Stanley's conjecture on the Schur positivity of distributive lattices

TL;DR

This paper addresses Stanley's 1998 conjecture on the Schur positivity of distributive lattices by constructing examples that are not Schur positive and some that are nice but not Schur positive, thus resolving an open problem.

Contribution

The paper provides the first explicit constructions of distributive lattices that are not Schur positive and also those that are nice but not Schur positive, answering a longstanding open question.

Findings

Constructed distributive lattices that are not Schur positive.

Identified distributive lattices that are nice but not Schur positive.

Resolved an open problem posed by Stanley in 1998.

Abstract

In this paper we solve an open problem on distributive lattices, which was proposed by Stanley in 1998. This problem was motivated by a conjecture due to Griggs, which equivalently states that the incomparability graph of the boolean algebra $B_n$ is nice. Stanley introduced the idea of studying the nice property of a graph by investigating the Schur positivity of its corresponding chromatic symmetric functions. Since the boolean algebras form a special class of distributive lattices, Stanley raised the question of whether the incomparability graph of any distributive lattice is Schur positive. Stanley further noted that this seems quite unlikely. In this paper, we construct a family of distributive lattices which are not nice and hence not Schur positive. We also provide a family of distributive lattices which are nice but not Schur positive.