Classification of homogeneous Fourier matrices
Modular data are commonly studied in mathematics and physics. A modular datum defines a finite-dimensional representation of the modular group SL₂(Z). In this paper, we show that there is a one-to-one correspondence between Fourier matrices associated to modular data and self-dual C-algebras that s...
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Singh, G. 2023-02-28T19:14:51Z 2023-02-28T19:14:51Z 2019 Classification of homogeneous Fourier matrices / G. Singh // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 1. — С. 75–84. — Бібліогр.: 7 назв. — англ. 1726-3255 2010 MSC: Primary 05E30; Secondary 05E99, 81R05. https://nasplib.isofts.kiev.ua/handle/123456789/188424 Modular data are commonly studied in mathematics and physics. A modular datum defines a finite-dimensional representation of the modular group SL₂(Z). In this paper, we show that there is a one-to-one correspondence between Fourier matrices associated to modular data and self-dual C-algebras that satisfy a certain condition. We prove that a homogenous C-algebra arising from a Fourier matrix has all the degrees equal to 1. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Classification of homogeneous Fourier matrices Article published earlier |
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Classification of homogeneous Fourier matrices |
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Classification of homogeneous Fourier matrices |
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Classification of homogeneous Fourier matrices |
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Classification of homogeneous Fourier matrices |
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Modular data are commonly studied in mathematics and physics. A modular datum defines a finite-dimensional representation of the modular group SL₂(Z). In this paper, we show that there is a one-to-one correspondence between Fourier matrices associated to modular data and self-dual C-algebras that satisfy a certain condition. We prove that a homogenous C-algebra arising from a Fourier matrix has all the degrees equal to 1.
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Classification of homogeneous Fourier matrices / G. Singh // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 1. — С. 75–84. — Бібліогр.: 7 назв. — англ. |
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“adm-n1” — 2019/3/22 — 12:03 — page 75 — #83
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 27 (2019). Number 1, pp. 75–84
c© Journal “Algebra and Discrete Mathematics”
Classification of homogeneous Fourier matrices
Gurmail Singh
Communicated by V. V. Kirichenko
Abstract. Modular data are commonly studied in mathe-
matics and physics. A modular datum defines a finite-dimensional
representation of the modular group SL2(Z). In this paper, we show
that there is a one-to-one correspondence between Fourier matrices
associated to modular data and self-dual C-algebras that satisfy a
certain condition. We prove that a homogenous C-algebra arising
from a Fourier matrix has all the degrees equal to 1.
1. Introduction
The set of columns of a Fourier matrix with entrywise multiplication
and usual addition generate a fusion algebra over C, see [4]. But a two-
step rescaling on Fourier matrices gives rise to the self-dual C-algebras.
A C-algebra with all equal degrees except the trivial degree is called
a homogeneous C-algebra and we call the associated Fourier matrix a
homogenous Fourier matrix. We use the properties of self-dual C-algebras
to classify homogeneous Fourier matrices, and to establish a one-to-one
correspondence between Fourier matrices and the self-dual C-algebras.
In Section 2, we collect the definitions and introduce a two-step rescal-
ing of Fourier matrices to construct the C-algebras from Fourier matrices.
In Section 3, we establish some properties of C-algebras that arise from
Fourier matrices. Also, we show that there is a one-to-one correspondence
between Fourier matrices and self-dual C-algebras that satisfy a certain
2010 MSC: Primary 05E30; Secondary 05E99, 81R05.
Key words and phrases: modular data, Fourier matrices, fusion rings,
C-algebras.
“adm-n1” — 2019/3/22 — 12:03 — page 76 — #84
76 Homogeneous Fourier matrices
integrality condition. In Section 4, we prove that every homogeneous
C-algebra arising from a Fourier matrix has all the degrees equal to 1, and
such a C-algebra with nonnegative structure constants is a group algebra.
2. C-algebras arising from Fourier matrices
A scaling of the rows of a Fourier matrix gives the basis containing
identity of a Fusion algebra. Here we introduce the two-step rescaling on
Fourier matrices and construct C-algebras from them.
