The lattices $\textbf m\times\textbf 2$ and $\textbf m\times\textbf 3$ are not Schur positive

TL;DR

This work addresses Schur positivity for distributive lattices, focusing on the lattices $ extbf{m} imes extbf{2}$ and $ extbf{m} imes extbf{3}$. It combines Wang–Wang's explicit formula for Schur coefficients of chromatic symmetric functions with Pieri rules to translate coefficient computations into stable composition counts on related graphs, notably $H_m^n$ and the incomparability graphs of the lattices. The authors prove non-Schur-positivity for both lattices when $m\ge 8$, with a concrete negative coefficient $[s_{(m-2)^24}]X_{ ext{inc}( extbf{m} imes extbf{2})}$ and a comparable negative coefficient for $ extbf{m} imes extbf{3}$, thereby confirming a conjecture in this regime; they also show $ extbf{m} imes extbf{3}$ is not strongly nice for $m\ge 44$. These results provide explicit counterexamples to universal Schur positivity among distributive lattices and clarify the interplay between Schur positivity and niceness properties in incomparability graphs. The methods integrate symmetric function theory with detailed graph-combinatorial counts, offering insights into where Schur positivity fails in distributive lattice families.

Abstract

We prove that the lattices $\textbf m\times\textbf 2$ and $\textbf m\times\textbf 3$ are not Schur positive for $m\ge 8$. This confirms a conjecture of Li, Qiu, Yang, and Zhang, as an extension of counterexamples to a comment of Stanley on the universal Schur positivity of distributive lattices. Our main tools include Pieri's rules, and Wang and Wang's combinatorial formula for computing any Schur coefficient of the chromatic symmetric function of a graph in terms of special ribbon tabloids. We further show that the lattice $\textbf m\times\textbf 3$ is not strongly nice for $m\ge 44$.

Figures (6)

• Figure 1: The lattice $\textbf{m}\times\textbf{2}$ and the graph $G_m$.
• Figure 2: The set $\mathcal{T}_{(m-2)^24}=\{\kappa_1, \dots, \kappa_6\}$.
• Figure 3: The set $\{\nu_1, \dots, \nu_4\}$ of partitions such that $[s_\lambda]s_{\nu_i}s_1^2\ne 0$. The shaded circles indicate the removed cells.
• Figure 4: The lattice $\textbf{m}\times\textbf{3}$ and the graph $H_m^3$.
• Figure 5: The decompositions of $X$, $Y$ and $Z$.

Theorems & Definitions (34)

• Conjecture 1.1
• Theorem 1.2: LQYZ24X
• Conjecture 1.3: LQYZ24X
• Theorem 2.1: WW20
• Proposition 2.2: LLYZ24X
• Proposition 2.3: Pieri's rules
• Lemma 2.4
• proof
• Lemma 2.5
• proof