7 SUBFACTORS AND HADAMARD MATRICES
WES CAMP AND REMUS NICOARA
Abstract. To any complex Hadamard matrix H one associates a spin
model commuting square, and therefore a hyperfinite subfactor. The stan-
dard invariant of this subfactor captures certain ”group-like” symmetries of
H . To gain some insight, we compute the first few relative commutants of
such subfactors for Hadamard matrices of small dimensions. Also, we show
that subfactors arising from Dita type matrices have intermediate subfac-
tors, and thus their standard invariants have some extra structure besides
the Jones projections.
1. Introduction
A complex Hadamard matrix is a matrix H ∈ Mn(C) having all entries of
absolute value 1 and all rows mutually orthogonal. Equivalently, 1√
H is a
unitary matrix with all entries of the same absolute value. For example, the
Fourier matrix Fn = (ω
ij)1≤i,j≤n, ω = e
2πi/n, is a Hadamard matrix.
In the recent years, complex Hadamard matrices have found applications
in various topics of mathematics and physics, such as quantum information
theory, error correcting codes, cyclic n-roots, spectral sets and Fuglede’s con-
jecture. A general classification of real or complex Hadamard matrices is not
available. A catalogue of most known complex Hadamard matrices can be
found in [TZ]. The complete classification is known for n ≤ 5 ([H]) and for
self-adjoint matrices of order 6 ([BeN]).
The connection between Hadamard matrices and von Neumann algebras
arose from an observation of Popa ([Po2]): a unitary matrix U is of the form
H , H Hadamard matrix, if and only if the algebra of n×n diagonal matrices
Dn is orthogonal onto UDnU∗, with respect to the inner product given by the
trace on Mn(C). Equivalently, the square of inclusions:
C(H) =
Dn ⊂ Mn(C)
C ⊂ UDnU∗
is a commuting square, in the sense of [Po1],[Po2]. Here τ denotes the trace
on Mn(C), normalized such that τ(1) = 1.
http://arxiv.org/abs/0704.1128v1
2 WES CAMP AND REMUS NICOARA
Such commuting squares are called spin models, the name coming from sta-
tistical mechanical considerations (see [JS]). By iterating Jones’ basic con-
struction, one can construct a hyperfinite, index n subfactor from H (see
for instance [JS]). The subfactor associated to H can be used to capture
some of the symmetries of H , and thus to classify H to a certain extent (see
[BHJ],[Jo2],[BaN]).
Let N ⊂ M be an inclusion of II1 factors of finite index, and let N ⊂ M
e2⊂ M2 ⊂ ... be the tower of factors constructed by iterating Jones’ basic
construction (see [Jo1]), where e1, e2, ... denote the Jones projections. The
standard invariant GN,M is then defined as the trace preserving isomorphism
class of the following sequence of commuting squares of inclusions of finite
dimensional ∗-algebras:
C = N ′ ∩N ⊂ N ′ ∩M ⊂ N ′ ∩M1 ⊂ N ′ ∩M2 ⊂ ...
∪ ∪ ∪
M ′ ∩M ⊂ M ′ ∩M1 ⊂ M ′ ∩M2 ⊂ ...
The Jones projections e1, e2, ..., en are always contained in N
′ ∩Mn. If the
index of the subfactor N ⊂ M is at least 4, they generate the Temperley-Lieb
algebra of order n, denoted TLn. In a lot of situations the relative commutant
N ′ ∩Mn has some interesting extra structure, besides TLn. For instance, the
five non-equivalent real Hadamard matrices of order 16 yield different dimen-
sions for the second relative commutant N ′ ∩ M1, and thus are classified by
these dimensions ([BHJ]).
In this paper we investigate the relation between Hadamard matrices and
their subfactors. We look at Hadamard matrices of small dimensions or of
special types. The paper is organized as follows: in the first section we recall,
in our present framework, several results of [Jo2],[JS] regarding computations
of standard invariants for spin models.
In section 2 we study the subfactors associated to Hadamard matrices of
Dita type. These are matrices that arise from a construction of [Di], which is
a generalization of a construction of Haagerup ([H]). Most known parametric
families of Hadamard matrices are of Dita type. We show that the associated
subfactors have intermediate subfactors.
In the last section we present a list of computations of the second and third
relative commutants N ′ ∩ M1, N ′ ∩ M2, for complex Hadamard matrices of
small dimensions. We make several remarks and conjectures regarding the
structure of the standard invariant. Most of the computations included were
done using computers, with the help of the Mathematica and GAP softwares.
We would like to thank Teodor Banica, Kyle Beauchamp and Dietmar Bisch
for fruitful discussions and correspondence. Wes Camp was supported in part
SUBFACTORS AND HADAMARD MATRICES 3
by NSF under Grant No. DMS 0353640 (REU Grant), and Remus Nicoara
was supported in part by NSF under Grant No. DMS 0500933.
2. Subfactors associated to Hadamard matrices
Let H be a complex n × n Hadamard matrix and let U = 1√
H . U is a
unitary matrix, with all entries of the same absolute value. One associates to
U the square of inclusions:
C(H) =
Dn ⊂ Mn(C)
C ⊂ UDnU∗
where Dn is the algebra of diagonal n×n matrices and τ is the trace onMn(C),
normalized such that τ(1) = 1.
Since H is a Hadamard matrix, C(H) is a commuting square in the sense of
[Po1],[Po2], i.e. EDnEUDnU∗ = EC. The notation EA refers to the τ -invariant
conditional expectation from Mn(C) onto the ∗-subalgebra A.
Recall that two complex Hadamard matrices are said to be equivalent if
there exist unitary diagonal matrices D1, D2 and permutation matrices P1, P2
such that H2 = P1D1H1D2P2. It is easy to see that H1, H2 are equivalent if
and only if C(H1),C(H2) are isomorphic as commuting squares, i.e. conjugate
by a unitary from Mn(C).
We denote by Ct(H) the commuting square obtained by flipping the upper
left and lower right corners of C(H):
t(H) =
UDnU∗ ⊂ Mn(C)
C ⊂ Dn
We have: Ct(H) = Ad(U)C(H∗). Thus, Ct(H) and C(H) are isomorphic as
commuting squares if and only if H,H∗ are equivalent as Hadamard matrices.
We now recall the construction of a subfactor from a commuting square. By
iterating Jones’ basic construction ([Jo1]), one obtains from Ct(H) a tower of
commuting squares of finite dimensional ∗-algebras:
UDnU∗ ⊂ Mn(C)
⊂ ...
∪ ∪ ∪ ∪
C ⊂ Dn
⊂ ...
together with the extension of the trace, which we will still denote by τ , and
Jones projections gi+2 ∈ Yi, i = 1, 2, ....
4 WES CAMP AND REMUS NICOARA
Let MH be the weak closure of ∪iXi, with respect to the trace τ , and let
NH be the weak closure of ∪iYi. NH ,MH are hyperfinite II1 factors, and the
trace τ extends continuously to the trace of MH , which we will still denote by
τ . It is well known that NH ⊂ MH is a subfactor of index n, which we will
call the subfactor associated to the Hadamard matrix H .
The standard invariant of NH ⊂ MH can be expressed in terms of commu-
tants of finite dimensional algebras, by using Ocneanu’s compactness argument
(5.7 in [JS]). Consider the basic construction for the commuting square C(H):
Dn ⊂ Mn(C)
e3⊂ P1
e4⊂ P2
e5⊂ ...
∪ ∪ ∪ ∪
C ⊂ UDnU∗
e3⊂ Q1
e4⊂ Q2
e5⊂ ...
Ocneanu’s compactness theorem asserts that the first row of the standard
invariant of NH ⊂ MH is the row of inclusions:
D′n ∩ UDnU∗ ⊂ D′n ∩ Q1 ⊂ D′n ∩ Q2 ⊂ D′n ∩ Q3 ⊂ ...
More precisely, if
NH ⊂ MH
e3⊂ MH,1
e4⊂ MH,2
e5⊂ ...
is the Jones tower obtained from iterating the basic construction for the inclu-
sion NH ⊂ MH , then:
D′n ∩ Qi = N ′H ∩MH,i, for all i ≥ 1.
Thus, the problem of computing the standard invariant of the subfactor
associated to H is the same as the computation of D′n ∩ Qi. However, such
computations seem very hard, and even for small i and for matrices H of
small dimensions they seem to require computer use. Jones ([Jo2]) provided a
diagrammatic description of the relative commutants D′n ∩ Qi (see also [JS]),
which we express below in the framework of this paper.
Let P0 = Mn(C) and let (ei,j)1≤i,j≤n be its canonical matrix units. Let
i,j=1
ei,j .
It is easy to check that e2 is a projection. Moreover: < Dn, e2 >= Mn(C)
and e2xe2 = EDn(x)e2 for all x ∈ Mn(C). Thus, e2 is realizing the basic
construction
C ⊂ Dn
e2⊂ Mn(C)
Let ek,l ⊗ ei,j denote the n2 × n2 matrix having only one non-zero entry,
equal to 1, at the intersection of row (i − 1)n + k and column (j − 1)n + l.