Definition 1. Let A be a finite dimensional and commutative algebra
over C with distinguished basis B = {b0 := 1A, b1, . . . , br−1}, and an
R-linear and C-conjugate linear involution ∗ : A → A. Let δ : A → C be
an algebra homomorphism. Then the triple (A,B, δ) is called a C-algebra
if it satisfies the following properties:
(i) for all bi ∈ B, (bi)
∗ = bi∗ ∈ B;
(ii) for all bi, bj ∈ B, we have bibj =
∑
bk∈B
λijkbk, for some λijk ∈ R;
(iii) for all bi, bj ∈ B, λij0 6= 0 ⇐⇒ j = i∗;
(iv) for all bi ∈ B, λii∗0 = λi∗i0 > 0;
(v) for all bi ∈ B, δ(bi) = δ(bi∗) > 0.
The algebra homomorphism δ is called a degree map, and δ(bi), for
all bi ∈ B, are called the degrees. If δ(bi) = λii∗0, for all bi ∈ B, we
say that B is a standard basis. The order of a C-algebra is denfined as
n := δ(B+) =
∑r−1
i=0 δ(bi). A C-algebra is called symmetric if bi∗ = bi, for
all i. A C-algebra with rational structure constants is called a rational
C-algebra. The readers interested in C-algebras are directed to [1], [3]
and [6].
To keep the generality, in the following definition of Modular data
we assume the structure constants to be integers instead of nonnegative
integers, see [4].
Definition 2. Let r ∈ Z+ and I an r × r identity matrix. A pair (S, T )
of r × r complex matrices is called a modular datum if
(i) S is a unitary and symmetric matrix, that is, SS̄T = 1, S = ST ,
(ii) T is diagonal matrix and of finite multiplicative order,
(iii) Si0 > 0, for 0 6 i 6 r− 1, where S is indexed by {0, 1, 2, . . . , r− 1},
(iv) (ST )3 = S2,
(v) Nijk =
∑
l SliSljS̄lkS
−1
l0 ∈ Z, for all 0 6 i, j, k 6 r − 1.
Definition 3. A matrix S satisfying the axioms (i), (iii) and (v) of
Definition 2 is called a Fourier matrix.
“adm-n1” — 2019/3/22 — 12:03 — page 77 — #85
G. Singh 77
Let S be a Fourier matrix. Let s = [sij ] be the matrix with entries
sij = Sij/Si0, for all i, j, and we call it an s-matrix associated to S (briefly,
s-matrix). Since S is a unitary matrix, ss̄T = diag(d0, d1, . . . , dr−1) is a
diagonal matrix, where di =
∑
j sij s̄ij . The numbers di are called norms
of the s-matrix. The principal norm d0 (= S−2
00 ) is also known as the size
of the modular datum, see [4, Definition 3.8]. The relation sij = Sij/Si0
implies the structure constants Nijk =
∑
l slisljslkd
−1
l , for all i, j, k. Since
the structure constants Nijk generated by the columns of S under entrywise
multiplication are integers, the numbers Sij/Si0 are algebraic integers, see
[4, Section 3]. Therefore, if S has only rational entries then the entries of
s-matrix are rational integers, and such s-matrices are known as integral
Fourier matrices, see [4, Definition 3.1]. Cuntz studied the integral Fourier
matrices, see [4]. In this paper, we consider the broader class of s-matrices
with algebraic integer entries. Note that an s-matrix satisfies all the axioms
of the definition of an integral Fourier matrix except the entries may not
be integers. For example, the first eigenmatrix (character table) of every
group algebra of a group of prime order is an s-matrix but not an integral
Fourier matrix.
There is an interesting row-and-column operation (two-step rescaling)
procedure that can be applied to a Fourier matrix S that results in the
first eigenmatrix, the character table, of a C-algebra, see Theorem 1. The
steps of the procedure are reversed to obtain the Fourier matrix S from
the first eigenmatrix. The explanation of the procedure is as follows. Let
S = [Sij ] be a Fourier matrix indexed with {0, 1, . . . , r−1}. We divide each
row of S with its first entry and obtain the s-matrix. The multiplication
of each column of the s-matrix with its first entry gives the P -matrix
associated to S (briefly, P -matrix), the first eigenmatrix of the C-algebra.
That is, sij = SijS
−1
i0 and pij = sijs0j , for all i, j, where pij denotes the
(i, j)-entry of the P -matrix. Conversely, to obtain the s-matrix from a
P -matrix, divide each column of the P -matrix with the squareroot of its
first entry. Further, the Fourier matrix S is obtained from the s-matrix
by dividing the ith row of s-matrix by
√
di, where di =
∑
j |sij |2. That is,
sij = pij/
√
p0j , and Sij = sij/
√
di, ∀ i, j. Since the entries of an s-matrix
are algebraic integers, the entries of P -matrix are also algebraic integers.
