# Fusion symmetric spaces and subfactors

Hans Wenzl  Department of Mathematics
University of California
San Diego, California<EMAIL_ADDRESS>
###### Abstract.

We construct analogs of the embedding of orthogonal and symplectic groups into
unitary groups in the context of fusion categories. At least some of the
resulting module categories also appear in boundary conformal field theory. We
determine when these categories are unitarizable, and explicitly calculate the
index and principal graph of the resulting subfactors.

This paper is a sequel of our previous paper [W4], where we introduced a
$q$-deformation of Brauer’s centralizer algebra for orthogonal and symplectic
groups; this algebra had already appeared more or less before in [Mo], see
also discussion in [W4]. It is motivated by finding a deformation of
orthogonal or symplectic subgroups of a unitary group which is compatible with
the standard quantum deformation of the big group. This has been done before
on the level of coideal subalgebras of Hopf algebras by Letzter. However, our
categorical approach also allows us to extend this to the level of fusion
tensor categories, where we find finite analogs of symmetric spaces related to
the already mentioned groups. Moreover, we can establish $C^{*}$ structures,
necessary for the construction of subfactors, in this categorical setting;
this is not so obvious to see in the setting of co-ideal algebras.

It is well-known how one can use a subgroup $H$ of a (for simplicity here)
finite group $G$ to construct a module category of the representation category
${\rm Rep\ }G$ of $G$. This module category also appears in the context of
subfactors of II1 von Neumann factors as follows: Let $R$ be the hyperfinite
II1 factor, and let $\mathcal{N}=R^{G}\subset\mathcal{M}=R^{H}$ be the fixed
points under outer actions of $G$ and $H$. Then the category of
$\mathcal{N}-\mathcal{N}$ bimodules is equivalent to ${\rm Rep\ }G$, and the
module category is given via the $\mathcal{M}-\mathcal{N}$ bimodules of the
inclusion $\mathcal{N}\subset\mathcal{M}$; its simple objects are labeled by
the irreducible representations of $H$. In particular, an important invariant
called the principal graph of the subfactor is determined by the restriction
rules for representations from $G$ to $H$. Important examples of subfactors
were constructed from fusion categories whose Grothendieck semirings are
quotients of the ones of semisimple Lie groups. So a natural question to ask
is whether one can perform a similar construction in this context. More
precisely, can we find restriction rules for type $A$ fusion categories which
describe a subfactor as before, and which will approach in the classical limit
the usual restriction rules from $U(N)$ to $O(N)$.

We answer this question in the positive in this paper via a fairly elementary
construction. We show that certain semisimple quotients of the $q$-Brauer
algebras have a $C^{*}$ structure and contain $C^{*}$-quotients of Hecke
algebras of type $A$. The subfactor is then obtained as the closure of
inductive limits of such algebras. Due to its close connection to Lie groups,
we can give very explicit general formulas for its index and its first
principal graph. Observe that the Lie algebra $\mathfrak{s}l_{N}$ decomposes
as an $\mathfrak{s}o_{N}$ module into the direct sum
$\mathfrak{s}o_{N}\oplus\mathfrak{p}$, where $\mathfrak{p}$ is a simple
$\mathfrak{s}o_{N}$-module. Then the index can be expressed explicitly in
terms of the weights of $\mathfrak{p}$, see Theorem 3.4. As before in the
group case, it can be interpreted as the quotient of the dimension of the
given fusion category by the sum of the squares of $q$-dimensions of
representations of orthogonal or symplectic subgroups whose labels are in the
alcove of a certain affine reflection group; however in our case, there is no
corresponding tensor category for the denominator, and the $q$-dimensions
differ from the ones of the corresponding quantum groups. Also, the
restriction rules for the corresponding bimodules of this subfactor, the first
principal graph, can be derived from the classical restriction rules via an
action of the already mentioned affine reflection group, similarly as it was
done before for tensor product rules for fusion categories. However, in our
case, the affine reflection comes from the highest short root of the
corresponding Lie algebra in the non-simply laced case; it is also different
from the one for fusion categories in the even dimensional orthogonal case.

In the case corresponding to $O(2)$, we obtain the Goodman-de la Harpe-Jones
subfactors for Dynkin diagrams $D_{n}$. We similarly also obtain a series of
subfactors for other even-dimensional orthogonal groups. Perhaps a little
surprisingly, there does not seem to be an analogous construction which would
correspond to the group $SO(N)$ with $N$ even. Our examples for the odd-
dimensional orthogonal group and for symplectic groups seem to be closely
related to examples constructed by Feng Xu [X] and Antony Wassermann [Wa] by
completely different methods. Several explicit cases are discussed in detail
at the end of the paper, as well as their connections to other approaches,
coming from boundary conformal field theory, subfactors and tensor categories.

The first chapter mostly contains basic material from subfactor theory which
will be needed later. In the second chapter we review and expand material on
the $q$-Brauer algebra as defined in [W4], see also [Mo]. In particular, we
define $C^{*}$-structures for certain quotients and use that to construct
subfactors. The third chapter is mainly concerned with the finer structure of
these subfactors, such as explicit closed formulas for the index and
calculation of the first principal graph. The same techniques would also
extend to other examples, such as the ones in [X].

$Acknowledgments:$ It is a pleasure to thank Antony Wassermann, David Jordan,
Viktor Ostrik and Feng Xu for useful references, and Fred Goodman also for
technical advice.

##  1\. II1 factors

###  1.1. Periodic commuting squares

We will construct subfactors using the set-up of periodic commuting squares as
in [W1]. More precisely, we assume that we have increasing sequences of finite
dimensional $C^{*}$ algebras $A_{1}\subset A_{2}\subset\ ...$ and
$B_{1}\subset B_{2}\subset\ ...$ such that $A_{n}\subset B_{n}$ for all
$n\in{\mathbb{N}}$. Let $\Lambda_{n}$ resp $\tilde{\Lambda}_{n}$ be labeling
sets for the simple components of $B_{n}$ and $A_{n}$ respectively. Let
$G_{n}$ be the inclusion matrix for $A_{n}\subset B_{n}$. If we write a
minimal idempotent $p_{\mu}\in A_{n,\mu}$ as a sum of minimal mutually
commuting idempotents of $B_{n}$, then the entry $g_{\lambda\mu}$ of $G_{n}$
denotes the number of those idempotents which are in $B_{n,\lambda}$. We say
that our sequences of algebras are periodic with period $d$, if there exists
an $n_{0}\in{\mathbb{N}}$ such that for any $n>n_{0}$ we have bijections $j$
between $\Lambda_{n}$ and $\Lambda_{n+d}$ as well as between
$\tilde{\Lambda}_{n}$ and $\tilde{\Lambda}_{n+d}$ which do not change the
inclusion matrices for $A_{n}\subset B_{n}$ as well as for $A_{n}\subset
A_{n+1}$ and $B_{n}\subset B_{n+1}$. This means, in particular, that
$g_{j(\lambda)j(\mu)}=g_{\lambda\mu}$ for all $\lambda\in\Lambda_{n}$,
$\mu\in\tilde{\Lambda}_{n}$, $n>n_{0}$.

The trace functional defines inner products on the algebras $A_{n}$ and
$B_{n}$ by $(b_{1},b_{2})=tr(b_{1}^{*}b_{2})$. Let $e_{A_{n+1}}$ and
$e_{B_{n}}$ be the orthogonal projections onto the subspaces $A_{n+1}$ and
$B_{n}$ of $B_{n+1}$. Then the commuting square condition says that
$e_{A_{n+1}}e_{B_{n}}=e_{A_{n}}=e_{B_{n}}e_{A_{n+1}}$ for all
$n\in{\mathbb{N}}$. Finally, we also note that the trace $tr$ is uniquely
determined on $A_{n}$ and $B_{n}$ by its weight vectors ${\bf a}_{n}$ and
${\bf b}_{n}$ which are defined as follows: Let $p_{\mu}$ be a minimal
idempotent in the simple component of $A_{n}$ labeled by $\mu$. Then we define
$a_{n,\mu}=tr(p_{\mu})$, and ${\bf a}_{n}=(a_{n,\mu})_{\mu}$, where $\mu$ runs
through a labeling set of the simple components of $A_{n}$. The weight vector
${\bf b}_{n}$ for $B_{n}$ is defined similarly. The following proposition
follows from [W1], Theorem 1.5 (where the matrix $G=(g_{\lambda\mu})$ defined
here would correspond to the matrix $G^{t}$ in [W1]):

######  Proposition 1.1.

Under the given conditions, we get a subfactor $\mathcal{N}\subset\mathcal{M}$
whose index $[\mathcal{M}:\mathcal{N}]$ is equal to $\|{\bf
a}_{n}\|^{2}/\|{\bf b}_{n}\|^{2}$ for any sufficiently large $n$. Moreover, we
have $\sum g_{\lambda\mu}a_{n,\lambda}=[\mathcal{M}:\mathcal{N}]b_{n,\mu}$.

###  1.2. Special periodic algebras

In general, it can be quite hard to determine finer invariants of the
subfactors, the so-called higher relative commutants (or centralizers) from
the generating sequence of algebras. However, under certain circumstances,
this can become quite easy. We describe such a set-up. It is a moderate
abstraction of an approach which has already been used before by a number of
authors. The reader familiar with tensor categories and module categories
should think of the algebras $A_{n}={\rm End}_{\mathcal{C}}(X^{\otimes n})$
and the algebras $B_{n}={\rm End}_{\mathcal{D}}(Y\otimes X^{\otimes n})$ for
$X$ an object in a $C^{*}$ tensor category $\mathcal{C}$, and $Y$ an object in
a module category ${\mathcal{D}}$ over $\mathcal{C}$. In the following, we
will make the following assumptions beyond the ones in the previous
subsection:

  1. 1.

The algebras $A_{n}$ will be monoidal $C^{*}$-algebras. This means we have
canonical embeddings of $C^{*}$ algebras $A_{m}\otimes A_{n}\to A_{n+m}$ with
multiplicativity of the trace, i.e. $tr(a_{1}\otimes
a_{2})=tr(a_{1})tr(a_{2})$.

  2. 2.

We have canonical embeddings $B_{m}\otimes A_{n}\to B_{n+m}$, again with
multiplicativity of the trace.

  3. 3.

We have the commuting square condition for the sequences of algebras
$A_{n}\subset B_{n}$ and $1\otimes A_{n-1}\subset A_{n}$.

  4. 4.

There exists $d\in{\mathbb{N}}$ and a projection $p\in A_{d}$ such that
$(1_{m}\otimes p)A_{m+d}(1\otimes p)\cong A_{m}$ and $(1_{m}\otimes
p)B_{m+d}(1\otimes p)\cong B_{m}$ for all $m\in{\mathbb{N}}$.

Examples for this set-up will be given at the end of this section and in
Section 2. Moreover, any finite depth subfactor
$\mathcal{N}\subset\mathcal{M}$ (see e.g. [GHJ], [EK] for definitions)
produces algebras for such a set-up as follows: Let $\mathcal{M}^{\otimes
n}=\mathcal{M}\otimes_{\mathcal{N}}\mathcal{M}\otimes_{\mathcal{N}}\ ...\
\otimes\mathcal{M}$ ($n$ factors). Obviously, $\mathcal{M}^{\otimes n}$ is an
$\mathcal{N}-\mathcal{N}$ as well as an $\mathcal{M}-\mathcal{N}$ bimodule.
One can check that for $A_{n}={\rm
End}_{\mathcal{N}-\mathcal{N}}\mathcal{M}^{\otimes n}\subset B_{n}={\rm
End}_{\mathcal{M}-\mathcal{N}}\mathcal{M}^{\otimes n+1}$ the axioms above are
satisfied; here the embedding is defined by letting the elements of $A_{n}$
act on the second to $(n+1)-st$ factor of $\mathcal{M}^{\otimes n+1}$. It is
also possible to define these algebras in connection of relative commutants in
the Jones tower of relative commutants (see e.g. [Bi] for details). Recall
that for factors $\mathcal{N}\subset\mathcal{M}$ the relative commutant (or
centralizer) $\mathcal{N}^{\prime}\cap\mathcal{M}$ is defined to be the set
$\\{b\in\mathcal{M},ab=ba$ for all $a\in\mathcal{N}\\}$.

