Combinatorics and Braid Varieties

Algebraic combinatorics is a branch of mathematics that studies algebraic structures using combinatorial methods, and combinatorial structures using algebraic methods. Such structures and methods are often fundamental to many different scientific disciplines and show up in many different contexts---algebraic combinatorics therefore has diverse applications to subjects like cryptography, protein folding, high-energy physics, and quantum computing. This project will use algebraic objects and methods to produce new combinatorial results, leveraging braid varieties--a sort of algebraic space associated to a knot--as a unifying tool. Funds will additionally support training graduate students and outreach efforts, including work on an interactive online discrete mathematics textbook. In more detail, this proposal suggests a framework for producing combinatorial results using braid varieties over finite fields, Hecke algebra traces, rational Cherednik algebras, and a new relationship with noncrossing combinatorics. The framework has already proven successful in producing substantial new results: the PI's recent joint work with Galashin, Lam, and Trinh resolved two decades-long open problems in Coxeter-Catalan combinatorics, simultaneously producing the first definition of rational noncrossing Coxeter-Catalan objects, while also giving the first uniform enumeration of noncrossing objects. Connections to Macdonald theory--diagonal harmonics and q,t-combinatorics--are also expected when working over the complex numbers. At different levels of generality, different techniques become available. For finite Coxeter groups, it is possible to compute everything in a case-by-case manner using an explicit decomposition of the Hecke algebra, and there are many interesting combinatorial and representation-theoretic problems open for immediate attack. Special classes of elements in finite type have favorable representation-theoretic properties that allow for uniform approaches. For affine Weyl groups, the main tool is a trace formula for translations, due to Opdam. For example, the proposed framework recovers some Tessler matrix identities due to Haglund in this setting. For general Kac-Moody Weyl groups, we are reduced to general recursive and cluster-theoretic methods. These methods also apply in both the finite and affine cases, but will require software implementation before further exploration is possible. 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.