2-Groups, 2-Characters, and Burnside Rings

TL;DR

This paper develops a character theory for 2-representations of 2-groups using extended Burnside rings, establishing a key connection with the Ganter-Kapranov 2-character and providing a new proof of Osorno's formula.

Contribution

It introduces a novel approach linking 2-character theory with Burnside rings, offering new insights and proofs in the study of 2-representations.

Findings

Ganter-Kapranov 2-character as a mark homomorphism

Extended Burnside rings effectively analyze 2-representations

New proof of Osorno's formula for 2-characters

Abstract

We study 2-representations, i.e., actions of 2-groups on 2-vector spaces. Our main focus is character theory for 2-representations. To this end we employ the technique of extended Burnside rings. Our main theorem is that the Ganter-Kapranov 2-character is a particular mark homomorphism of the Burnside ring. As an application we give a new proof of Osorno formula for the Ganter-Kapranov 2-character of a finite group.