Research paper

Diagrammatics for dicyclic groups

A diagrammatic presentation for the representation theory of the type D finite subgroups of SU(2).

diagrammaticsdicyclic groupsTemperley-LiebMcKay/ADE
arXiv

Data

  • Title: Diagrammatics for dicyclic groups
  • Authors: Peter DeBello and Daniel Tubbenhauer
  • Status: preprint. Last update: Mon, 16 Dec 2024 22:11:58 UTC
  • arXiv: https://arxiv.org/abs/2412.12376

Abstract

Using that the dicyclic group is the type D subgroup of SU(2), we extend the Temperley–Lieb diagrammatics to give a diagrammatic presentation of the complex representation theory of the dicyclic group.

What is the point?

The finite subgroups of SU(2) sit inside the ADE story. The type A case is comparatively simple. Type D is the first place where the diagrammatics becomes genuinely interesting. This paper describes what has to be added to Temperley–Lieb diagrams to see the representation category of dicyclic groups.

ADE:
finite subgroups of SU(2).
Type D:
dicyclic groups.
Diagrams:
decorated Temperley–Lieb calculus.

A picture

One of the diagrammatic relations appearing in the presentation.

Dicyclic diagrammatic relation