Vanishing evaluations of simple functors

TL;DR

This paper explores when simple biset functors evaluate to zero at finite groups, providing conditions and a formula for non-zero evaluations, advancing understanding of their behavior.

Contribution

It introduces new conditions and a closed formula for evaluating simple biset functors at finite groups, clarifying when evaluations vanish or not.

Findings

Identifies situations where evaluations are zero

Provides a closed formula for evaluations under certain conditions

Clarifies the behavior of simple biset functors at finite groups

Abstract

The classification of simple biset functors is known, but the evaluation of a simple biset functor at a finite group G may be zero. We investigate various situations where this happens, as well as cases where this does not occur. We also prove a closed formula for such an evaluation under some restrictive conditions on G.