Collaborative Research: Nonsymmetric Plethysm and Atom Positivity

About This Grant

Combinatorics is an area of mathematics concerned with analyzing, organizing, and optimizing over discrete data. It is a fundamental tool in many scientific areas such as genomics, computer science, statistics, and physics. This project will develop combinatorial methods for attacking problems in Lie theory and symmetric function theory, areas with applications to probability, statistical mechanics, and quantum information theory. A mutually beneficial component is the further development of the SAGE open-source mathematics software, a crucial tool for this investigation. Also graduate students will be trained as part of this project, This project addresses combinatorial problems tied to representation theory, algebraic geometry, and physics, with a focus on Macdonald polynomials and Schubert calculus. Macdonald polynomials are a remarkable family of orthogonal polynomials which form a basis for the ring of symmetric functions. Since their introduction in the 80's, they have developed connections with many areas, including Hilbert schemes, the Calogero-Sutherland model in particle physics, and knot invariants. In the 90's, Garsia and Haiman studied transformed versions of Macdonald polynomials, which they connected to the representation theory of polynomial rings, generalizing classical results of Chevalley, Shephard-Todd, and Steinberg on reflection groups. A separate line of work initiated by Cherednik, Macdonald, and Opdam in the 90's investigated nonsymmetric versions of Macdonald polynomials, which clarified the theory and connected it to affine Hecke algebras. The PIs and collaborators recently discovered a way of transforming these nonsymmetric versions in the same style as Garsia and Haiman did for the symmetric case. Further study of these new polynomials will unearth new representation theoretic and combinatorial mysteries of nonsymmetric Macdonald polynomials. 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.