Burnside rings for Real $2$-representation theory: The linear theory

TL;DR

This paper develops a foundational framework for Real 2-representation theory of 2-groups, introducing new constructions and unifying previous approaches through a Real Burnside ring perspective and categorical character theory.

Contribution

It introduces a 2-equivariant Morita bicategory framework, constructs a Real Burnside ring variant, and unifies existing theories of 2-representations and Real structures.

Findings

Identification of the Grothendieck ring as a Real Burnside ring variant

Introduction of a novel induction construction in the bicategory

Advancement of Real categorical character theory

Abstract

This paper is a fundamental study of the Real $2$-representation theory of $2$-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a $2$-equivariant Morita bicategory, where a novel construction of induction is introduced. We identify the Grothendieck ring of Real $2$-representations as a Real variant of the Burnside ring of the fundamental group of the $2$-group and study the Real categorical character theory. This paper unifies two previous lines of inquiry, the approach to $2$-representation theory via Morita theory and Burnside rings, initiated by the first author and Wendland, and the Real $2$-representation theory of $2$-groups, as studied by the second author.