Parameterized diagrammatic categories and their decategorification

Determine how introducing a parameter analogous to the Temperley–Lieb parameter $\delta$ into the diagrammatic categories constructed from representation graphs changes their combinatorics, representation theory, and decategorifications.

Background

The Temperley–Lieb category admits a scalar parameter δ\delta, with δ=2\delta=2 yielding the category related to SU(2). The paper suggests investigating analogous parameterizations for the newly constructed diagrammatic categories.

The authors leave unresolved what structural effects such a parameter would have, explicitly mentioning changes to combinatorics, representation theory, and decategorification as possible objects of study.

References

What would introducing such a parameter to these diagrammatic categories change about the combinatorics or representation theory? For example, one might explore how these categories decategorify.

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

This suggests an interpolation of a functor $SO(3)_{q_N}\to GPA(\Gamma_N)$ is hiding somewhere. While the source of this hypothetical interpolation functor is clear and well-defined, the target remains a mystery to us.

— A graph planar algebra approach to near-group categories  (2609.26684 - Edie-Michell et al., 22 Sep 2026) in Remark following Theorem in Section 3, subsection “Embedding the $SO(3)_q$ vertex”