Klaus Lux

Peakword Condensation and Submodule Lattices

We describe a new condensation method for computing the submodule lattice of a module for a finite dimensional algebra over a finite field, which exploits the idea of condensation and extends it to the case of primitive idempotents. The method has been implemented in the new C version of the MEAT-AXE developed at Aachen, and we give several examples which have been analysed with our method.

Treating the exceptional cases of the MeatAxe

We show that the Holt–Rees extension of the standard MeatAxe procedure finds submodules of modules over finite algebras with positive probability in more cases than originally claimed. For the case when the Holt–Rees method fails we propose a further, but still simple and efficient extension.

The decomposition numbers of the hecke algebra of typeF 4

LetW be the finite Coxeter group of typeF4, andHr(q) be the associated Hecke algebra, with parameter a prime powerq, defined over a valuation ringR in a large enough extension field ofQ, with residue class field of characteristicr. In this paper, ther-modular decomposition numbers ofHR(q) are determined for allq andr such thatr does not divideq. The methods of the proofs involve the study of the generic Hecke algebra of typeF4 over the ringA = ℤ[u1/2,u-1/2] of Laurent polynomials in an indeterminateu1/2 and its specializations onto the ring of integers in various cyclotomic number fields. Substancial use of computers and computer program systems (GAP, MAPLE, Meat-Axe) has been made.

Distance-transitive representations of the sporadic groups

A permutation representation of a finite group is multiplicity-free if all the irreducible constituents in the permutation character are distinct. There are three main reasons why these representations are interesting: it has been checked that all finite simple groups have such permutation representations, these are often of geometric interest, and the actions on vertices of distance-transitive graphs are multiplicity-free. In this paper we classify the primitive multiplicity-free representations of the sporadic simple groups and their automorphism groups. We determine all the distance-transitive graphs arising from these representations. Moreover, we obtain intersection matrices for most of these actions, which are of further interest and should be useful in future investigations of the sporadic simple groups.

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.

Determination of socle series using the condensation method

We show how the condensation method introduced by R. A. Parker can be applied to determine the socle series of a finite-dimensional representation of a group over a finite field. We develop several new techniques for this approach and illustrate their power by the example of the socle series of all projective indecomposable representations of the sporadic simple Mathieu group M23in characteristic 2.

Computing Homomorphism Spaces between Modules over Finite Dimensional Algebras

We describe an algorithm to compute homomorphism spaces between modules of finite dimensional algebras over finite fields. The algorithm is implemented in the C-Meat-Axe.

The Brauer Tree of the Principal 19-Block of the Sporadic Simple Thompson Group

This paper completes the construction of the Brauer tree of the sporadic simple Thompson group in characteristic 19. Our main computational tool to arrive at this result is a new parallel implementation of the DirectCondense method.

The multiplicity-free permutation characters of the sporadic simple groups and their automorphism groups

In this paper, we classify all the multiplicity-free permutation characters of sporadic simple groups and their automorphism groups. This project is an application of the group theory system GAP, its character table library, and its library of tables of marks.

Computing decompositions of modules over finite-dimensional algebras

Based on our method for determining endomorphism rings [Lux and Szőoke 03], we describe an algorithm to compute decompositions of modules of finite-dimensional algebras over finite fields. The algorithm is implemented in the C-Meat-Axe [Ringe 94].