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Fusion Rings over Drinfeld Doubles

Wenqi Li

Abstract

The fusion rules in $\mathrm{Rep}_f D(G)$ for a finite group $G$ can be computed in terms of character inner products. Using an explicit formula for these fusion rules, we show that $\mathrm{Rep}_f D(G)$ is multiplicity free for two infinite families of finite groups: the Dihedral groups and the Dicyclic groups. In fact, we will compute all fusion rules in these categories. Multiplicity freeness is a desired property for modular tensor categories, since it greatly simplifies the computation of $F$-matrices. Furthermore, we observe that the fusion rules for Dihedral groups $D_{2n}$ with $n$ odd are extremely similar to the fusion rules of Type $B$ level $2$ fusion algebras of Wess-Zumino-Witten conformal field theories. Moreover, we give a proof of the fusion rule formula by using Mackey theory.

Fusion Rings over Drinfeld Doubles

Abstract

The fusion rules in for a finite group can be computed in terms of character inner products. Using an explicit formula for these fusion rules, we show that is multiplicity free for two infinite families of finite groups: the Dihedral groups and the Dicyclic groups. In fact, we will compute all fusion rules in these categories. Multiplicity freeness is a desired property for modular tensor categories, since it greatly simplifies the computation of -matrices. Furthermore, we observe that the fusion rules for Dihedral groups with odd are extremely similar to the fusion rules of Type level fusion algebras of Wess-Zumino-Witten conformal field theories. Moreover, we give a proof of the fusion rule formula by using Mackey theory.
Paper Structure (12 sections, 32 theorems, 56 equations)

This paper contains 12 sections, 32 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Modular Tensor Categories from Drinfeld Doubles
  3. A Character Formula for Fusion Rules
  4. Fusion Rules via Mackey Theory
  5. Multiplicity Freeness for Dihedral Groups: $n$ odd
  6. Multiplicty Freeness for Dihedral Groups: $n$ even
  7. Dicyclic Groups: $n$ even
  8. Dicyclic Groups: $n$ odd
  9. Further Observations
  10. Computations using Sage
  11. Other multiplicity free examples
  12. Connection to the type B level 2 Verlinde algebras

Key Result

Theorem 2.1

The category of finite dimensional representations of $D(G)$ is a modular tensor category. For $g \in G$, denote its conjugacy class by $\overline{g}$. The dual of a simple object is and the normalized $S$-matrix of this category is

Theorems & Definitions (56)

  • Theorem 2.1
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • Theorem 3.3
  • Lemma 3.4
  • proof
  • Corollary 3.5
  • proof
  • ...and 46 more