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.

Project Report

An important part of my work is concerned with using combinatorial structures and methods for computations related to the representation theory of Lie algebras. Representation theory is a fundamental tool for studying symmetry by realizing the elements of abstract groups/algebras as linear transformations (of some vector spaces). Lie algebras are an important class of algebras with many applications to physics. There are interesting related deformations, called quantum groups, which depend on a parameter q, interpreted as physical temperature. At "zero temperature'', the representations of quantum groups can be encoded in certain colored directed graphs called crystals. Representations of quantum groups, and therefore crystals, have important applications to statistical mechanics. In this area, a physical system is modeled by a grid graph with weights associated to vertices (which represent atoms or particles) and a given number of states for the edges (which represent the bonds). Such a discrete model is called a vertex model. The probability of the system being in any given state at a particular time, and hence the properties of the system, are determined by certain statistical sums. These can be computed as sums over crystals of the so-called energy function at a crystal vertex. The graph in the figure is a crystal. The energy is constant on the (vertices of the) 4 connected components determined by the edges labeled 1 and 2. For each edge labeled 0, except the last one in a sequence of such edges, there is a decrease by 1 of the energy. So the energy (suitably normalized) on the 4 components is 0, -1, -2, -3, from left to right. As such graphs can be very large, it is much more efficient to compute the energy only in terms of some combinatorial data associated to the vertices (in the figure, we use some fillings of diagrams with numbers). With my co-authors, I have developed a very general model (data associated to the vertices), called the alcove model, in terms of which we can efficiently calculate the energy, and also carry out other computations. The alcove model, together with other structures I developed, also led to solutions of certain problems in an area known as modern Schubert calculus. Classical Schubert calculus is concerned with enumerating geometric objects satisfying prescribed incidence constraints. The simplest non-trivial example is: given 4 lines in general position in 3-space, how many lines meet all of them? (The answer is 2.) The basic questions in enumerative geometry were reformulated in the framework of intersection theory of flag manifolds, and thus reduced to multiplication formulas in some associated algebraic structures (cohomology rings). I worked on extensions of Schubert calculus in which ordinary cohomology is replaced with more refined algebraic structures (K-theory, elliptic cohomology), and finite-dimensional flag manifolds with infinite-dimensional ones. The alcove model has already been used by several researchers, and it was implemented in the open-source computer algebra system Sage. The support from the this grants allowed me to disseminate my research and interact with my colleagues in several national and international conferences, as well as in workshops/seminars at mathematical institutes (in the US and abroad), and in departmental seminars. My support also enabled me to visit and host several collaborators. I supported two of my Ph.D. students as Research Assistants. One of them, Arthur Lubovsky, recently received a Distinguished Doctoral Dissertation Award, and presented our joint work in a plenary talk at the 2012 international conference in algebraic combinatorics (FPSAC).

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Tomek Bartoszynski
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Suny at Albany
United States
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