A Categorical Decomposition of $\mathbb C^{\times}$-fibered $p$-biset Functors

TL;DR

The paper extends Bouc's orthogonal idempotent decomposition from the double Burnside algebra to the $\mathbb{C}^{\times}$-fibered setting, building a truncated algebra $\mathcal{E}(G)$ and a refined system of idempotents that decompose evaluations of $\mathbb{C}^{\times}$-fibered $p$-biset functors. It constructs a full set of mutually orthogonal idempotents in the double fibered Burnside algebra and uses them to obtain a canonical block decomposition of the category $\mathcal{F}_{k,p}^{\mathbb{C}^{\times}}$, indexed by isomorphism classes of atoric $p$-groups, with vertices for indecomposable functors and Ext-vanishing for simples with distinct vertices. The framework yields a concrete description of composition factors for the monomial Burnside functor and provides a method to analyze fibered functors through their underlying biset structure, connecting fibered and classical biset theory. Overall, the results offer a robust categorical decomposition and vertex-based structure theory for fibered $p$-biset functors, with potential applications to representation-theoretic constructions that incorporate multiplicative character data.

Abstract

We generalize Bouc's construction of orthogonal idempotents in the double Burnside algebra to the setting of the double $\mathbb{C}^\times$-fibered Burnside algebra. This yields a structural decomposition of the evaluations of $\mathbb{C}^\times$-fibered biset functors on finite groups. We then construct a complete set of orthogonal idempotents in the category of $\mathbb{C}^\times$-fibered $p$-biset functors, leading to a categorical decomposition of this category into subcategories indexed by isomorphism classes of atoric $p$-groups. Furthermore, we introduce the notion of vertices for indecomposable functors and establish that the Ext-groups between simple functors with distinct vertices vanish. As an application, we describe a set containing composition factors of the monomial Burnside functor, thereby providing new insights into its structure. Additionally, we develop a technique for analyzing fibered biset functors via their underlying biset structures.