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Arbeitsgebiete

  • Algebraische Lie Theorie
  • Algebraische Gruppen
  • Darstellungstheorie
  • Endliche Gruppen vom Lie-Typ
  • Hyperebenenarrangements
  • Spiegelungsarrangements

  • 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 DFG GEPRIS).
     
  • DFG project: On the Cohomology of complements of complex reflection arrangements (within the DFG GEPRIS).
     
  • Publications on MathSciNet.

  • Recent preprints on the ArXiv.
     
  • Latest preprints:

    Invariants and semi-invariants in the cohomology of the complement of a reflection arrangement

    with J. Matthew Douglass and Götz Pfeiffer
    Abstract:
    Suppose V is a finite dimensional, complex vector space, A is a finite set of codimension one subspaces of V, and G is a finite subgroup of the general linear group GL(V) that permutes the hyperplanes in A. In this paper we study invariants and semi-invariants in the graded QG-module H^*(M(A)), where M(A) denotes the complement in V of the hyperplanes in A and H^* denotes rational singular cohomology, in the case when G is generated by reflections in V and A is the set of reflecting hyperplanes determined by G, or a closely related arrangement. Our first main result is the construction of an explicit, natural (from the point of view of Coxeter groups) basis of the space of invariants, H^*(M(A))^G. In addition to providing a conceptual proof of a conjecture due to Felder and Veselov for Coxeter groups, this result extends the latter to all finite complex reflection groups. Moreover, we prove that determinant-like characters of complex reflection groups do not occur in H^*(M(A)). This extends to all finite complex reflection groups a result proved for Weyl groups by Lehrer.
    math.RT/2009.12847

    Complete reducibility: Variations on a theme of Serre

    with Maike Gruchot and Alastair Litterick
    Abstract:
    In this note, we unify and extend various concepts in the area of G-complete reducibility, where G is a reductive algebraic group. By results of Serre and Bate--Martin--Röhrle, the usual notion of G-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of G. We show that other variations of this notion, such as relative complete reducibility and \sigma-complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of more general properties.
    math.GR/2004.14604

    A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups

    with T. Abe, C. Stump and M. Yoshinaga
    Abstract:
    Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration of that module. Our approach is based on a detailed analysis of a flat connection applied to the primitive vector field. This generalizes and unifies analogous results for real reflection groups.
    math.DG/1809.05026


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    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 (Volkswohl Bund)
    Nils Amend (INTER Krankenversicherung AG)
    Anne Schauenburg (Aldi International Services)
    Maike Gruchot
    Tilman Möller
    Falk Bannuscher


    Editorial Activity:

    Tbilisi Mathematical Journal .

    Computer Programs