Periodic cellular automata and Bethe ansatz

0.16 MB·

by  Atsuo Kuniba

PERIODIC CELLULAR AUTOMATA AND BETHE ANSATZ
ATSUOKUNIBAANDAKIRATAKENOUCHI
6
0
0
Abstract. We review and generalize the recent progress in a soliton cellular
2
automaton known as the periodic box-ball system. It has the extended affine

n Weyl group symmetry and admits the commuting transfer matrix method and
a the Bethe ansatz at q = 0. Explicit formulas are proposed for the dynamical
J periodandthenumberofstates characterizedbyconservedquantities.

7

1. Introduction
3
v
In [13], a class of periodic soliton cellular automata is introduced associated with
3
non-exceptionalquantumaffinealgebras. Thedynamicalperiodanda statecounting
1
0 formula are proposed by the Bethe ansatz at q =0 [11]. In this paper we review and
1 generalize the results on the A(1) case, where the associated automaton is known as
n
1 the periodic box-ball system [14, 19]. The box-ball system was originally introduced
5
on the infinite lattice without boundary [18, 17]. Here is a collision of two solitons
0
withamplitudes3and1interchanginginternaldegreesoffreedomwithaphaseshift:
/
h
1114221111131111111111111111
p
··· ···
- 1111114221113111111111111111
h ··· ···
1111111114221311111111111111
t ··· ···
a 1111111111114232111111111111
m ···1111111111111121432111111111···
··· ···
: 1111111111111112111432111111
v ··· ···
1111111111111111211111432111
Xi ··· ···
The system was identified [4, 3] with a solvable lattice model [1] at q = 0, which
r led to a direct formulation by the crystal base theory [5] and generalizations to the
a
soliton cellular automata with quantum group symmetry [7, 6]. Here we develop the
approach to the periodic case in [13] further by combining the commuting transfer
matrix method [1] and the Bethe ansatz [2] at q =0.
In section 2 we formulate the most general periodic automaton for g = A(1) in
n
terms of the crystal theory. A commuting family of time evolutions T(a) is intro-
{ j }
duced as commuting transfer matrices under periodic boundary condition. The asso-
ciatedconservedquantitiesformann-tupleofYoungdiagramsm=(m(1),...,m(n)),
which we call the soliton content.
Insection3weinvoketheBetheansatzatq =0[11]tostudythe Betheeigenvalue
Λ(r) relevantto T(r). The Betheequationis linearizedinto thestringcenterequation
l l
and the Λ(r) is shown to be a root of unity. We also recall an explicit weight mul-
l
tiplicity formula obtained by counting the off-diagonal solutions to the string center
equation [11]. It is a version of the fermionic formula called the combinatorial com-
pleteness of the string hypothesis at q =0. These results are parameterizedwith the
number of strings, which we call the string content.
1
2 ATSUOKUNIBAANDAKIRATAKENOUCHI
In section 4 two applications of the results in section 3 are presented under the
identification of the soliton and the string contents. First we relate the root of unity
in the Bethe eigenvalue Λ(r) to the dynamical period of the periodic A(1) automaton
l n
under the time evolution T(r). Second we connect each summand in the weight
l
multiplicityformula[11]tothenumberofstatescharacterizedbyconservedquantities.
In [13], similar results have been announced concerning the highest states. Our
approach here is based on conserved quantities and covers a wider class of states
without recourseto the combinatorialBethe ansatz at q =1 [10]. We expect parallel
results in generalg . In fact all the essentialclaims in this paper make sense also for
n
g = D(1) and E(1) . Our formulas (11) and (13) include the results in [19] proved
n n 6,7,8
by a different approachas the case g =A(1) with B =(B1,1)⊗L and l = . For the
n 1 ∞
standard notation and facts in the crystal theory, we refer to [5, 9, 6].
