Research of Harm Derksen

Invariant Theory

Suppose that V is a K-vector space on which a group G acts. Then G also acts on the ring K[V] of polynomial functions on V. The set of G-invariant polynomials form a subring denoted by K[V]G. These rings where already studied in the 19th century, especially where G is SL2 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. Nowadays, Invariant Theory is applied to construct moduli spaces of various objects in 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 14th problem (one of the problems he proposed at the beginning of the 20th century to keep everyone busy). Before, in 1890, he proved that for G=SLn, G=GLn 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 14th problem where G is not reductive.

I am interested in the constructive aspect of Invariant Theory. I have worked on algorithms for finding generators of invariant rings, and finding upper bounds for the degrees of a minimal system of generators. Together with Gregor Kemper I wrote a book on Computational Invariant Theory. More recently, I have studied Invariant Theory in relation to Complexity Theory, and the Graph Isomorphism Problem. Currently I am studying tensor invariants (a classical topic in Invariant Theory) in relation to wheeled PROPs (a structure recently introduced by Markl, Merkulov and Shadrin).

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 interested in the connection between quiver representations and the representation theory of GLn. Studying the behaviour of ``generic'' representations of quivers, one can obtain results about Littlewood-Richardson coefficients (tensor product multiplicities that appear in the representation theory of the general linear group). Quivers with (super)potentials appear in String Theory, but also in my joint work with Weyman and Zelevinsky on Cluster Algebras.

Subspace Arrangements

A subspace arrangement is a vector space with a collection of subspaces. With Jessica Sidman, a former graduate student at the University of Michigan, I studied free resolutions of ideals of subspace arrangements, proving a conjecture of mine. Similar ideas turned out to have applications in engineering. Large data sets obtained from images, and videos often are concentrated along a subspace arrangement. Polymatroids are a an abstraction of Subspace arrangements. I have an introduced an invariant G of Polymatroids. To graphs one can associate matroids, which are a special kind of polymatroids. So in particular, we have an invariant of graphs. This graph invariant specializes to the Tutte polynomial and several other known graph invariants. Alex Fink and I showed that the invariant G has a certain universal property. An interesting question is, whether one can define a similar invariant for knots.

Number Theory

Consider a sequence a0,a1,a2,... of complex numbers (or an other field of characteristic 0) that satisfies a linear recurrence relation. Consider the set S={n|an=0}. The Skolem-Mahler-Lech theorem states, that S must be the union of a finite set and finitely many infinite arithmetic progressions. This statement is false for positive characteristic. I have given a different description of these sets in positive characteristic. Together with David Masser I have studied generalizations of this results by studying linear equations over multipliative groups.

Automorphisms of Affine space

The algebraic automorphism group of Kn (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 Kn+1 for some affine variety X, then X must be isomorphic to Kn.

or this conjecture by Abhyankar and Sathaye:

Conjecture: If f:Kn--->K is a polynomial such that f-1(0) is isomorphic to Kn-1, then any fiber f-1(a) is isomorphic to Kn-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 Kn. 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.

Coding Theory

Let F2 be the field with two elements. Elements of the vector space F2n are called words. The Hamming distance d(x,y) between two word x and y is the number of positions in which the vectors x and y differ. A binary code of length n is a subset C of F2n. The minimum distance of C is the smallest integer d such that every two codewords (elements) of C have Hamming distance at least d. Codes are used to correct errors in noisy communication. A code with minimum distance d can correct up to [(d-1)/2] errors. Let A(n,d) be the maximum number of words that a binary code of minimum distance d can have. One can give good lower bounds for A(n,d) by construction "good" codes. Using a construction of nonlinear codes, I obtained strong lower bound that improved about 15% of the best known codes for small values of n and d.