Conventions Used on the AnyonWiki

This page collects some of the (pages on) conventions that will be used on the wiki. Its aim is on the one hand to settle conventions and on the other to clarify the meaning of some commonly used symbols and expressions.

Pages Explaining Conventions

These might later be (partially) imported in this page.

Indexing of Roots of Polynomials - Documentation on how polynomial roots are indexed and referenced
$S$ -matrix - Conventions for the S-matrix in modular fusion categories

Fusion Ring Naming

Formal Name Convention

Fusion rings follow the standardized notation:

\text{FR}^{r,m}_n

Components:

FR: Fusion Ring identifier
$r$ (superscript, first): Rank - number of simple objects
$m$ (superscript, second): Multiplicity structure parameter
$n$ (subscript): Serial number within the classification

Examples

Formal Name	Meaning
$\text{FR}^{1,0}_1$	Trivial ring, rank 1
$\text{FR}^{2,0}_1$	$\mathbb{Z}_2$ ring, rank 2, serial 1
$\text{FR}^{2,0}_2$	Fibonacci ring, rank 2, serial 2
$\text{FR}^{3,0}_1$	Ising ring, rank 3, serial 1
$\text{FR}^{5,4}_3$	Rank 5, multiplicity parameter 4, serial 3

Common Names

Many fusion rings have common names from physics or representation theory:

Trivial: $\text{FR}^{1,0}_1$
$\mathbb{Z}_n$ : Cyclic group rings
Fibonacci (Fib): $\text{FR}^{2,0}_2$
Ising: $\text{FR}^{3,0}_1$
$\text{Rep}(D_n)$ : Dihedral group representations
$\text{PSU}(2)_k$ : Quantum $SU(2)$ at level $k$

Fusion Category Naming

Formal Name Convention

Fusion categories use a seven-parameter code:

\text{FC}^{a,b,c}_{d,e,f,g}

Components:

FC: Fusion Category identifier
$(a,b,c)$ (superscript): Triple encoding categorical structure
$(d,e,f,g)$ (subscript): Quadruple for refined classification

Relation to Fusion Rings

The category naming relates to its underlying fusion ring:

Parameters encode rank, structure type, and categorification data
Multiple categories can share the same fusion ring

Software Usage

To access a category by formal code:

Anyonica (Mathematica):

FusionCategoryByCode[[a,b,c,d,e,f,g]]
(* Shorthand *)
FCBC[[a,b,c,d,e,f,g]]

TensorCategories.jl (Julia):

anyonwiki(a,b,c,d,e,f,g)

Formal Codes

Structure

Each fusion ring has a unique formal code as an integer array:

[\text{rank}, m_1, m_2, n]

Parameters:

rank: Number of simple objects
$m_1$ : Primary multiplicity indicator (usually 1 for multiplicity-free)
$m_2$ : Secondary structural parameter
$n$ : Serial number in classification

Examples

Fusion Ring	Formal Code
Trivial	`[1, 1, 0, 1]`
$\mathbb{Z}_2$	`[2, 1, 0, 1]`
Fibonacci	`[2, 1, 0, 2]`
Ising	`[3, 1, 0, 1]`
$\text{Rep}(D_3)$	`[3, 1, 0, 2]`

Purpose

Unique identification: No two fusion rings share the same formal code
Computational: Used in database queries and software
Classification: Organizes rings by structural properties

Barcodes

Definition

A barcode is an alternative unique identifier for fusion rings, represented as an integer array.

Structure

The barcode encodes:

Fusion rule structure
Derived from multiplication table
Canonical form independent of object labeling

Example

For Fibonacci ring:

barcode: [specific integer sequence]

Usage

Database indexing: Alternative to formal codes
Isomorphism checking: Compare barcodes to test if rings are isomorphic
DOI reference: Cited in research papers

Mathematical Notation

Objects and Morphisms

Simple Objects:

X_0 = \mathbf{1}, X_1, X_2, \ldots, X_{r-1}

Tensor Product:

X_i \otimes X_j = \bigoplus_k (X_k)^{\oplus N_{ij}^k}

Dual Objects:

X_i^* \text{ or } \overline{X_i}

Morphism Spaces:

\text{Hom}_{\mathcal{C}}(X, Y) \text{ or } \text{Hom}(X, Y)

Fusion Coefficients

Notation: $N_{ij}^k$ or $N^k_{ij}$

Meaning: Multiplicity of $X_k$ in $X_i \otimes X_j$

Fusion Matrix:

(N_i)_{jk} = N_{ij}^k

Dimensions

Frobenius-Perron Dimension:

d_i^{\text{FP}} \text{ or simply } d_i

Quantum Dimension (in categories):

\dim(X_i) \text{ or } d_i

Global Dimension:

\mathcal{D}^2 = \sum_{i=0}^{r-1} d_i^2

In Database: Listed as $\mathcal{D}_{\text{FP}}^2$

Structure Constants

F-Symbols

Notation:

F^{abc}_d \text{ or } \left[F^{ab}_c\right]_{d\mu\nu}^{e\rho\sigma}

Simplified (multiplicity-free):

F^{abc}_d \in \mathbb{C}

Pentagon Equation:

\sum_{\mu} F^{ijk}_{\ell}{}^{m\mu} F^{m\ell n}_{p}{}^{ij\nu} = \sum_{\rho} F^{k\ell n}_{p}{}^{i\rho} F^{ijn}_{p}{}^{k\nu\rho}