######  Lemma 1.2.

The subfactor $\mathcal{N}\subset\mathcal{M}$ generated from the sequences of
algebras $1_{m}\otimes A_{n}\subset B_{n+m}$ has relative commutant $B_{m}$.
The same statement also holds with $B_{n+m}$ and $B_{m}$ in the last sentence
replaced by $A_{n+m}$ and $A_{m}$.

$Proof.$ This is essentially the proof used for Theorem 3.7 in [W1]. Observe
that by induction on $r$ and assumption 4 above, we also have $(1_{m}\otimes
p^{\otimes r})X_{m+rd}(1_{m}\otimes p^{\otimes r})\cong X_{m}$ for $X=A,B$. It
follows from Theorem 1.6 of [W1] that the dimension of the relative commutant
$\mathcal{N}^{\prime}\cap\mathcal{M}$ is at most equal to the dimension of
$B_{m}$. The claim follows from the fact that $B_{m}\otimes 1_{n}$ commutes
with $1_{m}\otimes A_{n}$ for all $n$.

###  1.3. Bimodules and principal graphs

We calculate the first principal graph for subfactors constructed in our set-
up, using fairly elementary methods from [W1] as well as the bimodule
approach. The latter was first used in the subfactor context by Ocneanu, see
e.g. [EK]. For the connection between bimodules and principal graphs, see [Bi]
and for more details compatible with our notation, see also [EW]. While most
of this section has already appeared before implicitly or explicitly, the
presentation in our set-up might be useful also in other contexts.

Pick $k$ large enough so that $m=kd>n_{0}$. Hence the inclusion matrices for
$A_{rd}\subset B_{rd}$ coincide for all $r\geq k$ using the bijection of
simple components as described in Section 1.1. Let $\Lambda_{m}$ and
$\tilde{\Lambda}_{m}$ be labelling sets for the simple components of $B_{m}$
and $A_{m}$ respectively. Let $\mathcal{N}$ and $\mathcal{M}$ be the factors
generated by the increasing sequences of algebras $A_{n}$ and $B_{n}$
respectively, see Prop. 1.1 or Lemma 1.2, with the $m$ there equal 0. Both of
these factors have a subfactor $\tilde{\mathcal{N}}$ generated by the
subalgebras $1_{m}\otimes A_{n}\subset A_{n+m}\subset B_{n+m}$. We now define
for each $\lambda\in\tilde{\Lambda}_{m}$ an $\mathcal{N}-\tilde{\mathcal{N}}$
bimdule $N_{\lambda}$ as follows: It is the Hilbert space completion of
$\mathcal{N}p_{\lambda}$ with respect to the inner product induced by $tr$,
where $p_{\lambda}$ is a minimal idempotent in $A_{m,\lambda}$, the simple
component of $A_{m}$ labeled by $\lambda$, with obvious left and right actions
by $\mathcal{N}$ and $\tilde{N}$. To ease notation, we shall often refer to it
as an $\mathcal{N}-\mathcal{N}$ bimodule, using the isomorphism between
$\tilde{\mathcal{N}}$ and $\mathcal{N}$ given by the trace preserving maps
$a\in A_{n}\mapsto 1_{m}\otimes a\in A_{n+m}$.

Similarly, we define $\mathcal{M}-\tilde{\mathcal{N}}$ bimodules $M_{\mu}$ for
any $\mu\in\Lambda_{m}$ which are Hilbert space completions of
$\mathcal{M}p_{\mu}$, where $p_{\mu}$ is a minimal idempotent in the simple
component $B_{m,\mu}$ of $B_{m}$. Finally, we define the inclusion numbers
$b_{\mu}^{\lambda}$ for elements $\lambda\in\tilde{\Lambda}$ and
$\mu\in\Lambda_{m}$ as usual (see Section 1.1).

######  Lemma 1.3.

The bimodules $N_{\lambda}$ and $M_{\mu}$ are irreducible
$\mathcal{N}-\tilde{\mathcal{N}}$ resp $\mathcal{M}-\tilde{\mathcal{N}}$
bimodules. We have the decomposition
$M_{\mu}\cong\bigoplus_{\lambda}b^{\lambda}_{\mu}N_{\lambda}$ as
$\mathcal{N}-\tilde{\mathcal{N}}$ modules.

$Proof.$ This is well-known (see e.g. [EW] for more details). It follows from
Lemma 1.2 that the endomorphism ring of the $\mathcal{M}-\tilde{\mathcal{N}}$
bimodule $\mathcal{M}$ is given by $B_{m}$. Hence the
$\mathcal{M}-\tilde{\mathcal{N}}$ bimodules $M_{\mu}$ are simple, as $p_{\mu}$
was chosen to be a minimal idempotent in $B_{m}$. One shows similarly that
also the $N_{\lambda}$’s are simple $\mathcal{N}-\tilde{\mathcal{N}}$
bimodules.

Observe that $\dim_{\mathcal{N}}N_{\lambda}=tr(p_{\lambda})$ and
$\dim_{\mathcal{M}}M_{\mu}=tr(p_{\mu})$ (see e.g. [J]). Now if $p_{\lambda}$
is a minimal idempotent in $A_{m}$, it follows from the definitions that ${\rm
Ind}_{\mathcal{N}}^{\mathcal{M}}N_{\lambda}:=\mathcal{M}p_{\lambda}$ is
isomorphic as an $\mathcal{M}-\tilde{\mathcal{N}}$ bimodule to the direct sum
$\oplus b^{\lambda}_{\mu}M_{\mu}$. By Frobenius reciprocity, see e.g. [EK],
[Bi] it follows that the module $N_{\lambda}$ appears with multiplicity
$b^{\lambda}_{\mu}$ in $\mathcal{M}_{\mu}$, viewed as an
$\mathcal{N}-\tilde{\mathcal{N}}$ bimodule. Hence the
$\mathcal{N}-\tilde{\mathcal{N}}$ bimodule $M_{\mu}$ has a submodule which is
isomorphic to $\bigoplus_{\lambda}b^{\lambda}_{\mu}N_{\lambda}$. But as
$M_{\mu}$ has $\mathcal{N}$-dimension $[\mathcal{M}:\mathcal{N}]tr(p_{\mu})$,
it coincides with this submodule, by Proposition 1.1.

######  Theorem 1.4.

Let $\mathcal{N}\subset\mathcal{M}$ be the subfactor generated by sequences of
algebras $A_{n}\subset B_{n}$ satisfying the conditions in Section 1.2. Then
its first principal graph is given by the inclusion graph for $A_{kd}\subset
B_{kd}$ for sufficiently large $k$.

$Proof.$ It is well-known that the first principal graph is given by the
induction-restriction graph of $\mathcal{M}-\mathcal{N}$ and
$\mathcal{N}-\mathcal{N}$ bimodules appearing in the tensor products
$\mathcal{M}^{\otimes n}$, $n\in{\mathbb{N}}$, where $\mathcal{M}^{\otimes
n}=\mathcal{M}\otimes_{\mathcal{N}}\mathcal{M}\otimes\ ...\
\otimes_{\mathcal{N}}\mathcal{M}$ ($n$ factors), see [EK], [Bi]. Obviously,
this graph does not change if we replace all $X-\mathcal{N}$ bimodules $H$ in
this setting by $X-q\mathcal{N}q$ bimodules $Hq$, for
$X=\mathcal{M},\mathcal{N}$ and $q$ a nonzero projection in $\mathcal{N}$. The
claim can now be shown for $q=p^{\otimes k}$ where $k$ is chosen large enough
so that $kd>n_{0}$, using Lemma 1.3.

Recall that many examples come from module tensor categories, where
$A_{n}={\rm End}_{\mathbb{C}}(X^{\otimes n})$ and $B_{n}={\rm
End}(Y\otimes(X^{\otimes n})$ for an object $X$ in a tensor category
${\mathbb{C}}$ and an object $Y$ in the module category ${\mathcal{D}}$ over
${\mathbb{C}}$. In this setting, the weight vectors of our trace are given by
$a_{n,\lambda}=\tilde{d}_{\lambda}/x^{n}$ and $b_{n,\mu}=d_{\mu}/yx^{n}$ for
positive quantities $d_{\mu},\tilde{d}_{\lambda},x$ and $y$. Then we have

######  Corollary 1.5.

Assuming the conditions for the trace weights as just given, we have
subfactors $\mathcal{N}\subset\mathcal{M}_{\mu}$ with index
$[\mathcal{M}_{\mu}:\mathcal{N}]=d_{\mu}^{2}[\mathcal{M}:\mathcal{N}]$, with
$\mathcal{N}\subset\mathcal{M}$ as in Theorem 1.4.

######  Remark 1.6.

There is also a second important invariant for
$\mathcal{N}\subset\mathcal{M}$, the dual principal graph. It can be
analogously defined as an induction-restriction graph between irreducible
$\mathcal{M}-\mathcal{M}$ and $\mathcal{M}-\mathcal{N}$ bimodules appearing in
the tensor powers $\mathcal{M}^{\otimes n}$. Its calculation is more difficult
than the first principal graph. This is quite similar to the corresponding
problem for subfactors coming from conformal inclusions and related
constructions, see e.g [X1], [BEK], [EW]. We plan to study this problem in a
future publication via suitable adaptions of techniques in those papers.

###  1.4. The $GHJ$-construction

We give a well-known and well-studied example for our current set-up, which
was first constructed in [GHJ]. Let $G$ be a matrix with nonnegative integer
entries and norm less than 2. It is well-known that such matrices are
classified by Coxeter graphs of type $ADE$. We assume that the columns of $G$
are indexed by the even vertices, and the rows by the odd vertices. We define
$C^{*}$-algebras $B_{n}$ by $B_{0}={\mathbb{C}}^{v_{e}}$, and $B_{1}=\oplus
M_{d_{j}}$, where $v_{e}$ is the number of even vertices, and the summands of
$B_{1}$ are labelled by the odd vertices $j$, whose dimension $d_{j}$ is equal
to the number of even vertices to which $j$ is connected. The embedding
$B_{o}\subset B_{1}$ is given by the inclusion matrix $G$. Then we define
recursively $B_{n+1}$ via Jones’ basic construction [J] for $B_{n-1}\subset
B_{n}$. Here the trace on $B_{n}$ is the unique normalized trace whose values
on minimal idempotents are given by the Perron-Frobenius vector of $G^{t}G$ or
$GG^{t}$, depending on whether $n$ is even or odd, and the vector is
normalized such that $tr(1)=1$. Then the algebra $B_{n+1}$ is generated by
$B_{n}$, acting on itself via left multiplication, and the orthogonal
projection $e_{n}$ onto the subspace $B_{n-1}$ of $B_{n}$, with respect to the
inner product coming from the trace. The algebra $A_{n}$ is defined to be the
subalgebra generated by the identity 1 and the projections $e_{i}$, $1\leq
i<1$. It is well-known that these algebras satisfy the commuting square
condition, that they are periodic with periodicity 2, and that the Jones
projections $e_{i}$ satisfy the conditions of the projection $p$ in Section
1.2. This has already been shown in [GHJ].

