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arXiv · 2309.14887

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

Alan J. Cain·António Malheiro·Fátima Rodrigues·Inês Rodrigues
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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.

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Alan J. Cain, António Malheiro, Fátima Rodrigues, Inês Rodrigues. 2023-09-26. Structure of quasi-crystal graphs and applications to the combinatorics of quasi-symmetric functions. https://arxiv.org/abs/2309.14887

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