Journal of Algebraic Combinatorics 12 (2000), 59–72
c 2000 Kluwer Academic Publishers. Manufactured in The Netherlands.
°
General Form of Non-Symmetric Spin Models
TAKUYA IKUTA
Kobe Gakuin Women’s College, Hayashiyama, Nagata-ku, Kobe, 653-0861 Japan
ikuta@koin.or.jp
KAZUMASA NOMURA
nomura@tmd.ac.jp
College of Liberal Arts and Sciences, Tokyo Medical and Dental University, Kohnodai, Ichikawa, 272-0827 Japan
Received August 20, 1998; Revised October 14, 1998
Abstract. A spin model (for link invariants) is a square matrix W with non-zero complex entries which satisfies
certain axioms. Recently (Jaeger and Nomura, J. Alg. Combin. 10 (1999), 241–278) it was shown that t WW −1 is
a permutation matrix (the order of this permutation matrix is called the “index” of W ), and a general form was
given for spin models of index 2. In the present paper, we generalize this general form to an arbitrary index m. In
particular, we give a simple form of W when m is a prime number.
Keywords: spin model, association scheme, Bose-Mesner algebra
1.
Introduction
Spin models were introduced by Vaughan Jones [7] to construct invariants of knots and
links. A spin model is essentially a square matrix W with nonzero entries which satisfies two conditions (type II and type III conditions). In his definition of a spin model,
Jones considered only symmetric matrices. It was generalized to non-symmetric case by
Kawagoe-Munemasa-Watatani [8].
Recently, François Jaeger and the second author [6] introduced the notion of “index” of a
spin model. For every spin model W , the transpose t W is obtained from W by a permutation
of rows. Let σ denote the corresponding permutation of X = {1, . . . , n} (n is the size
of W ). Then the index m is the order of σ . In [6], it was shown that X is partitioned into
m subsets X 0 , X 1 , . . . , X m−1 such that W (x, y) = ηi− j W (y, x) holds for all x ∈ X i ,
y ∈ X j . Moreover, the case of m = 2 was deeply investigated, and a general form of spin
models of index 2 was given.
In the present paper, we investigate the structure of spin models of an arbitrary index m.
In Section 4, we show that W is decomposed into blocks Wi j , and Wi j splits into Kronecker
product of two matrices Si j and Ti j (Proposition 4.3). In Section 5, we give conditions on
Ti j (Propositions 5.1 and 5.5). In Section 6, we apply this general form to some special
cases (Propositions 6.1 and 6.2). In particular, we give a simple form of W when the index
m is a prime number (Corollary 6.3).
60
2.
IKUTA AND NOMURA
Preliminaries
In this section, we give some basic materials concerning spin models and association
schemes. For more details the reader can refer to [4–7].
Let X be a finite non-empty set with n elements. We denote by Mat X (C) the set of
square matrices with complex entries whose rows and columns are indexed by X . For
W ∈ Mat X (C) and x, y ∈ X , the (x, y)-entry of W is denoted by W (x, y).
A type II matrix on X is a matrix W ∈ Mat X (C) with nonzero entries which satisfies the
type II condition:
X W (a, x)
= nδa,b
W (b, x)
x∈X
(for all a, b ∈ X ).
(1)
Let W − ∈ Mat X (C) be defined by W − (x, y) = W (y, x)−1 . Then type II condition is
written as W W − = n I (I denotes the identity matrix). Hence, if W is a type II matrix,
then W is non-singular with W −1 = n −1 W − . It is clear that W −1 and t W are also type II
matrices.
A type II matrix W is called a spin model on X if W satisfies type III condition:
X W (a, x)W (b, x)
W (a, b)
=D
W (c, x)
W (a, c)W (c, b)
x∈X
(for all a, b, c ∈ X )
(2)
for some nonzeroP
complex number D. The number D is called the loop variable of W . Setting b = c in (2), x∈X W (a, x) = DW(b, b)−1 holds, so that the diagonal entries W (b, b)
is a constant, which is called the modulus of W .
For a spin model W with loop variable D, any nonzero scalar multiple λW is a spin
model with loop variable λ2 D. Usually W is normalized so that D 2 = n, but we allow any
nonzero value of D in this paper to simplify our arguments.
