RESEARCH
Main focusses in research
Constructive Invariant Theory of Finite Groups
In invariant theory of finite groups one investigates the following situation:
Let G be a finite group, V a finite dimensional KG -
modul over a field K and 
the (graded) symmetric algebra over K. Then the group G
acts on by graded algebra automorphisms. The G - fixedpoints
therein build a graded subalgebra 
, the `invariant ring'. One example is the ring of
symmetric polynomials which arises if G is the symmetric group on n
letters permuting a basis of a vector space V of dimension n. In this
case the classical theorem on symmetric functions states that is isomorphic to a polynomial ring generated by the elementary -
symmetric polynomials in degrees less or equal to n. For arbitrary groups the
situation is far more complicated. Main questions are about the constructive
complexity of , i.e. finding algebra generators of bounded degrees, and the structural
complexity , i.e. how far away is an invariant ring from being a `nice
ring' like a polynomial ring or a Cohen - Macaulay ring.
Constructive Aspects : A classical result of Emmy Noether
states that the ring of invariants of finite groups are finitely generated.
Noether gave in fact two different constructive proofs, originally stated for
the case of complex coefficients; later she gave a non - constructive proof of
finite generation, valid for all fields. Her constructive proofs in
characteristic zero also show that the graded ring can be generated in degrees less or equal than |G|. This
bound on the degrees of generators is usually referred to as ` Noether's bound
'. With some slight changes this proof can also be made to work if the
characteristic K is larger than the order |G| of the group. Until
recently it had been an open question whether Noether's bound also holds in the
case that char K is smaller but not dividing |G|. If |G|
is divisible by the characteristic of K then Noether's bound does not
hold any more, as can be seen easily. I recently found a proof for the Noether
bound to hold if the characteristic of K does not divide the group
order. (see The
Noether Bound in Invariant Theory of Finite Groups, Adv. Math. 2000); ).
If K divides the group order is called a modular invariant ring. At the moment all known
general degree bounds for modular invariant rings are of an exponential nature.
We know `good' modular degree bounds only for special classes of invariant
rings like the ones defined by so called `trivial source' modules or p -
permutation modules (e.g. projective modules) (see the preprint On Invariant
Theory of Finite Groups )
Structure of Modular Invariant Rings: It is known
that non - modular invariant rings of finite groups are Cohen - Macaulay rings.
This is important, because in this case the invariant ring is a finitely
generated free module over a polynomial subalgebra. In the modular situation,
this is no longer true, but the `degree' to which this property still holds is
measured by the depth of the invariant ring, which is the length of a
maximal regular sequence, or the cohomological co - dimension of the invariant
ring as a module over a homogeneous system of parameters.
In 1980 G. Ellingsrud and T. Skjelbred proved a celebrated result that contains
a lower bound for the depth of modular invariant rings and gives a precise
formula if G is a cyclic p - group. For almost two decades this
has been the only general result on the depth of modular invariant rings, which
remains to be one of their most interesting, but difficult to determine
parameters. Recent results show exciting fundamental connections between the
structure of modular group representations, group cohomology and the depth of
modular invariant rings.
In general modular invariant theory lacks explicitly computed examples; even
the invariant rings of cyclic groups 
in characteristic p and indecomposable modules are known
only in special cases. Thus I am also very interested in developing efficient
computer algebra methods to construct new significant examples of invariant
rings.
Recommended New Texts on Invariant Theory of Finite Groups :
H. Derksen, G. Kemper, Computational Invariant Theory,
Enc. of Math. Sciences Vol. 130, Springer, 268 pp.(2002).
ISBN 3-540-43476-3
M. Neusel, L. Smith , Invariant Theory of Finite Groups,
Math. Surveys and Monographs, Vol. 94, AMS, 371 pp. (2002).
ISBN: 0-8218-2916-5
Structure Theory of Finite Groups
After the classification of finite simple groups, a major task in group theory is, to derive 'global' properties of finite groups from 'local' ones, i.e. from properties of p - subgroups or p - elements where p is a divisor of the group order.
Currently I am interested in the consequences of fixed point free actions of individual elements of the group.
An example of that kind of results is the recent classification of
'primitive p - Frobenius groups', which I achieved in cooperation with
W. Lempken (
A p - Frobenius group is a primitive permutation group such that each
two point stabilizer is a p - group; so this is a p - local
generalization of the classical notion of 'Frobenius groups', where two - point
stabilizers are assumed to be trivial. For example, if G is a p -
Frobenius - group with solvable socle V, then G is a semidirect
product VH where H is a 'generalized Frobenius - complement, i.e.
H is a linear group acting on the finite vector space V such that
each 
- element has no non - trivial fixed point on 
(see The
primitive p-Frobenius Groups (with W. Lempken and P.H. Tiep) ; Proceedings of
the Amer. Math. Soc. 126(5) 1998, 1337-1343. ).
Representation Theory of Finite Groups
Representation theory of finite groups is concerned with the realization of
abstract groups as linear groups. Thus a linear representation of a finite
group G is a homomorphism into 
, where V is a finite dimensional vectorspace over a field K.
Depending on the characteristic of K (char K), the theory splits
into two different cases:
In classical representation theory one assumes that char K
does not divide the group order |G| (e.g. 
the field of complex numbers). In this case all finite dimensional
representations of G are completely reducible.
In modular representation theory one assumes that char K = p divides |G|. In this case representations are no longer completely reducible and the theory is almost completely different from the classical case.
My main research interests in this area concern classical and modular
representations of finite groups of Lie type and corresponding Hecke -
Algebras. A finite group of Lie type 
is a 'finite analogue' of a classical Lie group, and is defined
over a finite field 
. I am interested in 'modular Harish-Chandra theory' which
investigates the relations beween complex and modular representations of 
over fields of characteristic 
q.
Decomposition of Rational functions
This is a project I am currently pursuing in collaboration with Prof Mohamed Ayad (University of Littoral/Calais).
We have developed an improved version of an algorithm by Alonso, Gutierrez and Recio to decompose univariate
rational functions or show their indecomposibility. Here is a software – package of this algorithm, written for MAGMA V2.11-6.
\\Korky\imsweb\personal\pf10\calais\decomp and an synopsis thereof \\Korky\imsweb\personal\pf10\calais\decomp_synopsis.txt.
Here is the preprint explaining this algorithm. It also contains some new indecomposiblity criteria, which were motivated
by experiments using the MAGMA implementation: ..\calais\decomp.pdf
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