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arXiv · 2503.14782

Crystal skeletons: Combinatorics and axioms

Sarah Brauner·Sylvie Corteel·Zajj Daugherty·Anne Schilling
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Abstract

Crystal skeletons were introduced by Maas-Gariépy in 2023 by contracting quasi-crystal components in a crystal graph. On the representation theoretic level, crystal skeletons model the expansion of Schur functions into Gessel's quasisymmetric functions. Motivated by questions of Schur positivity, we provide a combinatorial description of crystal skeletons, and prove many new properties, including a conjecture by Maas-Gariépy that crystal skeletons generalize dual equivalence graphs. We then present a new axiomatic approach to crystal skeletons. We give three versions of the axioms based on $GL_n$-branching, $S_n$-branching, and local axioms in analogy to the local Stembridge axioms for crystals based on novel commutation relations.

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Sarah Brauner, Sylvie Corteel, Zajj Daugherty, Anne Schilling. 2025-03-18. Crystal skeletons: Combinatorics and axioms. https://arxiv.org/abs/2503.14782

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