Thus, ek,l ⊗ ei,j are matrix units of Mn(C) ⊗ Mn(C). In what follows, we
SUBFACTORS AND HADAMARD MATRICES 5
will assume that the embedding of Mn(C) into Mn(C)⊗Mn(C) is realized as
ek,l → ek,l ⊗ In, where ek,l ⊗ In =
i=1 ek,l ⊗ ei,i.
Lemma 2.1. Let P1 = Mn(C)⊗Dn, P2 = Mn(C)⊗Mn(C), e3 =
i=1 ei⊗ei ∈
P1 and e4 = In ⊗ e2 ∈ P2. Then
Dn ⊂ Mn(C)
e3⊂ P1
is a basic construction with Jones projection e3 and
Mn(C) ⊂ P1
e4⊂ P2
is a basic construction with Jones projection e4.
Proof. To show that Dn ⊂ Mn(C)
e3⊂ P1 is a basic construction it is enough to
check that < Mn(C), e3 >= P1 and e3 is implementing EP1Mn(C). First part is
clear, since ek,ie3ei,l = ek,l ⊗ ei,i are a basis for P1 = Mn(C) ⊗ Dn. To check
that e3 implements the conditional expectation, let X = (xi,j) ∈ Mn(C). We
have:
e3(X ⊗ In)e3 =
i,j=1
(Di ⊗Di)(X ⊗ In)(Dj ⊗Dj)
DiXDi ⊗Di
(DiXDi ⊗ In)e3
= EDn⊗In(X)e3
Since C ⊂ Dn
e2⊂ Mn(C) is a basic construction, after tensoring to the left
by Mn(C) it follows that Mn(C) ⊂ P1
e4⊂ P2 is a basic construction, with
e4 = In ⊗ e2.
Proposition 2.1. The algebras P1,P2,P3, ... constructed in (2) are given by
P2k = ⊗k+1i=1Mn(C), P2k+1 = P2k ⊗Dn
with the Jones projections
e2k+2 = ⊗ki=1In ⊗ e2, e2k+3 = ⊗ki=1In ⊗ e3
Proof. Follows from the previous lemma, by tensoring successively by Mn(C).
6 WES CAMP AND REMUS NICOARA
Proposition 2.2. Let H be a complex n×n Hadamard matrix, let U = 1√
i,j=1
ūi,jej,j ⊗ ei,i, U1 = UDU .
Then the algebras Q1,Q2,Q3, ... constructed in (2) are given by
Qk = UkPk−1U∗k , k ≥ 1
where Uk ∈ Pk are the unitary elements:
U2k+1 = Π
i=0(⊗iIn ⊗ U1 ⊗k−i In), U2k = U2k−1(⊗kIn ⊗ U), k ≥ 1.
Proof. The unitary U1 satisfies:
(AdU1)(Dn) = (AdU)(Dn)
since U∗U1 = DU ∈ Dn. Moreover, we have:
(AdU1)(e2) = (AdU)Ad(
i,j=1
ūi,jej,j ⊗ ei,i)(
k,l=1
ek,l)
= (AdU)(
i,k,l=1
ūi,kui,lek,l ⊗ ei,i)
= (AdU)(AdU∗(e3))
It follows that AdU1 takes the basic construction C ⊂ Dn
e2⊂ Mn(C) onto the
inclusion C ⊂ UDnU∗
e3⊂ U1Mn(C)U∗1 . Thus this is also a basic construction,
which shows thatQ1 = U1Mn(C)U∗1 . Moreover, it follows that each AdUi takes
the basic construction Pi−1 ⊂ Pi ⊂ Pi+1 onto Qi ⊂ Qi+1 ⊂ Qi+2, which ends
the proof.
The first relative commutant D′n∩UDnU∗ is equal to C, since the commuting
square condition implies Dn ∩ UDnU∗ = C. Thus the subfactor NH ⊂ MH
is irreducible. In the following proposition we describe the higher relative
commutants of the subfactor NH ⊂ MH as the commutants of some matrices
Pi, i ≥ 1.
Proposition 2.3. With the previous notations, let Pi denote the projection
Uiei+3U
i ∈ Pi+1, i ≥ 1. Then we have the following formula for the (i+ 1)-th
relative commutant:
D′n ∩ Qi = P ′i ∩ D′n ∩ Pi.
SUBFACTORS AND HADAMARD MATRICES 7
Proof. We have:
D′n ∩Qi = D′n ∩AdUi(Pi−1)
= D′n ∩AdUi(e′i+3 ∩ Pi)
= D′n ∩ P ′i ∩ AdUi(Pi)
= D′n ∩ P ′i ∩ Pi
We used the fact that Pi−1 ⊂ Pi
⊂ Pi+1 is a basic construction, and thus
e′i+3 ∩ Pi = Pi−1. �
Remark 2.1. The n2 × n2 matrix P1 = U1e4U∗1 can be written as
a,b,c,d=1
a,bea,b ⊗ ec,d, where p
a,b =
ua,iūb,iūc,iud,i.
This matrix is used in the theory of Hadamard matrices and it is called the
profile of H. It is a result of Jones ([Jo2]) that the matrices P2i+1, i ≥ 1,
depend only on P1. Indeed, one can check that
P2i+1 =
k1,l1,...,ki,li=1
k1,l1
a,b p
k2,l2
k1,l1
ki,li
ea,b ⊗ ek1,l1 ⊗ ek2,l2 ⊗ ...⊗ eki,li ⊗ ec,d.
Thus, all higher relative commutants of even orders are determined by P1.
Let ΓH denote the graph of vertices {1, 2, ..., n} × {1, 2, ..., n}, in which the
distinct vertices (a, c) and (b, d) are connected if and only if p
a,b 6= 0. The
second relative commutant can be easily described in terms of ΓH . We recall
this in the following Proposition, which is a reformulation of a result in [Jo2]
(see also [JS]).
Proposition 2.4. The second relative commutant of the subfactor NH ⊂ MH
is abelian, its minimal projections are in bijection with the connected compo-
nents of ΓH , and their traces are proportional to the sizes of the connected
components.
Proof. Let
i,j=1 λ
iei,i⊗ej,j, λ
i ∈ {0, 1}, be a projection in the second relative
commutant P ′1 ∩ (Dn ⊗Dn). We have:
a,b,c,d=1
a,bea,b⊗ec,d)(
i,j=1
iei,i⊗ej,j) = (
i,j=1
iei,i⊗ej,j)(
a,b,c,d=1
a,bea,b⊗ec,d)
Equivalently:
a,c,i,j=1
q,iea,i ⊗ ec,j =
b,d,i,j=1
i,b ei,b ⊗ ej,d.
8 WES CAMP AND REMUS NICOARA
By relabeling and identifying the set of indices, it follows:
(λca − λ
a,i = 0.
Thus, if the vertices (a, c) and (i, j) are connected then λca = λ
i . This ends
the proof. �
3. Matrices of Dita type
In this section we investigate the standard invariant of subfactors associ-
ated to a particular class of Hadamard matrices, obtained by a construction of
P.Dita ([Di]), which is a generalization of an idea of U.Haagerup ([H]). These
matrices have a lot of symmetries, and we show that for such matrices the sec-
ond relative commutant has some extra structure besides the Jones projection.
Let n be non-prime, n = kl with k, l ≥ 2. Let A = (ai,j) ∈ Mk(C) and
B1, ..., Bk ∈ Mm(C) be complex Hadamard matrices. It is possible to construct
an n × n Hadamard matrix from A,B1, ..., Bk by using an idea of [Di] (see
also[H],[Pe]). This construction is a generalization of the tensor product of
two Hadamard matrices:
(6) H =
a1,1B1 a1,2B2 ... a1,kBk
a2,1B1 a2,2B2 ... a2,kBk
ak,1B1 ak,2B2 ... ak,kBk
Let (fi,j)1≤i,j≤k be the matrix units of Mk(C). We identify Mn(C) with the
tensor product Mm(C)⊗Mk(C), with the same conventions as before. Thus:
i,j=1
ai,jBj ⊗ fi,j
One can use construct multi-parametric families of non-equivalent Hadamard
matrices, by replacing B1, ..., Bk by B1D1, ...BkDk, where D1, ..., Dk are diag-
onal unitaries. Some of the families of Hadamard matrices of small orders
considered in the next section arise from this construction.
Recall that the second relative commutant always contains the Jones projec-
tion e3 =
eii ⊗ eii. In the next proposition we show that the second relative
commutant of a Dita type subfactor contains another projection f ≥ e3, so it
has dimension at least 3.
