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The Spectrum of the Burnside Tambara Functor of a Cyclic Group

Maxine Calle, Sam Ginnett

Abstract

We derive a family of prime ideals of the Burnside Tambara functor for a finite group $G$. In the case of cyclic groups, this family comprises the entire prime spectrum. We include some partial results towards the same result for a larger class of groups.

The Spectrum of the Burnside Tambara Functor of a Cyclic Group

Abstract

We derive a family of prime ideals of the Burnside Tambara functor for a finite group . In the case of cyclic groups, this family comprises the entire prime spectrum. We include some partial results towards the same result for a larger class of groups.

Paper Structure

This paper contains 10 sections, 23 theorems, 93 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background: Tambara Ideals and the Burnside Spectrum
  3. Tambara functors and prime ideals.
  4. The Burnside ring, its prime spectrum, and its Tambara structure
  5. A Collection of Prime Ideals
  6. Describing The Ideals In Terms of Generators
  7. Ring-Theoretic Generators
  8. Tambara-Theoretic Generators
  9. Calculating the Spectrum for a finite Cyclic Group
  10. Directions for Further Investigation

Key Result

Theorem 1.1

Let $G$ be a finite group. Then for any $C\leq G$ and $p$ a prime or zero, $\mathfrak{p}_{C, p}$ is a prime ideal of $\underline{A}_G$, where $\mathfrak{p}_{C, p}$ is defined in def:the_ideals. If $G$ is Abelian, then we can establish some containment rules. For any $H,K\leq G$ and $p,q$ prime, we h where $O^p(H)$ is defined in thm:dress containment. Furthermore, if $G$ is cyclic, then

Figures (2)

  • Figure 1: Inclusion structure of the prime ideals of $\underline{A}_{C_{12}}$. Every ideal in $\operatorname{Spec}(\underline{A}_{C_{12}})$ is of the form $\mathfrak{p}_{C_i, p}$ for some $i\mid 12$ and $p$ prime or $0$. There is an arrow from $\mathfrak{p}_{C_i,p}$ to $\mathfrak{p}_{C_j, q}$ if $\mathfrak{p}_{C_i,p}\subseteq\mathfrak{p}_{C_j, q}$. In the case where $p=2,3$, some of the inclusions become equalities, as indicated by the red and blue arrows, respectively.
  • Figure 2: Inclusion structure of prime ideals of $\underline{A}_{C_n}$. There is an arrow from $\mathfrak{p}_{C_i,p}$ to $\mathfrak{p}_{C_j,q}$ if $\mathfrak{p}_{C_i,p}\subseteq\mathfrak{p}_{C_j,q}$. At the $0$ and $p'$ levels (for $p'\nmid n$), the inclusion structure is dual to the subgroup lattice of $C_n$. If $p\mid n$, then some of the inclusions become equalities, precisely when $O^p(C_i)=O^p(C_j)$.

Theorems & Definitions (60)

  • Theorem 1.1: \ref{['thm:PCp prime', 'thm:main_thm', 'thm:containment']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.1: nakaoka:2014
  • Definition 2.7
  • Definition 2.8
  • ...and 50 more