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# Disputation Georg Lehner

22.11.2021 | 16:00
Thema der Dissertation:
The passage from the integral to the rational group ring in algebraic K-theory
Thema der Disputation:
Modular Representation Theory and Berman's Theorem
Abstract: Modular representation theory is the study of representations of a finite group \$G\$ over a base field \$k\$ with non-zero characteristic \$p\$. Many results which hold in ordinary representation theory require that the base field is algebraically closed and of characteristic \$0\$, and are not true in the modular setting. One key statement needed for the study of character tables is that the number of irreducible \$G\$-representations over an algebraically closed field of character zero is equal to the number of conjugacy classes of \$G\$. The appropriate generalization is given by Berman's Theorem, which states that the number of irreducible \$G\$-representations over an arbitrary field \$k\$ is equal to the number of \$k\$-conjugacy classes of \$p\$-regular elements of \$G\$, where \$p\$ is the characteristic of \$k\$. We will give a sketch of a proof of Berman's theorem using Brauer characters and give some applications to the algebraic \$K\$-theory of the group algebras \$\mathbb{F}_p G, \mathbb{Z}_p G\$ and \$\mathbb{Q}_p G\$.

### Zeit & Ort

22.11.2021 | 16:00