Proposition 3.1. Let H = (ai,jBj)1≤i,j≤k ∈ Mn(C) be a Dita type ma-
trix, where A = (ai,j)1≤i,j≤k ∈ Mk(C) and B1, ..., Bk ∈ Mm(C) are complex
SUBFACTORS AND HADAMARD MATRICES 9
Hadamard matrices, n = mk. Then the second relative commutant of the
subfactor associated to H contains the projection:
1≤i,j≤n, i≡j(mod m)
ei,i ⊗ ej,j ∈ Mn2(C).
Proof. For 1 ≤ i ≤ n, let i0 = (i− 1)(mod m) + 1 and i1 = i−i0m + 1. We will
use similar notations for 1 ≤ j ≤ n. Thus, the (i, j) entry of H is:
hi,j = ai1,j1b
i0,j0
where btr,s is the (r, s) entry of Bt, for all 1 ≤ t ≤ k, 1 ≤ r, s ≤ m.
With these notations, the projection f can be written as
i,j=1
iei,i ⊗ ej,j
where λ
i = 1 if i0 = j0 and λ
i = 0 for all other i, j.
According to Proposition 2.4, showing that f is in the second relative com-
mutant is equivalent to showing that p
i,c = 0 whenever c0 6= d0. Using the
formula for the entries of P1 and the fact that i0 = j0 we obtain:
i,c =
ui,xūc,xūj,xud,x
hi,xh̄c,xh̄j,xhd,x
ai1,x1b
i0,x0
āc1,x1 b̄
c0,x0
āj1,x1 b̄
j0,x0
ad1,x1b
d0,x0
ai1,x1āc1,x1 b̄
c0,x0
āj1,x1ad1,x1b
d0,x0
(ai1,x1āc1,x1āj1,x1ad1,x1(
b̄x1c0,x0b
d0,x0
ai1,x1āc1,x1āj1,x1ad1,x1δ
whenever c0 6= d0. �
10 WES CAMP AND REMUS NICOARA
We show that in fact the subfactor NH ⊂ MH associated to the Dita matrix
H has an intermediate subfactor NH ⊂ RH ⊂ MH , and the projection f is the
Bisch projection (in the sense of [Bi]) corresponding to RH .
Proposition 3.2. Let H =
1≤i,j≤k ai,jBj ⊗ fi,j ∈ Mn(C) be a Dita type
matrix, where A = (ai,j)1≤i,j≤k ∈ Mk(C) and B1, ..., Bk ∈ Mm(C) are complex
Hadamard matrices, n = mk. Then:
(a). The commuting square C(H) can be decomposed into two adjacent sym-
metric commuting squares:
Dm ⊗Dk ⊂ Mm(C)⊗Mk(C)
Dm ⊗ Ik ⊂ U(Mm(C)⊗Dk)U∗
C ⊂ U(Dm ⊗Dk)U∗
(b). The commuting square Ct(H) can be decomposed into two adjacent
symmetric commuting squares:
U(Dm ⊗Dk)U∗ ⊂ Mm(C)⊗Mk(C)
U(Im ⊗Dk)U∗ ⊂ Dm ⊗Mk(C)
C ⊂ Dm ⊗Dk
SUBFACTORS AND HADAMARD MATRICES 11
Proof. (a). We first show that Dm ⊗ Ik ⊂ U(Mm(C) ⊗ Dk)U∗. Equivalently,
we check that U∗(Dm ⊗ Ik)U ⊂ (Mm(C)⊗Dk). Indeed, for D ∈ Dm we have:
U∗(D ⊗ Ik)U =
1≤i′,j′≤k
āi′,j′B
j′ ⊗ fj′,i′)(D ⊗ Ik)(
1≤i,j≤k
ai,jBj ⊗ fi,j)
1≤i,j,j′≤k
āi,j′ai,jB
j′DBj ⊗ fj′,j
1≤j,j′≤k
āi,j′ai,j)B
j′DBj ⊗ fj′,j
1≤j,j′≤k
j′DBj ⊗ fj′,j
1≤j≤k
B∗jDBj ⊗ fj,j ∈ (Mm(C)⊗Dk)
The lower square of inclusions is clearly a commuting square, since C(H) is
a commuting square. We check that
Dm ⊗Dk ⊂ Mm(C)⊗Mk(C)
Dm ⊗ Ik ⊂ U(Mm(C)⊗Dk)U∗
is a commuting square. For X ∈ Mm(C) and D ∈ Dk we have:
U(X ⊗D)U∗ = 1
1≤i,j≤k
ai,jBj ⊗ fi,j)(X ⊗D)(
1≤i′,j′≤k
āi′,j′B
j′ ⊗ fj′,i′)
1≤i,i′,j≤k
āi′,jai,jBjXB
j ⊗Dj,jfi,i′
12 WES CAMP AND REMUS NICOARA
Hence:
EDn(U(X ⊗D)U∗) = EDn(
1≤i,i′,j≤k
āi′,jai,jBjXB
j ⊗Dj,jfi,i′)
1≤i,i′,j≤k
EDm(āi′,jai,jBjXB
j )⊗Dj,jδi
i fi,i
1≤i,j≤k
Dj,jEDm(BjXB
j )⊗ fi,i
1≤j≤k
Dj,jEDm(BjXB
j )⊗ Ik ∈ Dm ⊗ Ik
The lower commuting square is symmetric, since the product of the dimensions
of its upper left and lower right corners equals the dimension of its upper right
corner. This also implies that the upper commuting square is symmetric, since
C(H) is symmetric.
(b). The proof is similar to the proof of part (a). �
Corollary 3.1. The subfactors associated to Dita matrices have intermediate
subfactors.
Proof. By iterating the basic construction for the decomposition of Ct(H) in
commuting squares, we obtain the towers of algebras:
U(Dm ⊗Dk)U∗ ⊂ Mm(C)⊗Mk(C)
e3⊂ X1
e4⊂ X2
e5⊂ ...
∪ ∪ ∪ ∪
U(Im ⊗Dk)U∗ ⊂ Dm ⊗Mk(C)
e3⊂ R1
e4⊂ R2
e5⊂ ...
∪ ∪ ∪ ∪
C ⊂ Dm ⊗Dk
e3⊂ Y1
e4⊂ Y2
e5⊂ ...
where Ri =< Ri−1, ei+2 >⊂ Xi. Let RH be the weak closure of ∪iRi. We
have NH ⊂ RH ⊂ MH and RH is a II1 factor since the subfactor NH ⊂ MH is
irreducible.
Remark 3.1. It is immediate to check that the projection f ∈ Mn(C)⊗Mn(C)
from Proposition 3.1 implements the conditional expectation from Mn(C)⊗In =
Mn(C) onto Dm ⊗ Mk(C). It follows that f is the Bisch projection for the
intermediate subfactor NH ⊂ RH ⊂ MH .
SUBFACTORS AND HADAMARD MATRICES 13
4. Matrices of small order
In this section we compute the second relative commutants of the subfactors
associated to Hadamard matrices of small dimensions. For some of the matrices
considered we also specify the dimension of the third relative commutant. Most
computations included were done with the help of computers, using GAP and
Mathematica.
It is well known in subfactor theory that the dimension of the second relative
commutant D′∩Q1 is at most n, with equality if and only if H is equivalent to
a tensor product of Fourier matrices. In this case the subfactor NH ⊂ MH is
well understood, being a cross-product subfactor. For this reason, we exclude
from our analysis tensor products of Fourier matrices.
Some of the matrices we present are parameterized and they yield contin-
uous families of complex Hadamard matrices. In such cases, the strategy for
computing the second relative commutant will be to determine which entries
of the profile matrix P1 depend on the parameters, and for what values of the
parameters are these entries 0. According to Proposition 2.4, the second rela-
tive commutant will not change as long as the 0 entries of P1 do not change.
Thus, to compute the second relative commutant for any other value of the
parameters, it is enough to compute it for some random value.
We will describe the second relative commutant by specifying its minimal
projections. Each such projection p corresponds to a subset S ⊂ {1, 2, ..., n2}:
p is the n2 × n2 diagonal matrix having 1 on diagonal positions i ∈ S and 0
on all other positions. Since the Jones projection e3 is always in the second
relative commutant, one of the subsets of our partitions will always be {1, n+
2, 2n+ 3, ..., kn+ k + 1, ..., n2}.
Complex Hadamard matrices of dimension 4. There exists, up to
equivalence, only one family of complex Hadamard matrices of dimension 4:
F4(a) =
1 1 1 1
1 a −1 −a
1 −1 1 −1
1 −a −1 a
, |a| = 1
The entries of P1 that depend on the parameter a are
. Thus, the second relative commutant is the same for all values of a
that are not roots of these equations.
The roots a = 1, a = −1 yield matrices that are tensor products of 2 ×
2 Fourier matrices. Thus the dimension of the second relative commutant
14 WES CAMP AND REMUS NICOARA
is 4, and its minimal projections are given by the partition {1, 6, 11, 16},
{2, 5, 12, 15}, {3, 8, 9, 14}, {4, 7, 10, 13}.
