On the matrices with constant determinant and permanent over roots of unity.
The set of square matrices over the cyclic group \(\mu_{m}\) of the \(m\)th roots of unity is considered. Some results concerning (1,-1) square matrices given by \textit{E. T. Wang} [Isr. J. Math. 18, 353--361 (1975; Zbl 0297.15007)]; \textit{R. Simion} and \textit{F. W. Schmidt} [Discrete Math. 46, 107--108 (1983; Zbl 0512.15008)] and \textit{A. R. Kräuter} and \textit{N. Seifter} [Isr. J. Math. 45, 53--62 (1983; Zbl 0517.15009)] are generalized to this framework. It is shown that if \(m\) is a prime power, then for each non-zero complex number \(\delta\) there exists at most a finite number of square matrices whose permanent or determinant are equal to \(\delta\). If \(m\geq2\) is not a prime power, an infinite family of matrices over \(\mu_{m}\) with determinant 1 is constructed. Another theorem states that if \(p\) is a prime, \(\alpha\) a positive integer and \(n=p^{\alpha}-1\), then there is no \(n\times n\) matrix over \(\mu_{p}\) with vanishing permanent.
scientific article; zbMATH DE number 2013056