A minimaj-preserving crystal on ordered multiset partitions

TL;DR

This paper constructs a crystal structure on ordered multiset partitions related to the Delta Conjecture, showing minimaj's constancy on components and proving Schur positivity and equidistribution with major index.

Contribution

It introduces a novel crystal structure on ordered multiset partitions that preserves the minimaj statistic and connects it to key algebraic and combinatorial properties.

Findings

Minimaj is constant on crystal components.

Provides a new proof of Schur positivity for $R_{n,k}$.

Shows minimaj and major index are equidistributed.

Abstract

We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture. This conjecture was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connected components of the crystal. In particular, this yields another proof of the Schur positivity of the graded Frobenius series of the generalization $R_{n,k}$ due to Haglund, Rhoades and Shimozono of the coinvariant algebra $R_n$. The crystal structure also enables us to demonstrate the equidistributivity of the minimaj statistic with the major index statistic on ordered multiset partitions.