Decomposition in 2d non-invertible gaugings

We extend the decomposition conjecture to 2d quantum field theories with a gauged $\text{Rep}(H)$ symmetry category for $H$ a finite-dimensional semisimple Hopf algebra with $\text{Rep}(G)$ trivially-acting and $\text{Vec}(\Gamma)$ the remaining symmetry, for $G,\Gamma$ finite groups. We check our extension by explicitly computing partition functions, and by verifying that previous results arise as special cases. Furthermore, we compute the topological operators responsible for enforcing the decomposition. Then, drawing from these results, we formulate a plausible decomposition conjecture for the even more general case of $\text{Rep}(H'')$ trivially-acting and $\text{Rep}(H')$ the remaining symmetry, for $H',H''$ Hopf algebras, not necessarily associated with groups.