On the nonnegativity of monomial immanants for hook partitions

Source 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.