Dual equivalence graphs II: Transformations on locally Schur positive graphs

TL;DR

This paper extends the framework of dual equivalence graphs to a larger family, introducing transformations that can convert these graphs into dual equivalence graphs, thereby broadening their applicability in proving Schur positivity.

Contribution

It defines new maps on a larger class of graphs that can transform them into dual equivalence graphs, expanding the scope of symmetric function theory.

Findings

Transformations can convert larger graphs into dual equivalence graphs

Broader applications in Schur positivity problems

Potential to solve long-standing conjectures in symmetric functions

Abstract

Dual equivalence graphs are a powerful tool in symmetric function theory that provide a general framework for proving that a given quasisymmetric function is symmetric and Schur positive. In this paper, we study a larger family of graphs that includes dual equivalence graphs and define maps that, in certain cases, transform graphs in this larger family into dual equivalence graphs. This allows us to broaden the applications of dual equivalence graphs and points the way toward a broader theory that could solve many important, long-standing Schur positivity problems.