##  2\. $q$-Brauer algebras

###  2.1. Definitions

Fix $N\in{\mathbb{Z}}$ and let $[N]=(q^{N}-q^{-N})/(q-q^{-1})$, where $q$ is
considered to be a complex number. We denote by $H_{n}(q^{2})$ the Hecke
algebra of type $A_{n-1}$. It is given by generators $g_{1},g_{2},\ ...\
g_{n-1}$ which satisfy the usual braid relations and the quadratic relation
$g_{i}^{2}=(q^{2}-1)g_{i}+q^{2}$. The $q$-Brauer algebra $Br_{n}(N)$ is the
complex algebra defined via generators $g_{1},g_{2},\ ...\ g_{n-1}$ and $e$
and relations

  1. (H)

The elements $g_{1},g_{2},\ ...\ g_{n-1}$ satisfy the relations of the Hecke
algebra $H_{n}(q^{2})$.

  2. (E1)

$e^{2}=[N]e$,

  3. (E2)

$eg_{i}=g_{i}e$ for $i>2$, $eg_{1}=q^{2}e$, $eg_{2}e=q^{N+1}e$ and
$eg_{2}^{-1}e=q^{-1-N}e$.

  4. (E3)

$g_{2}g_{3}g_{1}^{-1}g_{2}^{-1}e_{(2)}=e_{(2)}=e_{(2)}g_{2}g_{3}g_{1}^{-1}g_{2}^{-1}$,
where $e_{(2)}=e(g_{2}g_{3}g_{1}^{-1}g_{2}^{-1})e$.

It is easy to see that this algebra coincides with the algebra defined in [W4]
after substituting $q$ there by $q^{2}$, and $e$ there by $q^{1-N}e$ (with the
$q$ of this paper); this is also compatible with the different definition of
$[N]$ in [W4]. We have chosen this parametrization as it will make it easier
to define a $*$ structure on it. More precisely, if $|q|=1$, there exists a
complex conjugate antiautomorphism $b\mapsto b^{*}$ on $Br_{n}(N)$ defined by

(2.1) $e^{*}=e,\quad g_{i}^{*}=g_{i}^{-1},\ 1\leq i<n.$

It is easy to check at the relations that this operation is well-defined.

###  2.2. Molev representation

We give a representation of our algebra $Br_{n}(N)$ in ${\rm End}(V^{\otimes
n})$, where $V={\mathbb{C}}^{N}$. For this we use the matrices used by Molev
in [Mo] for the definition of his $q$-deformation of Brauers’ centralizer
algebra. His defining relations are slightly different from ours; but Molev
has informed the author that our algebra satisfies the relations of his
algebra. It turns out that also his matrices satisfy the relations of our
algebras, which we will outline here. Let $R$ be the well-known solution of
the quantum Yang-Baxter equation for type $A$. For simplicity we will use this
notation for what is often denoted as $\check{R}$. If $E_{ij}$ are the matrix
units for $n\times n$ matrices, we define the following elements in ${\rm
End}(V^{\otimes 2})$:

$R\ =\ \sum_{i}qE_{ii}\otimes E_{ii}\ +\ \sum_{i\neq j}E_{ij}\otimes E_{ji}\
+\ \sum_{i<j}(q-q^{-1})E_{ii}\otimes E_{jj},$

and

$Q=\sum_{i,j}q^{N+1-2i}\ E_{ij}\otimes E_{ij}.$

Moreover, if $A\in{\rm End}(V^{\otimes 2})$, we define the operator
$A_{i}\in{\rm End}(V^{\otimes n})$ by

$A_{i}\ =\ 1_{i-1}\otimes A\otimes 1_{n-1-i},$

where $1_{k}$ is the identity on $V^{\otimes k}$. Then we have the following
proposition, all of whose essential parts were already proved in [Mo].
However, the relations for our algebras are slightly different, so we give
some of the adjustments of the work in [Mo] to our context below.

######  Proposition 2.1.

The map $g_{i}\mapsto qR_{n-i}$ and $e\mapsto Q_{n-1}$ defines a
representation $\Phi$ of $Br_{n}(N)$. It specializes to the usual
representation of Brauer’s centralizer algebra in ${\rm End}(V^{\otimes n})$
for $q=1$.

$Proof.$ Most of the relations are already known or are easy to check. E.g. it
is well-known that the matrices $qR_{i}$ satisfy the relations of the Hecke
algebra $H_{n}(q^{2})$. Relation $(E1)$ is checked easily, and also the
relations in $(E2)$ are fairly straightforward to check. It suffices to check
$(E3)$ for $n=4$. For this observe that by [Mo], (4.16), we have

$Q_{3}R_{2}R_{3}R_{1}R_{2}Q_{3}=Q_{1}Q_{3}+q^{N+1}(q-q^{-1})Q_{3}(R_{1}+q^{-1}1),$

in our notation. Using the relation $R_{i}=R_{i}^{-1}+(q-q^{-1})1$ for the
second and third factor of the left hand side, one derives from this

$Q_{3}R_{2}^{-1}R_{3}^{-1}R_{1}R_{2}Q_{3}=Q_{1}Q_{3}.$

To check relation $(E3)$, observe that

$R_{1}R_{2}(v_{i}\otimes v_{i}\otimes v_{j}\otimes v_{j})\ =\
R_{3}R_{2}(v_{i}\otimes v_{i}\otimes v_{j}\otimes v_{j}),$

where $(v_{i})$ is the standard basis for ${\mathbb{C}}^{N}=V$. One derives
from this that $R_{2}^{-1}R_{1}^{-1}R_{3}R_{2}Q_{1}Q_{3}=Q_{1}Q_{3}$.
Moreover, the same calculations above also work with $R_{i}$ replaced by
$R_{i}^{-1}$ and $Q_{j}$ replaced by its transpose $Q_{j}^{T}$. Hence one can
show as before that
$R_{2}R_{3}R_{1}^{-1}R_{2}^{-1}Q_{1}^{T}Q_{3}^{T}=Q_{1}^{T}Q_{3}^{T}$.
Transposing this, using $R_{i}^{T}=R_{i}$ shows the last part of the claim.

###  2.3. Quotients

We can now rephrase the main results of [W4] in our notation as follows:

######  Theorem 2.2.

(a) There exists a well-defined functional $tr$ on $Br_{n}(N)$ defined
inductively by $tr(g_{1})=q^{N+1}/[N]$, $tr(e)=1/[N]$ and
$tr(bg_{n})=tr(b)tr(g_{n})$ for all $b\in Br_{n}(N)$.

(b) Let $\overline{Br}_{n}(N)=Br_{n}(N)/I_{n}$, where $I_{n}$ is the
annihilator ideal of $tr$. Then $\overline{Br}_{n}(N)$ is semisimple and the
inclusion $\overline{Br}_{n}(N)\subset\overline{Br}_{n+1}(N)$ is well-defined
for all $n$.

It is possible to explicitly describe the structure of the quotients
$\overline{Br}_{n}=\overline{Br}_{n}(N)$. To do so, we need the following
definitions for the labeling sets of simple representations. More
conceptually, the labeling sets $\Lambda(N,\ell)$ consist of all such diagrams
$\lambda$ for which the quantities $d_{\mu}(q)\neq 0$ for any subdiagrams
$\mu\subset\lambda$ including $\lambda$ itself, where $q^{2}$ is a primitive
$\ell$-th root of unity and the $d_{\mu}$’s are defined in Section 2.5.

######  Definition 2.3.

Fix integers $N$ and $\ell$ satisfying $1<|N|<\ell$.

(i) The set $\tilde{\Lambda}(N,\ell)$ consists of all Young diagrams with
$\leq N$ rows such that the first and $N$-th row differ by at most $\ell-N$
boxes for $N>0$. If $N<0$, the Young diagrams have at most $|N|$ columns,
where the first and $|N|$-th column differ by at most $\ell-|N|$ boxes.

(ii) The set $\Lambda(N,\ell)$ consists of all Young diagrams $\lambda$ with
$\lambda_{i}$ boxes in the $i$-th row and $\lambda_{j}^{\prime}$ boxes in the
$j$-th column which satisfy

(a) $\lambda_{1}^{\prime}+\lambda_{2}^{\prime}\leq N$ and
$\lambda_{1}\leq(\ell-N)/2$ if $N>0$ and $\ell-N$ even,

(b) $\lambda_{1}^{\prime}+\lambda_{2}^{\prime}\leq N$ and
$\lambda_{1}+\lambda_{2}\leq\ell-N$ if $N>0$ and $\ell-N$ odd,

(c) $\lambda_{1}\leq|N|/2$ and
$\lambda_{1}^{\prime}+\lambda_{2}^{\prime}\leq\ell-|N|$ if $N<0$ is even,

(d) $\lambda_{1}+\lambda_{2}\leq|N|$ and
$\lambda^{\prime}_{1}+\lambda^{\prime}_{2}\leq\ell-|N|$ if $N<0$ is odd.

Diagrams which miss one of these inequalities only by the quantity one are
called boundary diagrams of $\Lambda(N,\ell)$; e.g. in case (a) if
$\lambda_{1}^{\prime}+\lambda_{2}^{\prime}=N+1$.

######  Theorem 2.4.

([W4], Section 5) Let $q^{2}$ be a primitive $\ell$-th root of unity, and let
$N$ be an integer satisfying $1<|N|<\ell$. Then the simple components of
$\overline{Br}_{n}=\overline{Br}_{n}(N)$ are labeled by the Young diagrams in
$\Lambda(N,\ell)$ with $n,n-2,n-4,...$ boxes. If $V_{n,\lambda}$ is a simple
$\overline{Br}_{n}$ module for such a diagram $\lambda$, it decomposes as a
$Br_{n-1}$ module as

(2.2) $V_{n,\lambda}\ \cong\ \bigoplus_{\mu}V_{n-1,\mu},$

where $\mu$ runs through diagrams in $\Lambda(N,\ell)$ obtained by removing
or, if $|\lambda|<n$, also by adding a box to $\lambda$.

###  2.4. Path idempotents and matrix units

We will give some details about the proof of Theorem 2.4 which will also be
needed for further results. Observe that the restriction rule 2.2 implies that
a minimal idempotent $p_{\mu}$ in $\overline{Br}_{n-1,\mu}$ can be written as
a sum of minimal idempotents with exactly one in $\overline{Br}_{n,\lambda}$
for each diagram $\lambda$ in $\Lambda(N,\ell)$ which can be obtained by
adding or subtracting a box from $\mu$. This inductively determines a system
of minimal idempotents and matrix units of $\overline{Br}_{n}(q^{N},q)$
labeled by paths resp. pairs of paths in $\Lambda(N,\ell)$ of length $n$. Such
a path is defined to be a sequence of Young diagrams
$(\lambda^{(i)})_{i=0}^{n}$ where $\lambda^{(0)}$ is the empty Young diagram,
and $\lambda^{(i+1)}$ is obtained from $\lambda^{(i)}$ by adding or removing a
box. It follows from the restriction rule above that the dimension of
$V_{n,\lambda}$ is equal to the number of paths of length $n$ with
$\lambda^{(n)}=\lambda$, and that we can label a complete system of matrix
units for the simple component $\overline{Br}_{n,\lambda}$ by pairs of such
paths. We then have the following lemma:

######  Lemma 2.5.

For each pair of paths $t_{1},t_{2}$ in $\Lambda(N,\infty)$ with the same
endpoint we can define the matrix unit $E_{t_{1},t_{2}}$ as a linear
combination of products of generators over algebraic functions (rational for
path idempotents) in $q$ with poles only at roots of unity. More precisely,
the formula for $E_{t_{1},t_{2}}$ is well-defined for $q^{2}$ a primitive
$\ell$-th root of unity if both $t_{1}$ and $t_{2}$ are paths in
$\Lambda(N,\ell)$.

