The Spectrum of the Burnside Tambara Functor

Abstract Tambara functors are an equivariant generalization of commutative rings. In previous work, Nakaoka introduced the spectrum of prime ideals of a Tambara functor and computed the spectrum of the Burnside Tambara functor, the equivariant analogue of the Zariski spectrum of the integers, over cyclic $p$-groups. Subsequently, Calle and Ginnett computed the spectrum of the Burnside Tambara functor over arbitrary finite cyclic groups using a generalization of Dress’ ghost coordinates for Burnside rings. In this paper, 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 $GL_{3}(\mathbb{F}_{2})$.