Combinatorial Studies in Algebra, Geometry, and Topology

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This research plan is divided into three projects in algebraic combinatorics and its applications to other areas of mathematics. The first project is in combinatorial representation theory, and is mostly concerned with the development of the alcove path model; this is a simple combinatorial model (recently introduced by the investigator in collaboration with A. Postnikov) for the representation theory of complex semisimple Lie algebras and, more generally, of complex symmetrizable Kac-Moody algebras. One problem is to describe the way in which the alcove path model specializes to other models in this area, such as Kashiwara-Nakashima tableaux (in the classical types), and the Kyoto path model (in the affine Kac-Moody types). Other problems are related to the combinatorics of Kashiwara's crystals and an efficient construction of a monomial basis of an irreducible representation; the approach to these problems is based on the alcove path model. Unrelated to this model, the first part of the project also includes a combinatorial study of explicit constructions of irreducible representations based on lattices with a small number of covers. The second part of the project is concerned with modern Schubert calculus on generalized flag varieties. The main goal is to derive combinatorial multiplication formulas for Schubert classes (i.e., the natural basis elements) in the cohomology and K-theory of flag varieties. One such problem is a Chevalley-type multiplication formula (by a codimension 1 class) in the equivariant K-theory of flag varieties for Kac-Moody groups; this formula generalizes a similar formula in the finite case based on the alcove path model. More general such formulas will be investigated in cohomology, based on combinatorial structures such as certain monoids for the Bruhat order on the corresponding Weyl group. The third part of the project is concerned with combinatorial applications to algebraic topology. More precisely, it involves combinatorial formulas (based on trees) for the coefficients of certain formal group laws. There are applications to topological conjectures about classifying spaces of certain finite abelian groups.A unifying theme of the project outlined here 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. This research plan is part of the ongoing effort to perform concrete computations, based on combinatorial structures. Such structures are used to study representations (i.e., actions on vector spaces) of complex Lie algebras. They are also used to study the geometry of certain classical algebraic varieties, namely flag varieties; related applications exist, for instance, to enumerative geometry (such as counting the lines or planes satisfying a number of generic intersection conditions, which is equivalent to performing certain cohomology calculations). The representations of Lie algebras and the geometry of flag varieties are related to each other, and they play a fundamental role in several areas of mathematics and theoretical physics. They display remarkable combinatorial complexity, which is investigated in this project.