Character-weighted projector identity for twisted gauging

Establish that for gauging a finite group G via a strongly symmetric duality operator D^(g) intertwining the twisted Hamiltonian H^g and the dual Hamiltonian \tilde{H}, the operator identity D^(g) (D^(g))^\dagger = \sum_{\hat{g} \in \mathrm{Rep}(G)} \chi_{\hat{g}}(g) \; \tilde{\mathcal{O}}_{\hat{g}} holds in \mathrm{End}(\tilde{\mathscr{H}}), where \chi_{\hat{g}}(g) is the character of the irreducible representation \hat{g} evaluated at g and \tilde{\mathcal{O}}_{\hat{g}} are the generators of the \mathrm{Rep}(G) symmetry of the dual model.

Background

The paper studies non-invertible dualities in (1+1)-dimensional quantum lattice models realized by gauging a finite group G. For strongly symmetric duality operators, the authors derive \mathcal{D}^\dagger\mathcal{D} = \sum_{g\in G}\mathcal{O}g and \mathcal{D}\mathcal{D}^\dagger = \sum{\hat{g}\in\mathrm{Rep}(G)}\mathrm{dim}(\hat{g})\,\tilde{\mathcal{O}}_{\hat{g}} as projector relations without twists.

They then conjecture a generalization to twisted sectors labeled by elements g\in G, asserting that in the dual theory the corresponding sandwiched projector is a character-weighted sum over \mathrm{Rep}(G). Proving this identity would clarify how twisted boundary conditions map to dual symmetry projectors and would systematize sector decomposition under gauging in lattice models.