CYCLIC, SEPARABLE AND SEMISIMPLE MATRICES IN THE SPECIAL LINEAR GROUPS OVER A FINITE FIELD

Abstract

A matrix $A$ with minimum polynomial $m_A$ and characteristic polynomial $c_A$ is said to be cyclic if $m_A = c_A$ , semisimple if $m_A$ has no repeated factors, and separable if it is both cyclic and semisimple. For any set $T$ of matrices, we write $C_T$ for the proportion of cyclic matrices in $T, SS_T$ for the proportion of semisimple matrices, and $S_T$ for the proportion of separable matrices. We will write $C_{{\rm GL}(\infty,q)}$ for $\lim_{d\rightarrow\infty}C_{{\rm GL}(d,q)}$ , and so on.