Representation Theory of Finite Groups over Fields of Positive Characteristic
The notion of a group is pervasive throughout much of modern mathematics, as a group essentially describes the symmetry of a particular mathematical object. A representation of a group is way to realize an abstract group in a more concrete setting, in such a way that all of the tools of linear algebra can be used to study the group. We study finite groups and their representations over both the “characteristic zero” fields – the rational numbers, the real numbers, or the complex numbers – and also over fields of prime characteristic p, which give new insight into the group structure. Much of the recent work in representation theory has been studying how the representations over characteristic zero and the representations in characteristic p interact, and we are working to extend many of the classical results, such as the Fong-Swan theorem for finite solvable groups and the Alperin weight conjecture for symmetric groups.