The roots a = i, a = −i yield the 4 × 4 Fourier matrix, thus the minimal
projections are {1, 6, 11, 16}, {2, 7, 12, 13}, {3, 8, 9, 14}, {4, 5, 10, 15}.
Any other values of a, |a| = 1, yield relative commutants of dimension 3:
{1, 6, 11, 16}, {2, 4, 5, 7, 10, 12, 13, 15}, {3, 8, 9, 14}. This is not surprising, since
this matrix is of Dita type (see Proposition 3.1).
The dimension of the third relative commutant is 10, and the dimension of
the fourth relative commutant is 35 unless a is a primitive root of order 8 of
unity, in which case the dimension is 36. Based on this evidence, we conjecture
that the principal graph of the subfactor associated to F4(a) is D
2k if a is a
primitive root of order 2k of unity, and D
∞ otherwise.
Complex Hadamard matrices of dimension 6. The Fourier matrix F6
is part of an affine 2-parameter family of Dita matrices:
F6(a, b) =
1 1 1 1 1 1
1 a e
π b e
π −1 a
π 1 e
1 −a b −1 a −b
π 1 e
−1 a e 2 i3 π b e i3 π
The entries of P1 that depend on a, b are: 2 (1 + a
−2 + b−2), 2 +
2 (−1)
2 (−1)
, 2 − 2 (−1)
2 (−1)
, 2 (1 + a2 + b2), 2 + 2 (−1)
3 a2 − 2 (−1)
3 b2, 2 −
2 (−1)
3 a2 + 2 (−1)
3 b2.
Making one of these entries 0 yields the following possibilities: a = −1
3, b = −1
3 or a = −1
3, b = −1
3 or a = 1
3, b =
3 or a = 1
3, b = 1
3 or a = −1
3, b = 1
or a = −1
3, b = 1
3 or a = 1
3, b = −1
3 or a =
3, b = −1
3 or a = −1, b = −1 or a = 1, b = 1 or a = −1, b = 1
or a = 1, b = −1.
In each of these cases the matrix F6(a, b) is a tensor product of Fourier
matrices.
For all other pairs (a, b) satisfying |a| = |b| = 1, the second relative commu-
tant has dimension 4: {1, 8, 15, 22, 29, 36}, {2, 4, 6, 7, 9, 11, 14, 16, 18, 19, 21, 23, 26,
28, 30, 31, 33, 35}, {3, 10, 17, 24, 25, 32}, {5, 12, 13, 20, 27, 34}.
SUBFACTORS AND HADAMARD MATRICES 15
The following family of self-adjoint, non-affine, complex Hadamard matrices
was obtained in [BeN], one of the motivations being the search for Hadamard
matrices of small dimensions that might yield subfactors with no extra struc-
ture in their relative commutants, besides the Jones projections.
BN6(θ) =
1 1 1 1 1 1
1 −1 x̄ −y −x̄ y
1 x −1 t −t −x
1 −ȳ t̄ −1 ȳ −t̄
1 −x −t̄ y 1 z̄
1 ȳ −x̄ −t z 1
where θ ∈ [−π,−arcos(−1+
)] ∪ [arcos(−1+
), π] and the variables x, y, z, t
are given by:
y = exp(iθ), z =
1 + 2y − y2
y(−1 + 2y + y2)
1 + 2y + y2 −
1 + 2y + 2y3 + y4
1 + 2y − y2
1 + 2y + y2 −
1 + 2y + 2y3 + y4
−1 + 2y + y2
The entries of BN6 do not depend linearly on the parameters, thus this is
not a Dita-type family. The corresponding subfactors have the second relative
commutant generated by the Jones projection. We conjecture thatBN6(θ) give
supertransitive subfactors, i.e. all the relative commutants of higher orders are
generated by the Jones projections.
There are other interesting complex Hadamard matrices of order 6, such
as the one found by Tao in connection to Fuglede’s conjecture ([T]), or the
Haagerup matrix ([H],TZ). We computed the second relative commutant for
these matrices, and it is generated by the Jones projection.
Complex Hadamard matrices of dimension 7. The following one-
parameter family was found in [Pe], providing a counterexample to a conjecture
of Popa regarding the finiteness of the number of complex Hadamard matrices
of prime dimension.
16 WES CAMP AND REMUS NICOARA
P7(a) =
1 1 1 1 1 1 1
1 a e
π −1 −1 e i3 π
π −1 e−i3 π −1 e i3 π
π −1 e
1 −1 e−i3 π 1
1 −1 −1 e i3 π e i3 π e−2 i3 π e−i3 π
π −1 −1 e−i3 π e−2 i3 π
The second relative commutant of the associated subfactors is generated
by the Jones projection, for all |a| = 1. For a = 1 we also computed the
third relative commutant, and it is just the Temperley-Lieb algebra TL2. We
conjecture that P7(a) yield subfactors with no extra structure in their higher
order relative commutants, besides the Jones projections.
Complex Hadamard matrices of dimension 8. The following 5-parameter
family of Hadamard matrices contains the Fourier matrix and is of Dita type:
F8(a, b, c, d, z) =
1 1 1 1 1 1 1 1
1 a e
π i b c e
π −1 a
−i b c
1 i d −1 −i d 1 i d −1 −i d
π z −i b c e
i b c z
1 −1 1 −1 1 −1 1 −1
i b c
−1 a e i4 π −i b c e 3 i4 π
1 −i d −1 i d 1 −i d −1 i d
−i b c z
−1 e 3 i4 π z i b c e
The list of possible values of a, b, c, d, z that yield 0 entries for P1 is very
long and we do not include it here. Outside these values, the second rela-
tive commutant has dimension 4 and it is given by {1, 10, 19, 28, 37, 46, 55, 64},
{2, 4, 6, 8, 9, 11, 13, 15, 18, 20, 22, 24, 25, 27, 29, 31, 34, 36, 38, 40, 41, 43, 45, 47, 50,
52, 54, 56, 57, 59, 61, 63}, {3, 7, 12, 16, 17, 21, 26, 30, 35, 39, 44, 48, 49, 53, 58, 62},
{5, 14, 23, 32, 33, 42, 51, 60}.
We analysed several other complex Hadamard matrices besides those in-
cluded in this paper, such as those found by [MRS],[Sz]. We tried to cover
SUBFACTORS AND HADAMARD MATRICES 17
most known examples of complex Hadamard matrices of dimensions 2, 3, ..., 11.
We draw some conclusions:
(1) Matrices of Dita type yield subfactors with intermediate subfactors, and
thus the second relative commutant has some extra structure besides
the Jones projection. We note that parametric families of Dita type
exist for every n non-prime, and they contain the Fourier matrix Fn.
(2) All non-Dita, non-Fourier matrices we tested have the second relative
commutant generated by the Jones projection. The third relative com-
mutant is also generated by the first two Jones projections for all cases
we could compute. It remains an open problem whether there exist
such complex Hadamard matrices that admit symmetries of higher or-
References
[BHJ] R. Bacher, P. de la Harpe, V. Jones, Carres commutatifs et invariants de structures
combinatoires, Comptes rendus Acad. Sci. Math., 1995, vol. 320, no9, 1049-1054
[Bi] D. Bisch, A note on intermediate subfactors, Pacific J. Math. 163 (1994), no. 2, 201-216.
[BaN] T. Banica and R. Nicoara, Quantum groups and Hadamard matrices, preprint,
math.OA/0610529
[BeN] K. Beauchamp and R. Nicoara, Orthogonal maximal abelian ∗-subalgebras of the
6× 6 matrices, preprint, math.OA/0609076.
[Di] P. Dita, Some results on the parametrization of complex Hadamard matrices, J. Phys.
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[GHJ] F. M. Goodman, P.de la Harpe and V.F.R.Jones,Coxeter Graphs and Towers of
Algebras, Math. Sciences Res. Inst. Publ. Springer Verlag 1989
[H] U. Haagerup, Orthogonal maximal abelian *-subalgebras of the n × n matrices and
cyclic n-roots, Operator Algebras and Quantum Field Theory (ed. S.Doplicher et al.),
International Press (1997),296-322
[HJ] P. de la Harpe and V. F. R. Jones, Paires de sous-algebres semi-simples et graphes
fortement reguliers, C.A. Acad. Sci. Paris 311, serie I (1990), 147-150
[Jo1] V. F. R. Jones, Index for subfactors, Invent. Math 72 (1983), 1–25.
[Jo2] V. F. R. Jones, Planar Algebras I, math.QA/9909027
[JS] V. F. R. Jones and V. S. Sunder, Introduction to subfactors, Cambridge University
press, (1997).
[MRS] M. Matolcsi, J. Reffy, F. Szollosi, Constructions of complex Hadamard matrices via
tiling abelian groups, preprint quant-ph/0607073.
[Ni] R. Nicoara, A finiteness result for commuting squares of matrix algebras, J. Operator
Theory 55 (2006), no. 2, 295–310.