R-Symbols

Notation:

R^{ab}_c \text{ or } [R^a_b]_{c\mu}^{\nu}

Braiding:

c_{X,Y}: X \otimes Y \xrightarrow{R} Y \otimes X

Hexagon Equations: Relate $R$ and $F$ symbols

S-Matrix

Notation:

S = (S_{ij})_{0 \leq i,j < r}

Definition (modular categories):

S_{ij} = \frac{1}{\mathcal{D}} \text{tr}(c_{X_j, X_i} \circ c_{X_i, X_j})

Properties:

Symmetric: $S_{ij} = S_{ji}$
Unitary: $S S^\dagger = I$
$S^2 = C$ (charge conjugation)

T-Matrix

Notation:

T = (T_{ij})_{0 \leq i,j < r}

Diagonal:

T_{ij} = \delta_{ij} \theta_i

where $\theta_i$ is the twist (topological spin) of $X_i$ .

Character Notation

Character Map:

\chi: \mathcal{R} \to \mathbb{C}

Character Table: Matrix $(\chi_i(X_j))$

Indexed:

Rows: Characters $\chi_0, \chi_1, \ldots, \chi_{r-1}$
Columns: Simple objects $X_0, X_1, \ldots, X_{r-1}$

Database Fields

JSON Structure

Fusion Ring Entry:

{
  "formal_code": [r, m1, m2, n],
  "barcode": [array of integers],
  "texnames": ["LaTeX name 1", "LaTeX name 2", ...],
  "mult_tab": [[fusion rules]],
  "numeric_characters": [[character values]],
  "numeric_frobenius_perron_dimensions": [d_0, d_1, ...],
  "numeric_frobenius_perron_dimension": D_FP^2,
  "tensor_product_decompositions": {...},
  "sub_fusion_rings": {...},
  "references": ["ref1", "ref2"],
  "software": ["software used"],
  "info": "description"
}

Field Descriptions

texnames: Array of alternative LaTeX names

First entry: Primary common name
Additional entries: Alternative notations

mult_tab: Multiplication table

mult_tab[i][j] = array of indices $k$ where $N_{ij}^k \neq 0$
Multiplicity-free: each entry is 0 or 1 occurrence

numeric_characters: Character table matrix

Rows: Characters
Columns: Simple objects
Values: Complex numbers (real for unitary)

File Naming

Downloaded Files

Structure Constants:

pentsol_*.txt: Pentagon equation solutions (F-symbols)
hexsol_*.txt: Hexagon equation solutions (R-symbols)
pivsol_*.txt: Pivotal structure data

Format (tab-separated):

i  j  k  l  m  n  Re(F)  Im(F)

First 6 columns: Object labels
Last 2 columns: Real and imaginary parts

LaTeX Rendering

In Documentation

Inline Math: Use $...$ or $...$

Display Math: Use $$...$$ or \[...\]

Common Symbols

Symbol	LaTeX	Meaning
$\mathbf{1}$	`\mathbf{1}`	Identity object
$\otimes$	`\otimes`	Tensor product
$\oplus$	`\oplus`	Direct sum
$\mathcal{C}$	`\mathcal{C}`	Category
$\mathcal{R}$	`\mathcal{R}`	Ring
$\mathcal{D}_{\text{FP}}^2$	`\mathcal{D}_{\text{FP}}^2`	FP dimension squared
$\mathbb{Z}_n$	`\mathbb{Z}_n`	Cyclic group

Abbreviations

Category Types

Abbreviation	Full Name
FC	Fusion Category
PFC	Pivotal Fusion Category
UFC	Unitary Fusion Category
BFC	Braided Fusion Category
MFC	Modular Fusion Category
UMTC	Unitary Modular Tensor Category

Other Terms

Abbreviation	Full Name
FP	Frobenius-Perron
QD	Quantum Dimension
TQFT	Topological Quantum Field Theory
6j	Six-j symbol (F-symbol)
DOI	Digital Object Identifier

Version and Updates

Data Version

AnyonWiki data follows semantic versioning:

Major: Significant structural changes
Minor: New entries added
Patch: Corrections and updates

Citing Data

When citing a specific fusion ring or category:

FR^{r,m}_n from AnyonWiki [version]
DOI: [if available]

Manual → - User guide
Data Format → - Technical specifications
API Documentation → - Programmatic access

References

For more on naming conventions, see:

Liu, Palcoux, Ren (2022) - Classification paper
Slingerland, Vercleyen (2022) - Low rank fusion rings
AnyonWiki GitHub repository - Implementation details

Conventions Used on the AnyonWiki

Pages Explaining Conventions

Fusion Ring Naming

Formal Name Convention

Examples

Common Names

Fusion Category Naming

Formal Name Convention

Relation to Fusion Rings

Software Usage

Formal Codes

Structure

Examples

Purpose

Barcodes

Definition

Structure

Example

Usage

Mathematical Notation

Objects and Morphisms

Fusion Coefficients

Dimensions

Structure Constants

F-Symbols

R-Symbols

S-Matrix

T-Matrix

Character Notation

Database Fields

JSON Structure

Field Descriptions

File Naming

Downloaded Files

LaTeX Rendering

In Documentation

Common Symbols

Abbreviations

Category Types

Other Terms

Version and Updates

Data Version

Citing Data

Related Pages

References