Diagonal torsion matrices associated with modular data
Modular data are commonly studied in mathematics and physics. A modular datum defines a finite-dimensional representation of the modular group SL₂(Z). Cuntz (2007) defined isomorphic integral modular data. Here we discuss isomorphic integral and non-integral modular data as well as non-isomorphic bu...
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Singh, G. 2023-03-12T18:30:58Z 2023-03-12T18:30:58Z 2021 Diagonal torsion matrices associated with modular data / G. Singh // Algebra and Discrete Mathematics. — 2021. — Vol. 32, № 1. — С. 127–137. — Бібліогр.: 7 назв. — англ. 1726-3255 DOI:10.12958/adm1368 2020 MSC: Primary 05E40; Secondary 05E99, 81R05. https://nasplib.isofts.kiev.ua/handle/123456789/188721 Modular data are commonly studied in mathematics and physics. A modular datum defines a finite-dimensional representation of the modular group SL₂(Z). Cuntz (2007) defined isomorphic integral modular data. Here we discuss isomorphic integral and non-integral modular data as well as non-isomorphic but closely related modular data. In this paper, we give some insights into diagonal torsion matrices associated to modular data. The author would like to thank Professor Allen Herman whose valuable suggestions helped him to improve this paper. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Diagonal torsion matrices associated with modular data Article published earlier |
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Diagonal torsion matrices associated with modular data |
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Diagonal torsion matrices associated with modular data Singh, G. |
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Diagonal torsion matrices associated with modular data |
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Diagonal torsion matrices associated with modular data |
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Diagonal torsion matrices associated with modular data |
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Diagonal torsion matrices associated with modular data |
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diagonal torsion matrices associated with modular data |
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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). Cuntz (2007) defined isomorphic integral modular data. Here we discuss isomorphic integral and non-integral modular data as well as non-isomorphic but closely related modular data. In this paper, we give some insights into diagonal torsion matrices associated to modular data.
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Diagonal torsion matrices associated with modular data / G. Singh // Algebra and Discrete Mathematics. — 2021. — Vol. 32, № 1. — С. 127–137. — Бібліогр.: 7 назв. — англ. |
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“adm-n3” — 2021/11/8 — 20:27 — page 127 — #129
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 32 (2021). Number 1, pp. 127–137
DOI:10.12958/adm1368
Diagonal torsion matrices associated
with modular data
G. Singh
Communicated by Yu. A. Drozd
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). Cuntz (2007) defined
isomorphic integral modular data. Here we discuss isomorphic inte-
gral and non-integral modular data as well as non-isomorphic but
closely related modular data. In this paper, we give some insights
into diagonal torsion matrices associated to modular data.
Introduction
Diagonal torsion matrices are a fundamental ingredient of modular
data. Modular data is a basic component of rational conformal field theory.
Further, rational conformal field theory has important applications in
physics. In particular, it has nice applications to string theory, statistical
mechanics, and condensed matter physics, see [4] and [7]. Modular data
give rise to fusion rings, C-algebras and C∗-algebras, see [2], [3], [5] and [6].
These rings and algebras are interesting topics of study in their own right.
A modular datum is a pair (S, T ), where S is a Fourier matrix and T
is a diagonal torsion matrix that satisfy certain properties. In this article,
we investigate diagonal torsion matrices associated with integral modular
data and non-integral modular data as well as isomorphic modular data
and non-isomorphic but closely related modular data in certain cases. This
2020 MSC: Primary 05E40; Secondary 05E99, 81R05.
Key words and phrases: Fourier matrices, diagonal torsion matrices, fusion
rings, C-algebras.
https://doi.org/10.12958/adm1368
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128 Diagonal torsion matrices
article is inspired from Cuntz’s example of non-isomorphic integral Fourier
matrices that are related to the character table of an elementary abelian
group of order 4, see [2, pp. 365].
In section 2, we collect the definitions and notations. In section 3,
we show how to find the diagonal torsion matrices associated to Fourier
matrices from the diagonal torsion matrices that either form isomorphic
modular data or closely related modular data. We show how to find the
whole chain of modular data from just one modular datum of that chain.
