Prof. Dr. Gerhard Röhrle

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Arbeitsgebiete

  • Algebraische Lie Theorie
  • Algebraische Gruppen
  • Darstellungstheorie

  • 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").
     
  • Past EPSRC-funded research projects.
     
  • Publications on MathSciNet.

  • Recent preprints on the ArXiv.
     
  • Latest preprints:

    On supersolvable reflection arrangements
    with Torsten Hoge
    Abstract:
    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, a notion due to Stanley. Jambu and Terao showed that every supersolvable arrangement is inductively free, a notion due to Terao. So this is a natural subclass of this particular class of free arrangements. Suppose that W is a finite, unitary reflection group acting on the complex vector space V. Let A = (A(W), V) be the associated hyperplane arrangement of W. In a recent paper, we determined all inductively free reflection arrangements. The aim of this note is to classify all supersolvable reflection arrangements. Moreover, we characterize the irreducible arrangements in this class by the presence of modular elements of rank 2 in their intersection lattice.
    math.GR/1209.1919

    On commuting varieties of nilradicals of Borel subalgebras of reductive Lie algebras
    with Simon Goodwin
    Abstract:
    Let G be a connected reductive algebraic group defined over an algebraically closed field of characteristic zero. We consider the commuting variety C(u) of the nilradical u of the Lie algebra b of a Borel subgroup B of G. In case B acts on u with only a finite number of orbits, we verify that C(u) is equidimensional and that the irreducible components are in correspondence with the distinguished B-orbits in u. We observe that in general C(u) is not equidimensional, and determine the irreducible components of C(u) in the minimal cases where there are infinitely many B-orbits in u.
    math.RT/1209.1289

    On inductively free reflection arrangements
    with T. Hoge
    Abstract:
    Suppose that W is a finite, unitary reflection group acting on the complex vector space V. Let A = A(W) be the associated hyperplane arrangement of W. Terao has shown that each such reflection arrangement A is free. There is the stronger notion of an inductively free arrangement. In 1992, Orlik and Terao conjectured that each reflection arrangement is inductively free. It has been known for quite some time that the braid arrangement as well as the Coxeter arrangements of type B and type D are inductively free. Barakat and Cuntz completed this list only recently by showing that every Coxeter arrangement is inductively free. Nevertheless, Orlik and Terao's conjecture is false in general. In a recent paper, we already gave two counterexamples to this conjecture among the exceptional complex reflection groups. In this paper we classify all inductively free reflection arrangements. In addition, we show that the notions of inductive freeness and that of hereditary inductive freeness coincide for reflection arrangements. As a consequence of our classification, we get an easy, purely combinatorial characterization of inductively free reflection arrangements A in terms of exponents of the restrictions to any hyperplane of A.
    math.GR/1208.3131v1

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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 (University of Kaiserslautern)
    Peter Mosch
    Nils Amend


    Editorial Activity: Tbilisi Mathematical Journal .

    Computer Programs