Observe that, for any spin models Wi on X i with loop variable Di (i = 1, 2), their tensor
(Kronecker) product W1 ⊗ W2 is a spin model with loop variable D = D1 D2 . Conversely,
it is not difficult to show that, if W1 ⊗ W2 and W1 are spin models, then W2 must be a spin
model.
A (class d) association scheme on X is a partition of X × X with nonempty relations
R0 , R1 , . . . , Rd , where R0 = {(x, x) | x ∈ X } which satisfy the following conditions:
(i) For every i in {0, 1, . . . , d}, there exists i 0 in {0, 1, . . . , d} such that Ri 0 = {(y, x) |
(x, y) ∈ Ri }.
(ii) There exist integers pikj (i, j, k ∈ {0, 1, . . . , d}) such that for every (x, y) ∈ Rk , there
are precisely pikj elements z such that (x, z) ∈ Ri and (z, y) ∈ R j .
(iii) pikj = p kji for every i, j in {0, 1, . . . , d}.
Let Ai denote the adjacency matrix of the relation Ri , so Ai ∈ Mat X (C) is a {0, 1}-matrix
whose (x, y)-entry is equal to 1 if and only if (x, y) ∈ Ri . Clearly A0 = I , Ai ◦ A j = δi, j Ai
NON-SYMMETRIC SPIN MODELS
61
Pd
P
(entry-wise product), i=0
Ai = J (all 1’s matrix), and Ai A j = dk=0 pikj Ak hold. The
linear span A of {A0 , A1 , . . . , Ad } becomes a subalgebra of Mat X (C), called the BoseMesner algebra of the association scheme. Observe that A is closed under entry-wise
product, A is closed under transposition A 7→ t A, and A contains I , J .
3.
Associated permutation
Let W be a spin model on X . Then there exists an association scheme R0 , . . . , Rd on X
such that the corresponding Bose-Mesner algebra A contains W ([5] Theorem 11). In [6], it
was shown that t WW −1 = As (the adjacency matrix of Rs ) for some s ∈ {0, 1, . . . , d}, and
moreover As is a permutation matrix ([6] Proposition 2). Let σ denote the corresponding
permutation on X , so that As (x, y) = 1 if y = σ (x) and As (x, y) = 0 otherwise. The
order m of σ is called the index of W .
Observe that m = 1 if and only if W is symmetric. Also observe that, for two spin models
Wi of index m i (i = 1, 2), the index of W1 ⊗ W2 is equal to the least common multiple of
m 1 and m 2 . In particular, tensor product of a spin model of index m with any symmetric
spin model has index m.
Lemma 3.1
(i) W (x, σ (x)) = W (y, σ (y)) (x, y ∈ X ).
(ii) W (y, x) = W (σ (x), y) (x, y ∈ X ).
(iii) Every orbit of σ has length m.
Proof:
Pd
ti Ai , so
(i) Observe that, since W ∈ A, W is written as a linear combination W = i=0
W (x, y) = ti for (x, y) ∈ Ri . Since (x, σ (x)) ∈ Rs (for every x ∈ X ), it holds that
W (x, σ (x)) = ts = W (y, σ (y)).
(ii) W (y, x) = t W (x, y) = (As W )(x, y) = W (σ (x), y).
(iii) Pick any i (0 < i < m). Since Ais is a linear combination of A0 , . . . , Ad and since Ais
is a permutation matrix, we get Ais = A j for some j 6= 0. Observe that the diagonal
entries of A j are all zero since j 6= 0. This means that σ i (which corresponds the
permutation matrix A j ) has no fixed point on X . We have shown that σ i fixes no point
2
(0 < i < m). Thus every orbit of σ must have length m.
Lemma 3.2
m − 1})
There is a partition X = X 0 ∪ · · · ∪ X m−1 such that ( for all i, j ∈ {0, . . . ,
W (x, y) = ηi− j W (y, x)
( for all x ∈ X i , y ∈ X j ),
(3)
where η denotes a primitive m-root of unity. Moreover, for every i, σ (X i ) = X j holds for
some j.
