, multiple flag varieties, and quiver representations.
Representation theory is the theory of symmetric objects. Since symmetry is usually the key to obtaining exact solutions in complicated situations, representation theory is one of the most powerful tools in mathematics, physics, and chemistry. The investigator's work seeks to describe various kinds of symmetry in explicit combinatorial terms. Put simply, the aim is to understand the objects in n-dimensional (and infinite-dimensional) space, which, like the circle or sphere, have a large, continuous family of symmetries, and to encode this symmetry in discrete combinatorics.
