I am searching for either (1) a reference of what's currently known or (2) a general outline to the approach to answering the following question. (I will also accept special cases, such as the case $d=1$, though I don't know if that really helps.)

Let $\mathbb{F}_q$ denote the finite field of $q=p^d$ elements. The root question is: what is the size of the conjugacy class of a given matrix $A \in \mathrm{GL}_n(\mathbb{F}_q)$?

Besides writing out the generic form of a conjugate of $A$ and trying to count them, I've made attempts by switching to centralizers by the orbit-stabilizer theorem. If $C(A)$ is the centralizing subgroup for $A$, we have that the size of the conjugacy class of $A$ is the index $[\mathrm{GL}_n(\mathbb{F}_q): C(A)]$. Since the order of $\mathrm{GL}_n(\mathbb{F}_q)$ is known, this gives my answer if only I can compute the order of $C(A)$. This is where I'm stuck. I've also tried thinking about canonical forms, but encounter field problems. It seems like working out who commutes with $A$ should be something known, but maybe this is deceptively difficult. Or, as in my usual style, I have missed something trivial or obvious.

Any references to the literature or paths to approach this question are appreciated.

  • 1
    $\begingroup$ Isn't this in J A Green's paper on the characters of this group? Or in Chaper IV of Macdonald's book on Symmetric Functions? $\endgroup$ May 28, 2019 at 16:21
  • $\begingroup$ @ancientmathematician I've no idea, hence my question. Do you have the title of the Green paper handy? $\endgroup$
    – Randall
    May 28, 2019 at 17:48
  • $\begingroup$ Found it as "Green, J. A. The characters of the finite general linear groups. Trans. Amer. Math. Soc. 80 (1955), 402--447." but I can't tell that I know how it helps me. $\endgroup$
    – Randall
    May 28, 2019 at 17:51
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    $\begingroup$ The conjugacy classes in ${\rm GL}_n(\Bbb F_q)$ ought to correspond to isomorphism classes of $\Bbb F_q[T]$-modules which are $n$-dimensional over $\Bbb F_q$, which are classified by the Fundamental Theorem of Finitely Generated Modules over PIDs. Centralizers correspond to automorphism groups associated with these isomorphism classes, and we can characterize these automorphisms as certain block matrices. $\endgroup$
    – anon
    May 28, 2019 at 23:05
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    $\begingroup$ There is an explicit formula for the orders of the centralisers in the version in Ian Macdonald's book, and he claims to be following Green in Ch IV of his book. $\endgroup$ May 29, 2019 at 6:40


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