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IKUTA AND NOMURA
Proof: The existence of such a partition follows from [6] Proposition 3. As in the proof
of Lemma 3.1(i), we have (x, σ (x)) ∈ Rs and W (x, σ (x)) = ts for all x ∈ X . Then there
exists s 0 such that (σ (x), x) ∈ Rs 0 , so that W (σ (x), x) = ts 0 . Now pick any x ∈ X i . Then
σ (x) ∈ X j for some j. On the other hand, W (x, σ (x)) = ηi− j W (σ (x), x). These imply
ηi− j = ts ts−1
0 . This means that j is independent of the choice of x ∈ X i , so that σ (X i ) = X j .
2
We fix a primitive m-root of unity η, and let X 0 , . . . , X m−1 be the partition of X given in
Lemma 3.2. We identify the index set {0, 1, . . . , m − 1} with Zm = Z/mZ. By Lemma 3.2,
there is a permutation π on Zm such that σ (X i ) = X π(i) (i ∈ Zm ). Let t denote the order
of π, and set k = m/t.
Lemma 3.3
π(i) − i = π( j) − j for all i, j ∈ Zm .
Proof: Pick any x ∈ X i , y ∈ X j . We have σ (x) ∈ X π(i) , σ (y) ∈ X π( j) . By Lemma 3.2,
W (x, σ (x)) = ηi−π(i) W (σ (x), x) and W (y, σ (y)) = ηi−π( j) W (σ (y), y). On the other
hand, W (x, σ (x)) = W (y, σ (y)) by Lemma 3.1(i), and also W (σ (x), x) = W (x, x) =
(the modulus of W) = W (y, y) = W (σ (y), y) by Lemma 3.1(ii). These imply ηi−π(i) =
2
η j−π( j) .
Lemma 3.4 There exists an automorphism ϕ of the additive group Zm such that π(ϕ(i)) =
ϕ(i + k) for all i ∈ Zm . Moreover, W (x, y) = (ηϕ(1) )i− j W (y, x) for every x ∈ X ϕ(i) ,
y ∈ X ϕ( j) .
Proof: Set k 0 = π(0). Then π(i) = i + k 0 (i ∈ Zm ) by Lemma 3.3. Thus k 0 Zm =
{0, k 0 , 2k 0 , . . . , (t − 1)k 0 } is an orbit of π . Note that every orbit of π has length t, and
hence the number of orbits of π is equal to k = m/t (in particular, k must be an integer).
Clearly k 0 Zm is the unique subgroup of Zm of order t, so k 0 Zm = kZm . Hence there is an
automorphism ϕ of the additive group kZm such that ϕ(k) = k 0 .
We claim that ϕ can be extended to an automorphism of Zm . In fact, for any cyclic
group G and for any subgroup H of G, any automorphism of H can be extended to an
automorphism of G. This fact can be easily shown when G is a cyclic p-group. For general
case, decompose G into the Sylow subgroups.
Now we have an automorphism ϕ of Zm such that ϕ(k) = k 0 . Since π(i) = i + k 0 for all
i ∈ Zm , we get π(ϕ(i)) = ϕ(i) + k 0 = ϕ(i) + ϕ(k) = ϕ(i + k).
Let x ∈ X ϕ(i) , y ∈ X ϕ( j) . Then, by Lemma 3.2, W (x, y) = ηϕ(i)−ϕ( j) W (y, x) holds for
all x ∈ X ϕ(i) , y ∈ X ϕ( j) . Here ϕ(i) − ϕ( j) = ϕ(i · 1) − ϕ( j · 1) = iϕ(1) − jϕ(1) =
2
ϕ(1)(i − j). Hence W (x, y) = (ηϕ(1) )i− j W (y, x).
Thus, by reordering the indices {0, 1, . . . , m − 1} by ϕ, and by replacing η with ηϕ(1) ,
we may assume that
π(i) = i + k
(i ∈ Zm ).
(4)
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NON-SYMMETRIC SPIN MODELS
4.
General form of W
We use the notation of the previous section. We also use the notation:
γk (`, i) = η−`i−(k/2)`(`−1) .
(5)
Proposition 4.1 Let i, j ∈ Zm and x ∈ X i , y ∈ X j . Then for `, `0 ∈ Z,
0
W (σ ` (x), σ ` (y)) = γk (` − `0 , i − j) W (x, y).
