Crystal skeletons: Combinatorics and axioms

2025/03/18 by Sarah Brauner, Brauner, Sarah, Sylvie Corteel +5
Chemistry · #05E05 #05E10 #17B37 #20G42 #Combinatorics (math.CO) #FOS: Mathematics #History and advancements in chemistry #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2503.14782

openalex publication_date 2025/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Quasicrystal Structure of Fundamental Quasisymmetric Functions, and Skeleton of Crystals
• Crystal skeleton polynomials with major index, charge and depth
• Structure of quasi-crystal graphs and applications to the combinatorics of quasi-symmetric functions
• Axioms for shifted tableau crystals
• A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs