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

From Groupprops
Jump to: navigation, search

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.

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_is 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_1s occur first, then the \lambda_2s, 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_is measure sizes of clusters of r_js 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_is equal 1.

Some particular cases for the partition of n as a sum of r_is, 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.

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]