(6)
Proof: Assume ` ≥ 0 and `0 ≥ 0. First we consider the case of `0 = 0. We proceed
by induction on `. Obviously (6) holds for ` = 0. By Lemma 3.1(ii) and Lemma 3.2,
W (y, x) = W (σ (x), y) and W (y, x) = η j−i W (x, y). Hence W (σ (x), y) = η j−i W (x, y),
so (6) holds for ` = 1. Now assume ` > 1. Noting σ (x) ∈ X π(i) = X i+k and using induction,
W (σ ` (x), y) = W (σ `−1 (σ (x)), y)
= γk (` − 1, (i + k) − j)W (σ (x), y)
= γk (` − 1, (i + k) − j)η j−i W (x, y)
= γk (`, i − j)W (x, y).
0
Hence (6) holds for `0 = 0. Now suppose `0 > 0. Noting σ ` (y) ∈ X j+`0 k and using
Lemma 3.2,
0
0
W (σ ` (x), σ ` (y)) = γk (`, i − ( j + `0 k))W (x, σ ` (y))
0
0
= γk (`, i − ( j + `0 k))ηi−( j+` k) W (σ ` (y), x)
0
= γk (`, i − ( j + `0 k))ηi−( j+` k) γk (`0 , j − i)W (y, x)
0
= γk (`, i − ( j + `0 k))ηi−( j+` k) γk (`0 , j − i)η j−i W (x, y)
= γk (` − `0 , i − j)W (x, y).
Thus (6) holds for non-negative integers `, `0 .
Since σ −` (x) ∈ X i−`k ,
W (x, y) = W (σ ` (σ −` (x)), y) = γk (`, (i − `k) − j)W (σ −` (x), y).
Hence
W (σ −` (x), y) = γk (`, i − `k − j)−1 W (x, y)
= η`(i−`k− j)+(k/2)`(`−1) W (x, y)
= η`(i− j)+(k/2)`(`+1) W (x, y)
= γk (−`, i − j)W (x, y).
0
Since σ −` (y) ∈ X j−`0 k ,
0
0
0
W (x, y) = W (x, σ ` (σ −` (y))) = γk (−`0 , i − ( j − `0 k))W (x, σ −` (y)).
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IKUTA AND NOMURA
Hence
0
W (x, σ −` (y)) = γk (−`0 , i − j + `0 k)−1 W (x, y)
= γk (`0 , i − j)W (x, y).
0
Since σ ` (y) ∈ X j+`0 k ,
0
0
W (σ −` (x), σ ` (y)) = γk (−`, i − ( j + `0 k))W (x, σ ` (y))
= γk (−`, i − j − `0 k)γk (−`0 , i − j)W (x, y)
= γk (−` − `0 , i − j)W (x, y).
Similarly, we can show that
0
W (σ ` (x), σ −` (y)) = γk (` + `0 , i − j)W (x, y),
and
0
W (σ −` (x), σ −` (y)) = γk (−` + `0 , i − j)W (x, y).
This completes the proof of (6).
Lemma 4.2
2
If m is even, then k is even.
Proof: We apply Proposition 4.1 for ` = m, `0 = 0 and i = j. Then (6) implies
γk (m, 0) = 1, and this becomes (η−m/2 )k(m−1) = 1. Observe that η−m/2 = −1, since η
is a primitive m-root of unity and m is even. Hence (−1)k(m−1) = 1, so that k must be
even.
2
For i ∈ Zm , set
1i =
t−1
[
X i+hk .
h=0
Observe that |1i | = t (n/m) = tn/(kt) = n/k, and that
X=
k−1
[
1i ,
i=0
Since σ (1i ) = 1i , 1i is partitioned into σ -orbits Yαi :
1i =
r
[
α=1
Yαi
(i = 0, . . . , k − 1),
65
NON-SYMMETRIC SPIN MODELS
where r = |1i |/m = n/(mk). Observe that |Yαi | = m and |Yαi ∩ X i | = k. We choose
representative elements
yαi ∈ Yαi ∩ X i
(i = 0, . . . , k − 1, α = 1, . . . , r ).
Then
ª
© ¡ ¢¯
X = σ ` yαi ¯ i = 0, . . . , k − 1, α = 1, . . . , r, ` = 0, . . . , m − 1 ,
(7)
¡
¡ ¡ ¢ 0 ¡ j ¢¢
j¢
W σ ` yαi , σ ` yβ = γk (` − `0 , i − j) W yαi , yβ
(8)
and
for `, `0 ∈ Zm , i, j = 0, . . . , k − 1 and α, β = 1, . . . , r .
