[Paper Review] Proportions of Cyclic Matrices in Maximal Reducible Matrix Groups and Algebras

Scott W. Brown, Cheryl E. Praeger+1 more|arXiv (Cornell University)|May 20, 2011

Finite Group Theory Research17 references3 citations

TL;DR

This paper investigates the limiting proportion of cyclic matrices in maximal reducible matrix groups and algebras over finite fields, using generating functions and representation theory. It proves that as dimension n → ∞ with fixed r-dimensional invariant subspace, the proportion of cyclic matrices in the stabilizer of an r-dimensional subspace in GL(n,q) approaches 1 − q⁻² + O(q⁻³), a significant shift from the 1 − q⁻³ + O(q⁻⁴) limit in irreducible settings.

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 $q^{-2} + O(q^{-3})$.

Motivation & Objective

• To determine the limiting proportion of cyclic matrices in maximal reducible matrix groups and algebras over finite fields, particularly when the invariant subspace has fixed dimension r.
• To extend the Cyclic Meataxe algorithm beyond irreducible matrix algebras to handle large reducible cases by analyzing cyclic matrix distributions.
• To provide a rigorous generating function-based analysis of cyclic matrix proportions in stabilizers of r-dimensional subspaces in GL(n,q) and M(n,q).
• To construct explicit families of noncyclic matrices in maximal reducible groups to establish lower bounds on their proportion.
• To compare the asymptotic behavior of cyclic matrix proportions in reducible vs. irreducible settings, highlighting the change in leading-order term from q⁻³ to q⁻².

Proposed method

• Adapts G.E. Wall’s generating function approach to the context of maximal reducible groups and algebras, focusing on stabilizers of fixed r-dimensional subspaces.
• Uses representation-theoretic decomposition of matrices into generalized eigenspaces and analyzes the action on complementary subspaces U and V.
• Applies the Principle of Inclusion-Exclusion to estimate the size of the union of sets Tλ (matrices with a specific eigenvalue λ in the invariant subspace).
• Employs bounds on the number of tuples (w₁, w₂, x₁, x₂, T) with distinct eigenvalues to estimate |Tλ ∩ Tγ|, leading to an upper bound of q^{n² + r² − nr − 6} on the intersection size.
• Derives explicit generating functions for small r and uses asymptotic analysis to extract the leading terms of the limiting proportion.
• Constructs a family of noncyclic matrices by considering matrices with eigenvalues λ ∈ F_q^* acting on the invariant subspace, showing their proportion is q⁻² + O(q⁻³).

Experimental results

Research questions

• RQ1 What is the limiting proportion of cyclic matrices in the stabilizer of an r-dimensional subspace in GL(n,q) as n → ∞ with r fixed?
• RQ2 How does the asymptotic proportion of cyclic matrices in maximal reducible groups compare to that in irreducible groups or full matrix algebras?
• RQ3 Can a family of noncyclic matrices be explicitly constructed in maximal reducible matrix groups, and what is their asymptotic proportion?
• RQ4 What is the role of the invariant subspace dimension r in shaping the asymptotic distribution of cyclic matrices in reducible matrix groups?
• RQ5 How do generating function techniques adapted from Wall’s work apply to the reducible case, and what modifications are needed?

Key findings

• The limiting proportion of cyclic matrices in the stabilizer of an r-dimensional subspace in GL(n,q) is 1 − q⁻² + O(q⁻³), with the leading correction term now at q⁻² instead of q⁻³.
• In contrast to the irreducible case (1 − q⁻³ + O(q⁻⁴)), the reducible setting exhibits a different asymptotic behavior due to the presence of invariant subspaces.
• A family of noncyclic matrices is constructed whose proportion in the maximal reducible group is q⁻² + O(q⁻³), matching the leading-order term of the noncyclic proportion.
• The size of the intersection |Tλ ∩ Tγ| for distinct eigenvalues λ, γ is bounded by q^{n² + r² − nr − 6}, which is crucial for bounding the inclusion-exclusion terms.
• The analysis confirms that the proportion of cyclic matrices in maximal reducible matrix algebras over F_q also tends to 1 − q⁻² + O(q⁻³) as n → ∞ with r fixed.
• The results support the extension of the Cyclic Meataxe algorithm to reducible matrix groups by establishing quantitative bounds on cyclic matrix density in such settings.