Ehud Meir (University of Aberdeen) – The complex representation theory of general linear groups over Z/p^r- a combinatorial approach
I will talk about a joint work with Tyrone Crisp and Uri Onn about the complex representation theory of the groups GL_n(Z/p^r). Following the work of Zelevinsky, we know that when r=1 the complex representation theory of the groups GL_n(Z/p) decomposes nicely into a combinatorial part and an arithmetic part. The representation theory of GL_n(Z/p^r) where r>1 seems to be much more complicated. We conjecture that a similar decomposition exists there. In this talk I will explain how we tackle this conjecture, using tools from symmetric monoidal categories and from classical representation theory. The main insight from symmetric monoidal categories is that the languages of finite groups and finite rings intertwine here, and that studying the representation theory of GL_n(Z/p^r) naturally contains the study of groups of the form Aut_R(M) for general finite rings R and finite modules M. This much more general framework, together with classical tools from representation theory, gives much better insight into the conjecture.