Herbert Pahlings

Computational aspects of representation theory of finite groups

In many applications of representation theory of finite groups numerical computations for particular groups are called for. Although there are cases where one has to construct matrices for representations, in the majority of cases it is sufficient to work with characters, in fact this seems to be the only way to deal with many problems for larger groups.

Computing with characters of finite groups

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.

Diagonal-complete Latin squares

We call a Latin square A = (aij) of order n, aij ∈ {1, 2, ..., n}, right-diagonal-complete if {(aij, ai+1,j+1) : 1 ≤ i,j ≤ n} = {(i,j) : 1 ≤ i,j ≤ n} where the indices are periodic mod n. Left-diagonal-completeness is defined similarly. A Latin square is called diagonal-complete if it is right- and left-diagonal-complete. It is shown that diagonal-complete Latin squares of order n = 4m always exist; for n = 4m + 2 no diagonal-complete-Latin squares based on a group exist. For n ∈ {21, 25, 27, 49, 81} we found diagonal-complete Latin squares by computer search. This search also showed that for n = 9 there exist right-diagonal-complete Latin squares but no diagonalcomplete Latin squares based on a group. For n ∈ {2, 3, 5, 6, 7}, there is no Latin square which is right-diagonal-complete or left-diagonal-complete.