Some simple biset functors

TL;DR

This paper determines the dimensions of simple biset functors evaluated at finite groups and introduces a Green biset functor related to conjugacy classes of p-elementary subgroups, providing new insights into biset functor structure.

Contribution

It provides explicit formulas for the dimensions of simple biset functors and introduces a new Green biset functor related to p-elementary subgroup conjugacy classes.

Findings

Dimension formula for simple biset functors evaluated at any finite group.

Introduction of the Green biset functor E_p as a quotient of the Burnside functor.

E_p(G) is a free abelian group with rank equal to p-elementary subgroup conjugacy classes.

Abstract

Let $p$ be a prime number, let $H$ be a finite $p$-group, and let $\mathbb{F}$ be a field of characteristic 0, considered as a trivial $\mathbb{F} \mathrm{Out}(H)$-module. The main result of this paper gives the dimension of the evaluation $S_{H,\mathbb{F}}(G)$ of the simple biset functor $S_{H,\mathbb{F}}$ at an arbitrary finite group $G$. A closely related result is proved in the last section: for each prime number $p$, a Green biset functor $E_p$ is introduced, as a specific quotient of the Burnside functor, and it is shown that the evaluation $E_p(G)$ is a free abelian group of rank equal to the number of conjugacy classes of $p$-elementary subgroups of $G$.