PreprintCC BY-NC-ND 4.0arXiv
Schur positivity of the lattices m×n
Abstract
We prove that the product of chains m×n is not Schur positive whenever n≥4 and m≥3n−1. Writing m=n+k, we exhibit a negative Schur coefficient indexed by (2n+k−1,2n+k−3,…,k+5,k,k,4) and evaluate it explicitly as (n−2)! times a polynomial of degree three in k. Together with earlier results, this establishes the two-case interpretation of the conjecture of Li, Qiu, Yang, and Zhang: these products are not Schur positive when n=2 and m≥8, or when n≥3 and m≥ n+5.
Related
BibTeX
@misc{zhang2026schur,
title = {{Schur positivity of the lattices $\mathbf m\times\mathbf n$}},
author = {Kai Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01617},
eprint = {2610.01617},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01617},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Schur positivity of the lattices m×n
AU - Kai Zhang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.01617
UR - https://arxiv.org/abs/2610.01617
AB - We prove that the product of chains m×n is not Schur positive whenever n≥4 and m≥3n−1. Writing m=n+k, we exhibit a negative Schur coefficient indexed by (2n+k−1,2n+k−3,…,k+5,k,k,4) and evaluate it explicitly as (n−2)! times a polynomial of degree three in k. Together with earlier results, this establishes the two-case interpretation of the conjecture of Li, Qiu, Yang, and Zhang: these products are not Schur positive when n=2 and m≥8, or when n≥3 and m≥ n+5.
ER -