Research in Algebraic Combinatorics

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The research supported by this grant is in algebraic combinatorics, an area of mathematics that seeks to develop connections between combinatorics (the science of counting, arranging and analyzing concrete discrete configurations) and fields of pure mathematics that involve sophisticated abstract algebraic structures. The idea is to use these connections to gain deeper insights and solve problems in combinatorics and in the other fields. The discrete configurations that are studied in combinatorics arise in various fields of mathematics, computer science, physics, biology and engineering; DNA sequences, phylogenetic trees, and communications networks are all examples of such discrete configurations. Combinatorial methods are playing an increasing role in these fields. The proposed research is comprised of three related projects involving generalizations and variations of classical combinatorial objects such as chromatic polynomials, Eulerian polynomials, partition lattices, and free Lie algebras. The first project deals with a refinement of Stanley's chromatic symmetric functions that was introduced in work of John Shareshian and the PI. The refinement, the chromatic quasisymmetric functions, generalizes Eulerian polynomials as well as chromatic polynomials. An algebro-geometric approach to settling the longstanding Stanley-Stembridge e-positivity conjecture, involving a connection between the chromatic quasisymmetric functions and the Hessenberg varieties of De Mari, Procesi, and Shayman, was presented in this work. This approach initiated significant interaction between combinatorialists working with symmetric functions and algebraic geometers working on Hessenberg varieties. Further study of the chromatic quasisymmetric functions, as well as its connection with Hessenberg varieties is proposed. The second project involves a new polynomial graph invariant and a more general symmetric function graph invariant introduced by the PI and her former student Rafael Gonzalez D'Leon. These graph invariants generalize variations of Eulerian polynomials such as the classical Narayana polynomials and Haiman's parking function symmetric functions. This project is focused on settling unimodality conjectures, e-positivity conjectures and other concrete conjectures for these graph invariants. The third project arose in theoretical physics. It deals with an n-ary generalization of a Lie algebra known as a Filippov algebra. The study of the representation of the symmetric group on the multilinear component of the free Filippov algebra was initiated in work of Friedmann, Hanlon, Stanley, and the PI. Although this work has already produced some significant results, there is much that remains to be done.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.