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

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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_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}.

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 giving size Size of conjugacy class Degree of polynomial giving size
1 1 q - 1 q - 1 1 1 0
2 2 (q^2 - 1)(q^2 - q) (q^2 - 1)(q^2 - q) 4 1 0
2 1 + 1 (q^2 - 1)(q^2 - q) (q - 1)^2 2 q(q + 1) 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
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
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
n n \prod_{i=0}^{n-1} (q^n - q^i) \prod_{i=0}^{n-1} (q^n - q^i) n^2 1 0
n 1 + 1 + \dots + 1 \prod_{i=0}^{n-1} (q^n - q^i) (q - 1)^n n q^n(q + 1)(q^2 + q + 1)(q^3 + q^2 + q + 1) \dots (q^{n-2} + q^{n-1} + \dots + q + 1) n(n - 1)