Prof. Dr. Gerhard Röhrle

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Arbeitsgebiete

  • Algebraische Lie Theorie
  • Algebraische Gruppen
  • Darstellungstheorie

  • Isaac Newton Institute for Mathematical Sciences Workshop: Perspectives in Algebraic Lie Theory.
     
  • 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:

    A computational approach to Coxeter Arrangements and Solomon's descent algebra, II: groups of rank five and six
    with M. Bishop, J. M. Douglass, and G. Pfeiffer
    Abstract:
    In recent papers we have refined a conjecture of Lehrer and Solomon expressing the character of a finite Coxeter group $W$ acting on the $p$th graded component of its Orlik-Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of $W$. Our refined conjecture relates the character above to the $p$th graded component of a decomposition of the regular character of $W$ related to Solomon's descent algebra of $W$. A consequence of our conjecture is that both the regular character of $W$ and the character of $W$ acting on the Orlik-Solomon algebra have parallel, graded decompositions as sums of characters induced from linear characters of centralizers of elements of $W$, one for each conjugacy class of $W$. The refined conjecture has been proved for symmetric and dihedral groups, as well as finite Coxeter groups of rank three and four. In this paper, the second in a series of three dealing with groups of rank up to eight (and in particular, all exceptional Coxeter groups), we prove the conjecture for finite Coxeter groups of rank five and six, further developing the algorithmic tools described in the previous article. The techniques developed and implemented in this paper provide previously unknown decompositions of the regular and Orlik-Solomon characters of the groups considered.
    math.RT/1201.4775

    On the irreducibility of symmetrizations of cross-characteristic representations of finite classical groups
    with Kay Magaard and Donna Testerman
    Abstract:
    Let W be a vector space over an algebraically closed field k. Let H be a quasisimple group of Lie type of characteristic p different from char(k) acting irreducibly on W. Suppose that G is a classical group with natural module W. Suppose also that G is a classical group with natural module, chosen minimally with respect to containing the image of H under the associated representation. We consider the question of when H can act irreducibly on a G-constituent of the e-fold tensor product of W with itself for some e and study its relationship to the maximal subgroup problem for finite classical groups.
    math.GR/1201.2057

    The strong Centre Conjecture: an invariant theory approach
    with M. Bate and B. Martin
    Abstract:
    The aim of this paper is to describe an approach to a a strengthened form of J. Tits' Centre Conjecture for spherical buildings. This is accomplished by generalizing a fundamental result of G. R. Kempf from Geometric Invariant Theory and interpreting this generalization in the context of spherical buildings. We are able to recapture the conjecture entirely in terms of our generalization of Kempf's notion of a state. We demonstrate the utility of this approach by proving the Centre Conjecture in some special cases.
    math.GR/1005.3212

    Coxeter arrangements and Solomon's descent algebra
    with J. Matthew Douglass and  Goetz Pfeiffer
    Abstract:
    We refine a conjecture by Lehrer and Solomon on the structure of the Orlik--Solomon algebra of a finite Coxeter group W and relate it to the descent algebra of W. As a result, we claim that both the group algebra of W, as well as the Orlik--Solomon algebra of W can be decomposed into a sum of induced one-dimensional representations of element centralizers, one for each conjugacy class of elements of W. We give a uniform proof of the claim for symmetric groups.
    math.RT/1101.2075

    Computations for Coxeter arrangements and Solomon's descent algebra: Groups of rank three and four
    with Marcus Bishop, J. Matthew Douglass and Goetz Pfeiffer
    Abstract:
    In recent papers we have refined a conjecture of Lehrer and Solomon expressing the character of the representation of a finite Coxeter group $W$ on the $p$th graded piece of its Orlik-Solomon algebra as a sum of characters induced from linear characters of centralizers of elements of $W$. Our refined conjecture relates the character of $W$ on the $p$th graded piece of its Orlik-Solomon algebra with the descent algebra of $W$. A consequence of our conjecture is that both the regular character of $W$ and the character of $W$ acting on its Orlik-Solomon algebra have parallel, graded decompositions as sums of characters induced from linear characters of centralizers of elements of $W$, one for each conjugacy class of elements of $W$. The refined conjectures have been proved for symmetric and dihedral groups. In this paper we develop algorithmic tools to prove the conjectures computationally for a given $W$ and we use these tools to verify the claim for all finite Coxeter groups of rank three and four. The techniques developed and implemented in this paper provide previously unknown decompositions of the regular characters and the Orlik-Solomon characters of the Coxeter groups of types $B_3$, $H_3$, $B_4$, $D_4$, $F_4$, and $H_4$ as sums of induced representations indexed by the set of conjugacy classes of $W$.
    math.RT/1101.5893

 

Former and current PhD students:
Simon Goodwin (University of Birmingham)
Michael Bate (University of York)
Russell Fowler (Npower)
Glenn Ubly (NHS)
Sebastian Herpel
Peter Mosch


Editorial Activity: Tbilisi Mathematical Journal .

Computer Programs