Doctor of Philosophy
Université de Strasbourg
Let $G$ be a finite group. The ring $R_\KK(G)$ of virtual characters of $G$ over the field $\KK$ is a $\lambda$-ring; as such, it is equipped with the so-called $\Gamma$-filtration, first defined by Grothendieck. In the first half of this thesis, we explore the properties of the associated graded ring $R^*_\KK(G)$, and present a set of tools to compute it through detailed examples. In particular, we use the functoriality of $R^*_\KK(-)$, and the topological properties of the $\Gamma$-filtration, to explicitly determine the graded character ring over the complex numbers of every group of order at most $8$, as well as that of dihedral groups of order $2p$ for $p$ prime. In the second half, we study the interplay between the graded character ring of a group and those of its subgroups: while restriction of representations gives rise to a well-defined graded ring homomorphism, induction does not preserve the $\Gamma$-filtration, thus $R^*_\KK(-)$ is not a Mackey functor. We introduce a modified filtration that remedies this, and explore ways to compute the associated graded ring. We then turn to tensor induction of representations, and show that in the case of complex characters of abelian groups, both inductions preserve the filtration. Therefore, the restriction of $R^*_\CC(-)$ to abelian groups is a Tambara functor.
Chetard, Beatrice Isabelle, "Graded Character Rings, Mackey Functors and Tambara Functors" (2019). Electronic Thesis and Dissertation Repository. 6236.
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.