[Pe] M. Petrescu, Existence of continuous families of complex Hadamard matrices of certain
prime dimensions and related results, PhD thesis, University of California Los Angeles,
1997.
[Po1] S. Popa, Classification of subfactors : the reduction to commuting squares, Invent.
Math., 101(1990),19-43
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http://arxiv.org/abs/math/0609076
http://arxiv.org/abs/math/9909027
http://arxiv.org/abs/quant-ph/0607073
18 WES CAMP AND REMUS NICOARA
[Po2] S. Popa, Orthogonal pairs of *-subalgebras in finite von Neumann algebras, J. Operator
Theory 9, 253-268 (1983)
[Sz] F. Szöllősi , Parametrizing Complex Hadamard Matrices, preprint, math.CO/0610297
[TZ] W. Tadej and K. Zyczkowski, A concise guide to complex Hadamard matrices Open
Systems & Infor. Dyn.,13(2006), 133-177.
[T] T. Tao, Fuglede’s conjecture is false in 5 and higher dimensions, Math. Res. Letters 11
(2004), 251
W.C.: Department of Mathematics, Vanderbilt University, 1326 Stevenson
Center, Nashville, TN 37240, USA
E-mail address : wes.camp@vanderbilt.edu
R.N.: Department of Mathematics, Vanderbilt University, 1326 Stevenson
Center, Nashville, TN 37240, USA
E-mail address : remus.nicoara@vanderbilt.edu
http://arxiv.org/abs/math/0610297
	1. Introduction
	2. Subfactors associated to Hadamard matrices
	3. Matrices of Dita type
	4. Matrices of small order
	References

  To any complex Hadamard matrix H one associates a spin model commuting
square, and therefore a hyperfinite subfactor. The standard invariant of this
subfactor captures certain "group-like" symmetries of H. To gain some insight,
we compute the first few relative commutants of such subfactors for Hadamard
matrices of small dimensions. Also, we show that subfactors arising from Dita
type matrices have intermediate subfactors, and thus their standard invariants
have some extra structure besides the Jones projections.

Introduction
A complex Hadamard matrix is a matrix H ∈ Mn(C) having all entries of
absolute value 1 and all rows mutually orthogonal. Equivalently, 1√
H is a
unitary matrix with all entries of the same absolute value. For example, the
Fourier matrix Fn = (ω
ij)1≤i,j≤n, ω = e
2πi/n, is a Hadamard matrix.
In the recent years, complex Hadamard matrices have found applications
in various topics of mathematics and physics, such as quantum information
theory, error correcting codes, cyclic n-roots, spectral sets and Fuglede’s con-
jecture. A general classification of real or complex Hadamard matrices is not
available. A catalogue of most known complex Hadamard matrices can be
found in [TZ]. The complete classification is known for n ≤ 5 ([H]) and for
self-adjoint matrices of order 6 ([BeN]).
The connection between Hadamard matrices and von Neumann algebras
arose from an observation of Popa ([Po2]): a unitary matrix U is of the form
H , H Hadamard matrix, if and only if the algebra of n×n diagonal matrices
Dn is orthogonal onto UDnU∗, with respect to the inner product given by the
trace on Mn(C). Equivalently, the square of inclusions:
C(H) =
Dn ⊂ Mn(C)
C ⊂ UDnU∗
is a commuting square, in the sense of [Po1],[Po2]. Here τ denotes the trace
on Mn(C), normalized such that τ(1) = 1.
http://arxiv.org/abs/0704.1128v1
2 WES CAMP AND REMUS NICOARA
Such commuting squares are called spin models, the name coming from sta-
tistical mechanical considerations (see [JS]). By iterating Jones’ basic con-
struction, one can construct a hyperfinite, index n subfactor from H (see
for instance [JS]). The subfactor associated to H can be used to capture
some of the symmetries of H , and thus to classify H to a certain extent (see
[BHJ],[Jo2],[BaN]).
Let N ⊂ M be an inclusion of II1 factors of finite index, and let N ⊂ M
e2⊂ M2 ⊂ ... be the tower of factors constructed by iterating Jones’ basic
construction (see [Jo1]), where e1, e2, ... denote the Jones projections. The
standard invariant GN,M is then defined as the trace preserving isomorphism
class of the following sequence of commuting squares of inclusions of finite
dimensional ∗-algebras:
C = N ′ ∩N ⊂ N ′ ∩M ⊂ N ′ ∩M1 ⊂ N ′ ∩M2 ⊂ ...
∪ ∪ ∪
M ′ ∩M ⊂ M ′ ∩M1 ⊂ M ′ ∩M2 ⊂ ...
The Jones projections e1, e2, ..., en are always contained in N
′ ∩Mn. If the
index of the subfactor N ⊂ M is at least 4, they generate the Temperley-Lieb
algebra of order n, denoted TLn. In a lot of situations the relative commutant
N ′ ∩Mn has some interesting extra structure, besides TLn. For instance, the
five non-equivalent real Hadamard matrices of order 16 yield different dimen-
sions for the second relative commutant N ′ ∩ M1, and thus are classified by
these dimensions ([BHJ]).
In this paper we investigate the relation between Hadamard matrices and
their subfactors. We look at Hadamard matrices of small dimensions or of
special types. The paper is organized as follows: in the first section we recall,
in our present framework, several results of [Jo2],[JS] regarding computations
of standard invariants for spin models.
In section 2 we study the subfactors associated to Hadamard matrices of
Dita type. These are matrices that arise from a construction of [Di], which is
a generalization of a construction of Haagerup ([H]). Most known parametric
families of Hadamard matrices are of Dita type. We show that the associated
subfactors have intermediate subfactors.
In the last section we present a list of computations of the second and third
relative commutants N ′ ∩ M1, N ′ ∩ M2, for complex Hadamard matrices of
small dimensions. We make several remarks and conjectures regarding the
structure of the standard invariant. Most of the computations included were
done using computers, with the help of the Mathematica and GAP softwares.
We would like to thank Teodor Banica, Kyle Beauchamp and Dietmar Bisch
for fruitful discussions and correspondence. Wes Camp was supported in part
SUBFACTORS AND HADAMARD MATRICES 3
by NSF under Grant No. DMS 0353640 (REU Grant), and Remus Nicoara
was supported in part by NSF under Grant No. DMS 0500933.
2. Subfactors associated to Hadamard matrices
Let H be a complex n × n Hadamard matrix and let U = 1√
H . U is a
unitary matrix, with all entries of the same absolute value. One associates to
U the square of inclusions:
C(H) =
Dn ⊂ Mn(C)
C ⊂ UDnU∗
where Dn is the algebra of diagonal n×n matrices and τ is the trace onMn(C),
normalized such that τ(1) = 1.
Since H is a Hadamard matrix, C(H) is a commuting square in the sense of
[Po1],[Po2], i.e. EDnEUDnU∗ = EC. The notation EA refers to the τ -invariant
conditional expectation from Mn(C) onto the ∗-subalgebra A.
Recall that two complex Hadamard matrices are said to be equivalent if
there exist unitary diagonal matrices D1, D2 and permutation matrices P1, P2
such that H2 = P1D1H1D2P2. It is easy to see that H1, H2 are equivalent if
and only if C(H1),C(H2) are isomorphic as commuting squares, i.e. conjugate
by a unitary from Mn(C).
We denote by Ct(H) the commuting square obtained by flipping the upper
left and lower right corners of C(H):
t(H) =
UDnU∗ ⊂ Mn(C)
C ⊂ Dn
We have: Ct(H) = Ad(U)C(H∗). Thus, Ct(H) and C(H) are isomorphic as
commuting squares if and only if H,H∗ are equivalent as Hadamard matrices.
We now recall the construction of a subfactor from a commuting square. By
iterating Jones’ basic construction ([Jo1]), one obtains from Ct(H) a tower of
commuting squares of finite dimensional ∗-algebras:
UDnU∗ ⊂ Mn(C)
⊂ ...
∪ ∪ ∪ ∪
C ⊂ Dn
⊂ ...
together with the extension of the trace, which we will still denote by τ , and
Jones projections gi+2 ∈ Yi, i = 1, 2, ....
4 WES CAMP AND REMUS NICOARA
Let MH be the weak closure of ∪iXi, with respect to the trace τ , and let
NH be the weak closure of ∪iYi. NH ,MH are hyperfinite II1 factors, and the
trace τ extends continuously to the trace of MH , which we will still denote by
τ . It is well known that NH ⊂ MH is a subfactor of index n, which we will
call the subfactor associated to the Hadamard matrix H .
The standard invariant of NH ⊂ MH can be expressed in terms of commu-
tants of finite dimensional algebras, by using Ocneanu’s compactness argument
(5.7 in [JS]). Consider the basic construction for the commuting square C(H):
Dn ⊂ Mn(C)
e3⊂ P1
e4⊂ P2
e5⊂ ...
