Frank Lübeck

Numbers of Conjugacy Classes in Finite Groups of Lie Type

This page provides for some series of finite groups of Lie type explicit numbers of certain conjugacy classes or elements.

Each series of groups is parameterized by a parameter q, which either runs through the set of all prime powers (the cardinalities of all finite fields) or for which q² runs through all odd powers of 2, respectively 3. It turns out that all the numbers given here can be described by a finite number of polynomials which must be evaluated at the parameter q.

Many of the numbers given here can be extracted from published papers, sometimes they are explicitly given and sometimes they can in principle be computed from given data. Actually, all given numbers were obtained independently with computer programs written by the page author. These are based on the CHEVIE package for GAP 3.

The Series of Groups

Data on the following series of groups are available, choose for which of them you want to get some information:
(select all listed groups)
Suzuki Groups Sz(q²) = ²B₂(q²), q² = 2^2m+1 ,
Ree Groups ²G₂(q²), q² = 3^2m+1,
G₂(q),
Ree Groups ²F₄(q²), q² = 2^2m+1,
³D₄(q),
F₄(q),
E₆(q)_sc of simply connected type,
E₆(q)_ad of adjoint type,
²E₆(q)_sc of simply connected type,
²E₆(q)_ad of adjoint type,
E₇(q)_sc of simply connected type,
E₇(q)_ad of adjoint type,
E₈(q).

Available Data

For all series of groups listed above the following data are available, choose which you want to see:
(select all available data)
The total number of conjugacy classes,
the number of semisimple conjugacy classes,
the number of semisimple elements,
the number of regular semisimple conjugacy classes,
the number of regular semisimple elements,
references on conjugacy classes.

Polynomial Display

Many numbers provided here are given in form of polynomials evaluated at the parameter q. You have several choices how these polynomials should be displayed.

HTML (Web page format)
GAP/Maple input
LaTeX input
images

Last updated: Wed Jul 21 00:20:31 2004 (CET)