We define square matrices Ti j of size r and Si j of size m (i, j = 0, . . . , k − 1) by
¡
j¢
Tij (α, β) = W yαi , yβ
0
(α, β = 1, . . . , r ),
0
Si j (`, ` ) = γk (` − ` , i − j)
(`, `0 = 0, . . . , m − 1).
For subsets A, B of X , let W | A×B denote the restriction (submatrix) of W on A × B. For
two matrices S, T , we denote the Kronecker product by S ⊗ T .
Proposition 4.3
For i, j = 0, . . . , k − 1,
W |Y i ×Y j = Ti j (α, β) Si j
α
β
(α, β = 1, . . . , r ),
and
W |1i ×1 j = Si j ⊗ Ti j .
Proof:
(9)
2
Clear.
Thus W decomposes into blocks Wij = W |1i ×1 j (i, j = 0, . . . , k − 1), and each block
has the form Wi j = Si j ⊗ Ti j (i, j = 0, . . . , k − 1).
5.
Type II and Type III conditions
Let m, k, t, r be positive integers with m = kt.
Let Ti j (i, j = 0, . . . , k − 1) be any matrices of size r with nonzero entries, and let Si j
(i, j = 0, . . . , k − 1) be the matrix of size m defined by
Si j (`, `0 ) = γk (` − `0 , i − j)
(`, `0 = 0, . . . , m − 1),
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IKUTA AND NOMURA
where γk is defined by (5) for a primitive m-root of unity η. Now set
Wi j = Si j ⊗ Ti j
(i, j = 0, . . . , k − 1),
and let W be the matrix of size n = kmr whose (i, j) block is Wij (i, j = 0, . . . , k − 1).
We index the rows and the columns of W by the set:
X = {[i, `, α] | 0 ≤ i ≤ k − 1, 0 ≤ ` ≤ m − 1, 1 ≤ α ≤ r },
so that
W ([i, `, α], [ j, `0 , β]) = Si j (`, `0 )Ti j (α, β).
(10)
Proposition 5.1 W is a type II matrix if and only if Ti j is a type II matrix for all i,
j ∈ {0, . . . , k − 1}.
Proof:
The type II condition (1) for a = [i 1 , `1 , α1 ], b = [i 2 , `2 , α2 ] becomes
r
k−1 m−1
X
XX
W ([i 1 , `1 , α1 ], [i, `, α])
= nδi1 ,i2 δ`1 ,`2 δα1 ,α2 .
W ([i 2 , `2 , α2 ], [i, `, α])
i=0 `=0 α=1
(11)
Using (10), we rewrite the left-hand-side as follows:
l.h.s. =
r
k−1 m−1
X
XX
γk (`1 − `, i 1 − i)Ti1 i (α1 , α)
γ (` − `, i 2 − i)Ti2 i (α2 , α)
i=0 `=0 α=1 k 2
= η−`1 ii +`2 i2 −(k/2)(`1 −`2 )(`1 +`2 −1)
k−1
X
η(`1 −`2 )i
r
X
X
Ti i (α1 , α) m−1
i
α=1
i=0
Ti2 i (α2 , α) `=0
η(i1 −i2 +k(`1 −`2 ))` .
Observe that, since η is a primitive m-root of unity,
m−1
X
`=0
η((i1 −i2 )+k(`1 −`2 ))` =
½
m
0
if (i 1 − i 2 ) + k(`1 − `2 ) ≡ 0
otherwise.
(mod m),
Observe that (i 1 − i 2 ) + k(`1 − `2 ) ≡ 0 (mod m) if and only if i 1 = i 2 and `1 ≡ `2
(mod t), since 0 ≤ i 1 , i 2 ≤ k − 1 and m = kt.
Now suppose that Ti j are type II (i, j = 0, . . . , k − 1). We must show that the l.h.s. of
(11) becomes zero for [i 1 , `1 , α1 ] 6= [i 2 , `2 , α2 ]. By the above observation, we may assume
that i 1 = i 2 and `1 ≡ `2 (mod t). We set `1 − `2 = ts. If α1 6= α2 , then l.h.s. of (11)
vanishes by type II condition for Ti1 i . Hence we may assume α1 = α2 . Thus we have
i 1 = i 2 , α1 = α2 , `1 − `2 ≡ 0 (mod t) and `1 6= `2 . Hence
l.h.s. = mr η−`1 i1 +`2 i2 −(k/2)(`1 −`2 )(`1 +`2 −1)
k−1
X
i=0
ηtsi .
