# Difference between revisions of "Conjugacy class size formula in general linear group over a finite field"

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! Item !! Value !! Degree as polynomial in <math>q</math> || Largest power of <math>q</math> in polynomial for value !! Largest power of <math>q - 1</math> in polynomial for value | ! Item !! Value !! Degree as polynomial in <math>q</math> || Largest power of <math>q</math> in polynomial for value !! Largest power of <math>q - 1</math> in polynomial for value | ||

|- | |- | ||

− | | Order of centralizer || <math>q^{\sum_{i=1}^k \binom{r_i}{2}} \prod_{i=1}^k \prod_{j=0}^{r_i} (q^{r_i - j} - 1) || <math>\sum_{i=1}^k r_i^2</math> || <math>\sum_{i=1}^k \binom{r_i}{2} = \frac{1}{2} \left(\sum_{i=1}^k r_i^2 \right) - \frac{n}{2} || <math>n</math> | + | | Order of centralizer || <math>q^{\sum_{i=1}^k \binom{r_i}{2}} \prod_{i=1}^k \prod_{j=0}^{r_i} (q^{r_i - j} - 1)</math> || <math>\sum_{i=1}^k r_i^2</math> || <math>\sum_{i=1}^k \binom{r_i}{2} = \frac{1}{2} \left(\sum_{i=1}^k r_i^2 \right) - \frac{n}{2}</math> || <math>n</math> |

|- | |- | ||

− | | Size of conjugacy class (obtained as order of <math>GL(n,q)</math> divided by order of centralizer) || (complicated polynomial, but it is the <math>q</math>-analogue of the binomial coefficient and is written as <math>\binom{n}{r_1,r_2,\dots,r_k}_q</math>) || <math>\frac{1}{2}\left(n^2 - \sum_{i=1}^k r_i^2 \right)</math> || 0 | + | | Size of conjugacy class (obtained as order of <math>GL(n,q)</math> divided by order of centralizer) || (complicated polynomial, but it is the <math>q</math>-analogue of the binomial coefficient and is written as <math>\binom{n}{r_1,r_2,\dots,r_k}_q</math>) || <math>n^2 \ sum_{i=1}^k r_i^2</math> || <math>\frac{1}{2}\left(n^2 - \sum_{i=1}^k r_i^2 \right)</math> || 0 |

|- | |- | ||

− | | Number of such conjugacy classes || <math>\binom{q - 1}{k} \binom{k}{s_1, s_2, \dots, s_n}</math || <math>k</math> || 0 || 1 | + | | Number of such conjugacy classes || <math>\binom{q - 1}{k} \binom{k}{s_1, s_2, \dots, s_n}</math> || <math>k</math> || 0 || 1 |

|} | |} | ||

## Revision as of 16:17, 8 July 2019

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.

## Case of semisimple elements

### Elements diagonalizable over

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 .

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

Order of centralizer | ||||

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 | ||

Number of such conjugacy classes | 0 | 1 |

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

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:

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

Order of centralizer | 0 | |||

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

Number of such conjugacy classes | (where is a necklace polynomial and is the characteristic function at 1; see explanation below) |

Thenecklace polynomial is also the number of irreducible monic polynomials of degree over .

The formula for is as follows:

The definition of is as follows: is the function that takes the value 1 at 1 and 0 elsewhere.

The number of irreducible polynomials corresponding to entries in is thus:

In other words, it equals when and when .

The reason we need to subtract 1 in the special case is in order to eliminate the case of the irreducible polynomial , whose root is 0. For , the root of the irreducible polynomial must be nonzero and hence invertible, so there is no need to subtract 1.

Some particular values of necklace polynomials for a few cases of are below:

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

1 | ||

a prime number | ||

a power of a prime |

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 |

### General semisimple case

The general semisimple case combines ideas from both the preceding cases (the diagonalizable over case, and the regular semisimple case), and generalizes both 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 formula for the order of the centralizer:

The size of the conjugacy class can be calculated by dividing by the order of the centralizer as given above.

The formula for the number of conjugacy classes can now be given in terms of necklace polynomials :

Here, we define the necklace polynomial as follows:

## 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

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

### Most general case

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