arXiv:1208.1500v1 [math.QA] 7 Aug 2012
Near-group fusion categories and their doubles
David E. Evans
School of Mathematics, Cardiff University,
Senghennydd Road, Cardiff CF24 4AG, Wales, U.K.
Department of Mathematics, University of Alberta,
Edmonton, Alberta, Canada T6G 2G1
August 8, 2012
A near-group fusion category is a fusion category C where all but 1 sim-
ple objects are invertible. Examples of these include the Tambara-Yamagami
categories and the even sectors of the D(1)
are infinitely many others. We classify the near-group fusion categories, and
compute their doubles and the modular data relevant to conformal field theory.
Among other things, we explicitly construct over 40 new finite depth subfac-
tors, with Jones index ranging from around 6.85 to around 14.93. We expect
all of these doubles to be realised by rational conformal field theories.
and E6 subfactors, though there
2 The near-group systems
2.1The numerical invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3First class: Near-group categories with n′= n − 1 . . . . . . . . . . . . . . . . . . . .
2.4The remaining class: n′a multiple of n . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1Which finite group module categories are near-group? . . . . . . . . . . . . . . . . .
3.2The type G + 0 classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3The near-group categories for the trivial group G . . . . . . . . . . . . . . . . . . . .
3.4 The type G + n − 1 C∗-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5At least as many T’s as S’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4Tube algebras and modular data
4.1The tube algebras of near-group systems . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 The first class near-group C∗-categories: n′= n − 1 . . . . . . . . . . . . . . . . . . .
4.3The tube algebra in the second class . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 The modular data for the double of G + n when n is odd
46. . . . . . . . . . . . . . .
Considerable effort in recent years has been directed at the classification of subfactors
of small index. Subfactors of index ≤ 5 are now all known (see e.g. ). The
classification for index ≤ 4 was established some time ago. The Haagerup subfactor
 with Jones index (5+√13)/2 ≈ 4.30278, the Asaeda-Haagerup subfactor  with
index (5 +√17)/2 ≈ 4.56155, and the extended Haagerup subfactor  with index
≈ 4.37720, arose in Haagerup’s classification  of irreducible finite depth subfactors
of index between 4 and 3 +√3 ≈ 4.73205. A Goodman-de la Harpe-Jones subfactor
, coming from the even sectors of the subfactor corresponding to the A1,10⊂ C2,1
conformal embedding, has index 3 +√3. Then comes the Izumi-Xu subfactor 2221
 with index (5+√21)/2 ≈ 4.79 and principal graph in Figure 1, coming from the
G2,3⊂ E6,1conformal embedding.
The punchline is that, at least for small index, there are unexpectedly few sub-
factors. Does this continue with higher index? Are the aforementioned subfactors
exotic, or can we put them into sequences? In , Izumi realised the Haagerup and
Izumi-Xu subfactors using endomorphisms in Cuntz algebras, and suggested that his
construction may generalise. More precisely, to any abelian group G of odd order,
Izumi wrote down a nonlinear system of equations; any solution to them corresponds
to a subfactor of index (|G| + 2 +?|G|2+ 4)/2. He showed the Haagerup subfactor
solutions for the next several G, explained that the number of these depends on the
prime decomposition of |G|2+4, and argued that the Haagerup subfactor belongs to
an infinite sequence of subfactors and so should not be regarded as exotic.
Izumi in  also associated a second nonlinear system of equations to each finite
abelian group; to any solution of this system he constructs a subfactor of index
(|G| + 2 +?|G|2+ 4|G|)/2 and with principal graph 2|G|1, i.e. a star with one edge
an example). Izumi then found solutions for G = Zn(n ≤ 5) and Z2× Z2. G = Z1
and Z2correspond to the index < 4 subfactors A4and E6, respectively; his solution
for Z3 provides his construction for Izumi-Xu. An alternate construction of 2221,
involving the conformal embedding G2,3⊂ E6,1, is due to Feng Xu as described in
the appendix to  (see also ). As we touch on later in the paper, there may be
a relation between the series containing the Haagerup subfactor, and that containing
the Izumi-Xu subfactor.
