The spectrum of the Burnside Tambara functor

We compute the spectrum of prime ideals in the Burnside Tambara functor over an arbitrary finite group. Our proof uses recent advances in the commutative algebra of Tambara functors, as well as a Tambara functor analogue of ghost coordinates which works over arbitrary finite groups and clarifies some previous computations. As examples, we explicitly compute the spectrum of the Burnside Tambara functor over all dihederal groups, the quaternion group $Q_8$, the alternating group $A_4$, and the general linear group $GL_3(F_2)$.

💡 Research Summary

The paper “The Spectrum of the Burnside Tambara Functor” provides a complete description of the prime‑ideal spectrum of the Burnside Tambara functor A_G for an arbitrary finite group G. The authors begin by recalling the classical theory of the Burnside ring A(G), Dress’s mark homomorphisms ϕ_H^G, and the description of the Zariski spectrum of A(G) in terms of kernels of these marks. They then introduce Tambara functors, emphasizing the three families of structure maps (restriction, transfer, norm) and the way a Tambara functor is determined by its values on orbits G/H.

The central technical innovation is the construction of a “ghost” Tambara functor (A_G) built from the tables of marks of all subgroups of G. The ghost map χ : A_G → (A_G) is shown to be a morphism of Tambara functors and a levelwise integral extension. This resolves a gap in earlier work where the Tambara nature of the ghost was not proved. Because χ_* : Spec((A_G)) → Spec(A_G) is surjective, the problem of determining the spectrum of A_G reduces to computing the spectrum of the ghost functor.

For each subgroup H ≤ G and each prime integer p (or p = 0) the authors define a prime ideal p_{H,p} ⊂ A_G. In the ghost world they construct corresponding prime ideals P_{H,p} ⊂ (A_G) and prove that these exhaust all prime ideals of (A_G) (Theorem 4.2, Corollary 4.11, Proposition 4.12). Pulling back via χ gives the full list of primes in A_G: \