On hereditary irreducibility of some monomial matrices over local rings

Abstract

We consider monomial matrices over a commutative local principal ideal ring $R$ of type $M(t,k,n)=\Phi\left(\begin{smallmatrix}I_k&0\\0\,\,&tI_{n-k}\end{smallmatrix}\right)$, $0<k<n$, where $t$ is a generating element of Jacobson radical $J(R)$ of $R$, $\Phi$ is the companion matrix to $\lambda^n-1$ and $I_k$ is the identity $k\times k$ matrix. In this paper, we indicate a criterion of the hereditary irreducibility of $M(t,k,n)$ in the case $t^{\left[\frac{k\cdot(n-k)}{n}\right]+1}\not=0$.