Representation Theory and Schubert Calculus: Combinatorics and Interactions

Suny At Albany, Albany NY

Investigators

Abstract

A unifying theme of this project is the emphasis on combinatorics and computation. During the last decades, computation has gained an important role in mathematical research. This stimulated the development of combinatorics, as the various discrete structures it studies (such as graphs and partially ordered sets) are particularly well suited for encoding and manipulating complex mathematical objects. The PI will use combinatorial techniques in representation theory (which is a fundamental tool for studying group symmetry, and which has important applications in mathematics and beyond, e.g., to theoretical physics), and in Schubert calculus (which has its origins in enumerative geometry, e.g., counting the lines or planes satisfying a number of generic intersection conditions, but is currently related to modern areas such as quantum cohomology). By developing and studying combinatorial models for certain representations and related algebraic varieties, the PI will pursue efficient related computations; furthermore, this work is expected to lead to a better understanding of the mentioned mathematical objects and of the subtle connections between them. Some of the models developed in the project will be implemented in the open source computer algebra system SAGE. This research project uses combinatorial structures and methods to solve problems in two main areas: the representation theory of Lie algebras and modern Schubert calculus on flag manifolds. An important avenue of this research is concerned with the alcove model, developed by the PI in collaboration with A. Postnikov; this is a combinatorial model for integrable highest weight representations of symmetrizable Kac-Moody algebras, as well as for certain multiplication formulas in the K-theory of flag manifolds. New applications of the alcove model were given in recent work of the PI with S. Naito, D. Sagaki, A. Schilling, and M. Shimozono. These highlight interesting connections between Kirillov-Reshetikhin modules for affine Lie algebras, Macdonald polynomials, and the quantum K-theory of flag manifolds. The PI will further explore some of these connections. He will also pursue applications of the alcove model to formulas for Whittaker functions on p-adic groups, which are a basic tool in the theory of automorphic forms. In modern Schubert calculus, the PI will work on: unifying and extending various combinatorial models for the Schubert structure constants in ordinary cohomology, Schubert calculus beyond K-theory (e.g., in elliptic cohomology), and combinatorial aspects of the geometric Satake correspondence (which gives a geometric construction of irreducible Lie algebra representations, based on the affine Grassmannian).