Overview
This research note details an investigation into the construction of near-group fusion categories. The approach focuses on generating explicit representations of these categories within specific graph planar algebras. A key aspect of this method involves developing a presentation for a cyclic near-group fusion category, utilizing a Q-system as a generating morphism. This presentation subsequently yields a rational system of equations. Solutions to this system correspond to faithful embeddings of near-group fusion categories into a particular 2-coloured graph planar algebra. The note concludes by presenting solutions for these systems of equations pertaining to odd cyclic groups up to order 13.
Research Context
The study operates within the mathematical domain of near-group fusion categories and graph planar algebras. Near-group fusion categories are a specific type of algebraic structure. Graph planar algebras are mathematical constructs that provide a framework for representing and manipulating certain algebraic objects. The work specifically addresses the challenge of constructing these near-group fusion categories. This construction method offers an alternative to the existing Evans-Gannon construction for the corresponding near-group fusion categories.
Approach
The methodological approach involved several stages:
- Investigation of Construction: The primary objective was to investigate the construction of near-group fusion categories.
- Representation Building: This construction was achieved by building explicit representations of these categories.
- Medium of Representation: These explicit representations were built within certain graph planar algebras.
- Presentation Development: A presentation for a cyclic near-group fusion category was developed. This presentation is characterized as 'subfactor-centric'.
- Generating Morphism: A Q-system was utilized as one of the generating morphisms within this presentation.
- Equation Derivation: The developed presentation was used to obtain a rational system of equations.
- Embedding Correspondence: Solutions to this rational system of equations correspond to a faithful embedding of a near-group fusion category into a specific 2-coloured graph planar algebra.
Findings
- A specific presentation for a cyclic near-group fusion category was achieved, notable for its 'subfactor-centric' nature.
- This presentation utilized a Q-system as a generating morphism.
- A rational system of equations was derived from this presentation.
- Solutions to this rational system of equations correspond to a faithful embedding of a near-group fusion category into a certain 2-coloured graph planar algebra.
- Solutions were provided for these systems of equations for odd cyclic groups up to order 13.
- The derived solutions offer an alternate construction method for the corresponding near-group fusion categories, distinct from the Evans-Gannon construction.
Why This Matters
The provision of an alternate construction method for near-group fusion categories, specifically for odd cyclic groups up to order 13, expands the available tools and techniques within abstract algebra. This new method, which leverages graph planar algebras and Q-systems, offers a distinct pathway to understand and build these complex mathematical structures, complementing existing methodologies like the Evans-Gannon construction.
The explicit solutions provided for a range of odd cyclic groups up to order 13 demonstrate the practical applicability of the developed rational system of equations and the underlying subfactor-centric presentation. These concrete examples validate the theoretical framework established in the note.
The focus on building explicit representations within specific graph planar algebras may provide new perspectives on the interconnections between fusion categories and graphical algebra structures, potentially facilitating further research in related fields.