- 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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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.
Ziegler's Multi-Reflection Arrangements are free
with T. Hoge
In 1989, Ziegler introduced the concept of a multi-arrangement. One natural example is the reflection arrangement of a unitary reflection group with multiplicity given by the number of reflections associated with each hyperplane. For all but three irreducible groups, Ziegler showed that each such multi-reflection arrangement is free. We complete Ziegler's example by confirming these outstanding cases.
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.
Supersolvable restrictions of reflection arrangements
with N. Amend and T. Hoge
Let A = (A,V) be a complex hyperplane arrangement and let L(A) denote its intersection lattice. The arrangement A is called supersolvable, provided its lattice L(A) is supersolvable. For X in L(A), it is known that the restriction A^X is supersolvable provided A is. Suppose that W is a finite, unitary reflection group acting on the complex vector space V. Let A(W) = (A(W), V) be its associated hyperplane arrangement. In earlier work by the last two authors, we classified all supersolvable reflection arrangements. Extending this work, the aim of this note is to determine all supersolvable restrictions of reflection arrangements. It turns out that apart from the obvious restrictions of supersolvable reflection arrangements there are only a few additional instances. Moreover, in our previous work, we classified all inductively free restrictions A(W)^X of reflection arrangements A(W). Since every supersolvable arrangement is inductively free, the supersolvable restrictions A(W)^X of reflection arrangements A(W) form a natural subclass of the class of inductively free restrictions A(W)^X. Finally, we characterize the irreducible supersolvable restrictions of reflection arrangements by the presence of modular elements of dimension 1 in their intersection lattice. This in turn shows that reflection arrangements as well as their restrictions are of fiber type if and only if they are strictly linearly fibered.
On inductively free Restrictions of Reflection Arrangements
with N. Amend and T. Hoge
Let W be a finite complex reflection group acting on the complex vector space V and let A(W) = (A(W), V) be the associated reflection arrangement. In an earlier paper by the last two authors, we classified all inductively free reflection arrangements A(W). The aim of this note is to extend this work by determining all inductively free restrictions of reflection 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)