Also, we determine the number of diagonal torsion matrices associated to
certain type of modular data.
1. Preliminaries
To keep the generality, in the following definition of Modular data
we assume the structure constants to be integers instead of nonnegative
integers, see [2].
Definition 1. Let r ∈ Z
+ and I an r × r identity matrix. A pair (S, T )
of r × r complex matrices is called modular datum if
(1) S is a unitary and symmetric matrix, that is, SS̄T = 1, S = ST ;
(2) T is diagonal matrix and of finite multiplicative order;
(3) Si0 > 0, for 0 6 i 6 r− 1, where S is indexed by {0, 1, 2, . . . , r− 1};
(4) (ST )3 = S2;
(5) Nijk =
∑
l SliSljS̄lkS
−1
l0 ∈ Z, for all 0 6 i, j, k 6 r − 1.
Definition 2. A matrix S satisfying the axioms (i), (iii) and (v) of
Definition 1 is called a Fourier matrix. A matrix T satisfying the axioms
(ii) and (iv) of Definition 1 is called a diagonal torsion matrix.
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). If an s-matrix has integral entries then the s-matrix is called
integral Fourier matrix and the pair (s, T ) is called an integral modular
datum, see [2, Definition 3.1]. A Fourier matrix S is called a homogeneous
Fourier matrix if all the entries of first row of its associated s-matrix
are equal to 1, otherwise, S is called a non-homogenous Fourier matrix,
see [5] and [6]. Let (S, T ) and (S0, T0) be two modular data. Then (S, T )
is called isomorphic to (S0, T0) if there exists a permutation matrix P
with P00 = 1 such that P tSP = S0 and P tTP = T0, otherwise (S, T ) and
(S0, T0) are called non-isomorphic modular data, also see the definition
of isomorphic integral modular data [2, Definfition 3.1]. Throughout the
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G. Singh 129
paper, M t denotes the transposed matrix of a matrix M , and ζk denotes
the kth primitive root of unity.
2. Diagonal torsion matrices associated to modular data
In this section, we establish some criteria to find the diagonal torsion
matrices associated to certain type of a Fourier matrix whose rows and/or
columns permutations result in a Fourier matrix that has an associated
diagonal torsion matrix. We show that the permutations of diagonal
entries of diagonal torsion matrix associated to a Fourier matrix result
again a diagonal torsion matrix associated to same Fourier matrix under
some conditions. Also, we determine number of diagonal torsion matrices
associated to certain type of Fourier matrices.
The matrix consists of the entries of the character table of a finite
dimensional group algebra is called the first eigenmatrices of the group
algebra. So, we use the terms character table of a group and the first
eigenmatrix of the group algebra interchangeably. Bannai and Bannai
classify the diagonal torsion matrices corresponding to Fourier matrices
whose associated s-matrices are the first eigenmatrices of group algebras of
finite cyclic groups, see [1, Theorem 1]. The tensor product of two modular
data is a modular datum, [2]. Let S be a Fourier matrix whose associated
s-matrix is a character table of an abelian group. Note that, character
table of an abelain group can be written as a tensor product of character
tables of cyclic groups. Therefore, the tensor products of the diagonal
torsion matrices corresponding to tensor factors of S are diagonal torsion
matrices corresponding to the Fourier matrix S. Our results enable us
to find the diagonal torsion matrices associated to Fourier matrices of
any type, not only integral homogeneous Fourier matrices that are tensor
product of Fourier matrices of smaller rank.
Let (S, T ) be a modular datum. Since (Sζ3T )
3 = (ST )3 = S2, (S, ζ3T )
is also a modular datum. Hence, in a modular datum, corresponding to
a Fourier matrix there must be at least three different corresponding
diagonal torsion matrices. For the remaining section we investigate the
properties of Fourier matrices and permutation matrices to establish the
results that help to find the additional diagonal torsion matrices from a
given diagonal torsion matrix T in a modular datum (S, T ). In particular,
the following results are useful when an s-matrix is not a tensor product
of the s-matrices of lower rank but can be obtained from such a matrix
by permuting its rows and/or columns.
