Convention for fusion ring names

To classify fusion rings it is essential that there exists a naming scheme that covers all fusion rings and assigns a unique name to each ring. This page describes the convention that is used on this wiki.

Naming scheme for multiplicity-free fusion rings

Multiplicity-free fusion rings are denoted by \(\text{FR}_i^{r,n}\) where

\(r\) denotes the rank of the fusion ring (which equals the number of particles including the vacuum).
\(n\) denotes the number of particles that are not self-dual, i.e. not their own antiparticle. Mathematically these are the elements whose square does not contain the identity.
\(i\) denotes the position of the ring in the list of multiplicity-free fusion rings of rank \(r\) with \(n\) non self-dual particles. The position of a ring in this list is determined by the canonical order described below.

Canonical order of multiplicity-free fusion rings

Given a rank \(r\) and \(n\) non self-dual particles the multiplicity-free fusion rings are ordered on amount of non-zero structure constants and, if there is still ambiguity, according to the following scheme:

First the elements of the rings themselves are sorted. All self-dual particles are grouped together and the non-self-dual particles are grouped in dual pairs. The self-dual particles appear before the pairs and are sorted by increasing quantum dimension. The pairs are sorted on increasing maximum quantum dimension and within the pairs the particles are sorted on quantum dimension.
Let \(N_{a,b}^c\) denote the structure constants of a multiplication table of a fusion ring. For each permutation of the elements of the ring (where the identity is kept fixed, and the previous ordering is kept fixed) a list of digits can be created by ordering the structure constants \(N_{a,b}^c\) lexicographically on \(a,b,c\) and expressing them in a number base which is higher than the rank of the ring: \(N_{1,1}^1,N_{1,1}^2,\ldots,N_{1,2}^1,N_{1,2}^2,\ldots,N_{r,r}^r\).
By regarding each of these lists as a number with digits given by the list elements, a unique number is assigned to each permutation of the same ring. Moreover these numbers are unique for each different fusion ring.
Taking the maximum value of the numbers generated per ring, a unique number is then assigned to each fusion ring.
Using these numbers, the lists of multiplicity-free fusion rings of rank \(r\) with \(n\) non self-dual particles can be ordered in a canonical way.

Note that

a single notation for any fusion ring is then obtained by setting \(\text{FR}_i^{r,1,n} \equiv \text{FR}_i^{r,n}\).
the ordering as described above is chosen with eye on computational efficiency, rather than simplicity. Most rings can already be ordered by looking at the rank and amount of non-self-dual particles

Example

Consider for example the ring \(\text{SU}(2)_3\) with multiplication table

\[\begin{array}{|llll|} \hline \mathbf 1 & \mathbf 2 & \mathbf 3 & \mathbf 4 \\[4pt] \mathbf 2 & \mathbf 1 & \mathbf 4 & \mathbf 3 \\[4pt] \mathbf 3 & \mathbf 4 & \mathbf 1 + \mathbf 4 & \mathbf 2 + \mathbf 3 \\[4pt] \mathbf 4 & \mathbf 3 & \mathbf 2 + \mathbf 3 & \mathbf 1 + \mathbf 4 \\[4pt] \hline \end{array}\]

and quantum dimensions

\[\left\{ 1, 1, \frac{1}{2}+\frac{\sqrt{5}}{2}, \frac{1}{2}+\frac{\sqrt{5}}{2} \right\}\]

If we sort the elements according to (1.) then we obtain either \(( \mathbf 1, \mathbf 2, \mathbf 3, \mathbf 4 )\) or \(( \mathbf 1, \mathbf 2, \mathbf 4, \mathbf 3 )\). The digits obtained from the lists of structure constants are the following:

\[d_1 = 1000010000100001010010000001001000100001100101100001001001101001\]

and

\[d_2 = 1000010000100001010010000001001000100001101001010001001001011010\]

of which \(d_2\) is the largest one and thus labels the fusion ring.

Naming scheme for rings with multiplicity

Fusion rings with multiplicity are denoted by \(\text{FR}_i^{r,m,n}\) where

\(r\) denotes the rank of the fusion ring.
\(m\) equals the multiplicity of the fusion ring, i.e. the largest structure constant.
\(n\) denotes the amount of particles that are not self-dual.
\(i\) denotes the position of the ring in the list of fusion rings of rank \(r\), multiplicity \(m\), and \(n\) non self-dual particles. The position of a ring in this list is determined by the canonical order described below.

Canonical order of fusion rings with multiplicity