The A -fibered Burnside Ring as A -Fibered Biset Functor in Characteristic Zero
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The A -fibered Burnside Ring as A -Fibered Biset Functor in Characteristic Zero Robert Boltje1
· Deniz Yılmaz1
Received: 16 September 2019 / Accepted: 14 September 2020 / © Springer Nature B.V. 2020
Abstract Let A be an abelian group such that torn (A) is finite for every n 1 and let K be a field of characteristic zero containing roots of unity of all orders equal to finite element orders in A. A In this paper we prove fundamental properties of the A-fibered Burnside ring functor BK as an A-fibered biset functor over K. This includes a description of the composition factors A and the lattice of subfunctors of B A in terms of what we call B A -pairs and a poset of BK K structure on their isomorphism classes. Unfortunately, we are not able to classify B A -pairs. The results of the paper extend results of Cos¸kun and Yılmaz for the A-fibered Burnside ring functor restricted to p-groups and results of Bouc in the case that A is trivial, i.e., the case of the Burnside ring functor as a biset functor over fields of characteristic zero. In the latter case, B A -pairs become Bouc’s B-groups which are also not known in general. Keywords Burnside ring · Monomial Burnside ring · Biset functors · Fibered biset functors Mathematics Subject Classification (2010) 19A22 · 20C15
1 Introduction Let A be a finite group and let k be a commutative ring. An A-fibered biset functor F over k is, informally speaking, a functor that assigns to each finite group G a k-module G F (G) together with maps resG H : F (G) → F (H ) and indH : F (H ) → F (G), whenG ever H G, called restriction and induction, maps infG/N : F (G/N ) → F (G) and defG G/N : F (G) → F (G/N ), whenever N is a normal subgroup of G, called inflation and deflation, and maps isof : F (G) → F (H ), whenever f : G → H is an isomorphism.
Presented by: Kenneth Goodearl Robert Boltje
[email protected] Deniz Yılmaz [email protected] 1
Department of Mathematics, University of California, Santa Cruz, CA 95064, USA
R. Boltje, D. Yılmaz
Moreover, the abelian group G∗ := Hom(G, A) acts k-linearly on F (G) for every finite group G. These operations satisfy natural relations. Standard examples are various representation rings of KG-modules, for a field K and A = K × . In this case, G∗ is the group of one-dimensional KG-modules acting by multiplication on these representation rings. In [3] the simple A-fibered biset functors were parametrized. If A is the trivial group then one obtains the well-established theory of biset functors, see [5] as special case. A-fibered biset functors over k can also be interpreted as the modules over the Green biset functor BkA , where BkA (G) is the A-fibered Burnside ring of G over k (also called the K-monomial Burnside ring of G over k, when A = K × for a field K, introduced by Dress in [8]). Another natural example of A-fibered biset functors (without deflation) is the unit group functor G → B A (G)× . This structure was established in a recent paper by Bouc and Mutlu, see [6] and generalizes the biset functor structure on the unit group B(
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