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130 Diagonal torsion matrices
In the following theorem we see if a permutation matrix commute with
a Fourier matrix then conjugation of the associated diagonal torsion matrix
with the permutation matrix results in diagonal torsion matrix associated
with the given Fourier matrix. Also, we see that simultaneous permutation
of rows and columns of a Fourier matrix and associated diagonal matrix
result in modular datum provided the first row and column are fixed. Note
that, we don’t require s-matrix to be an integral Fourier matrix.
Theorem 1. Let (S, T ) be a modular datum. Let P be a permutation
matrix.
(1) If SP = PS then (S, P tTP ) is a modular datum.
(2) If P00 = 1 then (P tSP, P tTP ) is a modular datum.
Proof. (1) (S, T ) is a modular datum, therefore (ST )3 = S2. Since SP =
PS, (S(P tTP ))3 = P t(ST )3P = P tS2P = S2. Note that, P tTP is a
diagonal torsion matrix of multiplicative order equal to the multiplicative
order of T . Hence (S, P tTP ) is a modular datum.
(2) The simultaneous row and column permutation of a unitary and
symmetric matrix results in a unitary and symmetric matrix. Thus P tSP
is a unitary and symmetric matrix. The assumption P00 = 1 assures
that the entries of the first row and column of P tSP matrix are positive
real numbers. Note that, the set of structure constants generated by the
columns of P tSP matrix under entrywise multiplication is same as the
set of structure constants generated by the columns of S matrix under
entrywise multiplication. Therefore, P tSP is a Fourier matrix.
Since (S, T ) is a modular datum, (ST )3 = S2. The inverse of a permu-
tation matrix is its transpose, therefore ((P tSP )(P tTP ))3 = P t(ST )3P
= P tS2P = P tSPP tSP = (P tSP )2. Also, P tTP is a diagonal matrix
with finite multiplicative order. Thus (P tSP, P tTP ) is a modular da-
tum.
Note that, if S is a homogeneous Fourier matrix of rank r then s =
√
rS.
Therefore, s-matrix is a symmetric matrix. Let P be a permutation matrix
of rank r. Then SP (PS, respectively) is a symmetric matrix if and only
if sP (Ps, respectively) is a symmetric matrix. Also, SP = PS if and
only if sP = Ps. In the following example, we apply Theorem 1 and
Proposition 1 to demonstrate how to relate the diagonal torsion matrices
of two non-isomorphic modular data.
Example 1. Consider the Fourier matrix S =
1√
2
[
1 1
1 −1
]
. Let T =
diag(x, y) be the corresponding diagonal matrix, where x, y ∈ C and have
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G. Singh 131
finite multiplicative order. Since (ST )3 = S2, we have
T ∈
{
diag(ζ724, ζ
13
24 ), diag(ζ1524 , ζ
21
24 ), diag(ζ2324 , ζ
5
24),
diag(ζ24, ζ
19
24 ), diag(ζ924, ζ
3
24), diag(ζ1724 , ζ
11
24 )
}
.
The s̃-matrix, the character table of an elementary abelian group of order 4,
is obtained from the s-matrix as follows.
s̃ = s⊗ s =
1 1 1 1
1 −1 1 −1
1 1 −1 −1
1 −1 −1 1
.
The tensor product of two modular data is a modular datum, see [2]. Thus,
corresponding Fourier matrix S̃ has associated diagonal torsion matrices
T̃ in the set below obtained from the tensor product of T matrices from
the above set. Therefore,
T̃ ∈
{
diag(ζ1424 , ζ
20
24 , ζ
20
24 , ζ
2
24), diag(ζ2224 , ζ
4
24, ζ
4
24, ζ
10
24 ), diag(ζ624, ζ
12
24 , ζ
12
24 , ζ
18
24 ),
diag(ζ824, ζ
2
24, ζ
14
24 , ζ
8
24), diag(ζ1624 , ζ
10
24 , ζ
22
24 , ζ
16
24 ), diag(1, ζ1824 , ζ
6
24, 1),
diag(ζ824, ζ
14
24 , ζ
2
24, ζ
8
24), diag(ζ1624 , ζ
22
24 , ζ
10
24 , ζ
16
24 ), diag(1, ζ624, ζ
18
24 , 1),
diag(ζ224, ζ
20
24 , ζ
20
24 , ζ
14
24 ), diag(ζ1024 , ζ
4
24, ζ
4
24, ζ
22
24 ),
diag(ζ1824 , ζ
12
24 , ζ
12
24 , ζ
6
24)
}
.