$Proof.$ This was proved in [W4], Section 5. As the result is not explicitly
stated as such, we give some details here. One observes that the two-sided
ideal generated by the element $\bar{e}\in\overline{Br}_{n+1}$ is isomorphic
to Jones’ basic construction for the algebras
$\overline{Br}_{n-1}\subset\overline{Br}_{n}$ (or, strictly speaking, by
certain conjugated subalgebras which are denoted by $i_{1}(\overline{Br}_{n})$
and $i_{2}(\overline{Br}_{n-1})$, see Section 5.2 in [W4]). One can then
define path idempotents and matrix units inductively as it was done in [RW],
Theorem 1.4 using the formulas for the weights of traces, which will also be
reviewed in Section 2.5; this is closely related to what is also known in
subfactor theory as the Ocneanu-Sunder path model [Su]. The complement of this
ideal is a quotient of the Hecke algebra $\bar{H}_{n+1}$ for which matrix
units already were more or less defined in [W1], p. 366.

######  Lemma 2.6.

Let $p_{[1^{N}]}$ be the minimal idempotent in $H_{N}$ corresponding to its
one-dimensional sign representation. Then we have $\bar{p}_{[1^{N}]}^{\otimes
2}\overline{Br}_{m+2N}\bar{p}_{[1^{N}]}^{\otimes 2}\cong\overline{Br}_{m}$ for
all $m>0$.

$Proof.$ Observe that if $p\in\overline{Br}_{2N,\emptyset}(N)$, the simple
component labeled by the empty Young diagram $\emptyset$, then it follows from
the restriction rule 2.2 (see also the equivalent version below Theorem 2.4)
by induction on $m$ that $p\overline{Br}_{m+2N}p\cong\overline{Br}_{m}$ for
all $m\geq 0$. Hence it suffices to show that $p_{[1^{N}]}^{\otimes 2}$ is
such an idempotent.

If $q=1$ and $N>0$, $\Phi(Br_{n}(N))$ coincides with the commutant of the
action of the orthogonal group $O(N)$ on $V^{\otimes n}$, which is semisimple.
Moreover, the trace $tr$ is just a multiple of the pull-back of the natural
trace on ${\rm End}(V^{\otimes n})$, so
$\Phi(Br_{n}(N))\cong\overline{Br}_{n}(N)$ at $q=1$. As $\Phi(p_{[1^{N}]})$
projects onto the one-dimensional determinant representation in $V^{\otimes
N}$, the claim follows easily in that case, using Brauer duality, i.e. the
fact that $\Phi(Br_{n}(N))$ is equal to the commutant of $O(N)$ on $V^{\otimes
n}$ for all $n$.

We will now use the fact that we can also define $Br_{n}(N)$ over the field of
rational functions ${\mathbb{C}}(q)$, see [W4]. It follows from Lemma 2.5 that
we can also define the path idempotents for $\overline{Br}_{n}(N)$ over that
field for paths of length $n$ in $\Lambda(N,\infty)$. As the rank of an
idempotent is an integer, the claim follows as well for $q$ a variable, and
for $q\in{\mathbb{C}}$ not a root of unity. But as
$p_{t}\bar{p}_{[1^{N}]}^{\otimes 2}p_{t}=0$ for any path $t$ of length $2N$ in
$\Lambda(N,\ell)$ which ends in $\lambda\neq\emptyset$, we also get the rank 0
for $\bar{p}_{[1^{N}]}^{\otimes 2}$ at $q^{2}$ a primitive $\ell$-th root of
unity in $\overline{Br}_{2N,\lambda}(N)$. This finishes the proof for $N>0$.
The proof for the symplectic case $N<0$ even goes the same way.

###  2.5. Weights of the trace

Using the character formulas of orthogonal groups, one can calculate the
weights of $tr$ for the algebras $Br_{n}(N)$, i.e. its values at minimal
idempotents of $Br_{n}(N)$. We will need the following quantities for a given
Young diagram $\lambda$

(2.3) $d(i,j)\ =\begin{cases}\lambda_{i}+\lambda_{j}-i-j&\text{if $i\leq
j$,}\\\ -\lambda_{i}^{\prime}-\lambda_{j}^{\prime}+i+j-2&\text{if
$i>j$.}\end{cases}$

Moreover, we define $h(i,j)$ to be the length of the hook in the Young diagram
$\lambda$ whose corner is the box in the $i$-th row and $j$-th column. We can
now restate [W4], Theorem 4.6 in the notations of this paper as follows:

######  Theorem 2.7.

The weights of the Markov trace $tr$ for the Hecke algebra
$\bar{H}_{n}(q^{2})$ are given by
$\tilde{\omega}_{\lambda}=\tilde{d}_{\lambda}/[N]^{n}$, where $|\lambda|=n$,
and for $\overline{Br}_{n}(N)$ they are given by
$\omega_{\lambda,n}=d_{\lambda}/[N]^{n}$, where

$\displaystyle\tilde{d}_{\lambda}\ =\
\prod_{(i,j)\in\lambda}\frac{[N+j-i]}{[h(i,j)]},\quad d_{\lambda}\ =\
\prod_{(i,j)\in\lambda}\frac{[N+d(i,j)]}{[h(i,j)]},$

where $\lambda$ runs through all the Young diagrams in
$\tilde{\Lambda}(N,\ell)$ with $n$ boxes for $\bar{H}_{n}(q^{2})$, and through
all Young diagrams in $\Lambda(N,\ell)$ with $n,n-2,n-4,...$ boxes. for
$\overline{Br}_{n}$.

######  Lemma 2.8.

The weights $\omega_{\lambda,n}$ are positive for all
$\lambda\in\Lambda(N,\ell)$ if and only if $q^{2}=e^{\pm 2\pi i/\ell}$ with
$\ell>N$ and

(a) $N>0$ and $\ell-N$ even or

(b) $N<0$ odd.

$Proof.$ The weights can be rewritten for our choice of $q$ as

$\omega_{\lambda,n}\ =\ \frac{\sin^{n}(\pi/\ell)}{\sin^{n}(N\pi/\ell)^{n}}\
\prod_{(i,j)\in\lambda}\frac{\sin(N+d(i,j))\pi/\ell}{\sin(h(i,j)\pi/\ell)}.$

As $h(i,j)\leq h(1,1)=\lambda_{1}+\lambda_{1}^{\prime}-1<\ell$ for all boxes
$(i,j)$ of $\lambda$, it follows that all factors in the formula above are
positive for $N>0$ (negative for $N<0$) except possibly the ones in the
numerator under the product. If $N>0$ and $\ell-N$ odd, one checks that for
the diagram $[\ell-N+1)/2]$ we have $\omega_{\lambda,|\lambda|}<0$. By the
same argument, one shows that $\omega_{\lambda,|\lambda|}<0$ for
$\lambda=[(|N|+1)/2]$ and $N<0$. In the other two cases, one checks that
$0<|d(i,j)|<\ell$ for all boxes $(i,j)$ of a diagram
$\lambda\in\Lambda(N,\ell)$.

###  2.6. $C^{*}$-quotients

######  Proposition 2.9.

If the weights $\omega_{\lambda,n}$ are positive for all
$\lambda\in\Lambda(N,\ell)$, the star operation defined by $e^{*}=e$ and by
$g_{i}^{*}=g_{i}^{-1}$ makes the quotients $\overline{Br}_{n}$ into $C^{*}$
algebras.

$Proof.$ The proof goes by induction on $n$, with the claims for $n=1$ and
$n=2$ easy to check. By [W4], the two-sided ideal $I_{n+1}$ generated by $e$
in $\overline{Br}_{n+1}$ is isomorphic to Jones’ basic construction for
$\overline{Br}_{n-1}\subset\overline{Br}_{n}$, see also the remarks before
Lemma 2.6. In particular, this ideal is spanned by elements of the form
$b_{1}eb_{2}$, with $b_{1},b_{2}\in i_{1}(\overline{Br}_{n})$, where
$i_{1}(a)=\Delta_{n+1}a\Delta_{n+1}^{-1}$, with $\Delta=(g_{1}g_{2}\
...g_{n-1})(g_{1}\ ...\ g_{n-2})\ ...\ g_{1}$. By induction assumption and
properties of Jones’ basic construction, this ideal has a $C^{*}$ structure
given by $(b_{1}eb_{2})^{*}=b_{2}^{*}eb_{1}^{*}$. This coincides with the $*$
operation defined before algebraically. It was shown in [W4] that
$\overline{Br}_{n+1}\cong I_{n+1}\oplus\bar{H}_{n+1}$, where $\bar{H}_{n+1}$
is a semisimple quotient of the Hecke algebra $H_{n+1}$ whose simple
components are labeled by the Young diagrams $\lambda\in\Lambda(N,\ell)$ with
$n+1$ boxes. All these simple representations satisfy the $(k,\ell)$ condition
in [W1]. It follows from that paper that the map $g_{i}^{*}=g_{i}^{-1}$
induces a $C^{*}$ structure for any such representation. This finishes the
proof.

######  Theorem 2.10.

For each choice of $N$ and $\ell$ with $q^{2}=e^{\pm 2\pi i/\ell}$, and for
each nonnegative integer $m$ we obtain a subfactor
$\mathcal{N}\subset\mathcal{M}$ with
$\mathcal{N}^{\prime}\cap\mathcal{M}=\overline{Br}_{m}$ and with index

$[\mathcal{M}:\mathcal{N}]\ =\ [N]^{m}\
\frac{\sum_{\mu\in\tilde{\Lambda}(N,\ell)}\tilde{d}_{\mu}^{2}}{\sum_{\lambda\in\Lambda(N,\ell)}d_{\lambda}^{2}},$

with notations as in Section 2.5. Moreover, its first principal graph is given
by the inclusion graph for $\bar{H}_{2Nk}\subset\overline{Br}_{2nk+m}$ for any
sufficiently large $k$.

$Proof.$ Let us first check conditions 1-4 in Section 1.2 with
$A_{n}=\bar{H}_{n}$ and $B_{n}=\overline{Br}_{n}(N)$ for $q=e^{\pi i/\ell}$
and $1<|N|<\ell$. Condition 1 is well-known and was checked in e.g.[W1].
Similarly, Cond. 2 follows from the results in [W4], using the map $b\otimes
g_{i}\in\overline{Br}_{m}\otimes\bar{H}_{n}\mapsto bg_{m+i}$. Condition 3
means that the conditional expectation from $\overline{Br}_{n+1}$ to
$\overline{Br}_{n}$ maps $\bar{H}_{n+1}$ onto $\bar{H}_{n}$. But as any
element of $\bar{H}_{n+1}$ can be written as a linear combination of elements
of the form $ag_{n}b$, with $a,b\in\bar{H}_{n}$, we have for any
$c\in\overline{Br}_{n}$ that

$tr(ag_{n}bc)=tr(g_{n})tr(abc)=tr(E_{\bar{H}_{n}}(ag_{n}b)c).$

Hence the commuting square condition is satisfied for any four algebras of the
type above. Finally, Condition 4 follows for $d=2N$ and the projection
$p=p_{[1^{N}]}^{\otimes 2r}$ from Lemma 2.6.

The periodicity condition for $\bar{H}_{n}$ was shown in [W1] by proving that
$\bar{p}_{[1^{N}]}\bar{H}_{m+N}\bar{p}_{[1^{N}]}\cong\bar{H}_{m}$. This
induces an injective map
$\tilde{\Lambda}(N,\ell)_{m}\to\tilde{\Lambda}(N,\ell)_{m+N}$ by adding a
column of $N$ boxes to the given Young diagram which has to become surjective
for sufficiently large $m$ by definition of $\tilde{\Lambda}(N,\ell)$. The
$2N$ periodicity for the algebras $\overline{Br}_{n}(N)$ follows similarly
using Lemma 2.6, or see [W4].

