Crystal skeletons: Combinatorics and axioms

TL;DR

The paper develops crystal skeletons CS(λ) as a combinatorial contraction of quasi-crystal components to translate quasisymmetric expansions into Schur expansions, and proves CS-graphs generalize dual equivalence graphs. It provides two concrete edge-labelings (Dyck-pattern intervals and cycles) and vertices labeled by SYT(λ) or Des(T), establishing rich structural properties including self-similarity, Lusztig symmetry, and branching akin to symmetric-group representations. Three axiom systems—GL$_n$, S$_n$, and local—are shown to uniquely characterize CS-graphs, mirroring Stembridge-style crystal theory and enabling Schur-positivity-style reasoning for quasisymmetric inputs. The two-row case is worked out in detail, illustrating the path-model perspective, rectangular decompositions, fans, evacuation, and strongly-connected components, which clarifies the interaction between descent data and edge dynamics. Overall, the work provides a robust combinatorial and axiomatic framework for crystal skeletons that connects Schur and quasisymmetric function theory through explicit graph-theoretic and representation-theoretic structures.

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.

Figures (6)

• Figure 1: Left: Crystal $B(2,1)$ of type $A_2$ with two quasi-crystal components indicated with dotted lines and standard tableaux indicated by $*$. The descent composition is denoted $\alpha$. Right: Corresponding crystal skeleton.
• Figure 2: The crystal skeleton ${\mathsf{CS}}(3,2,1)$. The edge labels in terms of intervals and cycles are defined in Sections \ref{['section.Dyck pattern']} and \ref{['section:cycles']}. This example is further decorated by tableaux coloring to indicate the descent compositions as in Example \ref{['example.descentcomposition']}; thick arrows to indicate the crystal $B(3,2,1)_3$ in Theorem \ref{['theorem.B short']}; and gray components to indicate the branching in Theorem \ref{['theorem.branching']}.
• Figure 3: Left: The crystal skeleton $\mathsf{CS}(2,2,1)$. Right: The crystal skeleton $\mathsf{CS}(3,2)$.
• Figure 4: The strongly-connected crystal skeleton $\mathsf{CS}(3,3)$.
• Figure 5: Left: Crystal skeleton subgraph $\mathsf{CS}(3,2,1)_3$ of Figure \ref{['figure.CS321']}. Right: Crystal $B(3,2,1)_3$ of type $A_2$.

Theorems & Definitions (144)

• Theorem 1.1: Theorem \ref{['theorem.descent composition']} summary
• Theorem 1.2: Theorem \ref{['thm:DEinsideCS']}
• Theorem 1.3: Theorem \ref{['theorem.self similar']}
• Theorem 1.4: Theorem \ref{['theorem.branching']}
• Theorem 1.5: Theorem \ref{['theorem.B short']}
• Corollary 1.6: Corollary \ref{['corollary.lusztig']}
• Theorem 1.7: Summary of §\ref{['section.axioms']}
• Example 2.1
• Proposition 2.2
• Example 2.3