Let P(ijk) (P(ij), respectively) be a permutation matrix that permutes
rows i, j, k (i and j, respectively) of a matrix on left multiplication to
it. Therefore, P(243)s̃ = s̃P(243). Thus, by Theorem 1(1), we obtain the
following associated diagonal torsion matrices T ′ from T̃ .
T ′ ∈
{
diag(ζ1424 , ζ
2
24, ζ
20
24 , ζ
20
24 ), diag(ζ2224 , ζ
10
24 , ζ
4
24, ζ
4
24), diag(ζ624, ζ
18
24 , ζ
12
24 , ζ
12
24 ),
diag(ζ824, ζ
8
24, ζ
2
24, ζ
14
24 ), diag(ζ1624 , ζ
16
24 , ζ
10
24 , ζ
22
24 ), diag(1, 1, ζ624, ζ
18
24 ),
diag(ζ824, ζ
8
24, ζ
14
24 , ζ
2
24), diag(ζ1624 , ζ
16
24 , ζ
22
24 , ζ
10
24 ), diag(1, 1, ζ1824 , ζ
6
24),
diag(ζ224, ζ
14
24 , ζ
20
24 , ζ
20
24 ), diag(ζ1024 , ζ
22
24 , ζ
4
24, ζ
4
24),
diag(ζ1824 , ζ
6
24, ζ
12
24 , ζ
12
24 )
}
.
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132 Diagonal torsion matrices
Also, P(234)s̃ = s̃P(234). Therefore, by Theorem 1(1), we obtain the follow-
ing associated diagonal torsion matrices T ′
1 from T ′.
T ′
1 ∈
{
diag(ζ1424 , ζ
20
24 , ζ
2
24, ζ
20
24 ), diag(ζ2224 , ζ
4
24, ζ
10
24 , ζ
4
24), diag(ζ624, ζ
12
24 , ζ
18
24 , ζ
12
24 ),
diag(ζ824, ζ
14
24 , ζ
8
24, ζ
2
24), diag(ζ1624 , ζ
22
24 , ζ
16
24 , ζ
10
24 ), diag(1, ζ1824 , 1, ζ
6
24),
diag(ζ824, ζ
2
24, ζ
8
24, ζ
14
24 ), diag(ζ1624 , ζ
10
24 , ζ
16
24 , ζ
22
24 ), diag(1, ζ624, 1, ζ
18
24 ),
diag(ζ224, ζ
20
24 , ζ
14
24 , ζ
20
24 ), diag(ζ1024 , ζ
4
24, ζ
22
24 , ζ
4
24),
diag(ζ1824 , ζ
12
24 , ζ
6
24, ζ
12
24 )
}
.
Note that, P(23)s̃ = s̃P(23), thus, by Theorem 1(1), on switching the second
and third entry of each of the diagonal torsion matrix associated to S̃ we
obtain a diagonal torsion matrix that is also associated to S̃. Therefore,
S̃ has above 36 associated diagonal torsion matrices.
The following s1-matrix cannot be obtained from the tensor product
of integral Fourier matrices of smaller rank.
s1 =
1 1 1 1
1 1 −1 −1
1 −1 −1 1
1 −1 1 −1
.
However, s1 = P t
(24)s̃P(24). Therefore, by applying Theorem 1(2), we can
obtain the diagonal torsion matrices associated to the Fourier matrix
S1 from the above diagonal torsion matrices. Also, an application of
Theorem 1(2) gives the diagonal torsion matrices associated to Fourier
matrix P t
(34)S̃P(34). Note that, s1 = P(243)s̃. Therefore, we can also apply
Proposition 1 to find the diagonal torsion matrices.