##  3\. $S$-matrix

We will need certain well-known identities, which can be found in [Kc], except
for one case, which is a variation of the other ones. Because of this, we
review the material in more detail. This might also be useful to the non-
expert reader, as the identities needed here can be derived by completely
elementary methods.

###  3.1. Lattices

Let $M\subset L\subset{\mathbb{R}}^{k}$ be two lattices of full rank. This
means that they are isomorphic to ${\mathbb{Z}}^{k}$ as abelian groups, and
each of them spans ${\mathbb{R}}^{k}$ over ${\mathbb{R}}$. Moreover, we assume
that we have an inner product on ${\mathbb{R}}^{k}$ such that $({\bf x},{\bf
y})\in{\mathbb{Z}}$ for all ${\bf x},{\bf y}\in M$. We define the dual lattice
$M^{*}$ to be the set of all ${\bf y}\in{\mathbb{R}}^{k}$ such that $({\bf
x},{\bf y})\in{\mathbb{Z}}$ for all ${\bf x}\in M$; the dual lattice $L^{*}$
is defined similarly. Obviously $M\subset L$ implies $L^{*}\subset M^{*}$.
Finally, we also assume that $A=L/M$ is a finite abelian group. Then each
$\gamma\in M^{*}$ defines a character of $A$ via the map $e^{\gamma}:{\bf
x}\in L\mapsto e^{2\pi i(\gamma,{\bf x})}$. In particular, one can identify
the group dual of $A$ with $M^{*}/L^{*}$. Define the matrix
$\SS=\frac{1}{|L:M|^{1/2}}(e^{\gamma}({\bf x}))$, where $\gamma$ and ${\bf x}$
are representatives for the cosets $M^{*}/L^{*}$ and $L/M$. Then $\SS$ is the
character matrix of $A$ up to a multiple and one easily concludes that it is
unitary. More precisely, we can view it as a unitary operator between Hilbert
spaces $V$ and $V^{*}$ with orthonormal bases labeled by the elements of $L/M$
and $M^{*}/L^{*}$ respectively.

###  3.2. Weights of traces

We will primarily be interested in lattices related to root, coroot and weight
lattices of orthogonal and symplectic groups. We define the lattices

(3.1) $Q=\\{{\bf x}\in{\mathbb{Z}}^{k},\ 2|\sum x_{i}\\}\quad{\rm and}\quad
P={\mathbb{Z}}^{k}\cup(\varepsilon+{\mathbb{Z}}^{k}),$

where $\varepsilon$ is the element in ${\mathbb{R}}^{k}$ with all its
coordinates equal to $1/2$. Observe that $P^{*}=Q$ with respect to the usual
scalar product of ${\mathbb{R}}^{k}$. Moreover, one can identify coroot and
weight lattices of $\mathfrak{s}o_{2k}$ or $\mathfrak{s}o_{2k+1}$ with $Q$ and
$P$ respectively. In particular, we define for any $\gamma\in P$ the
functional $e^{\gamma}:{\mathbb{R}}^{k}\to{\mathbb{C}}$ by $e^{\gamma}({\bf
x})=e^{2\pi i(\gamma,{\bf x})}$. The Weyl group of type $B_{k}$ acts as usual
via permutations and sign changes on the coordinates. Let
$a_{W}=\sum_{w}\varepsilon(w)w$, where $\varepsilon(w)$ is the sign of the
element $w$. Then the characters $\chi_{\lambda}$ for $\mathfrak{s}o_{2k+1}$
resp for $\mathfrak{s}p_{2k}$ are given by
$\chi_{\lambda}=a_{w}(e^{\lambda+\rho})/a_{w}(e^{\rho})$, where
$\rho=(k+1/2-i)$ for $\mathfrak{s}o_{2k+1}$ and $\rho=(k+1-i)$ for
$\mathfrak{s}p_{2k}$, and $W$ is the Weyl group of type $B_{k}$.

We will also need the somewhat less familiar character formulas for the full
orthogonal group $O(N)$: Recall that the irreducible representations of $O(N)$
are labeled by Young diagrams $\lambda$ with at most $N$ boxes in the first
two columns. $O(N)$-modules labeled by Young diagrams
$\lambda\neq\lambda^{\dagger}$ restrict to isomorphic $SO(N)$-modules if and
only if $\lambda_{1}^{\prime}=N-(\lambda^{\dagger})_{1}^{\prime}$ and
$\lambda_{i}^{\prime}=(\lambda^{\dagger})_{i}^{\prime}$ for $i>1$. Hence if
$g=exp({\bf x})$ is an element in $SO(N)$, it suffices to consider the
quantities $\chi_{\lambda}(g)=\chi_{\lambda}({\bf x})$ for $\lambda$ with at
most $k$ rows for $N=2k$ or $N=2k+1$. We can now express the weights of
Theorem 2.7 in terms of these characters; in fact the formulas in Theorem 2.7
were derived from these characters, see [Ko] and [W4].

######  Lemma 3.1.

Let $d_{\lambda},\ \tilde{d}_{\lambda}$ be as in Theorem 2.7 for $q=e^{\pi
i/\ell}$. Moreover, we define for $|N|=2k$ or $N=2k+1$ the vector
${\rho\check{}}\in{\mathbb{R}}^{k}$ by ${\rho\check{}}=((|N|+1)/2-i)_{i}$. By
the discussion above, it suffices to evaluate
$\chi^{O(N)}({\rho\check{}}/\ell)$ for Young diagrams $\lambda$ with
$\lambda_{1}^{\prime}\leq N/2$, which will be assumed in the following.

(a) If $N=2k+1>0$, then
$d_{\lambda}=\chi_{\lambda}^{O(N)}({\rho\check{}}/\ell)=\chi_{\lambda}^{SO(N)}({\rho\check{}}/\ell)$.

(b) If $N=2k>0$ and $\lambda_{1}^{\prime}\leq k$, then
$d_{\lambda}=m(\lambda)\det(\cos(l_{j}{\rho\check{}}_{i})/\det(\cos(k-j){\rho\check{}}_{i})$,
where $l_{j}=(\lambda+\rho)_{j}=\lambda_{j}+k-j$ and where $m(\lambda)=2$ or
$1$, depending on whether $\lambda$ has exactly $k$ rows or not.

(c) If $N=-2k$, then
$d_{\lambda}=(-1)^{|\lambda|}\chi_{\lambda^{t}}({\rho\check{}}/\ell)$ for the
symplectic character labelled by the transposed diagram $\lambda^{T}$.

(d) We have $\tilde{d}_{\lambda}=\chi_{\lambda}^{SU(N)}(\rho/\ell)$ for $N>0$
and
$\tilde{d}_{\lambda}=(-1)^{|\lambda|}\chi_{\lambda^{T}}^{SU(N)}(\rho/\ell)$
for $N<0$, where $\rho=((|N|+1)/2-i)\in{\mathbb{R}}^{|N|}$.

$Proof.$ Observe that ${\rho\check{}}$ is the element $\rho$ of the Cartan
subalgebra of $\mathfrak{s}l_{N}$, viewed as an element of the Cartan
subalgebra of the Lie subalgebra $\mathfrak{s}o_{N}$ or $\mathfrak{s}p_{N}$,
depending on the case. The proof now goes as e.g the proof of Theorem 4.6 in
[W4], which is essentially the one of [Ko]. The fact that these arguments also
work for the special quotients $\overline{Br}_{n}$ follows from the proof of
[W4], Theorem 5.5.

######  Remark 3.2.

Let $\Delta_{+}$ be the set of positive roots of a semisimple Lie algebra and
$|\Delta_{+}|$ be its cardinality. As usual, we can express the Weyl
denominator in $\chi_{\lambda}({\rho\check{}}/\ell)$ in product form as

(3.2) $\Delta({\rho\check{}}/\ell)\ =\
\prod_{\alpha>0}(e^{(\alpha,{\rho\check{}})\pi
i/\ell}-e^{-(\alpha,{\rho\check{}})\pi i/\ell})\ =\ (-i)^{|\Delta_{+}|}\
\prod_{\alpha>0}2\sin((\alpha,{\rho\check{}})\pi/\ell).$

###  3.3. Usual $S$-matrices

As usual, we pick as dominant chamber $C_{+}$ the regions given by
$x_{1}>x_{2}>\ ...\ x_{k}>0$ for Lie types $B_{k}$ and $C_{k}$. We also choose
the fundamental domains $D$ with respect to the translation actions of
$M,M^{*},L,L^{*}$ such that it has 0 in its center; here the lattices $M$ and
$L$ will be certain multiples of the lattices $P$, $Q$ or ${\mathbb{Z}}^{k}$
to be specified later. Let $\bar{P}_{+}$ be the intersection of $M^{*}$ with
the fundamental alcove $D\cap C_{+}$.

Observe that we also obtain a representation of the Weyl group $W$ on the
vector spaces $V$ and $V^{*}$. Then it is easy to check that $a_{W}(V^{*})$
has an orthonormal basis $|W|^{-1/2}a_{w}(e^{\gamma})$, with
$\gamma\in\bar{P}_{+}$, and we can define a similar basis $a_{W}({\bf x})$ for
$a_{W}(V)$. Let $S$ be the matrix which describes the action of
$\SS_{|a_{W}(V)}$ with respect to that basis. Then it is not hard to check
(and we will do a slightly more complicated case below) that its coefficents
are given by

(3.3) $s_{\gamma,{\bf x}}=\frac{1}{|L:M|^{1/2}}\sum_{w}\varepsilon(w)e^{2\pi
i(w.\gamma,{\bf x})}.$

If $L$ is the weight lattice of a simple Lie algebra, the entry
$s_{\gamma,{\bf x}}$ is the numerator of Weyl’s character formula for the
dominant weight $\lambda=\gamma-\rho$, up to the factor $|L:M|^{-1/2}$. As the
columns of the unitary matrix $S$ have norm one, it follows that

(3.4) $\sum_{\lambda}\chi_{\lambda}^{2}({\bf x})=\frac{|L:M]}{\Delta^{2}({\bf
x})},$

where $\Delta$ is the Weyl denominator, and the summation goes over the
dominant weights $\lambda$ such that $\lambda+\rho\in\bar{P}_{+}$. We are now
in the position to prove some cases of the following proposition:

######  Proposition 3.3.

Let $\Lambda(N,\ell)_{ev}$ be the subset of $\Lambda(N,\ell)$ consisting of
Young diagrams with an even number of boxes. Then we have

$\sum_{\lambda\in\Lambda(N,\ell)_{ev}}d_{\lambda}^{2}\ =\
\frac{\ell^{k}}{b(N)}\
\prod_{\alpha>0}\frac{1}{4\sin^{2}(\alpha,{\rho\check{}})\pi/\ell},$

where ${\rho\check{}}=((|N|+1)/2-i)$ and $\alpha>0$ runs through the positive
roots of $\mathfrak{s}o_{N}$ for $N>0$ and of $\mathfrak{s}p_{|N|}$ for $N<0$
even, and $b(N)=2$ for $N=2k>0$, and $b(N)=1$ otherwise.

