The spectrum of the Burnside Tambara functor

TL;DR

This paper determines the prime ideal spectrum of the Burnside Tambara functor for any finite group, using advanced algebraic techniques and explicit computations for specific groups.

Contribution

It introduces a Tambara functor analogue of ghost coordinates and computes the spectrum for various finite groups, expanding understanding of Tambara functors.

Findings

Computed spectra for dihedral groups, Q_8, A_4, and GL_3(F_2)

Developed a Tambara functor ghost coordinate framework

Enhanced the algebraic understanding of Burnside Tambara functors

Abstract

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 $GL_3(F_2)$.