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

2023/09/26 by Cain, Alan J., Malheiro, António, Rodrigues, Fátima +1 · 1 citation
#05E05 (Primary) 05E10 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2309.14887

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-Gariépy 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.

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• Crystal skeleton polynomials with major index, charge and depth

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