Categorical formulation of the component-wise fusion structure

Develop a categorical formulation of the associative component-wise fusion-like algebra of H-invariant representation components, potentially as a monoidal category or fusion category equipped with an associativity isomorphism.

Background

The paper constructs an associative fusion-like algebra for individual H-invariant field components using Clebsch–Gordan coefficients. The H-invariant vector spaces themselves do not form an ordinary closed fusion algebra because projected multiplicities can be noninteger and need not satisfy the usual associativity relations.

The authors suggest that the component-wise structure may instead admit an associativity isomorphism between the two bracketings of a triple tensor product. Determining whether and how this structure can be organized as a monoidal or fusion category is explicitly identified as future work.

References

This suggests the existence of an associativity isomorphism a_{\alpha,\beta,\gamma}: (\tilde{\mathbf{r}{(\alpha)} \otimes \tilde{\mathbf{r}{(\beta)})\otimes \tilde{\mathbf{r}{(\gamma)} \to \tilde{\mathbf{r}{(\alpha)} \otimes ( \tilde{\mathbf{r}{(\beta)}\otimes \tilde{\mathbf{r}{(\gamma)})$, allowing one to formulate this structure in terms of a monoidal category or a fusion category, rather than a simple associative algebra. Exploring this categorical structure is an interesting theoretical direction that we leave for future work.

— Non-invertible Selection Rules from Generalized Discrete Gauging of Finite Non-Abelian Symmetries  (2609.11895 - Ohki et al., 10 Sep 2026) in Conclusion, Section 6