Character values of finite groups as eigenvalues of nonnegative integer matrices

Abstract:

Let ${C_1}$, ${C_2}, \ldots$, ${C_k}$ be the conjugacy classes of the finite group $G$ and choose ${x_i} \in {C_i}$, for $i = 1$, $2, \ldots$, $k$. For every complex character $\theta$ of $G$ there is a $k \times k$ matrix $M(\theta )$ whose entries are nonnegative integers such that ${X^{ - 1}}M(\theta )X = {\text {diag}}(\theta ({x_1}),\theta ({x_2}), \ldots ,\theta ({x_k}))$ where $X$ is the character table matrix of $G$. Some consequences are shown.