In the next lemma, we show that the multiplication of a Fourier matrix
with a permutation matrix result in a Fourier matrix, provided the first
row and column of the Fourier matrix are not permuted.
Lemma 1. Let S be a non-singular symmetric matrix and P a permutation
matrix. Let SP be a symmetric matrix. If S is a Fourier matrix and P00 = 1
then PS and SP are Fourier matrices.
Proof. Since S is a unitary matrix and P−1 = P t, (PS)(PS)
t
= PSS
t
P t
= I, where I is the identity matrix. Therefore, PS is a unitary matrix.
Since P00 = 1, the entries of first row and column of PS are positive
real numbers. Also, the set of the structure constants of generated by the
columns of PS under entrywise multiplication is equal to the set of the
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G. Singh 133
structure constants of generated by the columns of S under entrywise
multiplication. Therefore, PS is a Fourier matrix. Similarly, SP is a
Fourier matrix.
In the following proposition, we prove that for a permutation matrix
P of order 3 and modular datum (S, T ) if PS is symmetric matrix then
(PS, P TTP ) is a modular datum.
Proposition 1. Let (S, T ) be a modular datum. Let P be a permutation
matrix of order 3 such that P00 = 1.
(1) If PS is a symmetric matrix then (PS, P tTP ) is a modular datum.
(2) If SP is a symmetric matrix then (SP, PTP t) is a modula datum.
Proof. (1) Since S is a Fourier matrix, by Lemma 1, PS is a Fourier matrix.
Also (ST )3 = S2 and (PS)t = PS imply (PSP tTP )3 = P 2S2P = (PS)2.
The matrix P tTP is a diagonal torsion matrix whose multiplicative order
is equal to the multiplicative order of T . Hence (PS, P tTP ) is a modular
datum.
(2) Proof is similar to part (1) above.
An application of the above proposition can be found in Example 1.
Definition 3. Let S be a symmetric matrix of rank r and P1, P2, . . . , Pn
be permutation matrices of rank r and multiplicative order k, where n is
a positive integer determined by r. Let SP1, SP1P2, . . . , SP1 . . . Pn, and
P1S, P1P2S, . . . , P1 . . . PnS be symmetric matrices. Then we call the set
1) V := {P t
1SP1, P
t
2P
t
1SP1P2, . . . , P
t
n . . . P
t
1SP1 . . . Pn} a chain of sy-
mmetric matrices under the action of k-cycles;
2) V1 := {SP1, SP1P2, . . . , SP1 . . . Pn} a left chain of symmetric ma-
trices under the action of k-cycles;
3) V2 := {P1S, P1P2S, . . . , P1 . . . PnS}, a right chain of symmetric
matrices under the action of k-cycles.
The character table of an abelian group can be written as a tensor
product of character tables of cyclic groups of smaller order. Therefore,
for each non-prime rank there exists a chain (see Definition 3) that has
a Fourier matrix which is a tensor product of s-matrices of smaller rank.
However, it is not necessary that a chain corresponds uniquely to order k,
see examples 1 and 2. Also, a Fourier matrix can be a member of more
than one chain. For example, for the Fourier matrix S whose s-matrix
is a character table of an elementary abelian group of order 8, there are
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134 Diagonal torsion matrices
5 different chains for 28 matrices obtained from the character table by
permutating its rows and/or columns.
In the next theorem, we find the number of diagonal torsion matrices
corresponding to a Fourier matrix S whose associated s-matrix is the
character table of an elementary abelian group.
Theorem 2. Let n be a positive integer. Let S be a Fourier matrix of
rank 2n whose associated s-matrix is the character table of an elementary
abelian group. Then S has 3n × 2n associated diagonal torsion matrices.
Proof. We prove the result by induction on n. For n = 1 and 2, the result
is true, see Example 1. Suppose the result is true for n = k. Let n = k+1.
