- 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").
- DFG project: Inductive freeness and rank-generating functions for arrangements of ideal type: Two conjectures of Sommers and Tymoczko revisited (within the
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Arrangements of ideal type
In 2006 Sommers and Tymoczko defined so called arrangements of ideal type A_I stemming from ideals I in the set of positive roots of a reduced root system. They showed in a case by case argument that A_I is free if the root system is of classical type or G_2 and conjectured that this is also the case for all types. This was established only recently in a uniform manner by Abe, Barakat, Cuntz, Hoge and Terao. The set of non-zero exponents of the free arrangement A_I is given by the dual of the height partition of the roots in the complement of I in the set of positive roots, generalizing the Shapiro-Steinberg-Kostant theorem. Our first aim in this paper is to investigate a stronger freeness property of the A_I. We show that all A_I are inductively free, with the possible exception of some cases in type E_8. In the same paper, Sommers and Tymoczko define a Poincar\'e polynomial I(t) associated with each ideal I which generalizes the Poincar\'e polynomial W(t) for the underlying Weyl group W. Solomon showed that W(t) satisfies a product decomposition depending on the exponents of W for any Coxeter group W. Sommers and Tymoczko showed in a case by case analysis in type A, B and C, and some small rank exceptional types that a similar factorization property holds for the Poincar\'e polynomials I(t) generalizing the formula of Solomon for W(t). They conjectured that their multiplicative formula for I(t) holds in all types. Here we show that this conjecture holds inductively in almost all instances.
Localizations of inductively factored arrangements
with T. Möller
We show that the class of inductively factored arrangements is closed under taking localizations. We illustrate the usefulness of this with an application.
Divisionally free Restrictions of Reflection Arrangements
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.
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.
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.
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)