Year: 2026
Type: preprint
Abstract: Let $A=(a_{ij})$ be an $n\times n$ 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 $$ ϕ^λ[A] = \sum_{σ\in\mathfrak{S}_n}ϕ^λ(σ) \prod_{i=1}^{n}a_{iσ(i)} $$ 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 $ϕ^λ[A]\ge 0$? This paper solves the problem for all hook partitions.
Source: arXiv (Cornell University)
Authors: Dong, Xiangshuai, Wu, Tingzeng, Gao, Xing
Institution: Lanzhou University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G3795835246, G8429749869
OpenAlex ID: W7214892635