Remark 1. Throughout this paper, unless mentioned explicitly, the set
of the columns of P -matrix and s-matrix are denoted by B = {b0, b1, . . .,
br−1} and B̃ = {b̃0, b̃1, . . . , b̃r−1}, respectively. The structure constants
generated by the columns, with the entrywise multiplication, of P -matrix
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78 Homogeneous Fourier matrices
and s-matrix are denoted by λijk and Nijk, respectively. MT denotes the
transpose of a matrix M .
The columns of a Fourier matrix gives rise to a fusion algebra, see [4].
In the next theorem we show that corresponding to every Fourier matrix
there exists a C-algebra.
Theorem 1. Let S be a Fourier matrix. Then the vector space A := CB
is a C-algebra of order d0, B is the standard basis of A, and P -matrix is
the first eigenmatrix of A.
Proof. The C-conjugate linear involution ∗ on columns of S is given by
the involution on elements of S, defined as (Sij)
∗ = Sij∗ = S̄ij for all
i, j. Therefore, if Sj denotes the jth column of a Fourier matrix S then
the involution on Sj is given by (Sj)
∗ = Sj∗ = [S0j∗ , S1j∗ , . . . , S(r−1),j∗ ]
T .
Since bi = s0ib̃i, the structure constants generated by the basis B are
given by λijk = Nijks0is0js
−1
0k , for all i, j, k. As S is a unitary matrix,
therefore, Nij0 =
∑
l SliSljS̄l0S
−1
l0 6= 0 ⇐⇒ j = i∗ and Nii∗0 = 1 > 0, for
all i, j. Hence, λij0 6= 0 ⇐⇒ j = i∗ and λii∗0 > 0, for all i, j.
Define a C-conjugate linear map δ : A −→ C as δ(
∑
i aib̃i) =
∑
i āis0i.
Thus δ(bi) = δ(s0ib̃i) = s20i, hence δ is positive valued. The map δ is an
algebra homomorphism, to see:
δ(bibj) = s0is0j
∑
k
Nijkδ(b̃k)
= s0is0j [
s0is0j
d0
r−1
∑
k=0
s̄0ks0k] + s0is0j [
r−1
∑
l=1
slislj
dl
r−1
∑
k=0
s̄lks0k]
= s0is0j [
s0is0j
d0
d0] + s0is0j [0]
= s20is
2
0j = δ(bi)δ(bj).
In the third last equality we use the fact that
n
∑
k=1
s0ks̄0k = d0 and the
rows of s-matrix are orthogonal. Since bi∗ = s̄0ib̃i∗ = s0ib̃i∗ , (bibi∗)0 =
s0is0i(b̃ib̃i∗)0 = s20i = δ(bi). Therefore, δ is the positive degree map and B
is the standard basis. The value of a basis element bj under an irreducible
character (linear character) is given by pij , for all i, j.
By Theorem 1, every Fourier matrix gives rise to a C-algebra. The
algebra A in the above theorem is denoted by (A,B, δ), and we say (A,B, δ)
is arising from a Fourier matrix S. A C-algebra arising from a Fourier
matrix S of rank r is a symmetric C-algebra if and only if S ∈ Rr×r.
“adm-n1” — 2019/3/22 — 12:03 — page 79 — #87
G. Singh 79
3. General results on C-algebras arising from Fourier
matrices
Let s be an integral Fourier matrix. Then
√
djsij =
√
disji, for all i, j.
Therefore, d0 = diδ(bi) for all i, that is, the degrees and norms divide d0.
This can be generalized to s-matrices under a certain condition. In the
following proposition we prove that not only the list of the degrees of a
C-algebra arising from a Fourier matrix matches with list of multiplicities
but their indices also match. Also, we prove that a C-algebra arising from
an integral Fourier matrix has perfect square integral degrees and rational
structure constants.
Proposition 1. Let (A,B, δ) be a C-algebra arising from a Fourier ma-
trix S. Let mj be the multiplicity of A corresponding to the irreducible
character χj.
(i) The degrees of A exactly match with the multiplicities of A, that is,
mj = δ(bj), for all j.
(ii) If degrees of A are rational numbers then the degrees and norms are
integers, and both the degrees and norms divide the order of A.
