# 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 the original field

#### Setup

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

Further, let's say that among the s, there are 1s, 2s, and so on till s.

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 .

In particular, we have:

#### Summary information

Item | Value | Degree as polynomial in | Largest power of in polynomial for value | Largest power of in polynomial for value | Leading coefficient |
---|---|---|---|---|---|

Order of centralizer | 1 | ||||

Size of conjugacy class (obtained as order of divided by order of centralizer) | complicated polynomial, but it is the -analogue of the binomial coefficient and is written as | 0 | 1 | ||

Number of such conjugacy classes | 0 | 1 |

#### Examples of conjugacy classes diagonalizable over

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

Partition of | Order of centralizer of diagonal element | Degree as polynomial of (equals ) | Size of conjugacy class | Degree as polynomial of (equals ) | Number of conjugacy classes (degree polynomial in ) | |||
---|---|---|---|---|---|---|---|---|

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

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

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

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

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

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

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

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

#### Setup

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.

Further, let's say that among the s, there are 1s, 2s, and so on till s.

In particular, we have:

#### Summary information

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

Item | Value | Degree as polynomial in | Largest power of in polynomial for value | Largest power of in polynomial for value | Leading coefficient |
---|---|---|---|---|---|

Order of centralizer | 0 | 1 | |||

Size of conjugacy class (obtained as order of divided by order of centralizer) | (complicated polynomial) | 1 | |||

Number of such conjugacy classes | (where is a necklace polynomial and is the characteristic function at 1; see explanation below) | number of with nonzero , i.e., the number of distinct parts in the partition of . It is less than or equal to | (this is the fraction of elements in the symmetric group whose cycle decomposition matches the partition) |

#### Most elements are regular semisimple

As noted in the summary table above:

- For each partition of , the number of conjugacy classes is a polynomial of degree and the size of each conjugacy class is a polynomial of degree . In total, therefore, the number of elements corresponding to that partition is a polynomial of degree in .
- The leading coefficient of that polynomial is the product of the leading coefficient for the size of conjugacy class polynomial (which is 1) and the number of conjugacy classes (which is the fraction of elements in the symmetric group whose cycle decomposition corresponds to that partition).
- Therefore, the total number of regular semisimple elements is a polynomial of degree whose leading coefficient is 1 (this is the sum of "the fraction of elements in the symmetric group whose cycle decomposition corresponds to that partition" over all partitions).
- The polynomial giving the size of is also a polynomial of degree with leading coefficient 1.
- Thus, the number of elements that are not regular semisimple is the difference of the two polynomials, which is a polynomial of degree strictly less than .

Therefore, as , holding constant, the fraction of elements that are regular semisimple goes to 1. Or, in simple terms, *most elements are regular semisimple*.

This statement also has an algebraic interpretation when we consider the analogous statement over an infinite algebraic group. Being regular semisimple corresponds to the discriminant of the characteristic polynomial being nonzero. The discriminant of the characteristic polynomial can be written as a polynomial function of the coefficients of the characteristic polynomial (which are the elementary symmetric functions) and hence of the matrix entries. Therefore, the set of regular semisimple elements is an open and hence dense subset, and therefore comprises most elements.

For large enough and , the following is true: among the regular semisimple elements, the partition , which corresponds to the case of elements with irreducible characteristic polynomial, gives rise to the largest number of conjugacy classes. Its leading coefficient is . In contrast, the partition , corresponding to the case of elements diagonalizable over , gives the fewest number of conjugacy classes; the leading coefficient is .

Finally, note that not only are most elements regular semisimple, most conjugacy classes are also regular semisimple, for very similar reasons.

#### Derivation of number of conjugacy classes in the regular semisimple case

Step no. | Statement | Justification |
---|---|---|

1 | The necklace polynomial is the number of irreducible monic polynomials of degree over . The formula for is as follows: | Define . We have, by element-counting, that . Apply Mobius inversion to get and then rewrite in terms of |

2 | The number of irreducible polynomials of degree over with invertible roots is , where is the necklace polynomial as defined in step (1), and is the function that is 1 if and zero otherwise. | When , the irreducible polynomials include the polynomial with 0 as a root. This is not invertible, so in this case we need to subtract 1 from to get the number of polynomials with invertible roots. For , the roots are outside , hence all nonzero, and hence invertible, so nothing needs to be subtracted. |

3 | The total number of conjugacy classes is | For each , there are distinct irreducible polynomials with invertible roots (distinctness follows from the regularity assumption). In other words, we need to select elements out of a pool of possibilities. The number of ways to select these elements is . Multiplying this expression over gives the number of possible conjugacy classes. |

#### Example necklace polynomial values

Value of | ||
---|---|---|

1 | ||

a prime number | ||

a power of a prime |

#### Examples of regular semisimple conjugacy classes

Note that in the particular cases of the partition , this simplifies to ; also this special case is the only case that overlaps with the previous section.

