Fibered biset functors

Abstract

The theory of biset functors, introduced by Serge Bouc, gives a unified treatment of operations in representation theory that are induced by permutation bimodules. In this paper, by considering fibered bisets, we introduce and describe the basic theory of fibered biset functors which is a natural framework for operations induced by monomial bimodules. The main result of this paper is the classification of simple fibered biset functors.

Fibered p-biset Functor Structure of the Fibered Burnside Rings

“…In this section, we recall basic theory of fibered bisets from [4] and specialize certain results to the case of abelian groups with sufficiently large fiber groups.…”

“…This product allows us to decompose any A-fibered (G, H)-biset into basic ones, as in the case of ordinary bisets. We refer to [4] for further details. In this paper, we only need the decomposition of fibered bisets for abelian groups with sufficiently large fiber group A, which we discuss next.…”

“…In [4], Boltje and the first author combined these two notions and introduced A-fibered (G, H)-bisets. Our aim in the present paper is to extend the results of Bouc and Thevénaz on Burnside functors and to determine the composition factors of the A-fibered Burnside functor kB A of p-groups over a field k of characteristic q with q = p and a cylic fiber group A.…”

“…In Section 2, we review the theory of fibered bisets and fibered biset functors from [4], together with some specializations to the case of abelian groups. In the next section, we introduce fibered Burnside rings, recall the idempotent formula for its primitive idempotents form [1] and determine the action of basic fibered bisets on these idempotents.…”

The −+ and −+ constructions for biset functors

“…This should also be of interest independent of the theory of canonical induction formulas, since the + -construction yields various important biset functors (see Example 4.9: The Burnside functor is the − + -construction of the constant biset functor with values Z. For an abelian group A, the monomial (also called A-fibered) Burnside ring is the − + -construction applied to the biset functor mapping a finite group G to the free Z-module with basis Hom(G, A), see [D71], [Ba04], or [BC16] for instance. This was used in the known examples of canonical induction formulas for the representation rings mentioned above, where A is a subgroup of the unit group of an appropriate field.…”