Combinatorics of Hecke Algebras and Crystal Skeletons

Our world is governed by symmetry and measured by data. For example, symmetries play a fundamental role in the wavefunctions and vibrations of molecules, and are intricately linked to quantum states of physical systems. Symmetries can be complex, and a major goal in mathematics is to understand them by breaking them into smaller pieces. Similarly, mathematicians try to interpret sets of data by finding patterns, and formulas that characterize those patterns. This project aims to exploit the rich interplay between these perspectives: if one discovers an elegant formula, it may be detecting some underlying symmetry. On the other hand, decomposing symmetries requires uncovering patterns in complicated data. The goal of the proposed work is to (1) use tools from the mathematical study of symmetry, called representation theory, to find patterns and formulas in data, and (2) analyze symmetry by finding methods of distilling it into discrete data sets. The PI also intends to use techniques from machine learning to achieve these goals. The project will further build the STEM workforce by providing research and conference travel support for graduate students. Specifically, this project will tackle two broad topics arising at the interface of probability theory, representation theory and combinatorics. First, the PI and collaborators will use the representation theory of the Hecke algebra to study quantum deformations of classical results in combinatorics and probability, aiming to (1) develop the combinatorics of the Hecke algebra center, (2) build a Hecke analog of the theory of permutation factorizations, generalizing (for example) the celebrated Harer-Zagier formula, and (3) introduce and analyze Hecke deformations of important Markov chains on the symmetric group. Second, the PI and collaborators plan to find decompositions of symmetric functions using the theory of crystals and crystal skeletons. This work will develop and generalize the relationship between symmetric and quasisymmetric functions from a combinatorial and algebraic perspective. 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.