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 $$GL(n,q)$$.

See also element structure of general linear group over a finite field.

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.

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 $$n$$.

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.