Quotienting by Quasisymmetrics: Combinatorics and Geometry

University Of Hawaii, Honolulu

Investigators

Abstract

Algebraic combinatorics is a field of mathematics originating from the dynamic interplay between discrete combinatorial objects and abstract algebraic notions. This interdisciplinary aspect makes for an extremely fertile field of investigation with applications to theoretical computer science, economics, statistics, computational biology, and other subjects of mathematics. A crucial component to answering questions in this area involves distilling hard-to-understand geometric or algebraic information into hands-on combinatorial data which has the added benefit of casting new light on the original context. This project applies tools from combinatorics and geometry to study classical algebraic constructions such as quotients of polynomial rings. The PI will develop combinatorial tools and techniques to gain insight, with the long term goal of developing a firmer grasp on positivity questions in algebraic combinatorics. Furthermore, this project provides several research training opportunities for graduate students. This project aims to advance our understanding of the quotient of the polynomial ring modulo the ideal of quasisymmetric polynomials. This quotient is an quasisymmetric analog of the coinvariant algebra, an object with a storied history going back to the work of A. Borel. A basis for the coinvariant algebra with deep geometric and combinatorial relevance is given by Schubert polynomials of Lascoux-Schützenberger. Motivated by questions pertaining to the permutahedral variety, the PI will study a new basis for the quasisymmetric quotient that is intimately tied with Schubert polynomials. The overarching goal of this project is to gain further insight into the long-standing open question of multiplying Schubert polynomials combinatorially. On the geometric side, the PI will draw upon connections between the permutahedral variety and the quasisymmetric quotient to bring forth novel aspects of lattice point enumeration in permutahedra. 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.