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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:

    Overgroups of regular unipotent elements in reductive groups

    with Michael Bate and Ben Martin
    Abstract:
    We study reductive subgroups H of a reductive linear algebraic group G --- possibly non-connected --- such that H contains a regular unipotent element of G. We show that under suitable hypotheses, such subgroups are G-irreducible in the sense of Serre. This generalizes results of Malle, Testerman and Zalesski. We obtain analogous results for Lie algebras and for finite groups of Lie type. Our proofs are short, conceptual and uniform.
    math.GR/2107.01925

    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

    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 (Barmenia Krankenversicherung AG)
    Tilman Möller
    Falk Bannuscher


    Editorial Activity:

    Advanced Studies: Euro-Tbilisi Mathematical Journal .

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