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Describe the image of the Ind×DHR homomorphism for G-dualities

Determine a closed-form description of the image of the homomorphism Ind×DHR from the group of dualities on the symmetric quasi-local algebra A^G (for a finite group G acting in the regular on-site representation) to R× × Aut_br(Z(Rep(G))).

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Background

The work defines a map combining a numerical index (generalizing the GNVW index) with the induced braided autoequivalence on the DHR category, yielding a homomorphism Ind×DHR into R× × Aut_br(Z(Rep(G))). The corollary provides a classification of equivalence classes of G-dualities but does not identify which pairs (r, φ) in the codomain are realized.

The authors explicitly state that obtaining a closed-form description of the image of this homomorphism is left for future work.

References

The above corollary completely classifies the equivalence of $G$-dualities, including which dualities are trivial. However, it falls short of providing a closed form description of the image of $\text{Ind}\times \text{DHR}$ in $\mathbbm{R}{\times}\times \text{Aut}_{br}(\mathcal{Z}(\text{Rep}(G)))$. We leave this question to future work.

— Quantum cellular automata and categorical dualities of spin chains (2410.08884 - Jones et al., 11 Oct 2024) in Introduction, after Corollary 1 (classificationcor)