The starting point for this project is a surprising relationship that we recently discovered between diagonal harmonics, Hopf algebras, and polytope theory. Studying the unexpected connections between these seemingly disparate fields requires a broad range of expertise across different areas, including algebraic combinatorics, discrete geometry, algebra, representation theory, and symmetric functions. We propose to forge novel connections between these areas in new and exciting ways. Our three main objectives are: 1. To introduce and study a Multi-Shuffle Conjecture in multivariate diagonal harmonics, and solve some open problems in rational Catalan combinaStorics; 2.To investigate innovative applications of Hopf algebras to multivariate diagonal harmonics and lattice theory; 3.To settle unsolved recent problems in polytope theory. These three goals will be achieved by attacking a selection of open problems that lie at the interface between diagonal harmonics, Hopf algebras, and polytope theory. This trilateral approach will open entirely new, unexplored avenues for future research. Our analysis will require the implementation of algorithms that will contribute to the development of free open source software.
|Effective start/end date||1/02/21 → 31/01/25|
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