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Stanley's conjecture on the Schur positivity of distributive lattices
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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.
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Cited by 1 Pith paper
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Exact thresholds for Schur positivity of the lattices $\mathbf m\times\mathbf 2$ and $\mathbf m\times\mathbf 3$
The chain-product lattices m×2 and m×3 are Schur positive exactly for m≤7 and m≤6, with explicit negative Schur coefficients proving non-Schur-positivity for m≥8.
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