- 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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Divisionally free Restrictions of Reflection Arrangements Abstract:
We study some aspects of divisionally free arrangements which were recently introduced by Abe. Crucially, Terao's conjecture on the combinatorial nature of freeness holds within this class. We show that while it is compatible with products, surprisingly, it is not closed under taking localizations. In addition, we determine all divisionally free restrictions of all reflection arrangements.
Nice reflection arrangements
with T. Hoge
The aim of this note is a classification of all nice and all inductively factored reflection arrangements. It turns out that apart from the supersolvable instances only the monomial groups G(r,r,3) for r at least 3 give rise to nice reflection arrangements. As a consequence of this and of the classification of all inductively free reflection arrangements from our earlier work, we deduce that the class of all inductively factored reflection arrangements coincides with the supersolvable reflection arrangements. Moreover, we extend these classifications to hereditarily factored and hereditarily inductively factored reflection arrangements.
Inductive and Recursive Freeness of Localizations of Multiarrangements
with T. Hoge and A. Schauenburg
The class of free multiarrangements is known to be closed under taking localizations. We extend this result to the stronger notions of inductive and recursive freeness. As an application, we prove that recursively free multiarrangements are compatible with the product construction for multiarrangements. In addition, we show how our results can be used to derive that some canonical classes of free multiarrangements are not inductively free.
Inductively free Multiderivations of Braid arrangements
with H. Conrad
The reflection arrangement of a Coxeter group is a well known instance of a free hyperplane arrangement. In 2002, Terao showed that equipped with a constant multiplicity each such reflection arrangement gives rise to a free multiarrangement. In this note we show that this multiarrangment satisfies the stronger property of inductive freeness in case the Coxeter group is of type A.
Cocharacter-closure and the rational Hilbert-Mumford Theorem
with M. Bate, S. Herpel, and B. Martin
For a field k, let G be a reductive k-group and V an affine k-variety on which G acts. Using the notion of cocharacter-closed G(k)-orbits in V, we prove a rational version of the celebrated Hilbert-Mumford Theorem from geometric invariant theory. We initiate a study of applications stemming from this rationality tool. A number of examples are discussed to illustrate the concept of cocharacter-closure and to highlight how it differs from the usual Zariski-closure.
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.
Restricting invariants of unitary reflection groups
with N. Amend, A. Berardinelli, and J. M. Douglass
Suppose that G is a finite, unitary reflection group acting on a complex vector space V and X is the fixed point subspace of an element of G. Define N to be the setwise stabilizer of X in G, Z to be the pointwise stabilizer, and C=N/Z. Then restriction defines a homomorphism from the algebra of G-invariant polynomial functions on V to the algebra of C-invariant functions on X. Extending earlier work by Douglass and Roehrle for Coxeter groups, we characterize when the restriction mapping is surjective for arbitrary unitary reflection groups G in terms of the exponents of G and C, and their reflection arrangements. A consequence of our main result is that the variety of G-orbits in the G-saturation of X is smooth if and only if it is normal.
Former and current PhD students:
Simon Goodwin (University of Birmingham)
Michael Bate (University of York)
Russell Fowler (Npower)
Glenn Ubly (NHS)
Sebastian Herpel (Barmenia Allgemeine Versicherungs-AG)
Peter Mosch (Gothaer Lebensversicherung AG)