The s-matrix is a character table of an elementary abelian group. Therefore,
Fourier matrix S can be expressed as S1 ⊗ S2, a tensor product of two
Fourier matrices S1 and S2 of rank 2k and 2, respectively. By induction,
there are 3k × 2k and 3 × 2 diagonal torsion matrices associated to S1
and S2, respectively. (Note that, the number of matrices are multiple of 3,
because if (S, T ) is a modular datum then (S, ζ3T ) is a modular datum.)
The tensor product of two modular data is a modular datum. Therefore,
corresponding to Fourier matrix S there are (3k×2k)×(3×2) = 3k+1×2k+1
diagonal matrices. Hence, the theorem is proved.
The following corollary is immediate from the above theorem.
Corollary 1. Let (S, T ) be a modular datum. Let s-matrix be the character
table of an elementary abelian group of order 2n. Then the chain has
(2n − 1)! Fourier matrices and (2n − 1)! × 3n × 2n associated diagonal
torsion matrices.
The following proposition has an application in Example 2.
Proposition 2. Let S be a Fourier matrix of rank r whose associated
s-matrix is a rows and/or columns permutation of the character table
of an elementary abelian group. Let Ti := diag(1, . . . ,−1, . . . , 1) be a
diagonal matrix with −1 at the ith position on its diagonal and all the
other diagonal entries equal to 1, where i ∈ {1, 2, . . . , r}.
(1) If (S, Ti) is a modular datum for an i then r = 4.
(2) If r = 4 then (S, Ti) is a modular datum for an i if and only if
sjj = 1, for all j.
(3) If r = 4 then (S, xTi) is a modular datum if and only if sjj = 1, for
all j, and x3 = 1, that is, x ∈ {1, ζ3, ζ23}.
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G. Singh 135
(4) If r = 4 then (S, y(−Ti)) is a modular datum if and only if sjj = 1,
for all j, and y3 = −1, that is, y ∈ {ζ6, ζ36 , ζ56 , }.
Proof. (1) Suppose T is a diagonal torsion matrix associated to Fourier
matrix S. Then (ST )3 = S2 = I, where I is an identity matrix of rank r.
Since S = r−1/2s, S2 = I if and only if s−1 = r−1s. Therefore, (ST )3 = I
if and only if (r−1/2sT )3 = I, that is, (sT )2 = r3/2T−1s−1 = r1/2T−1s.
Note that, s-matrix is a symmetric matrix and s2 = rI. Therefore, if Ti is
an associated diagonal torsion matrix then all the non-diagonal entries of
(sTi)
2 are ±2 and the diagonal entries are ±(r − 2). Since the entries of
s-matrix are only ±1, (sTi)
2 = r1/2T−1
i s implies r = 4, where T−1
i = Ti.
(2) Note that, by part (1), r = 4 and (sTi)
2 = 2Tis. We consider the
following two cases.
Case 1. Let i > 1.
Then (sTi)
2)0j = 2 for all j. The entries of s-matrix are ±1. Thus
(sTi)0i = −1 implies (sTi)
2
0i = 2 if and only if sii = 1. Since s-matrix is
a symmetric matrix, (sTi)
2
jj = 2 for all j 6= i. Therefore, (sTi)
2 = 2Tis if
and only if sjj = 1 for all j. Hence (S, Ti) is a modular datum if and only
if sjj = 1 for all j.
Case 2. Let i = 1.
Obviously, s00 = 1. The s-matrix and (sT1)
2 have ±1 and ±2 entries,
respectively. Also, s-matrix is a symmetric matrix. Thus, (sT1)
2
jj = 2 for
all j 6= 0. Therefore, (sT1)
2 = 2T1s if and only if sjj = 1 for all j. Hence
(S, T1) is a modular datum if and only if sjj = 1 for all j.
(3) By part (2), (S, Ti) is modular datum if and only if sjj = 1. Now
(S(xTi))
3 = x3(STi)
3 = x3I. Therefore, (S, xTi) is modular datum if and
only if sjj = 1 and x3 = 1, that is, x ∈ {1, ζ3, ζ23}.
(4) By part (2), (S, Ti) is modular datum if and only if sjj = 1.