2. Periodic A(1) automaton
n
Let Ba,j (1 a n,j Z≥1) be the crystal [9] of the Kirillov-Reshetikhin mod-
≤ ≤ ∈
ule W(a) over U (A(1)). Elements of Ba,j are labeled with semistandard tableaux
j q n
on an a j rectangular Young diagram with letters 1,2,...,n + 1 . For exam-
× { }
ple when n = 2, one has B1,1 = 1,2,3 ,B1,2 = 11,12,13,22,23,33 , B2,2 =
1 1 1 1 1 1 1 2 1 2 {2 2 } { }
2 2 , 2 3 , 3 3 , 2 3 , 3 3 , 3 3 as sets. Aff(Ba,j) = ζdb b Ba,j,d
{ | ∈ ∈
Zn denotestheaffinecrystal. ThecombinatoorialRistheisomorphismofaffinecrystals
A}ff(Ba,j) Aff(Bb,k) ∼ Aff(Bb,k) Aff(Ba,j) [16]. It has the form R(ζdb ζec) =
⊗ → ⊗ ⊗
ζe+Hc˜ ζd−H˜b, where H = H(b c) is the energy function. We normalize it so
⊗ ⊗
as to attain the maximum at H(ua,j ub,k) = 0, where ua,j Ba,j denotes the
⊗ ∈
classically highest element. We set B = Br1,l1 Br2,l2 BrL,lL and write
⊗ ⊗ ··· ⊗
Aff(Br1,l1) Aff(BrL,lL) simply as Aff(B). An element of B is called a state.
⊗···⊗
Givenastatep=b b B,regarditastheelementζ0b ζ0b Aff(B)
1 L 1 L
⊗···⊗ ∈ ⊗···⊗ ∈
and seek an element v Br,l such that ζ0v p (ζd1b′ ζdLb′ ) ζev un-
∈ ⊗ ≃ 1 ⊗···⊗ L ⊗
der the isomorphism Aff(Br,l) Aff(B) Aff(B) Aff(Br,l). If such a v exists and
⊗ ≃ ⊗
ζd1b′ ζdLb′ isuniqueevenifvisnotunique,wesaythatpis(r,l)-evolvableand
1⊗···⊗ L
writeT(r)(p)=b′ b′ B andE(r)(p)=e= d d . Otherwisewesay
l 1⊗···⊗ L ∈ l − 1−···− L
that p is not (r,l)-evolvable or Tl(r)(p)=0. We formally set Tl(r)(0)=0. El(r) ∈Z≥0
holdsunderthisnormalization. OurA(1) automatonisadynamicalsystemonB 0
n
⊔{ }
equipped with the family of time evolutions {Tl(r) | 1 ≤ r ≤ n,l ∈ Z≥1}. Tl(r) is the
q = 0 analogue of the transfer matrix in solvable vertex models. It is invertible and
weightpreservingonB. Usingthe Yang-Baxterequationofthe combinatorialR,one
can show (cf. [3, 6])
Theorem2.1. ThecommutativityT(a)T(b)(p)=T(b)T(a)(p)isvalidforany(a,j),(b,k)
j k k j
andp B,wherethebothsidesareeitherinB or0. IntheformercaseE(a)(T(b)(p))=
∈ j k
E(a)(p) and E(b)(T(a)(p))=E(b)(p) hold.
j k j k
Thus, {Ej(a) |1≤a≤n,j ∈Z≥1} is a family of conserved quantities.
Conjecture2.1. Forany1 a nandp B,thereexistsi 1suchthatT(a)(p)=0
≤ ≤ ∈ ≥ k 6
if and only if k i. The limit lim T(a)(p) B exists and E(a)(p) < E(a)(p) <
≥ k→∞ k ∈ i i+1
<E(a)(p)=E(a) (p)= holds for some j i.
··· j j+1 ··· ≥
PERIODIC CELLULAR AUTOMATA AND BETHE ANSATZ 3
LetS ,S ,...,S betheWeylgroupoperators[5]andprbethepromotionoperator
0 1 n
[16] acting on B component-wise. For instance for A(31), pr 23 23 34 ⊗ 1344 =
14 34 34 1124 B2,3 B1,4. They act on B as the extend(cid:16)ed affine Weyl gr(cid:17)oup
⊗ ∈ ⊗
W(g )=W(A(1))= pr,S ,S ,...,S .
n n 0 1 n
h i
Theorem 2.2. If T(a)(p) = 0, then for any w W(A(1)), the relations wT(a)(p) =
f f j 6 ∈ n j
T(a)(w(p)) and E(a)(w(p))=E(a)(p) are valid.
j j j
f
A state p B is called evolvable if it is (a,j)-evolvable for any (a,j). In Conjec-
∈
ture 2.1, we expect that the convergent limit T(a) equals a translation in W(A(1)).