Partition of | Order of centralizer of representative (degree polynomial in ) | Size of conjugacy class (degree polynomial in ) | Number of conjugacy classes (degree polynomial in ) | ||
---|---|---|---|---|---|

1 | 1 | 1 | |||

2 | 2 | ||||

2 | 1 + 1 | ||||

3 | 3 | ||||

3 | 2 + 1 | ||||

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

4 | 4 | ||||

4 | 3 + 1 | ||||

4 | 2 + 2 | ||||

4 | 2 + 1 + 1 | ||||

4 | 1 + 1 + 1 + 1 |

### Scalar over a field extension (i.e., the conjugacy class is semisimple and the characteristic polynomial is a power of an irreducible, also known as a primary polynomial)

#### Setup

For any , we can consider the case of a semisimple element where the matrix can be expressed as a block matrix, with all blocks of size with irreducible characteristic polynomial, and all blocks equal to each other. The characteristic polynomial in this case is the power of an irreducible polynomial of degree , and the minimal polynomial is that degree polynomial.

The centralizer is:

#### Summary table

Item | Value | Degree as polynomial in | Largest power of in polynomial for value | Largest power of in polynomial for value | Leading coefficient |
---|---|---|---|---|---|

Order of centralizer | 1 | ||||

Size of conjugacy class | 1 | ||||

Number of conjugacy classes | , where is the necklace polynomial and is the characteristic function for 1 | 1 |

#### Derivation of number of conjugacy classes in this case

This can be justified using steps 1 and 2 of the derivation of number of conjugacy classes in the regular semisimple case.

#### Overlap with previous cases

- The case is the "regular semisimple" case corresponding to the partition , and explicitly, it corresponds to the conjugacy classes where the characteristic polynomial is irreducible.
- The case where is the "diagonalizable over " case corresponding to the partition , and explicitly, it corresponds to the conjugacy classes of scalar matrices.

#### Examples

Of these examples, the first case not covered by either of the "diagonalizable over " case and the "regular semisimple" case is where .

Order of centralizer of representative (degree polynomial in ) | Size of conjugacy class (degree polynomial in ) | Number of conjugacy classes (degree polynomial in ) | |||
---|---|---|---|---|---|

1 | 1 | 1 | |||

2 | 1 | ||||

2 | 2 | 1 | |||

3 | 1 | ||||

3 | 3 | 1 | |||

4 | 1 | ||||

4 | 2 | ||||

4 | 4 | 1 |

### General semisimple case

The general semisimple case combines ideas from all the preceding cases (the diagonalizable over case, the regular semisimple case, and the scalar over a field extension case), and generalizes all of them.

For a given conjugacy class in , and for define to be the number of distinct cases where the conjugacy class contains exactly conjugate elements of (i.e., a multiplicity of for an extension of degree over ).

We have:

The centralizer is then described as follows:

#### How the previous three cases are special cases of this general case

- The "diagonalizable over " case is the special case where for . Moreover, in this general notation matches up with in the notation for that case.
- The "regular semisimple" case is the special case where for . Moreover, in this general notation matches in the notation for that case.
- The "scalar over a field extension" case is the special case where there is only one nonzero . The value of equals and equals .

#### Summary table

We can verify that restricting the formulas in the table below to the special cases of "diagonalizable over ", "regular semisimple", and "scalar over a field extension" gives the formulas for their respective summary tables.

Item | Value | Degree as polynomial in | Largest power of in polynomial for value | Largest power of in polynomial for value | Leading coefficient |
---|---|---|---|---|---|

Order of centralizer | 1 | ||||

Size of conjugacy class | complicated polynomial, obtained by dividing by the order of the centralizer as given above. | 1 | |||

Number of conjugacy classes | where is the necklace polynomial | Number of for which there is at least one nonzero |

#### Examples

The smallest example that is not covered by any of the three preceding special cases ("diagonalizable over ", "regular semisimple", and "scalar over a field extension") is where , with and all other equal to zero.

## 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.

Suppose the Jordan block sizes are .

The centralizer for each Jordan block of size is the invertible matrices in the subalgebra generated by the Jordan block. This corresponds to the units in , and the size of this group is .

The order of the centralizer is therefore:

The size of the conjugacy class is obtained by dividing the order of the group by this expression.

For each partition:

The number of conjugacy classes associated with that partition can be calculated as follows. Suppose that, among the s, there are 1s, 2s, and so on. Then, the total number of conjugacy classes is:

The total number of conjugacy classes corresponding to a choice of is therefore:

where the sum is across all the possible partitions.

Partition of | (number of parts) | Order of centralizer of representative (equals , a degree polynomial over ) | Size of conjugacy class (degree polynomial over ) | Number of conjugacy classes (degree polynomial over ) | ||
---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | |||

2 | 2 | 1 | ||||

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

3 | 3 | 1 | ||||

3 | 2 + 1 | 2 | ||||

3 | 1 + 1 + 1 | 3 |

### General case with all eigenvalues over

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### Most general case

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