The 2-character theory for finite 2-groups

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 G is equivalent to the Drinfeld center of the 2-group algebra Vec ₆, which categorifies the Fourier transform on finite abelian groups. After transferring the canonical nondegenerate braided monoidal structure from Z₁ (Vec ₆), we discover that irreducible 2-characters of 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 G. In particular, the fusion rule of 2Rep (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 B G.