67
NON-SYMMETRIC SPIN MODELS
Pk−1 t si
(η ) = 0, since s 6≡ 0 (mod k).
Observe that ηt is a primitive k-root of unity. So, i=0
We have shown that W is type II.
Next suppose that W is type II. Pick any distinct α1 , α2 ∈ {1, . . . , r }. From (11) at i 1 = i 2
and `1 ≡ `2 (mod t), we obtain
k−1
X
η(`1 −`2 )i
i=0
r
X
Tii i (α1 , α)
= 0.
T (α2 , α)
α=1 i 2 i
Setting
Ki =
r
X
Ti1 i (α1 , α)
T (α2 , α)
α=1 i 1 i
and considering the case `2 = 0, the above equation implies
k−1
X
¡
¢i
η `1 K i = 0
(`1 = 0, t, 2t, . . . , (k − 1)t),
i=0
or equivalently
k−1
X
(ηt )ei K i = 0
(e = 0, 1, . . . , k − 1).
i=0
Observe that (ηt )e (e = 0, 1, . . . , k − 1) are distinct, since ηt is a primitive k-root of unity.
Hence K i = 0 (i = 0, 1, . . . , k − 1) by Vandermonde determinant. Thus
r
X
Ti1 i (α1 , α)
=0
T (α2 , α)
α=1 i 1 i
(i = 0, . . . , k − 1),
2
so that Ti1 i is type II.
Lemma 5.2 Assume k is even when m is even. Then the matrix W satisfies the type III
condition (2) if and only if the following equation holds for all i 1 , i 2 , i 3 ∈ {0, . . . , k − 1}
and for all α1 , α2 , α3 ∈ {1, . . . , r }:
k−1
X
i=0
Ã
m−1
X
!Ã
η
−k`
γk (`, i − i 1 − i 2 + i 3 )
`=0
=D
Ti1 ,i2 (α1 , α2 )
.
Ti1 ,i3 (α1 , α3 )Ti3 ,i2 (α3 , α2 )
r
X
Ti1 ,i (α1 , α)Ti2 ,i (α2 , α)
Ti3 ,i (α3 , α)
α=1
!
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IKUTA AND NOMURA
Proof: The type III condition (2) for a = [i 1 , `1 , α1 ], b = [i 2 , `2 , α2 ], c = [i 3 , `3 , α3 ]
becomes
r
k−1 m−1
X
XX
γk (`1 − `, i 1 − i)γk (`2 − `, i 2 − i) Ti1 ,i (α1 , α)Ti2 ,i (α2 , α)
·
γk (`3 − `, i 3 − i)
Ti3 ,i (α3 , α)
i=0 `=0 α=1
=D
Ti1 ,i2 (α1 , α2 )
γk (`1 − `2 , i 1 − i 2 )
·
.
γk (`1 − `3 , i 1 − i 3 )γk (`3 − `2 , i 3 − i 2 ) Ti1 ,i3 (α1 , α3 )Ti3 ,i2 (α3 , α2 )
By a direct (but somewhat long) computation, we obtain
γk (`1 − `, i 1 − i)γk (`2 − `, i 2 − i) γk (`1 − `3 , i 1 − i 3 )γk (`3 − `2 , i 3 − i 2 )
·
γk (`3 − `, i 3 − i)
γk (`1 − `2 , i 1 − i 2 )
ˆ
ˆ i − î),
= η−k(`−`) γk (` − `,
where `ˆ = `1 + `2 − `3 , î = i 1 + i 2 − i 3 . So the type III condition becomes
!
!Ã
Ã
k−1
m−1
r
X
X
X
Ti1 ,i (α1 , α)Ti2 ,i (α2 , α)
ˆ
−k(`−`)
ˆ
η
γk (` − `, i − î)
Ti3 ,i (α3 , α)
i=0
`=0
α=1
=D
Ti1 ,i2 (α1 , α2 )
.
Ti1 ,i3 (α1 , α3 )Ti3 ,i2 (α3 , α2 )
To complete our proof, we must show that
m−1
X
`=0
ˆ
ˆ i − î) =
η−k(`−`) γk (` − `,
m−1
X
η−k` γk (`, i − î).
