Sector reconstruction via conjugacy-class/trivial-representation isomorphism

Construct an explicit isomorphism between the dual Hilbert subspace \tilde{\mathscr{H}}^{c} labeled by a conjugacy class c \subset G and the trivial-representation sector \mathscr{H}_g^{\hat{1}} of the Hilbert space of the twisted Hamiltonian H^g, for every g in c.

Background

Beyond projector identities, the authors analyze how dual Hilbert spaces decompose under gauging. They conjecture that sectors \tilde{\mathscr{H}}^{c} of the dual theory, labeled by conjugacy classes c of G, are isomorphic to the \hat{1} (trivial) representation sector of H^g for any g in c.

Establishing this sector-by-sector identification would provide a concrete lattice realization of the Tannaka–Krein duality picture and would rigorously justify how twisted sectors in the original model reconstruct the full dual Hilbert space.