∞ n
Compared with T in [13], the family T(a) here is more general and enjoys a larger
l { j }
symmetry W(A(1)). Define the subset of B by f
n
(1) P(m)= p B p:evolvable,E(a)(p)= min(j,k)m(a) .
f { ∈ | j k }
k≥1
X
Pictorially, m = (m(1),...,m(n)) is the n-tuple of Young diagrams and E(a) (resp.
k
m(a)) is the number of nodes in the first k columns (resp. number of length k rows)
k
in m(a). We call m the soliton content.
Remark 2.1. P(m) is W(A(1))-invariant due to Theorem 2.2.
n
Given p P(m), T(a)(p) P(m) is not necessarily valid. For instance p =
∈ jf ∈
112233 P(((22),(2))) B = (B1,1)⊗6 but T(2)(p) = 213213 is not evolvable since
∈ ⊂ 1
(T(2))2(p)=0. On the other hand one can show
1
Proposition 2.1. If p P(m) and (T(a))t(p)=0 for any t, then (T(a))t(p) P(m)
∈ j 6 j ∈
for any t.
Let W,Λ ,α be the Weyl group,the fundamental weightsandthe simple roots of
a a
A , respectively. We specify p(a) =p(a)(m) by (6) and set
n j j
n
(2) λ(m)= p(∞a)Λa, H ={(a,j)|1≤a≤n,j ∈Z≥1,m(ja) >0}.
a=1
X
Conjecture 2.2. P(m) = if and only if p(a) 0 for all (a,j) H. wtp p
6 ∅ j ≥ ∈ { | ∈
P(m) =Wλ(m).
}
The claim on the weights is consistent with the W(A(1))-invariance of P(m).
n
HereisanexampleoftimeevolutionsinB =B1,1 B1,1 B1,3 B1,1 B1,1 B1,1
⊗ ⊗ ⊗ ⊗ ⊗ ⊗
B1,2 with W(A(1)) symmetry. The leftmost columfn is p ,T(1)(p ),T(2)T(1)(p ) and
3 0 2 0 1 2 0
(3) (2) (1)
T T T (p )fromthe toptothe bottom. Ateachtime step,the statesconnected
2 1 2 0
by the Weyfl group actions S and S are shown, forming commutative diagrams. (
0 1
·
signifies .) All these states belong to P(((3111),(21),(1))).
⊗
2 1 233 4 1 2 12 S0 2 4 233 4 1 2 24 S1 1 4 133 4 1 2 14
· · · · · · 7→ · · · · · · 7→ · · · · · ·
1 2 123 3 4 1 22 4 2 234 3 4 1 22 4 1 134 3 4 1 12
· · · · · · · · · · · · · · · · · ·
1 2 112 3 2 3 24 1 2 244 3 2 3 24 1 2 144 3 1 3 14
· · · · · · · · · · · · · · · · · ·
2 3 112 4 2 3 12 2 3 244 4 2 3 12 2 3 144 4 1 3 11
· · · · · · · · · · · · · · · · · ·
4 ATSUOKUNIBAANDAKIRATAKENOUCHI
Remark 2.2. Let R (1 j L 1) be the combinatorial R that exchanges the j-th
j
≤ ≤ −
and (j+1)-st components in B =Br1,l1 BrL,lL and π(b1 bL)=bL b1
⊗···⊗ ⊗···⊗ ⊗ ⊗
···⊗bL−1. Together with R0 = π−1R1π, they act on ∪s∈SLBrs1,ls1 ⊗···⊗BrsL,lsL
as the extended affine Weyl group W(A(1) ) = π,R ,...,R . Theorem 2.2 is
L−1 h 0 L−1i
actually valid for w W(g =A(1)) W(A(1) ) as in [8]. In the homogeneous case
(r ,l ) = = (r ,∈l ), thne W(nA(1f)×) symLm−e1try shrinks down to the π-symmetry,
wh1ich1 is th·e··originLofLtfhe adjectiveL“−p1erfiodic”.
f
3. Bethe ansatz at q =0
Eigenvalues of row transfer matrices in trigonometric vertex models are given by
the analytic Bethe ansatz [15, 12]. Let Q (u)= sinhπ(u √ 1u(r)) be Baxter’s
r k − − k
Q-functionwhere u(a) satisfythe Bethe equationeq.(2.1)in[11]. We setq =e−2π~
{ j } Q
and ζ = e2πu. For the string solution ([11] Definition 2.3), the relevant quantity to
our T(r) is the top term of the eigenvalue Λ(r)(u) (cf.[12] (2.12)):
l l
(3) qli→m0QQrr((uu+−ll~~)) =ζ−El(r)Λl(r), Λl(r) := (−zj(rα))min(j,l).
