# 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 over a finite field with elements, which we denote by .

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

## Contents

## Case of semisimple elements

### Elements diagonalizable over

Suppose is diagonalizable over , with eigenvalues having multiplicities respectively (the s are all distinct). Note that .

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

In fact, if the diagonal entries are arranged so that all the s occur first, then the s, and so on, then the centralizer is the set of invertible block diagonal matrices with blocks of sizes .

The size of the conjugacy class is thus:

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

In all cases, this simplifies to a polynomial in , and its degree is .

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 .

The number of conjugacy classes corresponding to a partition of is given by an ordinary multinomial coefficient described as follows. Obtain a partition of as where where the s measure sizes of clusters of s with equal value. Then, the number of conjugacy classes is:

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

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

Partition of | Size of | Size of centralizer of diagonal element | Degree of polynomial | Size of conjugacy class | Degree of polynomial | Number of conjugacy classes (assuming ) | Degree of polynomial | |
---|---|---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | 0 | 1 | |||

2 | 2 | 4 | 1 | 0 | 1 | |||

2 | 1 + 1 | 2 | 2 | 2 | ||||

3 | 3 | 9 | 1 | 0 | 1 | |||

3 | 2 + 1 | 5 | 4 | 2 | ||||

3 | 1 + 1 + 1 | 3 | 6 | 3 | ||||

1 | 0 | 1 | ||||||

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

Some elements may be semisimple but not diagonalizable over , i.e., they can be diagonalized over a suitable field extension of . We begin by considering the *regular semisimple* case -- elements that can be diagonalized over some field extension of such that all their diagonal entries are pairwise distinct. We can show that these elements are precisely the ones that can be converted over to a block diagonal form for some partition , where the entry in block is diagonalizable with distinct diagonal entries over the field and no smaller field.

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

The size of the conjugacy class is thus:

For all the regular semisimple cases, the size of the centralizer is a degree polynomial and the size of the conjugacy class is a degree 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 .

Partition of | Size of | Size of centralizer of representative | Size of conjugacy class | Number of conjugacy classes | |
---|---|---|---|---|---|

1 | 1 | 1 | |||

2 | 2 | ||||

2 | 1 + 1 | ||||

3 | 3 | ||||

3 | 2 + 1 |

## Case of non-semisimple elements

### Regular elements with all eigenvalues over

We begin by considering a very easy class of non-semisimple elements: those where all the eigenvalues are over , 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.

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