Collaborative Research: Special Functions for Diagonal Harmonics and Schubert Calculus

Drexel University, Philadelphia PA

Investigators

Abstract

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 in symmetric function theory, an area with applications to probability, statistical mechanics, and quantum information theory. The project includes further development of the SAGE open-source mathematics software, a crucial tool for this investigation. Graduate students will be trained as part of this project. This project addresses combinatorial questions 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 that form a basis for the ring of symmetric functions. Macdonald polynomials also have connections with many other areas, including double affine Hecke algebras, the Calogero-Sutherland model in particle physics, and knot invariants. Studies of Macdonald polynomials also led to the space of diagonal harmonics, a graded representation of the symmetric group whose character is closely tied to Macdonald polynomials. Over the last two decades a beautiful picture has emerged connecting this story to the Hall algebra of an elliptic curve. This project ties Macdonald theory to Catalan functions, graded Euler characteristics of certain vector bundles on the flag variety. The investigators aim to apply tools from previous study of Catalan functions, particularly connections with Schubert calculus and crystals of quantum groups, to enrich the elliptic Hall algebra and its ties to diagonal harmonics with a new combinatorial framework. 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.