Gerhard Hiss

CHEVIE — A system for computing and processing generic character tables

CHEVIE is a computer algebra package which collects data and programs for the representation theory of finite groups of Lie type and associated structures. We explain the theoretical and conceptual background of the various parts of CHEVIE and we show the usage of the system by means of explicit examples. More precisely, we have sections on Weyl groups and Iwahori-Hecke algebras, generic character tables of series of finite groups of Lie type, and cyclotomic algebras.

Algorithmic Algebra and Number Theory

The aim of this article is to describe a computational approach to the study of the arithmetic of modular curves Xo(N) and to give applications of these computations.

CORRIGENDA: LOW-DIMENSIONAL REPRESENTATIONS OF QUASI-SIMPLE GROUPS

This paper contains corrections to the tables of low-dimensional representations of quasi-simple groups published in the paper, ‘Lowdimensional representations of quasi-simple groups’, LMS Journal of Computation and Mathematics4 (2001) 22‐63. In our paper ‘Low-dimensional representations of quasi-simple groups’ , we determine all the absolutely irreducible representations of quasi-simple groups of dimension at most 250, excluding those of groups of Lie type in their defining characteristic. Martin Liebeck has kindly pointed out to us three omissions in our tables: the 12- and 13dimensional representations of the group L3.3/, and the 248-dimensional representations of L4.5/ in characteristic 2. When checking our arguments and calculations we realized that in fact all the representations of L3.3/ were missing, as well as the representations of L4.5/ of dimension exceeding 247. The absolutely irreducible representations of L 3.3/ can be found in the modular Atlas [7]. This leads to the first part of Table 1 below.

Modular representation of finite groups of Lie type in non-defining characteristic

Let us consider a connected reductive algebraic group G, defined over the finite field F q with corresponding Frobenius morphism F. We are concerned here with properties of finite-dimensional modules for the finite group G F over a sufficiently large field k of characteristic l where l is a prime not dividing q.

Hermitian function fields, classical unitals, and representations of 3-dimensional unitary groups

Abstract We determine the elementary divisors, and hence the rank over an arbitrary field, of the incidence matrix of the classical unital.

3-blocks and 3-modular characters of G2(q)

This paper continues the investigations of the modular characters of the finite Chevalley groups G = G,(q). In a series of papers [7-10,5] the authors have investigated the p-blocks, the Brauer trees, and the p-modular characters in characteristics p different from 2 and 3. In the present paper we deal with the case p = 3 and q not divisible by 3. The case of p = 2 and odd q will be considered in a subsequent paper. We determine the distribution of the ordinary characters of G into 3-blocks, the defect groups, and for blocks with cyclic defect groups, the Brauer trees. For blocks with non-cyclic defect groups all but two of the irreducible Brauer characters are determined. The results are sufficient to find the minimal degree of a faithful 3-modular representation of G. Throughout our paper we have to distinguish between the two cases q = 1 (mod 3) and q E -1 (mod 3). The ordinary characters of G are taken from [4,2]. Our notation is that of Chang and Ree in [2]. The blocks are determined by using the method of central characters. Using lemmas from [8,9] the distribution into blocks and the exceptional characters are calculated. The Brauer trees for blocks with cyclic defect groups are then determined. It turns out that there is a unique block of maximal defect, the principal block. The defect groups of all the remaining blocks are abelian. 371

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.

Representations of Heeke Algebras and Finite Groups of Lie Type

R. Brauer posed in his famous lectures on modern mathematics 1963 more than 40 problems, questions and conjectures about the representation theory of finite groups, cf. [10]. These questions have had a big influence on the development of the theory since then. Most of the problems may be summarized under the following central question.

Autoequivalences of blocks and a conjecture of Zassenhaus

Abstract In this paper, we show that for every finite group with cyclic Sylow p-subgroups the principal p-block B is rigid with respect to the trivial simple module. This means that each autoequivalence which fixes the trivial simple module fixes the isomorphism class of each finitely generated B-module. As a consequence each augmentation preserving automorphism of the integral group ring of PSL(2, p), p a rational prime, is given by a group automorphism followed by a conjugation in QPSL(2, p). In particular this proves a conjecture of Zassenhaus for these groups. Finally we show the same statement for a couple of other simple groups by different methods.

The 5-modular characters of the sporadic simple fischer groups fi22 and fi23

The 5-modular decomposition numbers of the two simple Fischer groups Fi22and Fi23 and also those of the two maximal subgroups2. Fi22 and of Fi23 are determined. In an appendix Thomas Breuer comments on his construction of the ordinary character tables of the groups , which occur as maximal subgroups of 2. Fi22respectively Fi23- The results were obtained with the help of the computer algebra systems MOC [4] and GAP [10].