New Perspectives in Combinatorics for Lie Algebra Representations and Schubert Calculus

This project uses combinatorial models to solve problems and perform computations in algebra (more specifically, representation theory), as well as geometry and topology of flag manifolds; new connections between these areas are also revealed in this process. Combinatorics studies discrete structures (such as permutations and graphs), which are well suited for encoding complex mathematical objects. Representation theory is a fundamental tool for studying symmetry, by realizing the elements of abstract groups/algebras as linear transformations of some vector spaces. The PI studies representations of Lie algebras and quantum groups, which have many applications to physics, such as calculating the probability of a particle system being in a given state at a particular time. In geometry, the PI focuses on modern Schubert calculus on flag manifolds. This area has its origins in enumerative geometry (e.g., counting the lines or planes satisfying a number of generic intersection conditions), and is currently related to modern areas such as quantum cohomology/K-theory and elliptic cohomology. This project includes several research topics for graduate and undergraduate students. In representation theory, the PI will work on new applications and problems involving crystals, which are colored directed graphs encoding representations of quantum algebras when the quantum parameter goes to 0. One such problem is concerned with a refinement of the so-called atomic decomposition of crystals, due to the PI and C. Lecouvey; this refinement is related to the pre-canonical bases of Hecke algebras, recently defined by N. Libedinsky, L. Patimo, and D. Plaza, which interpolate between the standard basis and the Kazhdan-Lusztig basis. The PI will also extend and find new applications of his quantum alcove model, as a uniform combinatorial model for (tensor products of) Kirillov-Reshetikhin crystals, in all affine types. In modern Schubert calculus, the PI has two main projects. The first one involves an application of the Chevalley multiplication formula in the quantum K-theory of flag manifolds, recently proved by the PI, S. Naito, and D. Sagaki. The second one is concerned with the equivariant elliptic cohomology of flag manifolds, and more precisely with the combinatorics of the elliptic classes constructed by R. Rimányi and A. Weber. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.