Speaker: Peter Mosch

Title: Calculating Conjugacy Classes of Sylow p-Subgroups of Chevalley Groups

Abstract: Let \(G\) be a simple linear algebraic group defined over the finite field \(F_q\) of \(q = p^m\) elements. If \(F\) is a Frobenius map on \(G\), the group \(G^F\) of \(F\)-fixed points of \(G\) is called a finite group of Lie type, and if \(G\) is \(F\)-split, \(G^F\) is a finite Chevalley group.

We are interested in the number \(k(U^F)\) of \(U^F\)-conjugacy classes in \(U^F\), where \(U\) is the unipotent radical of an \(F\)-stable Borel subgroup \(B\) of \(G\). I will talk about how the calculation of the \(U^F\)-conjugacy classes is equivalent to counting so called minimal representatives of the adjoint \(U\)-orbits on its Lie algebra \(u\), which have coefficients in \(F_q\), based on work of Simon Goodwin.

Furthermore, I will give the value of \(k(U^F)\) for all types of G up to rank \(6\).