The 2-character theory of finite 2-groups

TL;DR

This paper develops a theory of 2-characters for finite 2-groups, establishing their properties, categorical equivalences, and connections to topological quantum field theories, thus extending classical character theory into higher categorical contexts.

Contribution

It introduces 2-characters for finite 2-groups, proves their properties, and relates them to the Drinfeld center, Lagrangian algebras, and TQFT, providing a higher categorical generalization.

Findings

2-characters exhibit conjugation invariance, additivity, multiplicativity, and orthogonality.

Category of class functors on a 2-group is equivalent to the Drinfeld center of its 2-group algebra.

Irreducible 2-characters correspond to full centers of 2-representations and Lagrangian algebras.

Abstract

In this work, we generalize the notion of character for 2-representations of finite 2-groups. The properties of 2-characters bear strong similarities to those classical characters of finite groups, including conjugation invariance, additivity, multiplicativity and orthogonality. With a careful analysis using homotopy fixed points and quotients for categories with 2-group actions, we prove that the category of class functors on a 2-group $\mathcal G$ is equivalent to the Drinfeld center of the 2-group algebra $\mathrm{Vec}_{\mathcal G}$, which categorifies the Fourier transform on finite abelian groups. After transferring the canonical nondegenerate braided monoidal structure from $\mathfrak Z_1(\mathrm{Vec}_{\mathcal G})$, we discover that irreducible 2-characters of $\mathcal G$ coincide with full centers of the corresponding 2-representations, which are in a one-to-one correspondence with Lagrangian algebras in the category of class functors on $\mathcal G$. In particular, the fusion rule of $2\mathrm{Rep}(\mathcal G)$ can be calculated from the pointwise product of Lagrangian algebras as class functors. From a topological quantum field theory (TQFT) point of view, the commutative Frobenius algebra structure on a 2-character is induced from a 2D topological sigma-model with target space $\lvert \mathrm{B} \mathcal G \rvert$.