∪ ∪ ∪ ∪
C ⊂ UDnU∗
e3⊂ Q1
e4⊂ Q2
e5⊂ ...
Ocneanu’s compactness theorem asserts that the first row of the standard
invariant of NH ⊂ MH is the row of inclusions:
D′n ∩ UDnU∗ ⊂ D′n ∩ Q1 ⊂ D′n ∩ Q2 ⊂ D′n ∩ Q3 ⊂ ...
More precisely, if
NH ⊂ MH
e3⊂ MH,1
e4⊂ MH,2
e5⊂ ...
is the Jones tower obtained from iterating the basic construction for the inclu-
sion NH ⊂ MH , then:
D′n ∩ Qi = N ′H ∩MH,i, for all i ≥ 1.
Thus, the problem of computing the standard invariant of the subfactor
associated to H is the same as the computation of D′n ∩ Qi. However, such
computations seem very hard, and even for small i and for matrices H of
small dimensions they seem to require computer use. Jones ([Jo2]) provided a
diagrammatic description of the relative commutants D′n ∩ Qi (see also [JS]),
which we express below in the framework of this paper.
Let P0 = Mn(C) and let (ei,j)1≤i,j≤n be its canonical matrix units. Let
i,j=1
ei,j .
It is easy to check that e2 is a projection. Moreover: < Dn, e2 >= Mn(C)
and e2xe2 = EDn(x)e2 for all x ∈ Mn(C). Thus, e2 is realizing the basic
construction
C ⊂ Dn
e2⊂ Mn(C)
Let ek,l ⊗ ei,j denote the n2 × n2 matrix having only one non-zero entry,
equal to 1, at the intersection of row (i − 1)n + k and column (j − 1)n + l.
Thus, ek,l ⊗ ei,j are matrix units of Mn(C) ⊗ Mn(C). In what follows, we
SUBFACTORS AND HADAMARD MATRICES 5
will assume that the embedding of Mn(C) into Mn(C)⊗Mn(C) is realized as
ek,l → ek,l ⊗ In, where ek,l ⊗ In =
i=1 ek,l ⊗ ei,i.
Lemma 2.1. Let P1 = Mn(C)⊗Dn, P2 = Mn(C)⊗Mn(C), e3 =
i=1 ei⊗ei ∈
P1 and e4 = In ⊗ e2 ∈ P2. Then
Dn ⊂ Mn(C)
e3⊂ P1
is a basic construction with Jones projection e3 and
Mn(C) ⊂ P1
e4⊂ P2
is a basic construction with Jones projection e4.
Proof. To show that Dn ⊂ Mn(C)
e3⊂ P1 is a basic construction it is enough to
check that < Mn(C), e3 >= P1 and e3 is implementing EP1Mn(C). First part is
clear, since ek,ie3ei,l = ek,l ⊗ ei,i are a basis for P1 = Mn(C) ⊗ Dn. To check
that e3 implements the conditional expectation, let X = (xi,j) ∈ Mn(C). We
have:
e3(X ⊗ In)e3 =
i,j=1
(Di ⊗Di)(X ⊗ In)(Dj ⊗Dj)
DiXDi ⊗Di
(DiXDi ⊗ In)e3
= EDn⊗In(X)e3
Since C ⊂ Dn
e2⊂ Mn(C) is a basic construction, after tensoring to the left
by Mn(C) it follows that Mn(C) ⊂ P1
e4⊂ P2 is a basic construction, with
e4 = In ⊗ e2.
Proposition 2.1. The algebras P1,P2,P3, ... constructed in (2) are given by
P2k = ⊗k+1i=1Mn(C), P2k+1 = P2k ⊗Dn
with the Jones projections
e2k+2 = ⊗ki=1In ⊗ e2, e2k+3 = ⊗ki=1In ⊗ e3
Proof. Follows from the previous lemma, by tensoring successively by Mn(C).
6 WES CAMP AND REMUS NICOARA
Proposition 2.2. Let H be a complex n×n Hadamard matrix, let U = 1√
i,j=1
ūi,jej,j ⊗ ei,i, U1 = UDU .
Then the algebras Q1,Q2,Q3, ... constructed in (2) are given by
Qk = UkPk−1U∗k , k ≥ 1
where Uk ∈ Pk are the unitary elements:
U2k+1 = Π
i=0(⊗iIn ⊗ U1 ⊗k−i In), U2k = U2k−1(⊗kIn ⊗ U), k ≥ 1.
Proof. The unitary U1 satisfies:
(AdU1)(Dn) = (AdU)(Dn)
since U∗U1 = DU ∈ Dn. Moreover, we have:
(AdU1)(e2) = (AdU)Ad(
i,j=1
ūi,jej,j ⊗ ei,i)(
k,l=1
ek,l)
= (AdU)(
i,k,l=1
ūi,kui,lek,l ⊗ ei,i)
= (AdU)(AdU∗(e3))
It follows that AdU1 takes the basic construction C ⊂ Dn
e2⊂ Mn(C) onto the
inclusion C ⊂ UDnU∗
e3⊂ U1Mn(C)U∗1 . Thus this is also a basic construction,
which shows thatQ1 = U1Mn(C)U∗1 . Moreover, it follows that each AdUi takes
the basic construction Pi−1 ⊂ Pi ⊂ Pi+1 onto Qi ⊂ Qi+1 ⊂ Qi+2, which ends
the proof.
The first relative commutant D′n∩UDnU∗ is equal to C, since the commuting
square condition implies Dn ∩ UDnU∗ = C. Thus the subfactor NH ⊂ MH
is irreducible. In the following proposition we describe the higher relative
commutants of the subfactor NH ⊂ MH as the commutants of some matrices
Pi, i ≥ 1.
Proposition 2.3. With the previous notations, let Pi denote the projection
Uiei+3U
i ∈ Pi+1, i ≥ 1. Then we have the following formula for the (i+ 1)-th
relative commutant:
D′n ∩ Qi = P ′i ∩ D′n ∩ Pi.
SUBFACTORS AND HADAMARD MATRICES 7
Proof. We have:
D′n ∩Qi = D′n ∩AdUi(Pi−1)
= D′n ∩AdUi(e′i+3 ∩ Pi)
= D′n ∩ P ′i ∩ AdUi(Pi)
= D′n ∩ P ′i ∩ Pi
We used the fact that Pi−1 ⊂ Pi
⊂ Pi+1 is a basic construction, and thus
e′i+3 ∩ Pi = Pi−1. �
Remark 2.1. The n2 × n2 matrix P1 = U1e4U∗1 can be written as
a,b,c,d=1
a,bea,b ⊗ ec,d, where p
a,b =
ua,iūb,iūc,iud,i.
This matrix is used in the theory of Hadamard matrices and it is called the
profile of H. It is a result of Jones ([Jo2]) that the matrices P2i+1, i ≥ 1,
depend only on P1. Indeed, one can check that
P2i+1 =
k1,l1,...,ki,li=1
k1,l1
a,b p
k2,l2
k1,l1
ki,li
ea,b ⊗ ek1,l1 ⊗ ek2,l2 ⊗ ...⊗ eki,li ⊗ ec,d.
Thus, all higher relative commutants of even orders are determined by P1.
Let ΓH denote the graph of vertices {1, 2, ..., n} × {1, 2, ..., n}, in which the
distinct vertices (a, c) and (b, d) are connected if and only if p
a,b 6= 0. The
second relative commutant can be easily described in terms of ΓH . We recall
this in the following Proposition, which is a reformulation of a result in [Jo2]
(see also [JS]).
Proposition 2.4. The second relative commutant of the subfactor NH ⊂ MH
is abelian, its minimal projections are in bijection with the connected compo-
nents of ΓH , and their traces are proportional to the sizes of the connected
components.
Proof. Let
i,j=1 λ
iei,i⊗ej,j, λ
i ∈ {0, 1}, be a projection in the second relative
commutant P ′1 ∩ (Dn ⊗Dn). We have:
a,b,c,d=1
a,bea,b⊗ec,d)(
i,j=1
iei,i⊗ej,j) = (
i,j=1
iei,i⊗ej,j)(
a,b,c,d=1
a,bea,b⊗ec,d)
Equivalently:
a,c,i,j=1
q,iea,i ⊗ ec,j =
b,d,i,j=1
i,b ei,b ⊗ ej,d.
8 WES CAMP AND REMUS NICOARA
By relabeling and identifying the set of indices, it follows:
(λca − λ
a,i = 0.
Thus, if the vertices (a, c) and (i, j) are connected then λca = λ
i . This ends
the proof. �
3. Matrices of Dita type
In this section we investigate the standard invariant of subfactors associ-
ated to a particular class of Hadamard matrices, obtained by a construction of
P.Dita ([Di]), which is a generalization of an idea of U.Haagerup ([H]). These
matrices have a lot of symmetries, and we show that for such matrices the sec-
ond relative commutant has some extra structure besides the Jones projection.