jα
Y
Herez(a) isthe centerofthe α-th stringhavingcolora andlengthj. Denote by m(a)
jα j
the number of such strings. We call the data m = (m(a)) the string content. The
j
product in (3) is taken over j ∈ Z≥1 and 1 ≤ α ≤ mj(r). El(r) is given by the same
expression as in (1) as the function of m. At q = 0 the Bethe equation becomes the
string center equation ([11] (2.36)):
m(b)
k
(4) (z(b))Aajα,bkβ =( 1)pj(a)+mj(a)+1,
kβ −
(b,Yk)∈HβY=1
(a) (a)
(5) A =δ δ δ (p +m )+C min(j,k) δ δ ,
ajα,bkβ ab jk αβ j j ab − ab jk
L
(6) p(a) = min(j,l )δ C min(j,k)m(b),
j i ari − ab k
Xi=1 (b,Xk)∈H
where (C ) is the Cartanmatrix ofA . To avoida notationalcomplexity we
ab 1≤a,b≤n n
temporally abbreviate the triple indices ajα to j, bkβ to k andaccordingly z(b) to z
kβ k
etc. Then (3) and (4) read
(7) Λ(r) = ( z )ρk, ( z )Aj,k =( 1)sj,
l − k − k −
k k
Y Y
whereρ isgivenbyρ =δ min(k,l)forkcorrespondingtobkβ,ands isaninteger.
k k br j
NotethatAj,k =Ak,j. Supposethattheq =0eigenvaluesatisfies(Λl(r))Pl(r) =±1for
genericsolutions to the string center equation1. It means thatthere existintegersξ
j
suchthat ξ A = (r)ρ ,orequivalentlyξ = (r)detA[j],whereA[j]denotesthe
j j j,k Pl k j Pl detA
matrix A=(A ) with its j-th column replaced by t(ρ ,ρ ,...). In view of the con-
P j,k 1 2
dition∀ξj ∈Z,the minimumintegerallowedforPl(r) isPl(r) =LCM 1, k′ddeteAtA[k] ,
whereLCMstandsforthe leastcommonmultiple and ′ meansthe(cid:16)unionovertho(cid:17)se
∪k S
1Pl(r) hereshouldnotbeconfusedwiththesymbolin(6).
PERIODIC CELLULAR AUTOMATA AND BETHE ANSATZ 5
k such that A[k] = 0. Back in the original indices, the determinants here can be
6
simplified (cf. [11] (3.9)) to those of matrices indexed with H:
detF
(8) (r) =LCM 1, ′ ,
Pl detF[b,k]
(cid:16) (b,[k)∈H (cid:17)
where the matrix F =(F ) is defined by
aj,bk (a,j),(b,k)∈H
(9) F =δ δ p(a)+C min(j,k)m(b).
aj,bk ab jk j ab k
The matrix F[b,k] is obtained from F by replacing its (b,k)-th column as
F (c,m)=(b,k),
aj,cm
(10) F[b,k] = 6
aj,cm
(δarmin(j,l) (c,m)=(b,k).
The union in (8) is taken over those (b,k) such that detF[b,k]=0.
6
(1) (1)
The LCM(8)canfurtherbe simplifiedforA ,r=1, r = l =1. We write p
1 ∀ i ∀i j
just as pj and parameterize the set H = {j ∈ Z≥1 | m(j1) > 0} as H = {(0 <)J1 <
<J . Setting i =min(J ,l) and i =0, one has
s k k 0
··· }
t
p p
(11) (1) =LCM 1, ′ ik+1 ik ,
Pl k=0 (ik+1−ik)pis!
