Graded character rings, Mackey functors and Tambara functors

2018/11/14 by Chetard, Beatrice I.
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1811.05946

Abstract

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

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