- 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:
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.
Spherical subgroups in simple algebraic groups
with F. Knop
Let G be a simple algebraic group. A closed subgroup H of G is called spherical provided it has a dense orbit on the flag variety G/B of G. Reductive spherical subgroups of simple Lie groups were classified by Kr\"amer in 1979. In 1997, Brundan showed that each example from Kr\"amer's list also gives rise to a spherical subgroup in the corresponding simple algebraic group in any positive characteristic. Nevertheless, there is no classification of all such instances in positive characteristic to date. The goal of this paper is to complete this classification. It turns out that there is only one additional instance (up to isogeny) in characteristic 2 which has no counterpart in Kr\"amer's classification.
Former and current PhD students:
Simon Goodwin (University of Birmingham)
Michael Bate (University of York)
Russell Fowler (Npower)
Glenn Ubly (NHS)
Sebastian Herpel (University of Kaiserslautern)