[
where 0 t s 1 is the maximum integer such that i >i .
t+1 t
≤ ≤ −
Let us turn to another Bethe ansatz result, the character formula called combina-
torial completeness of the string hypothesis at q =0 [11]:
L
(12) chBri,li = Ω(m)eλ(m),
i=1 m
Y X
(13) Ω(m)=detF 1 p(ja)+m(ja)−1 Z,
m(a) m(a) 1 ∈
(a,Yj)∈H j (cid:18) j − (cid:19)
where s = s(s 1) (s t+1)/t! and chBr,l is the character of Br,l. λ(m), p(a)
t − ··· − j
and F (cid:0)ar(cid:1)e defined by (2), (6) and (9). The sum in (12) extends over all m(ja) ∈ Z≥0
canceling out exactly leaving the character of B. (12) and (13) are the special cases
of eq.(5.13) and eq.(4.1) in [11], respectively. Ω(m) (denoted by R(ν,N) therein) is
the number ofoff-diagonalsolutions to the string center equationwith string content
m. It is known ([11] Lemma 3.7) that Ω(m) ∈ Z≥1 provided that p(ja) ≥ 0 for all
(a,j) H.
∈
4. Dynamical period and state counting
In (1), the soliton content m = (m(a)) is introduced as the conserved quantity
j
associated with the commuting transfer matrices. One the other hand, the m(a) in
j
the string content m=(m(a)) is the number of strings of color a and length j in the
j
Bethe ansatz in section 3. From now on we identify them motivated by the factor
ζ−El(r) in (3) andsome investigationofBethe vectorsatq =0. In view of Conjecture
2.2, the data of the form m=(m(a)) is defined to be a content if and only if p(a) 0
j j ≥
for all (a,j) H. Thus detF > 0 and λ(m) in (2) is a dominant weight for any
∈
content m.
6 ATSUOKUNIBAANDAKIRATAKENOUCHI
Conjecture 4.1. If p P(m) and (T(r))t(p)=0 for any t, the dynamical period of
∈ l 6
p under T(r) (minimum positive integer t such that (T(r))t(p) = p) is equal to (r)
l l Pl
(8) generically and its divisor otherwise.
In the situation under consideration, the whole T(r) orbit of p belongs to P(m)
l
due to Proposition 2.1. Naturally we expect (Λl(r))Pl(r) = 1, which can indeed be
(1) (1)
verified for A . Conjecture 4.1 has been checked, for example in A case, for
1 3
B =(B1,1)⊗3 B2,2 and B2,1 B2,1 B3,1 B3,2.
⊗ ⊗ ⊗ ⊗
LetuspresentmoreevidenceofConjecture4.1. Tosavethespace, = isdropped
· ⊗
when B = (B1,1)⊗L. In each table, the period under T(r) with maximum l is equal
l
to that under T(r).
∞
(1)
A , state =1221121122221, content =((321))
1
(r,l) LCM of = period
(1,1) 1, 13, 13, 13 13
(1,2) 1, 91, 91, 91 91
3 16 16
(1,3) 1, 91, 273, 273 273
16 107
A(1), state =122 112 12 1222 2 11111 1122 111, content =((4321))
1 · · · · · · ·
(r,l) LCM of = period
(1,1) 1, 2 2
(1,2) 1, 7, 7, 21, 42 42
2
(1,3) 1, 14, 7, 21, 21 42
4 2
(1,4) 1, 21, 21, 63, 126 126
2 8 29
A(1), state =134 34 1 134 23 1 13, content =((432),(31),(1))
3 · · · · · ·
(r,l) LCM of = period
(1,1) 1, 380, 95, 95, 380, 380, 380 380
39 6 6 31 27 29
(1,2) 1, 190, 95, 95, 190, 190, 190 190
39 12 12 31 27 29
(2,1) 1, 190, 95, 95, 190, 190, 190 190
13 4 4 137 9 73
(2,2) 1, 76, 38, 38, 76, 76, 76 76
5 3 3 41 21 31
(2,3) 1, 95, 95, 95, 95, 95, 95 95
6 11 11 34 48 41
(3,1) 1, 380, 95, 95, 380, 380, 380 380
13 2 2 137 9 263
1 2
A(1), state=233 2 3 1 1 1, content =((3),(3),(2))
3 · 3 4 · 3 4 ·
(r,l) LCM of = period
(1,1) 1, 11, 11, 22 22
2
(1,2) 1, 11, 11, 11 11
4 2
(1,3) 1, 11, 11, 22 22
6 3 3
(2,1) 1, 11, 33, 66 66
7 7
(2,2) 1, 11, 33, 33 33
2 14 7
(2,3) 1, 11, 11, 22 22
3 7 7
(3,2) 1, 11, 33, 33 33
7 20
PERIODIC CELLULAR AUTOMATA AND BETHE ANSATZ 7
Here, content=((3111),(44),(2)) for example means that m(1)=3,m(1)=m(3)=
1 3 2
1,m(2)=2 and the other m(a)’s are 0.
