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Author: Röhrle, Gerhard × End date: 2019 × Start date: 2019 × End date: 2013 × Start date: 2013 × Type: Text × Publisher: Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach ×

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    Calculating conjugacy classes in Sylow p-subgroups of finite Chevalley groups of rank six and seven
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Goodwi, Simon M.; Mosch, Peter; Röhrle, Gerhard
    Let G(q) be a finite Chevalley group, where q is a power of a good prime p, and let U(q) be a Sylow p-subgroup of G(q). Then a generalized version of a conjecture of Higman asserts that the number k(U(q)) of conjugacy classes in U(q) is given by a polynomial in q with integer coefficients. In [12], the first and the third authors developed an algorithm to calculate the values of k(U(q)). By implementing it into a computer program using GAP, they were able to calculate k(U(q)) for G of rank at most 5, thereby proving that for these cases k(U(q)) is given by a polynomial in q. In this paper we present some refinements and improvements of the algorithm that allow us to calculate the values of k(U(q)) for finite Chevalley groups of rank six and seven, except E7. We observe that k(U(q)) is a polynomial, so that the generalized Higman conjecture holds for these groups. Moreover, if we write k(U(q)) as a polynomial in q−1, then the coefficients are non-negative. Under the assumption that k(U(q)) is a polynomial in q−1, we also give an explicit formula for the coefficients of k(U(q)) of degrees zero, one and two.
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    G-complete reducibility in non-connected groups
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Bate, Michael; Herpel, Sebastian; Martin, Benjamin; Röhrle, Gerhard
    In this paper we present an algorithm for determining whether a subgroup H of a non-connected reductive group G is G-completely reducible. The algorithm consists of a series of reductions; at each step, we perform operations involving connected groups, such as checking whether a certain subgroup of G0 is G0-cr. This essentially reduces the problem of determining G-complete reducibility to the connected case.