$Proof.$ Let us consider the case $N=2k+1>0$, with $P$ and $Q$ as in 3.1. Let
$L=\ell^{-1}{\mathbb{Z}}^{k}$ and let $M_{1}=Q$ and $M_{2}={\mathbb{Z}}^{k}$.
Then we have $M_{1}^{*}=P$, and $M_{2}^{*}={\mathbb{Z}}^{k}$. Now observe that
$M_{1}^{*}$ is the weight lattice of $\mathfrak{s}o_{N}$, and the elements
$\gamma\in\bar{P}_{+}$ are in 1-1 correspondence with the dominant weights
$\lambda$ of $\mathfrak{s}o_{N}$ satisfying $\lambda_{1}\leq(\ell-N)/2$, via
the correspondence $\gamma=\lambda+\rho$. Moreover, $|L:M_{1}|=2\ell^{k}$.
Hence it follows from Eq 3.4 that
$\sum\chi_{\lambda}^{2}({\rho\check{}})=2\ell^{k}/\Delta^{2}({\rho\check{}})$.
Playing the same game for the lattice $M_{2}$, we now only get the sum over
the characters $\chi_{\lambda}^{2}$ for which $\lambda+\rho$ is in
${\mathbb{Z}}^{k}$, which is only half as large as before. Hence also the sum
over the characters $\chi_{\lambda}^{2}$ for which
$\lambda\in{\mathbb{Z}}^{k}$ has to have the same value. This sum coincides
with the right hand side of the statement for $N>0$ odd, by the restriction
rules for $O(N)$ to $SO(N)$ (see Lemma 3.1 and its preceding discussion).

The symplectic case $N=-2k<0$ goes similarly. Here we define $M\subset
L=\ell^{-1}P$, and with $L^{*}=\ell Q\subset M^{*}={\mathbb{Z}}^{k}$. Then it
follows that $\sum d_{\lambda}^{2}=2\ell^{k}/\Delta^{2}({\rho\check{}}/\ell)$,
where the summation goes over all diagrams $\lambda$ such that
$\lambda^{T}\in\Lambda(N,\ell)$. Playing the same game for $M=P$ and
$M^{*}=Q$, we get $\sum d_{\lambda}^{2}=\ell^{k}/\Delta({\rho\check{}}/\ell)$,
where now the summation goes over all even, or over all odd diagrams in
$\Lambda(N,\ell)$, depending on whether the sum of coordinates of
$\rho=(k+1-i)$ is odd or even. In each case, we obtain that
$\sum_{ev}d_{\lambda}^{2}=\ell^{k}/\Delta({\rho\check{}}/\ell)$. We have
proved the proposition except for the case $N=2k>0$, for which we need a
little more preparation.

###  3.4. Another $S$-matrix

We now consider a slight generalization of the above. Observe that we can
define a second sign function $\tilde{\varepsilon}$ for $W=W(B_{k})$ which
coincides with the usual sign function on its normal subgroup $W(D_{k})$,
while we have $\tilde{\varepsilon}(w)=-\varepsilon(w)$ for $w\not\in
W(D_{k})$. It is easy to see that also in this case we have
$\tilde{\varepsilon}(vw)=\tilde{\varepsilon}(v)\tilde{\varepsilon}(w)$ for all
$v,w\in W$. We define $\tilde{a}_{W}=\sum\tilde{\varepsilon}(w)w$, and also
denote the corresponding operators on the various (quotient) lattices and on
the vector spaces $V$ and $V^{*}$ by the same symbol. One observes that now we
get an orthonormal basis for $\tilde{a}_{W}(V^{*})$ of the form ${\bf
b}_{\gamma}=|Stab(\gamma)|^{-1/2}|W|^{-1/2}\tilde{a}_{W}(e^{\gamma})$, labeled
by the elements of $\bar{P}_{+}$ which now consist of the $\gamma\in D$ such
that $\gamma_{1}>\gamma_{2}>\ ...\ \gamma_{k}\geq 0$. Observe that
$|Stab(\gamma)|$ is equal to 1 or 2, depending on whether $\gamma_{k}>0$ or
$\gamma_{k}=0$. One similarly defines a basis for $\tilde{a}_{W}(V)$. Let
${\bf x}$ be such that $Stab({\bf x})=1$, i.e. $x_{k}>0$, and let $b_{\bf
x}=|W|^{-1/2}\tilde{a}_{W}({\bf x})$. Then, writing $M^{*}/\ell L^{*}$ as a
collection of $W$ orbits, we obtain

$\displaystyle\SS{\bf b}_{\bf x}\ $ $\displaystyle\
=|W|^{-1/2}\sum_{\lambda\in\bar{P}_{+}}\sum_{v,w\in
W}\frac{1}{|Stab_{W}(\gamma)|}\tilde{\varepsilon}(w)\tilde{s}_{v.\gamma,w.{\bf
x}}v.\gamma$ $\displaystyle\ =\
\sum_{\lambda\in\bar{P}_{+}}\sum_{v}(\sum_{w}\tilde{\varepsilon}(w)\tilde{s}_{w.\gamma,{\bf
x}}\frac{1}{|Stab_{W}(\lambda)|})\tilde{\varepsilon}(v)v.\gamma.\,$

where we replaced $\tilde{\varepsilon}(w)$ by
$\tilde{\varepsilon}(v)\tilde{\varepsilon}(w^{-1}v)$,
$\tilde{s}_{v.\lambda,w.{\bf x}}$ by $\tilde{s}_{w^{-1}v.\gamma,{\bf x}}$ and
finally also substituted $w^{-1}v$ by $w$. We see from this that the
coefficient of $v.\gamma$ is equal to 0 if $\gamma$ has a nontrivial
stabilizer except in the case when $\gamma_{k}=0$. Hence it follows that $\SS$
maps $a_{W}(V)$ into $a_{W}(V^{*})$. Taking bases
$(\tilde{a}_{W}(\gamma))_{\gamma\in P_{+}}$ and $(\tilde{a}_{W}({\bf x}))$, we
see that $\SS_{|a_{W}(V)}$ can be described by the matrix $S=(s_{\gamma,{\bf
x}})$ whose coefficients are given for ${\bf x}$ with trivial stabilizer by

(3.5) $s_{\gamma,{\bf
x}}=|Stab(\gamma)|^{-1/2}|L:M|^{-1/2}\sum_{w}\varepsilon(w)e^{2\pi
i(w.\gamma,{\bf x})}.$

###  3.5. Squares of characters

Using the discussion before and the formulas of Lemma 3.1 it is not hard to
see that for $N$ even and $\lambda_{1}^{\prime}\leq N/2$ we can write

$\chi_{\lambda}^{O(N)}=m(\lambda)\tilde{a}_{W}(e^{\lambda+\rho})/\tilde{a}_{W}(e^{\rho}),$

where $m(\lambda)=2$ or $1$ depending on whether $\lambda$ has exactly $k$
rows or not. In particular, applying this to the trivial representation, we
obtain $2\Delta(\rho)=\tilde{a}_{W}(e^{\rho})$.

Let $P$ and $Q$ be as in 3.1, and set $L=\ell^{-1}P$ and $M={\mathbb{Z}}^{k}$.
Then $L^{*}=\ell Q\subset M^{*}={\mathbb{Z}}^{k}$, and it is easy to see that
all of these lattices are $W=W(B_{k})$-invariant. Moreover, let
${\rho\check{}}/\ell=(k+1/2-i)/\ell\in\ell^{-1}P=M^{*}$. Then it follows for
$N=2k$ and $\ell$ even that

$\sum_{\lambda\in\Lambda(N,\ell)}\chi^{2}_{\lambda}({\rho\check{}}\ell)\ =\
\frac{1}{\Delta^{2}({\rho\check{}}\ell)}\sum_{\lambda_{k+1}=0,\lambda_{1}\leq(\ell-N)/2}(\tilde{a}_{W}(e^{\lambda+\rho})({\rho\check{}})\ell)^{2}\
=\
\frac{|L:M]}{2\Delta^{2}({\rho\check{}})}\sum_{\lambda}s_{\lambda,{\rho\check{}}/\ell}^{2}.$

Now observe that the matrix $S$ is unitary and that $[L:M]=2\ell^{k}$.
Moreover, by e.g. Prop. 1.1 and Theorem 2.10 the square sum over odd diagrams
must be equal to the square sum over even diagrams. Hence we obtain for $N>0$
even, and $\ell$ even that

(3.6) $\sum_{\lambda\in\Lambda(N,\ell)_{ev}}d_{\lambda}^{2}\ =\
\frac{\ell^{k}}{2\Delta^{2}({\rho\check{}})},$

where $\Lambda(N,\ell)_{ev}$ denotes the set of diagrams in $\Lambda(N,\ell)$
with an even number of boxes. This finishes the last case of the proof of
Proposition 3.3

###  3.6. Calculation of index

As usual, identify the Cartan algebra of $\mathfrak{s}l_{N}$ with the diagonal
$N\times N$ matrices with zero trace. The embedding of the Cartan algebras of
an orthogonal or symplectic subalgebra is given via diagonal matrices for
which the $N+1-i$-th entry is the negative of the $i$-th entry, for $1\leq
i\leq N/2$. Hence, if $\epsilon_{i}$ is the $\mathfrak{s}l_{N}$ weight given
by the projection onto the $i$-th diagonal entry, we have
$(\epsilon_{N+1-i})_{|\mathfrak{s}o_{N}}=(-\epsilon_{i})_{|\mathfrak{s}o_{N}}$,
with a similar identity also holding for symplectic subalgebras. Using our
description of coroot and weight lattices of orthogonal and symplectic Lie
algebras as sublattices of ${\mathbb{R}}^{k}$, and defining $\phi_{i}$ to be
the projection onto the $i$-coordinate, we see that
$(\epsilon_{N+1-i})_{|\mathfrak{s}o_{N}}=-\phi_{i}=(-\epsilon_{i})_{|\mathfrak{s}o_{N}}$.
This allows us to describe the decomposition of $\mathfrak{s}l_{N}$ as an
$\mathfrak{s}o_{N}$ resp. $\mathfrak{s}p_{N}$ module as follows: We have

(3.7) $\mathfrak{s}l_{N}=\mathfrak{s}o_{N}\oplus\mathfrak{p},\quad{\rm
resp.}\quad\mathfrak{s}l_{N}=\mathfrak{s}p_{N}\oplus\mathfrak{p}$

where $\mathfrak{p}$ is the nontrivial irreducible submodule in the
symmetrization of the vector representation of $\mathfrak{s}o_{N}$, resp.
$\mathfrak{p}$ is the nontrivial irreducible submodule in the
antisymmetrization of the vector representation of $\mathfrak{s}p_{N}$. The
nonzero weights $\omega>0$ of $\mathfrak{p}$ coming from positive roots of
$\mathfrak{s}l_{N}$ and the multiplicity $n(\mathfrak{p})$ of the weight 0 in
$\mathfrak{p}$ are given by

(a) $2\phi_{i},\phi_{i}$ and $\phi_{i}\pm\phi_{j}$ for $1\leq i<j\leq k$ with
$n(\mathfrak{p})=k$ for $\mathfrak{s}o_{N}$ with $N=2k+1$ odd,

(b) $2\phi_{i}$ and $\phi_{i}\pm\phi_{j}$ for $1\leq i<j\leq k$ with
$n(\mathfrak{p})=k-1$ for $\mathfrak{s}o_{N}$ with $N=2k$ even,

(c) $\phi_{i}\pm\phi_{j}$ for $1\leq i<j\leq k$ with $n(\mathfrak{p})=k-1$ for
$\mathfrak{s}p_{|N|}$ with $N=-2k<0$ even.

######  Theorem 3.4.

The index of the subfactor $\mathcal{N}\subset\mathcal{M}$ obtained from the
inclusions of algebras $\bar{H}_{n}(q)\subset\overline{Br}_{n}(q^{N},q)$ is
given by

$[\mathcal{M}:\mathcal{N}]=b(N)\ell^{n(\mathfrak{p})}\prod_{\omega>0}\frac{1}{4\sin^{2}(\omega,{\rho\check{}})\pi/\ell},$

where the product goes over the weights $\omega>0$ of $\mathfrak{p}$ coming
from positive roots of $\mathfrak{s}l_{N}$, as listed above, $n(\mathfrak{p})$
is the multiplicity of the zero weight in $\mathfrak{p}$, and $b(N)$ and
${\rho\check{}}$ are as in Prop. 3.3.

