# Number of conjugacy classes in unitriangular matrix group of fixed degree over a finite field is polynomial function of field size

From Groupprops

## Contents

## Statement

Suppose is a natural number. Then, there exists a polynomial function of degree such that, for any prime power , the number of conjugacy classes in the unitriangular matrix group (i.e., the unitriangular matrix group of degree over the finite field of size ) is .

## General observations

Below, we list some general observations about the polynomial in giving number of conjugacy classes in .

Item | Value |
---|---|

Degree of polynomial | |

Leading coefficient of polynomial | positive, value unclear |

Factors of polynomial | no common factors |

Coefficients of polynomial | Looks like all coefficients are integers |

## Particular cases

(degree of unitriangular matrix group) | (degree of polynomial) | polynomial of giving number of conjugacy classes in | More information |
---|---|---|---|

1 | 0 | 1 | trivial group |

2 | 1 | isomorphic to the additive group of the field, hence abelian of order | |

3 | 2 | See element structure of unitriangular matrix group of degree three over a finite field and linear representation theory of unitriangular matrix group of degree three over a finite field | |

4 | 3 | See element structure of unitriangular matrix group of degree four over a finite field and linear representation theory of unitriangular matrix group of degree four over a finite field | |

5 | 4 | See element structure of unitriangular matrix group of degree five over a finite field and linear representation theory of unitriangular matrix group of degree five over a finite field | |

6 | 5 | ? | |

7 | 6 | ? |

There is probably some general formula for these polynomials for all degrees without having to do the entire analysis of element structure and conjugacy classes, but it's unclear what this formula is.

## Related facts

### Stronger facts

- Degrees of irreducible representations of unitriangular matrix group of fixed degree over a finite field are all powers of field size and number of occurrences of each is polynomial function of field size
- Conjugacy class sizes of unitriangular matrix group of fixed degree over a finite field are all powers of field size and number of occurrences of each is polynomial function of field size

### Other facts about number of conjugacy classes

- Number of conjugacy classes in general linear group of fixed degree over a finite field is polynomial function of field size
- Number of conjugacy classes in special linear group of fixed degree over a finite field is PORC function of field size
- Number of conjugacy classes in projective general linear group of fixed degree over a finite field is PORC function of field size
- Number of conjugacy classes in projective special linear group of fixed degree over a finite field is PORC function of field size