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On the nonnegativity of monomial immanants for hook partitions
Abstract
Let A=(aij) be an n× n real matrix and let λ be a partition of n. Let ϕλ be the class function dual to the Young permutation character, and let ϕλ[A] = ∑σ∈Snϕλ(σ) ∏i=1naiσ(i) be the corresponding monomial immanant. Stembridge [Canad. J. Math. 44 (1992), pp. 1079-1099] posed the following open problem: If all minors of A of order at most r are nonnegative, and the partition λ has length at most r, is it true that ϕλ[A]≥ 0? This paper solves the problem for all hook partitions.
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BibTeX
@misc{dong2026nonnegativity,
title = {{On the nonnegativity of monomial immanants for hook partitions}},
author = {Xiangshuai Dong and Tingzeng Wu and Xing Gao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36665},
eprint = {2609.36665},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36665},
}
RIS
TY - UNPB
TI - On the nonnegativity of monomial immanants for hook partitions
AU - Xiangshuai Dong
AU - Tingzeng Wu
AU - Xing Gao
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36665
UR - https://arxiv.org/abs/2609.36665
AB - Let A=(a_ij) be an n× n real matrix and let λ be a partition of n. Let ϕ^λ be the class function dual to the Young permutation character, and let ϕ^λ[A] = ∑_σ∈S_nϕ^λ(σ) ∏_i=1^na_iσ(i) be the corresponding monomial immanant. Stembridge [Canad. J. Math. 44 (1992), pp. 1079-1099] posed the following open problem: If all minors of A of order at most r are nonnegative, and the partition λ has length at most r, is it true that ϕ^λ[A]≥ 0? This paper solves the problem for all hook partitions.
ER -