Structure of quasi-crystal graphs and applications to the combinatorics of quasi-symmetric functions

TL;DR

This paper explores the structure of quasi-crystal graphs related to the hypoplactic monoid and quasi-symmetric functions, introducing a new combinatorial structure and addressing key conjectures in the field.

Contribution

It provides an explicit description of quasi-crystal graph components using quasi-arrays and resolves two conjectures about their interactions with symmetric functions.

Findings

Confirmed a conjecture on fundamental quasi-symmetric and Schur functions interaction.

Disproved a conjecture on the arrangement of quasi-crystal components within crystal components.

Introduced quasi-arrays as a new combinatorial tool for studying quasi-crystal graphs.

Abstract

Crystal graphs are powerful combinatorial tools for working with the plactic monoid and symmetric functions. Quasi-crystal graphs are an analogous concept for the hypoplactic monoid and quasi-symmetric functions. This paper makes a combinatorial study of these objects. We explain a previously-observed isomorphism of components of the quasi-crystal graph, and provide an explicit description using a new combinatorial structure called a quasi-array. Then two conjectures of Maas-Gari\'epy on the interaction of fundamental quasi-symmetric functions and Schur functions and on the arrangement of quasi-crystal components within crystal components are answered, the former positively, the latter negatively.