Dual Equivalence Graphs and their Applications

Abstract

In 2007 Sami Assaf introduced dual equivalence graphs as a method for demonstrating that a quasisymmetric function is Schur positive. The method involves the creation of a graph whose vertices are weighted by Ira Gessel's fundamental quasisymmetric functions in such a way that the sum of the weights of a connected component is a single Schur function. The graphs are termed dual equivalence graphs, and this dissertation is the compilation of works that focus on the further development of the theory of said graphs. This work further includes applications to Macdonald polynomials, Hall-Littlewood polynomials, and Lascoux-Leclerc-Thibon polynomials. In joint work with Sara Billey, Zach Hamaker, and Benjamin Young, we also give a generalization of dual equivalence graphs to the Coxeter-Knuth graph of Lie type B and illustrate the relationship of these graphs to a newly defined type B Little bump. For the sake of completeness, we also include an appendix providing a proof of the original axiomatization of dual equivalence graphs as described by Sami Assaf.