Let n be non-prime, n = kl with k, l ≥ 2. Let A = (ai,j) ∈ Mk(C) and
B1, ..., Bk ∈ Mm(C) be complex Hadamard matrices. It is possible to construct
an n × n Hadamard matrix from A,B1, ..., Bk by using an idea of [Di] (see
also[H],[Pe]). This construction is a generalization of the tensor product of
two Hadamard matrices:
(6) H =
a1,1B1 a1,2B2 ... a1,kBk
a2,1B1 a2,2B2 ... a2,kBk
ak,1B1 ak,2B2 ... ak,kBk
Let (fi,j)1≤i,j≤k be the matrix units of Mk(C). We identify Mn(C) with the
tensor product Mm(C)⊗Mk(C), with the same conventions as before. Thus:
i,j=1
ai,jBj ⊗ fi,j
One can use construct multi-parametric families of non-equivalent Hadamard
matrices, by replacing B1, ..., Bk by B1D1, ...BkDk, where D1, ..., Dk are diag-
onal unitaries. Some of the families of Hadamard matrices of small orders
considered in the next section arise from this construction.
Recall that the second relative commutant always contains the Jones projec-
tion e3 =
eii ⊗ eii. In the next proposition we show that the second relative
commutant of a Dita type subfactor contains another projection f ≥ e3, so it
has dimension at least 3.
Proposition 3.1. Let H = (ai,jBj)1≤i,j≤k ∈ Mn(C) be a Dita type ma-
trix, where A = (ai,j)1≤i,j≤k ∈ Mk(C) and B1, ..., Bk ∈ Mm(C) are complex
SUBFACTORS AND HADAMARD MATRICES 9
Hadamard matrices, n = mk. Then the second relative commutant of the
subfactor associated to H contains the projection:
1≤i,j≤n, i≡j(mod m)
ei,i ⊗ ej,j ∈ Mn2(C).
Proof. For 1 ≤ i ≤ n, let i0 = (i− 1)(mod m) + 1 and i1 = i−i0m + 1. We will
use similar notations for 1 ≤ j ≤ n. Thus, the (i, j) entry of H is:
hi,j = ai1,j1b
i0,j0
where btr,s is the (r, s) entry of Bt, for all 1 ≤ t ≤ k, 1 ≤ r, s ≤ m.
With these notations, the projection f can be written as
i,j=1
iei,i ⊗ ej,j
where λ
i = 1 if i0 = j0 and λ
i = 0 for all other i, j.
According to Proposition 2.4, showing that f is in the second relative com-
mutant is equivalent to showing that p
i,c = 0 whenever c0 6= d0. Using the
formula for the entries of P1 and the fact that i0 = j0 we obtain:
i,c =
ui,xūc,xūj,xud,x
hi,xh̄c,xh̄j,xhd,x
ai1,x1b
i0,x0
āc1,x1 b̄
c0,x0
āj1,x1 b̄
j0,x0
ad1,x1b
d0,x0
ai1,x1āc1,x1 b̄
c0,x0
āj1,x1ad1,x1b
d0,x0
(ai1,x1āc1,x1āj1,x1ad1,x1(
b̄x1c0,x0b
d0,x0
ai1,x1āc1,x1āj1,x1ad1,x1δ
whenever c0 6= d0. �
10 WES CAMP AND REMUS NICOARA
We show that in fact the subfactor NH ⊂ MH associated to the Dita matrix
H has an intermediate subfactor NH ⊂ RH ⊂ MH , and the projection f is the
Bisch projection (in the sense of [Bi]) corresponding to RH .
Proposition 3.2. Let H =
1≤i,j≤k ai,jBj ⊗ fi,j ∈ Mn(C) be a Dita type
matrix, where A = (ai,j)1≤i,j≤k ∈ Mk(C) and B1, ..., Bk ∈ Mm(C) are complex
Hadamard matrices, n = mk. Then:
(a). The commuting square C(H) can be decomposed into two adjacent sym-
metric commuting squares:
Dm ⊗Dk ⊂ Mm(C)⊗Mk(C)
Dm ⊗ Ik ⊂ U(Mm(C)⊗Dk)U∗
C ⊂ U(Dm ⊗Dk)U∗
(b). The commuting square Ct(H) can be decomposed into two adjacent
symmetric commuting squares:
U(Dm ⊗Dk)U∗ ⊂ Mm(C)⊗Mk(C)
U(Im ⊗Dk)U∗ ⊂ Dm ⊗Mk(C)
C ⊂ Dm ⊗Dk
SUBFACTORS AND HADAMARD MATRICES 11
Proof. (a). We first show that Dm ⊗ Ik ⊂ U(Mm(C) ⊗ Dk)U∗. Equivalently,
we check that U∗(Dm ⊗ Ik)U ⊂ (Mm(C)⊗Dk). Indeed, for D ∈ Dm we have:
U∗(D ⊗ Ik)U =
1≤i′,j′≤k
āi′,j′B
j′ ⊗ fj′,i′)(D ⊗ Ik)(
1≤i,j≤k
ai,jBj ⊗ fi,j)
1≤i,j,j′≤k
āi,j′ai,jB
j′DBj ⊗ fj′,j
1≤j,j′≤k
āi,j′ai,j)B
j′DBj ⊗ fj′,j
1≤j,j′≤k
j′DBj ⊗ fj′,j
1≤j≤k
B∗jDBj ⊗ fj,j ∈ (Mm(C)⊗Dk)
The lower square of inclusions is clearly a commuting square, since C(H) is
a commuting square. We check that
Dm ⊗Dk ⊂ Mm(C)⊗Mk(C)
Dm ⊗ Ik ⊂ U(Mm(C)⊗Dk)U∗
is a commuting square. For X ∈ Mm(C) and D ∈ Dk we have:
U(X ⊗D)U∗ = 1
1≤i,j≤k
ai,jBj ⊗ fi,j)(X ⊗D)(
1≤i′,j′≤k
āi′,j′B
j′ ⊗ fj′,i′)
1≤i,i′,j≤k
āi′,jai,jBjXB
j ⊗Dj,jfi,i′
12 WES CAMP AND REMUS NICOARA
Hence:
EDn(U(X ⊗D)U∗) = EDn(
1≤i,i′,j≤k
āi′,jai,jBjXB
j ⊗Dj,jfi,i′)
1≤i,i′,j≤k
EDm(āi′,jai,jBjXB
j )⊗Dj,jδi
i fi,i
1≤i,j≤k
Dj,jEDm(BjXB
j )⊗ fi,i
1≤j≤k
Dj,jEDm(BjXB
j )⊗ Ik ∈ Dm ⊗ Ik
The lower commuting square is symmetric, since the product of the dimensions
of its upper left and lower right corners equals the dimension of its upper right
corner. This also implies that the upper commuting square is symmetric, since
C(H) is symmetric.
(b). The proof is similar to the proof of part (a). �
Corollary 3.1. The subfactors associated to Dita matrices have intermediate
subfactors.
Proof. By iterating the basic construction for the decomposition of Ct(H) in
commuting squares, we obtain the towers of algebras:
U(Dm ⊗Dk)U∗ ⊂ Mm(C)⊗Mk(C)
e3⊂ X1
e4⊂ X2
e5⊂ ...
∪ ∪ ∪ ∪
U(Im ⊗Dk)U∗ ⊂ Dm ⊗Mk(C)
e3⊂ R1
e4⊂ R2
e5⊂ ...
∪ ∪ ∪ ∪
C ⊂ Dm ⊗Dk
e3⊂ Y1
e4⊂ Y2
e5⊂ ...
where Ri =< Ri−1, ei+2 >⊂ Xi. Let RH be the weak closure of ∪iRi. We
have NH ⊂ RH ⊂ MH and RH is a II1 factor since the subfactor NH ⊂ MH is
irreducible.
Remark 3.1. It is immediate to check that the projection f ∈ Mn(C)⊗Mn(C)
from Proposition 3.1 implements the conditional expectation from Mn(C)⊗In =
Mn(C) onto Dm ⊗ Mk(C). It follows that f is the Bisch projection for the
intermediate subfactor NH ⊂ RH ⊂ MH .
SUBFACTORS AND HADAMARD MATRICES 13
4. Matrices of small order
In this section we compute the second relative commutants of the subfactors
associated to Hadamard matrices of small dimensions. For some of the matrices
considered we also specify the dimension of the third relative commutant. Most
computations included were done with the help of computers, using GAP and
Mathematica.
It is well known in subfactor theory that the dimension of the second relative
commutant D′∩Q1 is at most n, with equality if and only if H is equivalent to
a tensor product of Fourier matrices. In this case the subfactor NH ⊂ MH is
well understood, being a cross-product subfactor. For this reason, we exclude
from our analysis tensor products of Fourier matrices.
Some of the matrices we present are parameterized and they yield contin-
uous families of complex Hadamard matrices. In such cases, the strategy for
computing the second relative commutant will be to determine which entries
of the profile matrix P1 depend on the parameters, and for what values of the
parameters are these entries 0. According to Proposition 2.4, the second rela-
tive commutant will not change as long as the 0 entries of P1 do not change.
