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Diagrammatics for dicyclic groups

Peter DeBello, Daniel Tubbenhauer

TL;DR

The paper develops a complete diagrammatic framework for the representation theory of dicyclic groups by extending Temperley–Lieb diagrams to the dicyclic setting. It defines the infinite and finite dicyclic TL categories, proves a ribbon equivalence with the corresponding representation categories via Karoubi completion, and constructs dicyclic Jones–Wenzl projectors that project onto leading tensor-summands. It also furnishes explicit basis constructions for endomorphism algebras via affine type–D walks and provides Magma computations that illustrate the diagrammatic calculus and its algebraic realizations. This yields a robust, diagrammatic description of both $\mathrm{Dic}_{\infty}$ and $\mathrm{Dic}_{n}$ representation theories, paralleling the classical $SU(2)$ and TL theory while capturing the nuances of the dicyclic setting.

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.

Diagrammatics for dicyclic groups

TL;DR

The paper develops a complete diagrammatic framework for the representation theory of dicyclic groups by extending Temperley–Lieb diagrams to the dicyclic setting. It defines the infinite and finite dicyclic TL categories, proves a ribbon equivalence with the corresponding representation categories via Karoubi completion, and constructs dicyclic Jones–Wenzl projectors that project onto leading tensor-summands. It also furnishes explicit basis constructions for endomorphism algebras via affine type–D walks and provides Magma computations that illustrate the diagrammatic calculus and its algebraic realizations. This yields a robust, diagrammatic description of both and representation theories, paralleling the classical and TL theory while capturing the nuances of the dicyclic setting.

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.

Paper Structure

This paper contains 15 sections, 34 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Representation theory of dicyclic groups
  3. The type D subgroups $\text{Dic}_{\infty}$ and $\text{Dic}_{n}$
  4. Representation theory of dicyclic groups
  5. Diagrammatics for dicyclic groups
  6. The dicyclic Temperley--Lieb algebra $DiTL_k(\delta)$
  7. The dicyclic Temperley--Lieb category $\mathbf{DiTL}(\delta, \infty)$
  8. Infinite dicyclic categories
  9. An equivalence $\text{Kar}(\mathbf{DiTL}(-2,\infty)) \cong \mathbf{Rep}(\text{Dic}_{\infty})$
  10. Dicyclic Jones--Wenzl projectors
  11. Finite dicyclic categories
  12. The degree $n$ dicyclic Temperley--Lieb category $\mathbf{DiTL}(\delta,n)$
  13. An equivalence $\text{Kar}({\mathbf{DiTL}}(-2, n) \cong \mathbf{Rep}(\text{Dic}_n)$
  14. Dicyclic bases and walks
  15. Example computations in Magma

Key Result

Lemma 1

We have the following properties of $\text{Dic}_{\infty}$.

Theorems & Definitions (88)

  • Lemma 1
  • proof
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 78 more