######  Corollary 3.5.

If $q=e^{\pi i/\ell}\to 1$, the index $[\mathcal{M}:\mathcal{N}]$ goes to
$\infty$ with asymptotics $\ell^{\dim\mathfrak{p}}$.

$Proof.$ We use Theorem 2.10, where the denominator has been calculated in
Proposition 3.3. The numerator follows from a standard argument for
$S$-matrices for Lie type $A$, see [Kc], versions of which have also been used
in this section. For an elementary calculation, see [E].

######  Remark 3.6.

It is straightforward to adapt our index formula to subfactors related to
other fixed points $H=G^{\alpha}$ of an order two automorphism $\alpha$ of a
compact Lie group $G$, up to some integer (or perhaps rational) constant
$b(H,G)$. Again, $\mathfrak{p}$ would be the $-1$ eigenspace of the induced
action of $\alpha$ on the Lie algebra $\mathfrak{g}$, and the same $S$-matrix
techniques applied in this section would go through. E.g. our formulas for
$N=3$ and $\ell$ odd coincide with the ones at the end of [X] for even level
of $SU(3)$, up to a factor 3. This is to be expected as in our case only those
diagrams appear in the principal graph (see next section) which also label
representations of the projective group $PSU(3)$.

###  3.7. Restriction rules and principal graph

It follows from Theorem 1.4 that the principal graph of
$\mathcal{N}\subset\mathcal{M}$ is given by the inclusion matrix for
$\bar{H}_{2k}\subset\overline{Br}_{2k}$ for $k$ sufficiently large. This still
leaves the question how to explicitly calculate these graphs. Observe that in
the classical case $q=1$ these would be given by the restriction rules from
the unitary group $U(N)$ to $O(N)$, for $N>0$. Formulas for these restriction
coefficients have been well-known, see e.g. [Wy] (see Theorems 7.8F and 7.9C),
Littlewood’s formula (see e.g. [KT], Section 1.5, and the whole paper for
additional results). Another approach closely related to the setting of fusion
categories can also be found in [W5].

Let $b^{\lambda}_{\mu}(N)$ be the multiplicity of the simple $O(N)$-module
$V_{\mu}$ in the $U(N)$ module $F_{\lambda}$, for $N>0$, where $\lambda$,
$\mu$ are Young diagrams. It is well-known that for fixed Young diagrams
$\lambda$ and $\mu$, the number $b_{\mu}^{\lambda}(N)$ will become a constant
$b_{\mu}^{\lambda}$ for $N$ large enough. Fix now also $\ell>|N|$. We define
similar coefficients in our setting as follows: Recall that the simple
components of $\bar{H}_{n}$ are labeled by the diagrams in
$\tilde{\Lambda}(N,\ell)_{n}$ and the ones of $\overline{Br}_{n}$ by the
diagrams in $\Lambda(N,\ell)$. We then define for
$\lambda\in\tilde{\Lambda}(N,\ell)$ and $\mu\in\Lambda(N,\ell)$ the number
$b^{\lambda}_{\mu}(N,\ell)$ to be the multiplicity of a simple
$\bar{H}_{n,\lambda}$ module in a simple $\overline{Br}_{n,\mu}$ module.

In the following lemma the symbol $\chi_{\mu}$ will also be used for the
$O(N)$ character corresponding to the simple representation labeled by the
Young diagram $\mu$. Moreover, we also denote by $\overline{Br}_{\infty}$ the
inductive limit of the finite dimensional algebras $\overline{Br}_{n}$ under
their standard inclusions, for fixed $N$ and $\ell$.

######  Lemma 3.7.

(a) Each $g\in O(N)$ for which $\chi_{\mu}(g)=0$ for all boundary diagrams
$\mu$ of $\Lambda(N,\ell)$ defines a trace on $\overline{Br}_{\infty}$
determined by $tr(p_{\mu})=\chi_{\mu}(g)/\chi_{[1]}(g)^{n}$, where $p_{\mu}$
is a minimal projection of $\overline{Br}_{n,\mu}$.

(b) For given $\lambda\in\tilde{\Lambda}(N,\ell)_{n}$ the coefficients
$b^{\lambda}_{\mu}(N,\ell)$ are uniquely determined by the equations
$\chi^{U(N)}_{\lambda}(g)=\sum_{\mu}b^{\lambda}_{\mu}(N,\ell)\chi_{\mu}(g)$
for all $g$ as in (a), where the summation goes over all diagrams $\mu$ in
$\Lambda(N,\ell)$ with $n,n-2,\ ...$ boxes.

$Proof.$ The formula in statement (a) determines a trace on
$\overline{Br}_{n}$ for each $n$. To show that these formulas are compatible
with the standard embeddings we observe that a minimal idempotent
$p_{\mu}\in\overline{Br}_{n,\mu}$ is the sum of minimal idempotents
$e_{\lambda}\in\overline{Br}_{n+1,\lambda}$ where $\lambda$ runs through all
diagrams in $\Lambda(N,\ell)$ obtained by adding or removing a box to/from
$\lambda$, see Eq 2.2 and the remarks below that theorem. Evaluating the
traces of these idempotents and multiplying everything by
$\chi_{\lambda}(g)^{n+1}$, equality of the traces is equivalent to

$\chi_{\mu}(g)\chi_{[1]}(g)=\sum_{\lambda}\chi_{\lambda}(g).$

By the usual tensor product rule for orthogonal groups, the left hand side
would be equal to the sum of characters corresponding to $all$ diagrams
$\lambda$ which differ from $\mu$ by only one box. It is easy to check that
this differs from the sum above only by boundary diagrams, for which the
characters at $g$ is equal to 0. This shows (a).

For (b), we first show that
$tr(p_{\lambda})=\chi_{\lambda}^{U(N)}(g)/\chi^{U(N)}_{[1]}(g)^{n}$ for
$p_{\lambda}\in\bar{H}_{n,\lambda}$ a minimal idempotent and $tr$ a trace as
in (a). As the weight vector for $\overline{Br}_{n+2N}$ is a multiple of the
one of $\overline{Br}_{n}$, for $n$ large enough, the same must also hold for
the weight vectors of $\bar{H}_{n}$ and $\bar{H}_{n+2N}$, by periodicity of
the inclusions. Hence these weight vectors must be eigenvectors of the
inclusion matrix for $\bar{H}_{n}\subset\bar{H}_{n+2N}$. As this inclusion
matrix is just a block of the $2N$-th power of the fusion matrix of the vector
representation for the corresponding type $A$ fusion category, its entries
must be given by $U(N)$ characters of a suitable group element. To identify
these elements, it suffices to observe that the antisymmetrizations of the
vector representation, labeled by the Young diagrams $\lambda=[1^{j}]$, $1\leq
j\leq N$, remain irreducible as $O(N)$ modules. This means the corresponding
Hecke algebra idempotent remains a minimal idempotent also in
$\overline{Br}_{j}$. Hence $tr(p_{\lambda})=\chi_{\lambda}^{U(N)}(g)$ for
$\lambda=[1^{j}]$ and $1\leq j\leq N$. But as the antisymmetrizations generate
the representation ring of $U(N)$, and also of the corresponding fusion ring,
the claim follows for general $\lambda$. For more details, see e.g. [GW]

Recall that the coefficient $b^{\lambda}_{\mu}(N,\ell)$ can be defined as the
rank of $p_{\lambda}$ in an irreducible $\overline{Br}_{n,\mu}$
representation. So obviously the formula in the statement holds for any $g$ as
in (a). Examples for such $g$ come from $exp({\bf x})$ with ${\bf x}\in
M^{*}=\ell^{-1}Q$ for which the character is given by the expression
$\chi_{\lambda}({\bf x})$ as in Section 2.5. As the columns of the orthogonal
$S$-matrix are linearly independent, this would identify $SO(N)$
representations. If $N$ is odd, the two $O(N)$ representations which reduce to
the same $SO(N)$ representation are labeled by Young diagrams with opposite
parities. Hence only one of them can occur in the decomposition of a given
$U(N)$ representation. A similar argument also works in the symplectic case.

For $N$ even, we can have two diagrams $\lambda$ and $\lambda^{\dagger}$ with
the same $SO(N)$ character, where one of them, say $\lambda$ has less than $k$
rows. They can be distinguished by elements $g\in O(N)\backslash{}SO(N)$ for
which $\chi_{\lambda^{\dagger}}(g)=-\chi_{\lambda}(g)$. It is well-known that
such elements $g$ must have eigenvalues $\pm 1$, and $\chi_{\lambda}(g)$ is
given by the character formula for $Sp(2k-2)$ in the remaining $2k-2$
eigenvalues (see [Wy]). It follows from the invertibility of the $S$-matrix
for $Sp(2k-2)$ at level $\ell/2-k$ (see [Kc]) that we can identify those
diagrams $\lambda$ by evaluating $\chi_{\lambda}^{Sp(2k-2)}({\bf x}/\ell)$ for
${\bf x}\in{\mathbb{Z}}^{k-1}$ with $\ell/2>x_{1}>x_{2}>\ ...\ >x_{k-1}>0$,
and that those elements satisfy the boundary condition
$\chi^{Sp(2k-2)}_{\lambda}({\bf x})=0$ for any boundary diagram $\lambda$.

The lemma above is illustrated in the following section for a number of
explicit examples. We can also give a closed formula for the restriction
coefficients, using a well-known quotient map for fusion rings (even though in
our case, the quotient ring does not correspond to a tensor category as far as
we know). In the context of fusion rings, this is known as the Kac-Walton
formula; for type $A$ see also e.g. [GW]. In our case, we need to use a
slightly different affine reflection group $\mathcal{W}$. In the orthogonal
case $N=2k$ and $N=2k+1$ it is given by the semidirect product of
$\ell{\mathbb{Z}}^{k}$ with the Weyl group of type $B_{k}$. In the symplectic
case, it is given by the semidirect product of $\ell Q$ with the Weyl group of
type $B_{k}$. As usual, we define the dot action of $\mathcal{W}$ on
${\mathbb{R}}^{k}$ by $w.{\bf x}=w({\bf x}+\rho)-\rho$, where $\rho$ is half
the sum of the positive roots of the corresponding Lie algebra, with the roots
embedded into ${\mathbb{R}}^{k}$ as described above, and $\varepsilon$ is the
usual sign function for reflection groups. This can be extended to an action
on the labeling set of $O(N)$ representations by identifying a Young diagrams
with $\leq k$ rows with the corresponding vector in ${\mathbb{Z}}^{k}$, and by
using the restriction rule from $O(N)$ to $SO(N)$ in the other cases. See also
e.g. Lemma 1.7 in [W5] for more details.

######  Theorem 3.8.

With notations as above, the restriction multiplicity
$b^{\lambda}_{\mu}(N,\ell)$ for $N=2k+1>0$ and $N=-2k$ is given by

$b^{\lambda}_{\mu}(N,\ell)=\sum_{w\in\mathcal{W}}\varepsilon(w)b^{\lambda}_{w.\mu}(N).$

If $N=2k>0$, we have to replace $\varepsilon$ by $\tilde{\varepsilon}$ (see
Section 3.4) in the formula above.