Now (Sy(−Ti))
3 = −y3(STi)
3 = −y3I. Therefore, (S, y(−Ti)) is modular
datum if and only if sjj = 1 and y3 = −1, that is, y ∈ {ζ6, ζ36 , ζ56 , }.
In the next proposition, we show that for a Fourier matrix S and a
permutation matrix P of order 2, SP is a Fourier matrix only if the rank
of S is 4.
Proposition 3. Let S be a Fourier matrix whose s-matrix is the character
table of an elementary abelian group of order r and P a permutation matrix
of order 2. If SP is a Fourier matrix then r = 4.
Proof. By Lemma 1, SP is a Fourier matrix. Let R := SP . The symmetry
of R and S imply Rij = Rik, Rjj = Rkk, for all i 6= j, k. Note that,
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136 Diagonal torsion matrices
j, k > 1 and rows/columns of s-matrix are orthogonal, thus Rjk 6= Rkk and
Rkj 6= Rjj . The entries of s-matrix are ±1. Therefore, the orthogonality
of columns of s-matrix gives (r − 2)− 2 = 0 implies r = 4.
Example 2. Consider the s̃-matrix as described in Example 1. The
following matrix s2 = P(23)s̃ is obtained from s̃-matrix by interchanging
its second and third rows.
s2 =
1 1 1 1
1 1 −1 −1
1 −1 1 −1
1 −1 −1 1
.
Note that, the Fourier matrix S2 is not a tensor product of any lower rank
Fourier matrices. Let Ti be the diagonal matrices of rank 4 as defined
in Proposition 2. Since (s2)jj = 1, by Proposition 2, xT1, xT2, xT3, xT4,
y(−T1), y(−T2), y(−T3), and y(−T4) are the diagonal torsion matrices
associated to Fourier matrix S2, where x ∈ {1, ζ3, ζ23} and y ∈ {ζ6, ζ36 , ζ56}.
The following corollary is immediate from the proof of the above results
and the definition.
Corollary 2. Let V be a chain (left chain, right chain) of symmetric
matrices under the action of k-cycles, where k ∈ {2, 3}.
(1) If V has a Fourier matrix then every element of the chain is a
Fourier matrix.
(2) If V has a homogenous Fourier matrix then every element of the
chain is a homogeneous Fourier matrix.
(3) If an element S in the chain V forms a modular datum. Then each
element of V forms a modular datum and they have equal number of
associated diagonal torsion matrices that are completely determined
by the diagonal matrices for S.
Acknowledgment
The author would like to thank Professor Allen Herman whose valuable
suggestions helped him to improve this paper.
References
[1] E. Bannai and E. Bannai, Spin Models on Finite Cyclic Groups, Journal of
Algebraic Combinatorics, 3, 1994 , 243-259.
[2] Michael Cuntz, Integral modular data and congruences, J Algebr Comb, 29, 2009,
357-387.
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G. Singh 137
[3] Allen Herman and Gurmail Singh, Central torsion units of integral reality-based
algebras with a positive degree map, International Electronic Journal of Algebra,
21, 2017, 121-126.
[4] Martin Schottenholer, A Mathematical Introduction to Conformal Field Theory,
Springer-Verlag, Berlin, Heidelberg, 2nd edition, 2008.
[5] Gurmail Singh, Classification of homogeneous Fourier matrices, Algebra and
Discrete Mathematics, 27(1), 2019, 75-84.
[6] Gurmail Singh, Fourier matrices of small rank, Journal of Algebra Combinatorics
Discrete Structures and Applications, 5 (2), 2018, 51-63.
[7] Ali Zahabi, Applications of Conformal Field Theory and String Theory in Statistical
Systems, Ph.D. Dissertation, University of Helsinki, Helsinki, Finland, 2013.
Contact information
Gurmail Singh Department of Mathematics and Statistics,
University of Regina, Regina, Canada, S4S 0A2
E-Mail(s): Gurmail.Singh@uregina.ca
Received by the editors: 03.04.2019
and in final form 19.04.2021.
mailto:Gurmail.Singh@uregina.ca
G. Singh
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