Categorical representation theory of diagrammatic categories for finite subgroups of SU(2)

Classify and study the representations of the diagrammatic categories associated with the finite subgroups of SU(2), including their categorical representation theory and extensions of notions such as highest-weight modules, semisimplicity, and irreducible modules.

Background

The paper constructs diagrammatic categories from representation graphs and establishes, under specified hypotheses, equivalences between suitable quotients of these categories and monoidal subcategories of representation categories. The authors identify the representation theory of the resulting diagrammatic categories as a direction not addressed by the construction itself.

The unresolved problem concerns diagrammatic categories associated with finite subgroups of SU(2), asking whether standard representation-theoretic structures can be developed for them and how their categorical representation theory should be characterized.

References

Can we classify and study the representations of the diagrammatic categories associated to the finite subgroups of $SU(2)$? In particular, what is the categorical representation theory of these diagrammatic categories, and can we extend some notions such as highest weight module, semi-simplicity, irreducible modules, etc. to these categories?

— Diagrammatic Categories which arise from Representation Graphs  (2502.05005 - Reynolds, 7 Feb 2025) in Question in Section 1, Introduction