On the nonnegativity of monomial immanants for hook partitions

Abstract

Let $A=(a_{ij})$ be an $n\times n$ real matrix and let $\lambda$ be a partition of $n$. Let $\phi^\lambda$ be the class function dual to the Young permutation character, and let $$ \phi^\lambda[A] = \sum_{\sigma\in\mathfrak{S}_n}\phi^\lambda(\sigma) \prod_{i=1}^{n}a_{i\sigma(i)} $$ 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 $\lambda$ has length at most $r$, is it true that $\phi^\lambda[A]\ge 0$? This paper solves the problem for all hook partitions.