Proportions of Cyclic Matrices in Maximal Reducible Matrix Groups and Algebras

Abstract

A matrix is said to be {\it cyclic} if its characteristic polynomial is equal to its minimal polynomial. Cyclic matrices play an important role in some algorithms for matrix group computation, such as the Cyclic Meataxe developed by P. M. Neumann and C. E. Praeger in 1999. In that year also, G. E. Wall and J. E. Fulman independently found the limiting proportion of cyclic matrices in general linear groups over a finite field of fixed order q as the dimension n approaches infinity, namely $(1-q^{-5}) \prod_{i=3}^\infty (1-q^{-i}) = 1 - q^{-3} + O(q^{-4}).$ We study cyclic matrices in a maximal reducible matrix group or algebra, that is, in the largest subgroup or subalgebra that leaves invariant some proper nontrivial subspace. We modify Wall's generating function approach to determine the limiting proportions of cyclic matrices in maximal reducible matrix groups and algebras over a field of order q, as the dimension of the underlying vector space increases while that of the invariant subspace remains fixed. The limiting proportion in a maximal reducible group is proved to be $1 - q^{-2} + O(q^{-3})$; note the change of the exponent of q in the second term of the expansion. Moreover, we exhibit in each maximal reducible matrix group a family of noncyclic matrices whose proportion is $q^{-2} + O(q^{-3})$.