Determination of the tensor ideal giving the quotient equivalence

Determine the tensor ideal $\mathcal{I}$ of the diagrammatic category $\mathbf{Dgrams}_{R(V,G)}$, for a connected representation graph with no multiple parallel edges, by specifying relations that ensure the categorical Schur property and yield an equivalence with the corresponding full subcategory of $G$-modules.

Background

The paper proves that, after quotienting DgramsR(V,G)\mathbf{Dgrams}_{R(V,G)} by a tensor ideal satisfying the categorical analogue of Schur's lemma, the induced functor to the subcategory of irreducible-generated GG-modules is an equivalence.

The tensor ideal is not explicitly determined in general. The authors note that finding it requires deriving diagrammatic relations from the representation theory of the particular group and may be computationally nontrivial, so the problem is left unresolved on a case-by-case basis.

References

Of course, it remains to determine $\mathcal{I}$, for example, by giving a set of relations. This will presumably depend on the specifics of the representation theory of $G$ and would need to be determined on a case-by-case basis.

— Diagrammatic Categories which arise from Representation Graphs  (2502.05005 - Reynolds, 7 Feb 2025) in Immediately after the equivalence theorem in Section 3, subsection “The Induced Functor $\overline{\mathcal{H}_{R(V,G)}}$”