corresponds to G = Z3, and that there also is a solution for G = Z5. In  we found
of length 1 and |G| edges of length 2 radiating from the central vertex (Figure 1 is
Figure 1. The 231 principal graph
One of our tasks in this paper is to construct several more solutions to Izumi’s
second family of equations, strongly suggesting that this family also contains infinitely
many subfactors. But more important, in this paper we study a broad class of systems
of endomorphisms, the near-group fusion categories, including the Izumi-Xu series as
a special case. We obtain a system of equations, generalising those of Izumi, providing
necessary and sufficient conditions for their existence. We identify the complete list
of solutions to the first several of these systems, which permits us the construction
of over 40 new finite-depth subfactors of index < 15,
A fusion category C  is a C-linear semisimple rigid monoidal category with
finitely many simple objects and finite-dimensional spaces of morphisms, such that
the endomorphism algebra of the neutral object is C. The Grothendieck ring of a
fusion category is called a fusion ring. Perhaps the simplest examples are associated
to a finite group G: the objects are G-graded vector spaces ⊕gVg, with monoidal
product Vg⊗ V′
examples group categories. The category Mod(G) of finite-dimensional G-modules is
also a fusion category.
We’re actually interested in certain concrete realisations of fusion categories, which
we call fusion C∗-categories: the objects are endomorphisms (or rather sectors, i.e.
equivalence classes of endomorphisms under the adjoint action of unitaries) on some
infinite factor M, the spaces Hom(ρ,σ) of morphisms are intertwiners, and the prod-
uct is composition. Two fusion C∗-categories are equivalent iff they are equivalent as
fusion categories — all that matters for us is that the factor M exists, not which one
it is. Every finite-depth subfactor N ⊂ M gives rise to two of these, one correspond-
ing to the principal, or N-N, sectors and the other to the dual principal, or M-M,
ones. For example, given an outer action α of a finite group G on an infinite factor N,
we get a subfactor N ⊂ N×G = M coming from the crossed product construction:
the N-N system realises the group category for G, while the M-M system realises
Mod(G). Not all fusion categories can be realised as fusion C∗-categories (e.g. the
modular tensor categories associated to the so-called nonunitary Virasoro minimal
models are not fusion C∗-categories).
Perhaps the simplest nontrivial example of the extension of a fusion category is
when the category C has precisely 1 more simple object than the subcategory C0, and
the latter corresponds to a finite abelian group. More precisely, simple objects [g] in C0
correspond to group elements g ∈ G, with tensor product [g][h] = [gh] corresponding
to group multiplication. The simple objects of C consist of the [g], together with
h= (V ⊗ V′)gh. Its fusion ring is the group ring ZG. We call such
some object we’ll denote [ρ]. Then [ρ] must be self-conjugate, [g][ρ] = [ρ] = [ρ][g], and
of  must be 1).
We call these near-group categories of type G + n′. In this paper we restrict to
abelian G, and we reserve n always for the order of G. Examples of these have been
studied in the literature:
g∈G[g] (the multiplicities n′′
gin the second term must be independent of
g because of equivariance [g][ρ] = [ρ]; because [ρ] is its own conjugate, the multiplicity
• the Ising model and the module category of the dihedral group D4, which are
of type Z2+ 0 and Z2× Z2+ 0, respectively;
• more generally, the Tambara-Yamagami systems are by definition those of type
G + 0;
• the A4,E6and Izumi-Xu subfactors are of type G + n for G = Z1,Z2,Z3re-
• more generally, Izumi’s second hypothetical family would be of type G + n;
• the D(1)
Z2+ 1, Z2+ 1, and Z3+ 2, respectively;
subfactor and the module category of groups S3and A4are of type
• more generally, the representation category of the affine group Aff1(Fq)∼=
More precisely, Aff1(Fq) is the group of all affine maps x ?→ ax + b where a ∈ F∗
and b ∈ Fq. It has precisely q = pkconjugacy classes, with representatives (a,0)
and (1,1). It has precisely q − 1 1-dimensional representations, corresponding to the
characters of F∗
and is the nontrivial summand of the natural permutation representation of Aff1(Fq)
on Fqgiven by the affine maps: (a,b).x = ax + b.
