Asymptotic of Number of Similarity Classes of Commuting Tuples

Abstract: We have for positive integers $n$, $k$ and finite field $\mathbb{F}_q$, $c(n,k,q)$, as the number of simultaneous similarity classes of $k$-tuples of commuting $n\times n$ matrices over the $\mathbb{F}_q$. In this paper, it has been shown that $c(n,k,q)$ as a function of $k$ for fixed $n$ and $q$ is asymptotically $q^{m(n)k}$, where $m(n) = \left[\frac{n^2}{4}\right] + 1$, which is the dimension of the maximal commutative subalgebra of $M_n(\mathbb{F}_q)$ (the algebra of $n\times n$ matrices over $\mathbb{F}_q$).