On Wedderburn-Artin decomposition and property (T) for Aut(F_5)
In this lecture we will see how to use representation theory of finite groups to simplify the sum of squares decomposition problem. I will use G=SL(n,Z) with its Weyl group as a model and will guide you through the argument. Even though the representation theory of finite groups is considered “known”,there are still some simply stated open problems in group rings (of the “Weyl group”) which we encountered on the way.
This talk is based on joint work with Piotr Nowak (IMPAN) and Narutaka Ozawa (RIMS Kyoto) arXiv:1712.07167. Using the simplification we were able to prove that Aut(F_5) has property (T), which (at the time) was the first positive result in this direction.