Stanley’s Conjecture on the Schur Positivity of Distributive Lattices
Grace M. X. Li, Dun Qiu, Arthur L. B. Yang, Zhong-Xue Zhang · SIAM Journal on Discrete Mathematics · 2025
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 [Formula: see text] is nice. Stanley introduced the idea of studying the nice property of a graph by investigating the Schur positivity of its corresponding chromatic symmetric function. 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. 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.