The 2-character theory of finite 2-groups

Abstract

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 $|\mathrm{B}\mathcal{G}|$.

38 pages. Add section 4.3 on S^1-invariance and add more remarks. Comments are welcome

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