Graded character rings, Mackey functors and Tambara functors

Abstract

Let $G$ be a finite group and $\mathbb{K}$ a field of characteristic zero. the ring $R_{\mathbb{K}}(G)$ of virtual characters of $G$ over $\mathbb{K}$ is naturally endowed with a so-called Grothendieck filtration, with associated graded ring $R_{\mathbb{K}}^{\ast }(G)$. Restriction of representations to any $H\le G$ induces a homomorphism $R_{\mathbb{K}}^{\ast }(G)\to R_{\mathbb{K}}^{\ast }(H)$. We show that, when $G$ is abelian, induction of representations preserves the filtration, so $R_{\mathbb{C}}^{\ast }(-)$ is a Mackey functor; in the general case, we propose a modified filtration which turns $R_{\mathbb{K}}^{\ast }(-)$ into a Mackey functor. We then turn to tensor induction of representations, and show that in the abelian case $R_{\mathbb{C}}^{\ast }(-)$ is a Tambara functor.