Blocks of Fibered Burnside Rings
Benjamín García, Alberto G. Raggi-Cárdenas
TL;DR
This work characterizes the primitive idempotents and indecomposable factors (blocks) of the $A$-fibered Burnside ring $B^A_R(G)$ for a characteristic-zero domain $R$ with finite $\mathrm{Tor}_{\mathrm{exp}(G)}(A)$ and no invertible prime divisor of $|G|$. It elucidates the prime spectrum by extending Dress' construction, classifying prime ideals via $P$-equivalence and residue characteristic, and describing how the spectrum fibers over $\mathrm{Spec}(R[\zeta])$ with Galois action. The blocks are explicitly described: primitive idempotents $e^G_{[J]}$ sum over $(H,\Phi)$ with $O^s(H)=_G J$, and each block $B^A_R(G) e^G_{[J]}$ is isomorphic to the corresponding block for the Weyl group $W_G J$, with the solvable component given by $J=1$. These results provide bases for indecomposable factors, link fibered Burnside rings to solvable Weyl-group components, and refine Dress' prime-spectrum analysis in a broad, algebraic setting.
Abstract
In this article, we provide bases for the indecomposable factors of fibered Burnside rings in finite rank, and we give further characterizations of these as solvable components of fibered Burnside rings for certain Weyl groups. We revisit Dress' construction of prime spectra of fibered Burnside rings and its connected components for some rings of characteristic zero.