In this paper we classify the near-group C∗-categories G+n′, in the sense that we
obtain polynomial equations in finitely many variables, whose solutions correspond
bijectively to equivalence classes of the near-group C∗-categories. Given any near-
group C∗-category C with n′> 0, we identify a natural subfactor ρ(M) ⊂ M whose
even systems are both identified with C. We also work out the principal graph of the
closely related subfactor ρ(M) ⊂ MG. By contrast, we can realise some but not all
C with n′= 0, as the even sectors of a subfactor.
There is a fundamental dichotomy here: n′either equals n−1, or is a multiple of
n, where as always n = |G|. When n′< n, we have a complete classification:
Fact. Let G be any abelian group of order n.
(a) There are precisely two C∗-categories of type G + 0.
(b) When n′is not a multiple of n = |G|, the only C∗-categories of type G + n′are
Mod(Aff1(Fn+1)), except for n = 1,2,3,7 which have 1,2,1,1 additional C∗-categories.
In all cases here, n + 1 is a prime power, n′= n − 1, and G = Zn+1.
qof a finite field Fqis of type Zq−1+ (q − 2).
q. The remaining irrep is thus of dimension?q(q − 1) − q − 1 = q−1,
This is our Corollary 4 and Proposition 5 respectively, proven below. Type G+0
and type Zn+ n − 1 fusion categories were classified by Tambara-Yamagami 
and Etingof-Gelaki-Ostrik , respectively; we find that for these types, all fusion
categories can be realised as C∗-categories. Our proof of (a) is independent of and
much simpler than .
inequivalent subfactors with principal graph 2n1whose principal even sectors satisfy
the near-group fusions of type G + n.
For every nontrivial cyclic group G = Zn, there are at least 2
We have verified this for n ≤ 13. For those n, the complete classification is given
in Table 2 below. In the process, we construct dozens of new finite depth subfactors of
small index with principal even sectors of near-group type. This classification for n =
3 yields a uniqueness proof (up to complex conjugation) for the principal even sectors
of the Izumi-Xu 2221 subfactor; this can be compared to Han’s uniqueness proof 
of the 2221 subfactor. Again, our proof is independent of and both considerably
shorter and simpler than the original one. We do not yet feel confident speculating
on systems with n′> n; the corresponding subfactors would have principal graph as
in Figure 3 below.
Two morals can be drawn from this paper together with our previous one . One
is that there is surely a plethora of undiscovered finite-depth subfactors, of relatively
small index. This is in marked contrast to the observations of e.g. , who speak of
the ‘little desert’ in the interval 5 < [M : N] < 3+√5. The situation here is probably
very analogous to the classification of finite groups, which also is very tame for small
orders. The second moral is that, when the fusions are close to that of a group, a very
promising approach to the classification and construction of corresponding systems
of endomorphisms, equivalently C∗-categories, or the corresponding subfactors, is the
Cuntz algebra method developed in e.g.  and championed here. This approach
also makes the computation of the tube algebra and corresponding modular data
etc (to be discussed shortly) completely accessible. In contrast, the technique of
planar algebras is more robust, able to handle subfactors unrelated to groups, such
as Asaeda-Haagerup and the extended Haagerup. But planar algebra techniques
applied to e.g. the Haagerup fail to see that it (surely) lies in an infinite family. In
a few minutes the interested computer can construct several more subfactors of the
type described in Conjecture 1, using the Cuntz algebra method here, each of which
would be a serious challenge for the planar algebra method.
The underlying presence of groups here begs the question of K-theory realisations
of these fusion rings. For example, the fusion ring of the near-group C∗-categories
when n′= n − 1 can be expressed as KAff1(Fq)
in the other class, i.e. when n′∈ nZ?
An important class of fusion categories are the modular tensor categories ,
which are among other things braided and carry a representation (called modular
data) of the modular group SL2(Z) of dimension equal to the rank of the category,
from which e.g. the fusion coefficients can be computed. These arise from braided
systems of endomorphisms on an infinite factor, from representations of completely
(1). Is there a K-theoretic expression