# Research Profile of Harm Derksen

Let me just say a few words about the kind of math problems
I like to work on.
### Invariant Theory

Let G be an algebraic group and V is a (rational) representation of G. Then
G also acts on the coordinate ring K[V] (K is some field). The polynomial
functions on V which are invariant under G form a subring, the
invariant ring K[V]^{G}. These rings where already studied
(although not in this language) in the 19^{th} century, especially
where G is SL_{2} or an other classical group.
With David Hilbert, an new era for invariant theory arose
(and another era ended). His papers of 1890 and 1893 not only
proved one of the major results in invariant theory, but also build
the foundation of what is now commutative algebra and algebraic geometry.
One of the major problems is whether K[V]^{G} is always
finitely generated as an algebra over K. This is known
as Hilbert's 14^{th} problem (one of the problems he proposed
at the beginning of the 20^{th} century to keep
everyone busy). Before, in 1890, he proved that for G=SL_{n},
G=GL_{n} and other classical groups. It can be shown
that K[V]^{G} is finitely generated for any reductive
group. However, Nagata found a counterexample for Hilbert's 14^{th}
problem where G is not reductive.

I am very much interested in the constructive aspect of invariant theory.
One of my contributions is an algorithm for computing a finite set
of generating invariants for K[V]^{G} if G is linearly reductive.
Let b(V) be the smallest integer d such that K[V]^{G} is
generated by invariants of degree less or equal than d.
I also obtained some good upper bounds for b(V).

### Quiver representations

A quiver Q is just a graph. If we attach to each vertex a vector space
and to each arrow a linear map, then this is called a representation
of the quiver Q. Quiver representations give a natural generalization
of some linear algebra problems.
There are several reasons for studying quiver representations.
To each quiver one can associate its path algebra. Any
finite dimensional associative algebra is just a path algebra
modulo some ideal (up to Morita equivalence). So study of
finite dimensional associative algebras leads to the study of quivers.
Results of Kac, Lusztig and Nakajima also link quiver representations
to Kac-Moody algebras and Quantum groups etc.

I am particularly interested in the connection between quiver representations
and the representation theory of GL_{n}. Studying the behaviour
of ``generic'' representations of quivers, one can obtain results
about Littlewood-Richardson coefficients.

### Automorphisms of Affine space

The algebraic automorphism group of K^{n}
(K an algebraically closed field, for example the complex numbers)
is very complicated. There are still many open problems
relating to this automorphism group. For example,
the Zariski Cancellation conjecture:
**Conjecture:** If XxK is
isomorphic to K^{n+1} for some affine variety X,
then X must be isomorphic to K^{n}.

or this conjecture by Abhyankar and Sathaye:

**Conjecture:** If f:K^{n}--->K is a polynomial
such that f^{-1}(0) is isomorphic to K^{n-1},
then any fiber f^{-1}(a) is isomorphic to
K^{n-1} (or even stronger: f must be a polynomial
coordinate).

and then there is the linearization conjecture:

**Conjecture:** Let G be a linear algebraic group
acting rationally on K^{n}. Is the action of
G linearizable, i.e., is the action linear after a polynomial
change of coordinates?

To this last conjecture there are counterexamples
in positive characteristic (Asanuma), if G is
not commutative (Schwarz, Knop and others),
and in the holomorphic category (Kutzschebauch and myself).
However it is still open if G is abelian and K has
characteristic 0.

10/5/2000