(iii) If the associated s-matrix has integral entries then degrees of A are
perfect square integers, and A is a rational C-algebra.
(iv) If A is a rational C-algebra the degrees of A are integers.
(v) A has unique degree if and only if s-matrix has unique norm.
Proof. (i). The multiplicities, mj = δ(B+)/
r−1
∑
i=0
|χj(bi∗ )|
2
λii∗0
, see [2, Corol-
lary 5.6]. Therefore, mj = d0/
r−1
∑
i=0
|sji|2 = d0/dj , for all j. Since S is a
symmetric matrix, di|sji|2 = dj |sij |2. Thus d0/dj = s20j , for all j. Hence
mj = d0/dj = s20j = δ(bj), for all j.
(ii). The fact that the P -matrix has algebraic entries implies that the
degrees of the algebra A are integers, consequently d0 is an integer. By
the above part (i), dj = d0δ(bj)
−1, for all j. Therefore, dj are rational
numbers. The entries of s-matrix are algebraic integers, thus the norms dj
are rational integers. Hence both the degrees and norms divide the order
of A, because d0 = djδ(bj), for all j.
(iii). The entries of s-matrix are integers and
√
δ(bj) = s0j imply
√
δ(bj) ∈ Z, for all j. By the proof of Theorem 1, λijk = (Nijk
√
δ(bi)
√
δ(bj))/
√
δ(bk), for all i, j, k. Hence λijk ∈ Q, for all i, j, k.
(iv). Since A is a rational algebra, λijk ∈ Q. Therefore, δ(bi)=λii∗0∈Q,
because B is a standard basis. But the entries of P -matrix are algebraic
“adm-n1” — 2019/3/22 — 12:03 — page 80 — #88
80 Homogeneous Fourier matrices
integers and δ(bi) = p0i, for all i. Therefore, δ(bi) are integers. (Though
we remark that rationality of all the structure constants is not required.)
(v). The result follows from the fact that d0 = diδ(bi), for all i.
By Proposition 1 (i), a C-algebra arising from a Fourier matrix S is
a self-dual C-algebra. But every self-dual C-algebra not necessarily arise
from a Fourier matrix, see Example 1. Therefore, in general, the converse
is not true. In the next theorem we prove that the converse is also true
under a certain integrality condition.
Theorem 2. Let (A,B, δ) be a C-algebra with standard basis B =
{b0, b1, . . . , br−1}. Let λijk be the structure constants generated by the basis
B. Then A is self-dual and λijk
√
δ(bk)/
√
δ(bi)δ(bj) ∈ Z if and only if A
arises from a Fourier matrix S.
Proof. Suppose (A,B, δ) is a self-dual C-algebra and λijk
√
δ(bk)/(
√
δ(bi)
√
δ(bj)) ∈ Z, for all i, j, k. Let P be the first eigenmatrix of A and I an
identity matrix. Therefore, without loss of generality, assume PP̄ = d0I
and mj = δ(bj), where mj is the multiplicity of an irreducible character χj .
By [2, Theorem 5.5 (i)], pji/δ(bi) = pij/mj , implies pji/δ(bi) = pij/δ(bj),
for all i, j. Let L = diag(1/
√
δ(b0), 1/
√
δ(b1), . . . , 1/
√
δ(br−1), and s =
PL. Therefore,
ss̄T = diag(d0/δ(b0), d0/δ(b1), . . . , d0/δ(br−1)) = diag(d0, d1, . . . , dr−1),
where d0 = diδ(bi), di =
∑
j |sij |2 and sij = pij/
√
δ(bj), for all i, j. Also,
sij
√
dj = sij
√
di, because pji/δ(bi) = pij/δ(bj), for all i, j.. Hence s
is an s-matrix associated to a Fourier matrix S = [Sij ], where Sij =
sij/
√
di, for all i, j. Since Nijk ∈ Z, the other direction follows from the
Proposition 1(i).
Remark 2. By Pontryagin duality, every group algebra of a finite abelian
group is self-dual. A group algebra of a finite abelian group also satisfies
the integrality condition of the above theorem. On the other hand, a self-
dual C-algebra with a unique degree and nonnegative structure constants
is a group algebra, see Theorem 3. Hence a self-dual C-algebra with
nonnegative structure constants has a unique degree if and only if it is a
group algebra of a finite abelian group.
