Calculating conjugacy classes in Sylow p-subgroups of finite Chevalley groups of rank six and seven

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 an earlier paper, 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 E_7. 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.Comment: 16 page

On the coadjoint orbits of maximal unipotent subgroups of reductive groups

Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and \u its Lie algebra. We prove the separability of orbit maps and the connectedness of centralizers for the coadjoint action of U on (certain quotients of) the dual \u* of \u. This leads to a method to give a parametrization of the coadjoint orbits in terms of so-called minimal representatives which form a disjoint union of quasi-affine varieties. Moreover, we obtain an algorithm to explicitly calculate this parametrization which has been used for G of rank at most 8, except E8. When G is defined and split over the field of q elements, for q the power of a good prime for G, this algorithmic parametrization is used to calculate the number k(U(q), \u*(q)) of coadjoint orbits of U(q) on \u*(q). Since k(U(q), \u*(q)) coincides with the number k(U(q)) of conjugacy classes in U(q), these calculations can be viewed as an extension of the results obtained in our earlier paper. In each case considered here there is a polynomial h(t) with integer coefficients such that for every such q we have k(U(q)) = h(q).Comment: 14 pages; v2 23 pages; to appear in Transformation Group

The World House: Prophetic Protestantism and the Struggle for Environmental Justice

Though much has been rightly made of the destruction wrought by the oil spill in the Gulf of Mexico in April, 2010, it pales in comparison to the ongoing damage done to the environment by our unsustainable way of life. Global instability exacerbated by the economic crisis of 2008 to the present has compounded increasing ecological vulnerabilities, including the most basic needs of food and water among the world’s poorest. Because of these combined crises, it is vital that people of faith develop methods of collaboration that support and enrich each others' efforts towards sustainability sooner rather than later. Given the extent and magnitude of our economic and environmental problems, it is time for evangelicals and liberal Protestants to work together for the ecological good. Protestantism has been a religious movement for great good, but also the source of deep divisions. Born through a division with the Roman Catholic Church, Protestantism, in its engagement with modernity, forced a further division into evangelical and liberal communities. Now the urgent need to preserve and regenerate the global environment offers an ideal opportunity to work towards the repair of these older theological divisions through joining together in the common task of environment justice. In seeking to repair the earth as a common ethical task, Protestants can also enact an ecumenical vocation to heal the divisions within the broader world Christian movement

How To Reach This Broken World

...Every single person here this morning is in mission. You are not in remission. You are in mission. There isn\u27t one person here this morning who is not in mission. We are called into mission. We are not in a career. We are in a calling. A holy calling from the Creator God this morning. A holy calling. And every one of us here this morning is gifted. And our gifting and our calling are to be received, unwrapped and given back to the one who gave it to us