$Proof.$ Looking at the character formulas, we see that an action of an
element $w$ of the finite reflection group on $\lambda$ just changes the
character by the sign of $w$. Moreover, by definition of the elements ${\bf
x}$ we have that $\chi_{\lambda}({\bf x})=\chi_{\lambda+\mu}({\bf x})$ for any
$\mu\in M$. It follows that $\chi_{w.\lambda}({\bf
x})=\varepsilon(w)\chi_{\lambda}({\bf x})$ for all ${\bf x}\in M^{*}$ and
$w\in\mathcal{W}$. Hence summing over the $\mathcal{W}$-orbits, we obtain for
any ${\bf x}\in M^{*}$, $\lambda\in\tilde{\Lambda}(N,\ell)$ and
$\mu\in\Lambda(N,\ell)$ that

$\chi^{U(N)}_{\lambda}({\bf
x})=\sum_{\gamma}b^{\lambda}_{\gamma}(N)\chi_{\gamma}=\sum_{\mu}(\sum_{w}b^{\lambda}_{w.\mu}(N))\chi_{\mu}.$

The claim now follows from this and Lemma 3.7.

##  4\. Examples and other approaches

###  4.1.  The case $N=2$:

This corresponds to the Goodman-de la Harpe-Jones subfactors for type
$D_{\ell/2+1}$, where $\ell>2$ has to be even. It follows from our theorem
that the even vertices of the principal graph are labeled by the Young
diagrams $\lambda$ with an even number $n$ of boxes, at most two rows and with
$\lambda_{1}-\lambda_{2}\leq\ell-2$; there are $(\ell-2)/2$ such diagrams.
Their dimensions are given by $\tilde{d}_{k}=[2k+1]$, $0\leq k<(\ell-2)/2$.

Moreover, one checks that $\Lambda(2,\ell)$ consists of Young diagrams $[j]$
with $0\leq j\leq\ell/2-1$ and of $[1^{2}]$, one column with 2 boxes, with
dimensions $d_{[j]}=2\cos j\pi/\ell$ for $j>0$ and dimension equal to 1 for
the remaining cases (i.e. for $\emptyset$ and for $[1^{2}]$). The restriction
rule (i.e. principal graph) follows from writing the dimensions as

$\tilde{d}_{k}=2\cos\tilde{k}\pi/\ell\ +\ 2\cos(\tilde{k}-2)\pi/\ell+\ ...\
+1,$

where $\tilde{k}=$ min $\\{k,\ell/2-k\\}$. Indeed, this determines the graph
completely except for whether to pick the diagram $\emptyset$ or $[1^{2}]$ for
the object with dimension 1. It follows from the restriction rule $O(2)\subset
U(2)$ that we take $\emptyset$ for $j$ even, and $[1^{2}]$ for $j$ odd. To
calculate the index one can check by elementary means that $\sum_{\lambda\
even}d_{\lambda}^{2}=\ell/2$. Moreover, it is well-known that the sum
$\sum_{\lambda\ even}\tilde{d}_{\lambda}^{2}$ over even partitions for
$\mathfrak{s}l_{2}$ is equal to $\ell/4\sin^{2}\pi/\ell$. Hence we obtain as
index $[\mathcal{M}:\mathcal{N}]=1/2\sin^{2}\pi/\ell$.

###  4.2.  The case $N=3$

It is also fairly elementary to work out this case in detail. Recall that by
Weyl’s dimension formula we have

$\tilde{d}_{\lambda}=\frac{[\lambda_{1}-\lambda_{2}+1][\lambda_{2}-\lambda_{3}+1][\lambda_{1}-\lambda_{3}+2]}{[1]^{2}[2]}.$

Now observe that the product of two $q$-numbers is given by the tensor product
rules for $\mathfrak{s}l_{2}$, i.e. we have for $n\geq m$ that
$[n][m]=[n+m-1]+[n+m-3]+...+[n-m+1]$. As an example, we have

$\tilde{d}_{[4]}\ =\ \frac{[6][5]}{[2]}\ =\ \frac{[10]+[8]+[6]+[4]+[2]}{[2]}\
=\ [9]+[5]+[1],$

i.e. the fourth antisymmetrization of the vector representation of $U(3)$
decomposes as a direct sum of the one-, five- and nine-dimensional
representation of $SO(3)$. One similarly can show the well-known result that
the adjoint representation of $SU(3)$, labeled by the Young diagram $[2,1]$
decomposes into the direct sum of the three- and the five-dimensional
representation of $SO(3)$, i.e. $\mathfrak{p}$ is the five dimensional
representation of $SO(3)$. Hence we get from Theorem 3.4 that the index is
equal to

$[\mathcal{M}:\mathcal{N}]\ =\ \frac{\ell}{4^{2}\sin^{2}(2\pi/\ell)\
\sin^{2}(\pi/\ell)}.$

We note that here as well as in the other examples, the dimensions (i.e.
entries of the Perron-Frobenius vectors) are given by $|\tilde{d}_{\lambda}|$
for even vertices, and by $\sqrt{[\mathcal{M}:\mathcal{N}]}|d_{\mu}|$ for odd
vertices, with $\tilde{d}_{\lambda}$ and $d_{\mu}$ as in Lemma 3.1. To
consider explicit examples, the first nontrivial case for $N=3$ occurs for
$\ell=7$. We leave it to the reader to check that in this case the first
principal graph is given by the Dynkin graph $D_{8}$. A more interesting graph
is obtained for $\ell=9$, see Fig 4.1. Here we have the three invertible
objects of the $SU(3)_{6}$ fusion category, including the trivial object
(often denoted as $*$) on the left; they generate a group isomorphic to
${\mathbb{Z}}/3$. The vertices with the double edge are labeled by the object
corresponding to the 5-dimensional representation of $SO(3)$ and the diagram
$[4,2]$ for $SU(3)_{6}$. This is the only fixed point under the
${\mathbb{Z}}/3$ action given by the invertible objects (or, in physics
language, the currents). It would be interesting to see whether one can carry
out an orbifold construction in this context related to the one in [EK1].

Figure 4.1. SO(3) for $\ell=9$

###  4.3. The case $N=4$

This is the first nontrivial and, apparently, new case corresponding to even-
dimensional orthogonal groups. As we shall see, somewhat surprisingly, the
corresponding construction for $SO(4)$ does not seem to work. We do the case
with $\ell=8$ in explicit detail. It is not hard to check that we already get
the periodic inclusion matrix for $n=12$. As we consider an analog of the
restriction to $O(4)$ for which the determinant can be $\pm 1$, we should,
strictly speaking, consider a fusion category for $SU(4)\times\\{\pm 1\\}$. We
shall actually use the Young diagram notation for representations of $U(4)$.
For $n=12$ we have the invertible objects labeled by $[3^{4}]$, $[4^{3}]$,
$[5^{2}1^{2}]$ and $[62^{3}]$ (i.e. e.g. the last diagram has six boxes in the
first and two boxes in the second, third and fourth row). They generate a
subgroup isomorphic to ${\mathbb{Z}}/4$. It follows from the $O(4)$
restriction rules that $[3^{4}]$ and $[5^{2}1^{2}]$ contain the determinant
representation, and $[4^{3}]$ and $[62^{3}]$ contain the trivial
representation as one-dimensional $O(4)$ subrepresentations. This allows us to
calculate the restrictions for representations of each ${\mathbb{Z}}/4$ orbit
simultaneously. As usually for at least one element of each orbit the ordinary
restriction rules still hold, it makes the general calculations easier. The
principal graph can be seen in Fig. 4.2. As in the $N=3$ example, the one-
dimensional currents, including the trivial object $*$ appear as the left- and
right-most vertices in the graph. The lowest vertex corresponds to the
$O(4)$-object $[2]$ which is connected to the objects in the
${\mathbb{Z}}/4$-orbit $\\{[2,1^{2}],[3,1],[4,3,1],[3,3,2]\\}$. We also note
that we get the same graph for the $Sp(4)$ case $N=-4$ for $\ell=8$. However,
for other roots of unities, already the indices of the subfactors differ which
are given by

$O(4):\ \frac{2\ell}{4\sin^{2}(3\pi/\ell)\ 4\sin^{2}(2\pi/\ell)\
16\sin^{4}(\pi/\ell)}\hskip 30.00005ptSp(4):\
\frac{\ell}{4\sin^{2}(2\pi/\ell)\ 4\sin^{2}(\pi/\ell)}.$

It was originally thought that we should also be able to get fusion category
analogs for the restriction from $SU(N)$ to $SO(N)$ for $N$ even. It is easy
to check that this is not possible for $O(2)$. Some initial checks also seem
to suggest a similar phenomenon for higher ranks. E.g. using the same element
${\rho\check{}}$ in the $SO(N)$ character formula would give dimension
functions which are not invariant under the $D_{N}$ diagram automorphism.

Figure 4.2. O(4) and Sp(4) for $\ell=8$

###  4.4. Related results

We discuss several results related to our findings. Our original motivation
was to construct subfactors related to twisted loop groups. It was shown in R.
Verill’s PhD thesis [V] that it is not possible to construct a fusion tensor
product for representations of twisted loop groups. However, it seemed
reasonable to expect that representations of twisted loop groups could become
a module category over representations of their untwisted counterparts. Many
results, in particular about the combinatorics of such categories can be found
in the context of boundary conformal field theory in papers by Gaberdiel,
Gannon, Fuchs, Schweigert, diFrancesco, Petkova, Zuber and others (see e.g.
[GG], [FS], [PZ] and the papers cited therein).

In the mathematics literature, one can find closely related results in the
papers [X] and [Wa]. Here the authors construct module categories via a
completely different approach in the context of loop groups. E.g. the formulas
at the end of [X] for the special case $N=3$ differ only by a factor 3 (which
can be explained, see Remark 3.6) from our formulas for $N=3$ for even level
(together with Corollary 1.5), modulo misprints; similar formulas for the
symplectic case as well as restriction coefficients also appear at the end of
[Wa]. We can not get results corresponding to the odd level cases in [X]. The
combinatorics there suggests that this would require considering an embedding
of $Sp(N-1)$ into $SU(N)$ under which the vector representation would not
remain irreducible. In contrast, we can also construct module categories for
$\ell-N$ odd, which would correspond to odd level; however, these categories
are not unitarizable (which follows from Lemma 2.8) and they have different
fusion rules. However, we do get fairly general formulas for the index and
principal graphs of this type of subfactors in the unitary case, which was one
of the problems posed in [X]. These formulas were known to this author as well
as to Antony Wassermann at least back in 2008 when they had discussions about
their respective works in Oberwolfach and at the Schrödinger Institute.

We close this section by mentioning that while our results for $N>0$ odd and
$N<0$ even are in many ways parallel to results obtained via other approaches
in connection with twisted loop groups, there does not seem to be an obvious
analog for our results for $N>0$ even. E.g. the combinatorial results in [GG]
for that case seem to be different to ours.

###  4.5. Conclusions and further explorations

We have constructed module categories of fusion categories of type $A$ via
deformations of centralizer algebras of certain subgroups of unitary groups.
We have also classified when they are unitarizable, and we have constructed
the corresponding subfactors. These deformations are compatible with the
Drinfeld-Jimbo deformation of the unitary group but not with the Drinfeld-
Jimbo deformation of the subgroup. Most of the deformation was already done in
[W4] via elementary methods. In principle, at least, it should be possible to
use this elementary approach also for other inclusions. However, this might
become increasingly tedious.

As we have seen already in Section 2.2, it should be possible to get a
somewhat more conceptual approach using different deformations of the
subgroup, see [N], [Mo], [L1], [L2], [IK] and references therein. In
particular in the work of Letzter, such deformations via co-ideal algebras
have been defined for a large class of embeddings of a semisimple Lie algebra
into another one. At this point, it does not seem obvious how to define
$C^{*}$-structures in this setting, and additional complications arise as
these coideal algebras are not expected to be semisimple at roots of unity.
Nevertheless, the results in this and other papers such as [X], [Wa] would
seem to suggest that similar constructions might be possible also in a more
general setting. It would also be interesting to see whether this could lead
to a general solution of Problem 5.5 in [O].

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