In the next example, we show that the condition λijk
√
δ(bk)/(
√
δ(bi)
√
δ(bj)) ∈ Z cannot be removed.
“adm-n1” — 2019/3/22 — 12:03 — page 81 — #89
G. Singh 81
Example 1. Since the row sum of a character table is zero, the character
table of a C-algebra of rank 2 with basis B = {b0, bi} is given by P =
[
1 n
1 −1
]
, and the structure constants are given by b21 = nb0 + (n− 1)b1.
(Note that, for n ∈ Z+, P is the first eigenmatrix of an association scheme
of order n+ 1 and rank 2.) Apply the two-step scaling on the matrix P ,
we obtain S =
1√
n+ 1
[
1
√
n√
n −1
]
. But for 1 6= n ∈ R+, the condition
λijk
√
δ(bk)/
√
δ(bi)δ(bj) ∈ Z is not satisfied, because λ111/
√
n 6∈ Z, and
the matrix S is not a unitary matrix.
Cuntz proved that the size of every integral Fourier matrix with odd
rank is a square integer, [4, Lemma 3.7]. In the following lemma we
generalize the Cuntz’s result.
Lemma 1. Let (A,B, δ) be a C-algebra arising from a Fourier matrix S
with rank r. Let r be an odd integer. If determinant of the P -matrix is an
integer then the order of A is a square integer.
Proof. Let det(P ) denote the determinant of P -matrix. A is a self dual
C-algebra, thus PP̄ = nI implies (det(P ))2 = nr, where n is the order of
A. Since r is an odd integer,
√
n is an integer.
4. Homogeneous C-algebras arising from Fourier
matrices
In this section we show that every homogenous C-algebra arising from
a Fourier matrix has unique degree, and such a C-algebra under a certain
condition is a group algebra. In the following lemma we prove that if
a degree of a C-algebra arising from a Fourier matrix divides the all
nontrivial degrees then that degree might be equal to 1.
Lemma 2. Let (A,B, δ) be a C-algebra arising from a Fourier matrix S
such that the degrees δ(bi) ∈ Z, for all i. If for a given j, δ(bj) divides
δ(bi) for all i > 1, then δ(bj) = 1, for all j.
Proof. The degrees δ(bi) are integers, therefore, by Proposition 1(ii), dj are
integers. Since d0 = 1+
r−1
∑
i=1
δ(bi), (1 +
r−1
∑
i=1
δ(bi))δ(bj)
−1 = δ(bj)
−1+α ∈ Z,
where α ∈ Z. Hence δ(bj) = 1, for all j.
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82 Homogeneous Fourier matrices
Note that, the above lemma is true for any self-dual C-algebra with
integral norms and degrees. We can apply this lemma to recognize some C-
algebras not arising from s-matrices just by looking at the degree pattern
(or the first row of the character table) of a C-algebra. For example, the
character tables of the association schemes as5(2), as9(2), as9(3),
as9(8) and as9(9) violate the above result so they are not the character
tables of the adjacency algebras arising from Fourier matrices. For the
character tables of the association schemes see [5].
Definition 4. Let (A,B, δ) be a C-algebra arising from a Fourier matrix S.
Let t ∈ R+ and δ(bi) = t, for all 1 6 i 6 r − 1. Then A is called a
homogenous C-algebra with homogeneity degree t, and the associated
Fourier matrix S is called a homogeneous Fourier matrix.
In the next proposition, we prove that if a C-algebra arising from a
Fourier matrix S is either homogeneous, or of prime order with integral
degrees then each degree of the algebra is equal to 1.
Proposition 2. Let (A,B, δ) be a C-algebra arising from a Fourier ma-
trix S.
(i) If A is a homogenous C-algebra then A has unique degree, that is,
δ(bi) = 1, for all i.
(ii) If the order of A is a prime number and δ(bi) ∈ Z+, for all i, then
δ(bi) = 1, for all i.
Proof. (i). Let t be the homogeneity degree of A. If the rank of A is 2
then the result holds trivially, see Example 1. Suppose t ∈ R+\Q+. Let
bi, bj be two nonidentity elements of B such that bj 6= bi∗ . Therefore, the
support of b̃ib̃j does not contain the identity element, because λij0 = 0
implies Nij0 = 0. Note that the first entry of the column vector b̃ib̃j is t.
