On Matrices over Roots of Unity with Vanishing Permanent

Abstract

For any integer $m\ge 2$, let ${\mu }_m$ be the group of $m$th roots of unity. Let $p$ be a prime and $a$ a positive integer. For $m=p^{\alpha }$, it is shown that there is no $n\times n$ matrix over ${\mu }_m$ with vanishing permanent if $n<p$.