Combinatorics of Crystals, Macdonald Polynomials, and Schubert Calculus

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This research project uses combinatorial structures and methods to solve problems in two areas: the representation theory of Lie algebras and Schubert calculus. In representation theory, the investigator will continue his work on deriving explicit tableau formulas in classical Lie types from the Ram-Yip formula for Macdonald polynomials and the Yip formula for their product; these formulas are in terms of the alcove walk model, which was introduced by Gaussent-Littelmann and the investigator in collaboration with A. Postnikov, and was then developed by other mathematicians. One of the applications is an efficient computation of the energy function, which defines the affine grading on a tensor product of Kirillov-Reshetikhin crystals. This application is based on an interesting connection between Macdonald polynomials, quantum cohomology, and affine crystals. In order to better understand it, the investigator proposes a crystal-theoretic counterpart (based on the alcove model) of the "quantum=affine" phenomenon relating the quantum cohomology of flag varieties and the homology of the affine Grassmannian. Other projects in representation theory involve explicit constructions of certain representations, as well as the topology of crystals (as posets) and of a poset which encodes the structure of Mirkovic-Vilonen cycles. In Schubert calculus, the investigator has projects related to the cohomology, quantum cohomology, and quantum K-theory of generalized flag varieties. Most of these projects involve new approaches to positive combinatorial formulas for the Schubert structure constants, which express the product of Schubert classes in the basis of Schubert classes.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 it became clear that combinatorial structures are particularly well suited for encoding complex mathematical objects, while combinatorial methods are well suited for related computations. The investigator 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). Certain representations and the related algebraic varieties are modeled by graphs or partially ordered sets. By studying the structure of these discrete objects, which displays remarkable complexity, the investigator will be able to derive important algebraic and geometric information.