Combinatorial Representation Theory of Quantum Groups and Coinvariant Algebras

Combinatorics has been described as the nanotechnology of mathematics. It is concerned with counting discrete objects, which naturally arise in many applications. As one example, software development frequently requires choosing between different algorithms to solve a problem. Combinatorics allows one to count the number of steps each candidate algorithm takes and then choose the best solution. In this way, combinatorics provides a set of basic tools and a collection of argument prototypes that guide the solution of problems throughout STEM. One of the virtues of combinatorics research is that it provides students with concrete opportunities to develop problem-solving, software development, and other key skills. Algebraic combinatorics, more specifically, focuses on the combinatorial essence of highly structured and often advanced problems coming from topology, representation theory, particle physics, and other areas. Such problems are frequently reduced in some fashion to an intricate combinatorial analysis. One such algebraic problem is to understand quantum groups. These remarkable structures arose around 1980 from connections with integrable lattice models in quantum mechanics, and some of the technically deepest theories in pure mathematics and physics are in this area. One of the main focuses of the present project is to further develop certain combinatorial diagrams called web bases. These combinatorial objects encode the representation category of quantum groups and allow for efficient computations with powerful topological quantum invariants. They connect a remarkably diverse collection of topics, including total positivity, alternating sign matrices, plane partitions, crystal bases, dynamical algebraic combinatorics, and the geometry of the affine Grassmannian. Students will be involved in the research project. 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.