Closed-form description of the image of Ind × DHR for finite-group symmetric dualities

Determine a closed-form description of the image of the map Ind × DHR from the group of dualities on the symmetric operator algebra A^G of a one-dimensional spin chain with faithful on-site action by a finite group G in a self-dual representation to R^× × Aut_br(Z(Rep(G))).

Background

The authors define a homomorphism combining a numerical index (Ind), generalizing the GNVW index, with the action on the DHR category (DHR) to map dualities to R^× × Aut_br(Z(Rep(G))). They prove a classification criterion: two G-dualities are equivalent up to locally symmetric finite-depth circuits if and only if they have the same Ind value and induce the same braided autoequivalence on Z(Rep(G)).

Despite this equivalence classification, the authors do not provide a closed-form characterization of which pairs in R^× × Aut_br(Z(Rep(G))) actually occur as images of dualities under Ind × DHR, and they explicitly defer this precise description to future work.