Quasicrystal Structure of Fundamental Quasisymmetric Functions, and Skeleton of Crystals

TL;DR

This paper explores the structure of quasicrystals within tableaux crystals, providing new formulas, constructive proofs, and conjectures about their organization and applications to plethysm and basis transformations.

Contribution

It introduces the concept of quasicrystals as subgraphs of tableaux crystals, offers formulas for tableau counts and Kostka numbers, and proposes a conjecture relating the crystal skeleton to dual equivalence graphs.

Findings

Quasicrystals are isomorphic to specific tableaux crystals.

A formula for counting tableaux of shape λ with maximal entry n.

A constructive proof for Kostka number formulas.

Abstract

We use crystals of tableaux and descent compositions to understand the decomposition of Schur functions $s_\lambda$ into Gessel's fundamental quasisymmetric functions $F_\alpha$. The connected crystal of tableaux $B(\lambda)$, associated to $s_\lambda$, is shown to be partitionned into a disjoint union of connected induced subgraphs $B(T_\alpha)$ corresponding to the $F_\alpha$'s. We show that these subgraphs, which we call quasicrystals, are isomorphic (as graphs) to specific crystals of tableaux. This allows us to give a formula for the number of tableaux of shape $\lambda$ and maximal entry $n$. We also use this setting to give a constructive proof of a combinatorial formula for Kostka numbers $K^\lambda_\mu$. We study the position of the quasicrystals within the crystal $B(\lambda)$, and show that they appear in dually positionned pairs, with the crystal anti-automorphism between them being given by a generalization of Sch\"utzenberger's evacuation. We introduce the notion of skeleton of the crystal $B(\lambda)$ given by replacing each subgraph $B(T_\alpha)$ by the associated standard tableau of shape $\lambda$. We conjecture that its graph includes the dual equivalence graph for $\lambda$, introduced by Assaf, and that its subgraphs of tableaux with fixed number of descents have particular structures. Finally, we describe applications to plethysm, among which we give an algorithm to express any symmetric sum of fundamental quasisymmetric functions into the Schur basis, whose construction gives insight into the relationship between the two basis.