Thus, to compute the second relative commutant for any other value of the
parameters, it is enough to compute it for some random value.
We will describe the second relative commutant by specifying its minimal
projections. Each such projection p corresponds to a subset S ⊂ {1, 2, ..., n2}:
p is the n2 × n2 diagonal matrix having 1 on diagonal positions i ∈ S and 0
on all other positions. Since the Jones projection e3 is always in the second
relative commutant, one of the subsets of our partitions will always be {1, n+
2, 2n+ 3, ..., kn+ k + 1, ..., n2}.
Complex Hadamard matrices of dimension 4. There exists, up to
equivalence, only one family of complex Hadamard matrices of dimension 4:
F4(a) =
1 1 1 1
1 a −1 −a
1 −1 1 −1
1 −a −1 a
, |a| = 1
The entries of P1 that depend on the parameter a are
. Thus, the second relative commutant is the same for all values of a
that are not roots of these equations.
The roots a = 1, a = −1 yield matrices that are tensor products of 2 ×
2 Fourier matrices. Thus the dimension of the second relative commutant
14 WES CAMP AND REMUS NICOARA
is 4, and its minimal projections are given by the partition {1, 6, 11, 16},
{2, 5, 12, 15}, {3, 8, 9, 14}, {4, 7, 10, 13}.
The roots a = i, a = −i yield the 4 × 4 Fourier matrix, thus the minimal
projections are {1, 6, 11, 16}, {2, 7, 12, 13}, {3, 8, 9, 14}, {4, 5, 10, 15}.
Any other values of a, |a| = 1, yield relative commutants of dimension 3:
{1, 6, 11, 16}, {2, 4, 5, 7, 10, 12, 13, 15}, {3, 8, 9, 14}. This is not surprising, since
this matrix is of Dita type (see Proposition 3.1).
The dimension of the third relative commutant is 10, and the dimension of
the fourth relative commutant is 35 unless a is a primitive root of order 8 of
unity, in which case the dimension is 36. Based on this evidence, we conjecture
that the principal graph of the subfactor associated to F4(a) is D
2k if a is a
primitive root of order 2k of unity, and D
∞ otherwise.
Complex Hadamard matrices of dimension 6. The Fourier matrix F6
is part of an affine 2-parameter family of Dita matrices:
F6(a, b) =
1 1 1 1 1 1
1 a e
π b e
π −1 a
π 1 e
1 −a b −1 a −b
π 1 e
−1 a e 2 i3 π b e i3 π
The entries of P1 that depend on a, b are: 2 (1 + a
−2 + b−2), 2 +
2 (−1)
2 (−1)
, 2 − 2 (−1)
2 (−1)
, 2 (1 + a2 + b2), 2 + 2 (−1)
3 a2 − 2 (−1)
3 b2, 2 −
2 (−1)
3 a2 + 2 (−1)
3 b2.
Making one of these entries 0 yields the following possibilities: a = −1
3, b = −1
3 or a = −1
3, b = −1
3 or a = 1
3, b =
3 or a = 1
3, b = 1
3 or a = −1
3, b = 1
or a = −1
3, b = 1
3 or a = 1
3, b = −1
3 or a =
3, b = −1
3 or a = −1, b = −1 or a = 1, b = 1 or a = −1, b = 1
or a = 1, b = −1.
In each of these cases the matrix F6(a, b) is a tensor product of Fourier
matrices.
For all other pairs (a, b) satisfying |a| = |b| = 1, the second relative commu-
tant has dimension 4: {1, 8, 15, 22, 29, 36}, {2, 4, 6, 7, 9, 11, 14, 16, 18, 19, 21, 23, 26,
28, 30, 31, 33, 35}, {3, 10, 17, 24, 25, 32}, {5, 12, 13, 20, 27, 34}.
SUBFACTORS AND HADAMARD MATRICES 15
The following family of self-adjoint, non-affine, complex Hadamard matrices
was obtained in [BeN], one of the motivations being the search for Hadamard
matrices of small dimensions that might yield subfactors with no extra struc-
ture in their relative commutants, besides the Jones projections.
BN6(θ) =
1 1 1 1 1 1
1 −1 x̄ −y −x̄ y
1 x −1 t −t −x
1 −ȳ t̄ −1 ȳ −t̄
1 −x −t̄ y 1 z̄
1 ȳ −x̄ −t z 1
where θ ∈ [−π,−arcos(−1+
)] ∪ [arcos(−1+
), π] and the variables x, y, z, t
are given by:
y = exp(iθ), z =
1 + 2y − y2
y(−1 + 2y + y2)
1 + 2y + y2 −
1 + 2y + 2y3 + y4
1 + 2y − y2
1 + 2y + y2 −
1 + 2y + 2y3 + y4
−1 + 2y + y2
The entries of BN6 do not depend linearly on the parameters, thus this is
not a Dita-type family. The corresponding subfactors have the second relative
commutant generated by the Jones projection. We conjecture thatBN6(θ) give
supertransitive subfactors, i.e. all the relative commutants of higher orders are
generated by the Jones projections.
There are other interesting complex Hadamard matrices of order 6, such
as the one found by Tao in connection to Fuglede’s conjecture ([T]), or the
Haagerup matrix ([H],TZ). We computed the second relative commutant for
these matrices, and it is generated by the Jones projection.
Complex Hadamard matrices of dimension 7. The following one-
parameter family was found in [Pe], providing a counterexample to a conjecture
of Popa regarding the finiteness of the number of complex Hadamard matrices
of prime dimension.
16 WES CAMP AND REMUS NICOARA
P7(a) =
1 1 1 1 1 1 1
1 a e
π −1 −1 e i3 π
π −1 e−i3 π −1 e i3 π
π −1 e
1 −1 e−i3 π 1
1 −1 −1 e i3 π e i3 π e−2 i3 π e−i3 π
π −1 −1 e−i3 π e−2 i3 π
The second relative commutant of the associated subfactors is generated
by the Jones projection, for all |a| = 1. For a = 1 we also computed the
third relative commutant, and it is just the Temperley-Lieb algebra TL2. We
conjecture that P7(a) yield subfactors with no extra structure in their higher
order relative commutants, besides the Jones projections.
Complex Hadamard matrices of dimension 8. The following 5-parameter
family of Hadamard matrices contains the Fourier matrix and is of Dita type:
F8(a, b, c, d, z) =
1 1 1 1 1 1 1 1
1 a e
π i b c e
π −1 a
−i b c
1 i d −1 −i d 1 i d −1 −i d
π z −i b c e
i b c z
1 −1 1 −1 1 −1 1 −1
i b c
−1 a e i4 π −i b c e 3 i4 π
1 −i d −1 i d 1 −i d −1 i d
−i b c z
−1 e 3 i4 π z i b c e
The list of possible values of a, b, c, d, z that yield 0 entries for P1 is very
long and we do not include it here. Outside these values, the second rela-
tive commutant has dimension 4 and it is given by {1, 10, 19, 28, 37, 46, 55, 64},
{2, 4, 6, 8, 9, 11, 13, 15, 18, 20, 22, 24, 25, 27, 29, 31, 34, 36, 38, 40, 41, 43, 45, 47, 50,
52, 54, 56, 57, 59, 61, 63}, {3, 7, 12, 16, 17, 21, 26, 30, 35, 39, 44, 48, 49, 53, 58, 62},
{5, 14, 23, 32, 33, 42, 51, 60}.
We analysed several other complex Hadamard matrices besides those in-
cluded in this paper, such as those found by [MRS],[Sz]. We tried to cover
SUBFACTORS AND HADAMARD MATRICES 17
most known examples of complex Hadamard matrices of dimensions 2, 3, ..., 11.
We draw some conclusions:
(1) Matrices of Dita type yield subfactors with intermediate subfactors, and
thus the second relative commutant has some extra structure besides
the Jones projection. We note that parametric families of Dita type
exist for every n non-prime, and they contain the Fourier matrix Fn.
(2) All non-Dita, non-Fourier matrices we tested have the second relative
commutant generated by the Jones projection. The third relative com-
mutant is also generated by the first two Jones projections for all cases
we could compute. It remains an open problem whether there exist
such complex Hadamard matrices that admit symmetries of higher or-
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W.C.: Department of Mathematics, Vanderbilt University, 1326 Stevenson
Center, Nashville, TN 37240, USA
E-mail address : wes.camp@vanderbilt.edu
R.N.: Department of Mathematics, Vanderbilt University, 1326 Stevenson
Center, Nashville, TN 37240, USA
E-mail address : remus.nicoara@vanderbilt.edu
http://arxiv.org/abs/math/0610297
	1. Introduction
	2. Subfactors associated to Hadamard matrices
	3. Matrices of Dita type
	4. Matrices of small order
	References
