- Algebraische Lie Theorie
- Algebraische Gruppen
- Endliche Gruppen vom Lie-Typ
- DFG project: Serre's notion of Complete Reducibility and Geometric Invariant Theory (within the
DFG Priority Programme in Representation Theory).
- DFG project: Computational aspects of the Cohomology of Coxeter arrangements: On Conjectures of Lehrer-Solomon and Felder-Veselov (within the
DFG Priority Programme "Algorithmic and experimental methods in Algebra,
Geometry and Number Theory").
- DFG project: Arrangements of complex reflection groups: Geometry and combinatorics (joint with M. Cuntz (Kaiserslautern) within the
DFG Priority Programme "Algorithmic and experimental methods in Algebra, Geometry and Number Theory").
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- Latest preprints:
On the coadjoint orbits of maximal unipotent subgroups of reductive groups
with S. M. Goodwin and P. Mosch
Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and \u its Lie algebra. We prove the separability of orbit maps and the connectedness of centralizers for the coadjoint action of U on (certain quotients of) the dual \u* of \u. This leads to a method to give a parametrization of the coadjoint orbits in terms of so-called minimal representatives which form a disjoint union of quasi-affine varieties. Moreover, we obtain an algorithm to explicitly calculate this parametrization which has been used for G of rank at most 8, except E8. When G is defined and split over the field of q elements, for q the power of a good prime for G, this algorithmic parametrization is used to calculate the number k(U(q), \u*(q)) of coadjoint orbits of U(q) on \u*(q). Since k(U(q), \u*(q)) coincides with the number k(U(q)) of conjugacy classes in U(q), these calculations can be viewed as an extension of the results obtained in our earlier paper. In each case considered here there is a polynomial h(t) with integer coefficients such that for every such q we have k(U(q)) = h(q).
Computations for Coxeter arrangements and Solomon's descent algebra III: Groups of rank seven and eight
with M. Bishop, J. M. Douglass and G. Pfeiffer
In this paper we extend the computations in parts I and II of this series of papers and complete the proof of a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group W acting on the pth graded component of its Orlik-Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of W for groups of rank seven and eight. For classical Coxeter groups, these characters are given using a formula that is expected to hold in all ranks.
Addition-Deletion Theorems for Factorizations of Orlik-Solomon Algebras and nice Arrangements
with T. Hoge
We study the notion of a nice partition or factorization of a hyperplane arrangement due to Terao from the early 1990s. The principal aim of this note is an analogue of Terao's celebrated addition-deletion theorem for free arrangements for the class of nice arrangements. This is a natural setting for the stronger property of an inductive factorization of a hyperplane arrangement by Jambu and Paris. In addition, we show that supersolvable arrangements are inductively factored and that inductively factored arrangements are inductively free. Combined with our addition-deletion theorem this leads to the concept of an induction table for inductive factorizations. Finally, we prove that the notions of factored and inductively factored arrangements are compatible with the product construction for arrangements.