On fibered Burnside rings, fiber change maps and cyclic fiber groups
Benjamín García, Alberto G. Raggi-Cárdenas
TL;DR
The paper develops fiber change maps between fibered Burnside rings $B^A(G)$ and analyzes their functorial and naturality properties with respect to biset operations, providing new insight into the interaction between fiber group morphisms and the biset functor structure. Special emphasis is placed on the case where the fiber group is cyclic, $A=C_n$, where species and primitive idempotents simplify and become explicitly computable, enabling precise control of how idempotents transform under restriction, inflation, and deflation. A central contribution is a sharp formula for the conductors of primitive idempotents in the cyclic case under the condition $(n,\, ext{exp}(G)/n)=1$, namely $c^{G,n}_{H,h}=[N_G(H,hH^{[n]}):H^{[n]}][H^{[n]}:H']_0$, obtained via Boltje’s integrality criterion and Muller-type arguments. These results unify known extremes (Burnside and monomial representation rings) and have implications for understanding monomial representation rings over fields where $X^{\mathrm{exp}(G)}-1$ splits, thereby enriching the structural theory of $B^A(G)$ and its biset-theoretic applications.
Abstract
Fibered Burnside rings appear as Grothendieck rings of fibered permutation representations of a finite group, generalizing Burnside rings and monomial representation rings. Their species, primitive idempotents and their conductors are of particular interest in representation theory as they encode information related to the structure of the group. In this note, we introduce fiber change maps between fibered Burnside rings, and we present results on their functoriality and naturality with respect to biset operations. We present some advances on the conductors for cyclic fiber groups, and fully determine them in particular cases, covering a wide range of interesting examples.