Research in Algebraic Combinatorics

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The main thrust of the project is to continue joint work with John Shareshian on the interplay between symmetric function theory, enumerative combinatorics, and algebraic geometry. This project arose from the study of a class of symmetric functions that appears in various contexts such as in the work of Carlitz, Scoville and Vaughan on enumeration of Smirnov words, in the work of Procesi and Stanley on the representation of the symmetric group on the cohomology of the toric variety associated with the type A root system, and in the work of the investigator and Shareshian on a q-analog of the Eulerian polynomials. In an effort to understand the basis for the relationship between these structures, the investigator and Shareshian have proposed a far-reaching generalization of this relationship, which this project will explore. The conjectured generalization involves a quasisymmetric refinement of Stanley's chromatic symmetric functions, Tymoczko's representation of the symmetric group on the cohomology of the regular semisimple Hessenberg variety of type A, and a q-analog of a generalization (due to De Mari, Procesi and Shayman) of the Eulerian polynomials. The conjectured relationship would provide an algebro-geometric approach to attacking Stanley and Stembridge's long standing $e$-positivity conjecture for chromatic symmetric functions and a conjecture of the investigator and Shareshian on unimodality of their q-analog of the De Mari-Procesi-Shayman generalized Eulerian polynomials. The project will also continue the work of the investigator on the interplay between poset topology and enumerative combinatorics.The research supported by this grant is in algebraic combinatorics, which is 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 discrete configurations. Combinatorial methods are playing an increasingly important role in these fields.