Computing with characters of finite groups

Abstract

Character tables of finite groups are an important tool in Representation Theory and are particularly important also for applications. The character table of a finite group (in this paper this will always mean the table of irreducible characters over C, in contrast to Brauer-character tables to be discussed later) contains a wealth of information on the group usually in a relatively compact form. The latter is true in particular for simple groups; even for large groups where it is completely out of the question to compute with elements or even store them in a computer, the character table might still be easily manageable. For instance the Monster group (also called Fischer Griess group or ”friendly giant”) has about 10⁵⁴ elements but its character table is just a 194 × 194 matrix. To make full use of the information encoded in the character table it is usually necessary to perform calculations which for larger examples one certainly would not like to do without the help of a computer. So it is certainly useful to have character tables not only in book form but also stored on a computer.