`=0
To show this, it is enough to show that γk (` + m, j) = γk (`, j) holds for all j, `.
γk (` + m, j) = η−(`+m) j−(k/2)(`+m)(`+m−1)
= γk (`, j)η−m j−km` η−(k/2)m(m−1)
= γk (`, j)η−km(m−1)/2 .
When m is odd, (m − 1)/2 is an integer. When m is even, k is even by our assumption, and
2
so k/2 is an integer. Thus η−km(m−1)/2 = 1.
Lemma 5.3 For all u, s (0 ≤ u ≤ t − 1, 0 ≤ s ≤ k − 1),
γk (u + st, j) = ((−1)t−1 η−t j )s γk (u, j).
Proof:
We compute γk (u + st, j) as follows.
γk (u + st, j) = η−(u+st) j−(k/2)(u+st)(u+st−1)
= η−u j−(k/2)u(u−1) η−st j η−(kt)su η−(k/2)st (st−1)
= γk (u, j)η−st j η−(k/2)st (st−1) .
69
NON-SYMMETRIC SPIN MODELS
So it is enough to show that
η−(k/2)st (st−1) = (−1)(t−1)s .
(12)
If t is even, then m is even and so η(m/2) = −1. Hence, noting st − 1 is odd,
¡
¢(st−1)s ¡
¢s
η−(k/2)st (st−1) = η−(m/2)
= (−1)(st−1) = (−1)s ,
so (12) holds. Next assume t is odd. If s is even, then
η−(k/2)st (st−1) = η−(kt)(s/2)(st−1) = η−m(s/2)(st−1) = 1.
If s is odd, then st − 1 is even. Hence
η−(k/2)st (st−1) = η−(kt)s(st−1)/2 = (η−m )s(st−1)/2 = 1.
2
Therefore (12) holds in each case.
Lemma 5.4
(i) If t is odd, then
m−1
X
η−k` γk (`, j) =
`=0
 t−1
X

k
η−u j−ku(u+1)/2


0
(mod k),
otherwise.
(ii) If t and k are even, then
 t−1
X

m−1
k
X
η−u j−ku(u+1)/2
−k`
η γk (`, j) =
u=0


`=0
0
Proof:
if j ≡ 0
u=0
if j ≡
k
2
(mod k),
otherwise.
Using Lemma 5.3, we proceed as follows.
m−1
X
η−k` γk (`, j) =
`=0
k−1
t−1 X
X
η−k(u+st) γk (u + st, j)
u=0 s=0
=
k−1
t−1 X
X
η−(ku+ms) ((−1)t−1 η−t j )s γk (u, j)
u=0 s=0
Ã
=
k−1
X
((−1)t−1 η−t j )s
s=0
!Ã
t−1
X
!
η−ku γk (u, j) .
u=0
If t is odd, then the first factor becomes
(
k−1
k−1
X
X
k if j ≡ 0 (mod k),
−t j s
−t js
(η ) =
(η ) =
0 otherwise.
s=0
s=0
70
IKUTA AND NOMURA
Suppose t and k are even. In this case, m is also even, so that η(kt/2) = η(m/2) = −1. Hence
the first factor becomes
k−1
k−1
X
X
¡ (kt/2) −t j ¢s
((−1)η−t j )s =
η
η
s=0
s=0
=
k−1
X
(ηt )((k/2)− j)s
s=0
=

k

0
k
− j ≡0
2
otherwise.
if
(mod k),
Now the result follows by
η−ku γk (u, j) = η−u j−ku(u+1)/2 .
2
Proposition 5.5 Assume k is even when m is even. Then the matrix W satisfies the type
III condition (2) if and only if the following equation holds for all i 1 , i 2 , i 3 ∈ {0, . . . , k − 1}
and for all α1 , α2 , α3 ∈ {1, . . . , r }:
!
!Ã
Ã
t−1
r
X
X
T
(α
,
α)T
(α
,
α)
i
,i
1
i
,i
2
1
2
η−u(i −î)−ku(u+1)/2
Ti3 ,i (α3 , α)
u=0
α=1
= (D/k)
Ti1 ,i2 (α1 , α2 )
,
Ti1 ,i3 (α1 , α3 )Ti3 ,i2 (α3 , α2 )
where î = i 1 + i 2 − i 3 , and i denotes the integer in {0, . . . , k − 1} such that

 î (mod k)
if t is odd,
i≡
 î + k (mod k) if t is even.
