Gunter Malle

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.

Inverse Galois theory

I. The Rigidity Method.- II. Applications of Rigidity.- III. Action of Braids.- IV. Embedding Problems.- V. Rigid Analytic Methods.

Generation of classical groups

Each finite simple group other thanU3(3) can be generated by three of its involutions. In fact, each such group is generated by two elements, of which one is strongly real and the other is an involution.

Simple groups admit Beauville structures

We answer a conjecture of Bauer, Catanese and Grunewald showing that all finite simple groups other than the alternating group of degree 5 admit unmixed Beauville structures. We also consider an analog of the result for simple algebraic groups which depends on some upper bounds for character values of regular semisimple elements in finite groups of Lie type and obtain definitive results about the variety of triples in semisimple regular classes with product 1. Finally, we prove that any finite simple group contains two conjugacy classes C,D such that any pair of elements in C x D generates the group.

Weights of Markov traces and generic degrees

Abstract In this paper we prove the second authors conjecture about weights of Markov traces for Ariki-Koike algebras. As an application, we obtain a proof of the third authors conjecture on the generic degrees of these algebras. This yields explicit combinatorial formulas for the generic degrees which are important for several applications, for example, the construction of unipotent degrees for ‘spetses’.

Towards spetses I

We present a formalization, using data uniquely defined at the level of the Weyl group, of the construction and combinatorial properties of unipotent character sheaves and unipotent characters for reductive algebraic groups over an algebraic closure of a finite field. This formalization extends to the case where the Weyl group is replaced by a complex reflection group, and in many cases we get families of unipotent characters for a mysterious object, a kind of reductive algebraic group with a nonreal Weyl group, the “spets”.In this first part, we present the general results about complex reflection groups, their associated braid groups and Hecke algebras, which will be needed later on for properties of “spetses”. Not all irreducible complex reflection groups will give rise to a spets (the ones which do so are called “spetsial”), but all of them afford properties which already allow us to generalize many of the notions attached to the Weyl groups through the approach of “generic groups” (see [BMM1]).

Reduced words and a length function for G(e,1,n)☆

Abstract In the first part of this paper we study normal forms of elements of the imprimitive complex reflection group G(e,1,n). This allows to prove a conjecture of Broue on basis elements and the canonical symmetrizing form of the associated cyclotomic Hecke algebra. Secondly we introduce a root system for G(e,1,n) and study the associated length function. This has many properties in common with the usual length function for finite Weyl groups.

Products of conjugacy classes and fixed point spaces

We prove several results on products of conjugacy classes in finite simple groups. The first result is that there always exists a uniform generating triple. This result and other ideas are used to solve a 1966 conjecture of Peter Neumann about the existence of elements in an irreducible linear group with small fixed space. We also show that there always exist two conjugacy classes in a finite non-abelian simple group whose product contains every nontrivial element of the group. We use this to show that every element in a non-abelian finite simple group can be written as a product of two rth powers for any prime power r (in particular, a product of two squares).

The finite irreducible linear groups with polynomial ring of invariants

We prove the following result: LetG be a finite irreducible linear group. Then the ring of invariants ofG is a polynomial ring if and only ifG is generated by pseudoreflections and the pointwise stabilizer inG of any nontrivial subspace has a polynomial ring of invariants. This is well-known in characteristic zero. For the case of positive characteristic we use the classification of finite irreducible groups generated by pseudoreflections due to Kantor, Wagner, Zalesskiî and Serežkin. This allows us to obtain a complete list of all irreducible linear groups with a polynomial ring of invariants.

Die unipotenten charaktere von 2F4(q2)

.Alle endlichen Gruppen vom Lie-Typ besitzen eine ausgezeichnete Menge von irreduziblen komplexen Charakteren, die von Lusztig eingefuhrten unipotenten Charaktere. Diese werden fulr einen vorgegebenen Typ unabhangig vom zugrunde liegenden Korper klassifiziert. Aufgrund der ebenfalls von Lusztig bewiesenen Jordan-Zerlegung der irreduziblen komplexen Charaktere laiit sich die Charaktertafel einer jeden endlichen Gruppe G vom Lie-Typ im wesentlichen zusammenbauen, wenn man die unipotenten Charaktere der halbeinfachen Anteile der Zentralisatoren aller halbeinfachen Elemente in G kennt. Das Problem der Berechnung der Charaktertafel ist somit zuruckgefuhrt auf die Bestimmung aller Werte der unipotenten Charaktere der einfachen Gruppen vom Lie-Typ.