# Conjugacy class size formula in general linear group over a finite field

This article gives formula(s) for the conjugacy class sizes in a general linear group of finite degree $n$ over a finite field with $q$ elements, which we denote by $GL(n,q)$.

## Case of semisimple elements

### Elements diagonalizable over $\mathbb{F}_q$

Suppose $g \in GL(n,q)$ is diagonalizable over $\mathbb{F}_q$, with eigenvalues $\lambda_1, \dots, \lambda_k$ having multiplicities $r_1,r_2,\dots,r_k$ respectively (the $\lambda_i$s are all distinct). Note that $\sum_{i=1}^k r_i = n$.

Then, the centralizer of the diagonal representative of this conjugacy class is isomorphic to:

$GL(r_1,q) \times GL(r_2,q) \times GL(r_3,q) \times \dots \times GL(r_k,q)$

In fact, if the diagonal entries are arranged so that all the $\lambda_1$s occur first, then the $\lambda_2$s, and so on, then the centralizer is the set of invertible block diagonal matrices with blocks of sizes $r_1, r_2, \dots, r_k$.

The size of the conjugacy class is thus:

$\frac{|GL(n,q)|}{\prod_{i=1}^k |GL(r_i,q)|}$

This is the same as the $q$-analogue of the multinomial coefficient:

$\binom{n}{r_1,r_2,\dots,r_k}_q$

In all cases, this simplifies to a polynomial in $q$, and its degree is $q^{n^2 - \sum_{i=1}^k r_i^2}$.

The conjugacy class type with the largest degree polynomial describing its size is the one where all diagonal entries are distinct -- in this case, the degree is $n(n - 1)$.

The number of conjugacy classes corresponding to a partition $r_1 + r_2 + \dots + r_k$ of $n$ is given by an ordinary multinomial coefficient described as follows. Obtain a partition of $k$ as $k = s_1 + s_2 + \dots + s_l$ where where the $s_i$s measure sizes of clusters of $r_j$s with equal value. Then, the number of conjugacy classes is:

$\binom{q - 1}{s_1,s_2,\dots,s_k,q-1-k} = \binom{q-1}{k} \binom{k}{s_1,s_2,\dots,s_k}$

This is a polynomial of degree $k$ in $q$. Thus, for large enough $q$, this number is also maximum when all the $r_i$s equal $1$.

Some particular cases for the partition of $n$ as a sum of $r_i$s, and the corresponding sizes, are given below.

$n$ Partition of $n$ Size of $|GL(n,q)|$ Size of centralizer of diagonal element Degree of polynomial Size of conjugacy class Degree of polynomial Number of conjugacy classes (assuming $q > k$) Degree of polynomial
1 1 $q - 1$ $q - 1$ 1 1 0 $q - 1$ 1
2 2 $(q^2 - 1)(q^2 - q)$ $(q^2 - 1)(q^2 - q)$ 4 1 0 $q - 1$ 1
2 1 + 1 $(q^2 - 1)(q^2 - q)$ $(q - 1)^2$ 2 $q(q + 1)$ 2 $\binom{q - 1}{2} = (q - 1)(q - 2)/2$ 2
3 3 $(q^3 - 1)(q^3 - q)(q^3 - q^2)$ $(q^3 - 1)(q^3 - q)(q^3 - q^2)$ 9 1 0 $q - 1$ 1
3 2 + 1 $(q^3 - 1)(q^3 - q)(q^3 - q^2)$ $(q^2 - 1)(q^2 - q)(q - 1)$ 5 $q^2(q^2 + q + 1)$ 4 $2\binom{q-1}{2} = (q - 1)(q - 2)$ 2
3 1 + 1 + 1 $(q^3 - 1)(q^3 - q)(q^3 - q^2)$ $(q - 1)^3$ 3 $q^3(q^2 + q + 1)(q + 1)$ 6 $\binom{q-1}{3} = (q - 1)(q - 2)(q - 3)/6$ 3
$n$ $n$ $\prod_{i=0}^{n-1} (q^n - q^i)$ $\prod_{i=0}^{n-1} (q^n - q^i)$ $n^2$ 1 0 $q - 1$ 1
$n$ $1 + 1 + \dots + 1$ $\prod_{i=0}^{n-1} (q^n - q^i)$ $(q - 1)^n$ $n$ $q^{\binom{n}{2}}\prod_{i=1}^{n-1} \left[\left(\sum_{j=0}^i q^j \right)\right]$ $n(n - 1)$ $\binom{q - 1}{n}$ $n$

### Regular semisimple elements not diagonalizable over the original field

Some elements may be semisimple but not diagonalizable over $\mathbb{F}_q$, i.e., they can be diagonalized over a suitable field extension of $\mathbb{F}_q$. We begin by considering the regular semisimple case -- elements that can be diagonalized over some field extension of $\mathbb{F}_q$ such that all their diagonal entries are pairwise distinct. We can show that these elements are precisely the ones that can be converted over $\mathbb{F}_q$ to a block diagonal form for some partition $r_1 + r_2 + \dots + r_k = n$, where the entry in block $r_i$ is diagonalizable with distinct diagonal entries over the field $\mathbb{F}_{q^{r_i}}$ and no smaller field.

In this case, the centralizer of the element in this block diagonal form is:

$\mathbb{F}_{q^{r_1}}^\ast \times \mathbb{F}_{q^{r_2}}^\ast \times \dots \times \mathbb{F}_{q^{r_k}}^\ast$

The size of the conjugacy class is thus:

$\frac{|GL(n,q)|}{\prod_{i=1}^k (q^{r_i} - 1)}$

For all the regular semisimple cases, the size of the centralizer is a degree $n$ polynomial and the size of the conjugacy class is a degree $n(n-1)$ polynomial.

The number of conjugacy classes of this sort is more complicated to describe. However, it is easy to see that the degree of this polynomial is also $n$.

$n$ Partition of $n$ Size of $|GL(n,q)|$ Size of centralizer of representative Size of conjugacy class Number of conjugacy classes
1 1 $q - 1$ $q - 1$ 1 $q - 1$
2 2 $(q^2 - 1)(q^2 - q)$ $q^2 - 1$ $q(q - 1)$ $q(q - 1)/2$
2 1 + 1 $(q^2 - 1)(q^2 - q)$ $(q - 1)^2$ $q(q + 1)$ $(q - 1)(q - 2)/2$
3 3 $(q^3 - 1)(q^3 - q)(q^3 - q^2)$ $q^3 - 1$ $(q^3 - q)(q^3 - q^2) = q^3(q - 1)^2(q + 1)$ $(q^3 - q)/2$
3 2 + 1 $(q^3 - 1)(q^3 - q)(q^3 - q^2)$ $(q^2 - 1)(q - 1)$ $q^3(q^3 - 1)$ $q(q - 1)^2/2$

## Case of non-semisimple elements

### Regular elements with all eigenvalues over $\mathbb{F}_q$

We begin by considering a very easy class of non-semisimple elements: those where all the eigenvalues are over $\mathbb{F}_q$, and where all distinct Jordan blocks correspond to distinct eigenvalues, i.e., they are regular elements. This means that the minimal polynomial coincides with the characteristic polynomial.