2
Proof:
6.
This is a direct consequence of Lemmas 5.2 and 5.4.
Some special cases
We use the notation in Section 4.
Proposition 6.1 Suppose k = 1. Then m is odd, and
W = S ⊗ T,
where S is a spin model of size m and index m which is given by
0
0
S(`, `0 ) = η−(1/2)(`−` )(`−` −1)
(`, `0 = 0, 1, . . . , m − 1),
and T is a symmetric spin model of size n/m.
2
71
NON-SYMMETRIC SPIN MODELS
Proof: When k = 1, we have X = 10 and r = n/m. Setting S = S00 and T = T00 ,
W = W |10 = S ⊗ T by Proposition 4.3. Obviously S(`, `0 ) = S00 (`, `0 ) = γ1 (` − `0 , 0).
For α, β ∈ {1, . . . , r }, T (α, β) = W (yα0 , yβ0 ) = η0−0 W (yβ0 , yα0 ) = T (β, α), so that T is
symmetric.
Since m is odd by Lemma 4.2, S is a spin model on the cyclic group of order m, which
was constructed in [2] (see also [1, 3]). Since W = S ⊗ T with W , S are spin models, T
must be a spin model.
2
Proposition 6.2
Suppose k = m. Then
W | X i ×X j = Si j ⊗ Ti j
(i, j = 0, 1, . . . , m − 1),
and
0
Si j (`, `0 ) = η−(`−` )(i− j)
(`, `0 = 0, . . . , m − 1).
The matrices Ti j are type II matrices of size r = n/m 2 . Moreover the following equation
holds for all i 1 , i 2 , i 3 ∈ {0, . . . , m − 1} and for all α1 , α2 , α3 ∈ {1, . . . , r }:
r
X
Ti1 ,i2 (α1 , α2 )
Ti1 ,i (α1 , α)Ti2 ,i (α2 , α)
= (D/m)
,
Ti3 ,i (α3 , α)
Ti1 ,i3 (α1 , α3 )Ti3 ,i2 (α3 , α2 )
α=1
where i denotes the integer in {0, . . . , m − 1} such that i ≡ i 1 + i 2 − i 3
(mod m).
Proof: We have t = 1, r = n/(mk) = n/m 2 and X i = 1i (i = 0, . . . , m − 1). Hence
W | X i ×X j = Si j ⊗ Ti j by Proposition 4.3. Since η(m/2)`(`−1) = (ηm )`(`−1)/2 = 1, γm (`, i) =
0
η−`i . So Si j (`, `0 ) = η−(`−` )(i− j) . By Proposition 5.1, Ti j is a type II matrix. Now the
result follows from Proposition 5.1 and 5.5.
2
Corollary 6.3 Let W be a spin model on X of prime index m. Then one of the following
holds, where η denotes a primitive m-root of unity.
(i) W = S ⊗ T , where S is a spin model of size m with
0
0
S(`, `0 ) = η−(1/2)(`−` )(`−` −1)
(`, `0 = 0, 1, . . . , m − 1),
and T is a symmetric spin model of size |X |/m.
(ii) W decomposes into m 2 blocks Wi j (i, j = 0, . . . , m − 1) with Wi j = Si j ⊗ Ti j , where
Si j are matrices of size m defined by
0
Si j (`, `0 ) = η−(`−` )(i− j)
(`, `0 = 0, 1, . . . , m − 1),
and Ti j are type II matrices of size r = n/m 2 which satisfy the following equation for
all i 1 , i 2 , i 3 ∈ {0, . . . , m − 1} and for all α1 , α2 , α3 ∈ {1, . . . , r }:
r
X
Ti1 ,i2 (α1 , α2 )
Ti1 ,i (α1 , α)Ti2 ,i (α2 , α)
= (D/m)
,
Ti3 ,i (α3 , α)
Ti1 ,i3 (α1 , α3 )Ti3 ,i2 (α3 , α2 )
α=1
where i denotes the integer in {0, . . . , m − 1} such that i ≡ i 1 + i 2 − i 3
(mod m).
72
IKUTA AND NOMURA
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