Exact thresholds for Schur positivity of the lattices $\mathbf{m}\times 2$ and $\mathbf{m}\times 3$

Abstract

We determine the exact thresholds for Schur positivity in the two families of chain products $\mathbf{m}\times 2$ and $\mathbf{m}\times 3$: the former is Schur positive exactly for $m\le 7$, and the latter exactly for $m\le 6$. For $m\ge 8$, we prove non-Schur-positivity in both families by exhibiting explicit negative Schur coefficients obtained from Pieri's rules and stable-composition counts. The remaining finite cases are settled by exact SageMath computations; in particular, $7\times 3$ has a negative Schur coefficient. These results settle the $n=2$ and $n=3$ cases in the conjectural picture of Li, Qiu, Yang, and Zhang and sharpen the $n=3$ boundary by one. We also show that $\mathbf{m}\times 3$ is not strongly nice for $m\ge 44$.

21 pages, 8 figures