This is a database of character tables of endomorphism rings. Let G be a finite group, K a field and M a finite set on which G acts transitively. For a in M let M_{1}={a},M_{2},...,M_{r} be the distinct orbits of G_{a}, which have respective representatives a_{1}=a, a_{2},..., a_{r}. Let E_{i} be an orbital for G as a subset of MxM. For 1 <= i <= r let A_{i}[k,l] be the collapsed adjacency matrix for the orbital digraph (M,E_{i}). Therefore A_{i} is defined as the number of neighbours of a_{k} in M_{l} (see PrSoi for details).

Let R denote the endomorphism ring End_{KG}(KM) with Schurbasis S={S_{1},...,S_{k}}. Assume that R is abelian. Then the character table of R relative to S can be obtained by computing simultaneous eigenvectors of collapsed adjacency matrices.

The entries of a column of the character table are the eigenvalues of the corresponding orbital digraph (see PrSoi for details).

A regular graph X of degree k is called a Ramanujan graph if max { |b| | b is an eigenvalue of X with |b|< k } is less than or equal 2(k-1)^{1/2} (see CePoTeTrVe for details).

For rank up to 5 the collapsed adjacency matrices have been computed by Cheryl E. Praeger and Leonard H. Soicher (PrSoi). Several matrices (also for larger rank cases) can be found in IvLiLuSaSoi, where numerous further references are given.) The following matrices originally have been published in:

LLS: Fi_{23} with 2^{11}.M_{23}

Soi: Co_{1} with 2_{+}^{1+8}.O_{8}^{+}(2)

IM: J_{4} with 2^{11}.M_{24}

Nor: M with 2.BM.

All programs used to produce this data base are written in GAP4.

If you have suggestions for improving this data base please feel free to send an e-mail message to ines.hoehler@math.rwth-aachen.de. This may concern for example known character tables that are missing here, possible errors or the layout of the pages.

If you are interested in collapsed adjacency matrices, in a character table, in informations about the permutation character or in finding Ramanujan graphs, go to the section Table of Collapsed Adjacency Matrices, Character Tables and Ramanujan Graphs.

The methods how the collapsed adjacency matrices and the character tables were computed and the program which tests if a graph is a Ramanujan graph are described in the section Computing collapsed adjacency matrices and character tables. There you will also find links to the GAP-programs used.

You will also find GAP-readable data bases. You can use them to compute with the collapsed adjacency matrices and character tables presented here.

Some cases are left open. These are listed in the section Unsolved Cases.

The file also contains the numbers of the ordinary characters referring to the character table of the group and the character table itself. The lines of the character table are corresponding to the numbers of the ordinary characters arranged.

Finally, the columns which produce Ramanujan graphs are given. The numbers refer to the columns of the character table. More than one number in parenthesis means that the graph corresponding to the union of the columns with the shown numbers is considered. (For the subgroup 2

Some large character tables are printed in landscape format. In these cases a PostScript-file

B | Co_{1} |
Co_{2} |
Co_{3} |
Fi_{22}
| Fi_{23} |
Fi'_{24} |
He | |

HN | HS | J_{1} |
J_{2} |
J_{3} |
J_{4} |
Ly | M | |

M_{11} |
M_{12} |
M_{22} |
M_{23} |
M_{24} |
McL | O'N | Ru | Suz |

For the transitive actions of the sporadic simple groups and their automorphism groups with **rank 2**, a generic form of the collapsed adjacency matrices and the character tables exists. See these generic forms in a dvi-file.

For the transitive actions of the sporadic simple groups and their automorphism groups of **ranks 3 up to 5**, all collapsed adjacency matrices have been computed by Cheryl E. Praeger and Leonard H. Soicher and are given in PrSoi. For computing the collapsed adjacency matrices of **rank > 5** a permutation representation has to be found. For ** M _{11}, M_{12}, J_{1}, M_{22}, J_{2}, M_{23}, HS, J_{3}, M_{24}, McL, He and their automorphism groups (except HS.2)** the program MultFreeFromTOM computes a permutation representation of the groups with the help of the TableOfMarks, which is implemented in GAP. For the other cases the permutation representations can be found in Rob Wilson's Atlas of Group Representations, or have been computed by Jürgen Müller.

Using these permutation representation the program CompCAM computes the collapsed adjacency matrices. The input is the permutation representation and the number of points the group is acting on.

With the input of a list of the collapsed adjacency matrices the program CAMToChar computes the character table in each case. The program CharToLaTeXChar is a program which returns a LaTeX-readable character table.

The information on which graphs are Ramanujan graphs is computed by the GAP-program RamanujanTest. The input has to be a character table, the output is a list of all combinations of columns of the character table for which the corresponding graph is Ramanujan.

e.g. M11Database[3].sbgname is "3^2:8".

You can view the complete data base with all available informations, but you can also view data bases for singular groups:

Complete Data base |

B | Co_{1} |
Co_{2} |
Co_{3} |
Fi_{22}
| Fi_{23} |
Fi'_{24} |
He | |

HN | HS | J_{1} |
J_{2} |
J_{3} |
J_{4} |
Ly | M | |

M_{11} |
M_{12} |
M_{22} |
M_{23} |
M_{24} |
McL | O'N | Ru | Suz |

Great thanks to all people of Lehrstuhl D for fantastic help, especially Jürgen Müller for computing numerous permutation representations and Thomas Breuer for the programs MultFreeFromTOM and CompCAM and for his help with GAP-problems.

File created on 2nd May 2001

Last modified: 14th May 2001 by Ines Hoehler