4 j
Let us turn to another application of the Bethe ansatz results (12) and (13). We
introduce (P(m)) = n T(a)(p) p P(m) , which is the subset of B
T a=1 j≥1{ j | ∈ }
consisting of all kinds of one step time evolutions of P(m). Under Conjecture 2.1,
S S
any state p P(m) is (a,j)-evolvable for j sufficiently large. Thus from Proposition
∈
2.1, p is expressed as p = (T(a))k(p) for some k, showing that (P(m)) P(m). In
j T ⊇
general (P(m)) can contain non-evolvable states which do not belong to P(m).
T
Conjecture 4.2. For any content m such that (P(m)) = P(m), the following
T
relation holds:
P(m)
(14) Ω(m)= | | .
Wλ(m)
| |
In view of Remark 2.1 and Conjecture 2.2, the right hand side is the number of
statesinthe periodic A(1) automatonhavingthe contentm anda fixedweight. Thus
n
it is equal to ♯ p P(m) wtp = λ(m) . In case (P(m)) % P(m), we expect that
{ ∈ | } T
P(m)/Wλ(m) is a divisor of Ω(m).
| | | |
Let us present two examples of Conjecture 4.2. In the periodic A(1) automaton
3
with B = B1,2 B1,1 B1,2 B1,1, there are 1600 states among which 824 are
⊗ ⊗ ⊗
evolvable. They are classified according to the contents m in the following table.
m λ(m) Wλ(m) P(m) Ω(m)
| | | |
( , , ) (6,0,0,0) 4 4 1
∅ ∅ ∅
((1), , ) (5,1,0,0) 12 48 4
∅ ∅
((11), , ) (4,2,0,0) 12 24 2
∅ ∅
((2), , ) (4,2,0,0) 12 72 6
∅ ∅
((21), , ) (3,3,0,0) 6 24 4
∅ ∅
((3), , ) (3,3,0,0) 6 36 6
∅ ∅
((11),(1), ) (4,1,1,0) 12 96 8
∅
((22),(2), )∗ (2,2,2,0) 4 24 12
∅
((21),(1), ) (3,2,1,0) 24 432 18
∅
((111),(11),(1)) (3,1,1,1) 4 16 4
((211),(11),(1)) (2,2,1,1) 6 48 8
In the second column, (λ ,λ ,λ ,λ ) means λ(m) = (λ λ )Λ +(λ λ )Λ +
1 2 3 4 1 2 1 2 3 2
− −
(λ λ )Λ . In the last two cases, the subsets of P(m) having the dominant weight
3 4 3
−
λ(m) are given by
11 2 13 4, 12 3 14 1, 13 4 11 2, 14 1 12 3 form=((111),(11),(1)),
{ · · · · · · · · · · · · }
11 2 23 4, 12 2 13 4, 12 3 24 1, 13 4 12 2,
{ · · · · · · · · · · · ·
14 1 22 3, 22 3 14 1, 23 4 11 2, 24 1 12 3 form=((211),(11),(1)).
· · · · · · · · · · · · }
In the case of B = B2,1 B2,1 B2,2, there are 720 states among which 518 are
⊗ ⊗
evolvable.
8 ATSUOKUNIBAANDAKIRATAKENOUCHI
m λ(m) Wλ(m) P(m) Ω(m)
| | | |
( , , ) (4,4,0,0) 6 6 1
∅ ∅ ∅
( ,(1), ) (4,3,1,0) 24 72 3
∅ ∅
( ,(11),(1)) (4,2,1,1) 12 36 3
∅
( ,(2), ) (4,2,2,0) 12 48 4
∅ ∅
((1),(11), ) (3,3,2,0) 12 36 3
∅
((1),(11),(1)) (3,3,1,1) 6 72 12
((1),(21),(1)) (3,2,2,1) 12 240 20
((2),(22),(2))∗ (2,2,2,2) 1 8 32
Theassumption (P(m))=P(m)oftheconjectureisvalidforallthecontentsexcept
T
((22),(2), ) and ((2),(22),(2)) marked with .
∅ ∗
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Institute of Physics, University of Tokyo, Tokyo 153-8902,Japan
E-mail address: [email protected]
Institute of Physics, University of Tokyo, Tokyo 153-8902,Japan
E-mail address: [email protected]