To obtain t from the linear combinations of the nonidentity elements of
B̃ the structure constants Nijk must involve
√
t, because the first entry
of each column of s-matrix is
√
t except the first column. Therefore, the
structure constants Nijk cannot be integers, a contradiction. Thus t ∈ Q+.
The entries of the P -matrix are algebraic integers, thus t must be an
integer. Therefore, t divides δ(bi) for all i > 0. Hence by Lemma 2, t = 1,
that is, δ(bi) = 1, for all i.
(ii). By Proposition 1 (ii), δ(bi) divides d0, for all i. But d0 is a prime
number, therefore, δ(bi) = 1, for all i.
Michael Cuntz made a conjecture that is a generalization to his result,
see [4, Lemma 3.11]. The conjecture states that if s is an integral Fourier
“adm-n1” — 2019/3/22 — 12:03 — page 83 — #91
G. Singh 83
matrix with unique norm, then s ∈ {±1}r×r, see [4, Conjecture 3.10].
The conjecture cannot be extended to non-real s-matrices, the trivial
contradiction is the character tables of cyclic groups of prime order. Note
that if all the structure constants are nonnegative then |pij | 6 p0j (or
equivalently, |sij | 6 s0j) for all j, see [7, Proposition 4.1]. The next theorem
completely classify the s-matrices under the conditions of Proposition 2
and |pij | 6 p0j , for all j. It also shows that even for real s-matrices it is
not necessary to pre-assume the uniqueness of the norm to get all entries
of s-matrices equal to ±1.
Theorem 3. Let (A,B, δ) be a C-algebra arising from a Fourier matrix S.
Suppose A is either a homogeneous C-algebra, or A has prime order and
integral degrees. Let |sij | 6 s0j, for all i, j.
(i) The modulus of each entry of the s-matrix is 1, that is, |sij | = 1, for
all i, j.
(ii) The columns of the s-matrix form an abelian group under entrywise
multiplication.
(iii) If s-matrix is a real matrix then the columns of s-matrix form an
elementary abelian group.
Proof. (i). By Theorem 2, δ(bj) = 1, for all j. Therefore, dj = d0, for all j.
Since |sij | 6 1, |sij | = 1, for all i, j.
(ii). Let si be the i-th column of s-matrix and let sisj be the entrywise
multiplication of si and sj . For a column sk of s, each entry of sisjsk has
modulus value 1, and the sum of the entries of sisjsk cannot be −r because
s0l = 1, for all l. But the structure constants Nijk =
1
r
∑
l slisljslk are
integers. Therefore, Nijk are either 0 or r. Since sisj is a nonzero vector,
it cannot be orthogonal to all the columns of s-matrix. Thus sisj = sk for
some column sk of s. Therefore, the columns of s-matrix are closed under
entrywise multiplication. The first column of s-matrix serves as an identity
of the group of columns of s-matrix under entrywise multiplication.
(iii). Let s be a real matrix. Since |sij | = 1, sij = ±1. Thus, by Part (ii),
s is a group. Hence s is a character table of an elementary abelian group
of order r.
If an s-matrix of rank r has unique norm then the Fourier matrix
S = r−1/2s. Thus the above theorem also classifies the homogeneous
Fourier matrices under the same condition.
References
[1] Z. Arad, E. Fisman, and M. Muzychuk, Generalized table algebras, Israel J. Math.,
114, 1999, 29-60.
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84 Homogeneous Fourier matrices
[2] E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Ben-
jamin/Cummings, Menlo Park, CA, 1984.
[3] Harvey I. Blau, Table algebras, European J. Combin., 30(6), 2009, 1426-1455.
[4] Michael Cuntz, Integral modular data and congruences, J Algebr Comb., 29, 2009,
357-387.
[5] A. Hanaki and I. Miyamoto, Classification of Small Association Schemes.
(http://math.shinshu-u.ac.jp/ hanaki/as/)
[6] D. G. Higman, Coherent algebras, Linear Algebra Appl., 93, 1987, 209-239.
[7] Bangteng Xu, Characters of table algebras and applications to association schemes,
Journal of Combinatorial Theory, Series A 115, 2008, 1358-1373.
Contact information
G. Singh Department of Mathematics and Statistics,
University of Regina, Regina, Saskatchewan,
Canada, S4S 0A2
E-Mail(s): Gurmail.Singh@uregina.ca
Received by the editors: 14.04.2017
and in final form 19.02.2018.
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