肥覇 ¨

覇¨
肥
17 Junc 1996
PHYS!CS LETTERS A
Physics Lcttcrs A 216(199037-46
An alternative proof for the minimal dimension of the trace map:
The intersection of algebraic varieties
Kazumoto Iguchi
The Institute of Physical and Ch.emical Research
'
RIKEN), 2-l Hirosawa, Wako-shi, Saitama
35 t -01, Japan
Received 5 June 1995; revised manuscript received 7 March 1996; accepted for publication 18 March 1996
Communicared by A.R. Bishop
Abstract
Rcccntly,A宙 shai,Bcrend,and Glaubman IPhys.Rcv.Lctt.72(19901842]obmincd thc minimal dimcnsion 3″ -3 of
hc tracc maps in tlle thcoヮ
of quasipcrioliC and apcriodic lattices constructcd by dctcrlllinistic substitution schemcs of r
distinct lcttcrs.Wc irst present an altcmativc proof of thc minimal dimcnsion,which is g市 cn by:″ (′ +1)for lく ′<3
and 3′ -3 for 3<″ (■ hcorem l),using thc scqucnce of Grammians in thc tlTcc― dimcnsional vcctor space,sincc transfer
matnces can bc regardcd as three― dilncnsional vcctors. Sccond, we gencralizc tlle proof to systems with r vcctors in an
―
″
dimcnsional lincar vcctor space.助 c minimal dimcnsion is givcn asケ (″ +1)for lく ′
く′andれ ′一:れ (4-1)forれ <′
(■lcorem 2).Rnally,wc conclude tllat thc minimal dimcnsion is that of thc interscction of algcbraic varictics that aκ
deined as a scrics of Grammians in tllc systcm.
PACSf 02 10.Rn;0220-a;4753+n;6144+p;7110.+x
l.
Introduction
There has been much interest in quasiperiodic systems in recent years, since the discovery of quasicrystals Il]
and their one-dimensional modelings [Z]. fnls led to a deep mathematical study of Schrodinger equations with
arbitrary deterministic potential sequences in order to obtain the spectrum and the wave functions of the systems
[:]. Since the pioneering work on the Fibonacci lattice [4], it became clear that the so-called trace map with an
inuaiant surface plays an essential role in order to realize it. Following the line of thought of the trace map, we
have seen a notable progress in obtaining the minimal dimension of the trace map (i.e., the minimal number of
traces representing a trace map) for quasiperiodic and aperiodic lattice systems with an arbitrary number of
letters [5]. Henceforth, we call the number of distinct species of atoms in the system the rank of the system. We
denote the rank by r.
This problem was first addressed by Luck [6] who asked whether or not a trace map exists for systems with
more than two letters. Ko16r and Nori [7] claimed that the dimension grows faster than exponentially as
‐
E― mail:iguchi@poStmall五
kcll.go jp
C1375-9601/96/S12 Klll Copy五 ght◎ 1996 Publishcd by EIsevicr Sciencc B V All Hghts rcscⅣ
PfF S0375-9601(96)00271-X
cd
a
g″ ε
κ′
みj/Pり SiCS
Lι ″
`だ
A226イ 199θ ,7-イ 6
function of r.A宙 shai and Berend 18]reduced it to 2r_1,using the propemes of matl・ ices in SD(2,R).18μ chi
レ]К duced it hr■ er to:r(r+1),ushg■ e so― caled Frノ教 ιJα ″″ιs[101.MoК over,he found a perfect
`κ
ιlsι れヵηれψ ttα riο れS in mathemadcs[111.
′
ηg″ αれψ ttα ″οれS in our problem and Ⅳ′
equ市 alence between scα ι
Avisllai,Berend,and Glaubman(ABG)[5]reconSidCred the problcm and obtained the final answer that the
minimal dimension of the trace map is given by 3r-3 for r>1,using a new typc of constralnts of rank 4,
which must be satisied for any four madces bclonging to SL(2,R),where this constrdnt comes from the
four―
dimcnsional vector character of matrices in SL(2,R).Such constraints deine
αな
`ら
do Fricke idendties of ranks 2 and 3.
rα ′
ε υαri`″
`s[12],as
Stilnulated by the work of ABG,Iguchi was ablc tO prOve that such algebraic varleties can exist up to an
arbhary rank r and that the algebrttc varieties of rank r fom,in anケ (r+1)― dimensiOnal space,a new class
of noncompact surfaccs(1.e.,ゎ
pι だ′
r/aθ ι
S)that
arc let invariant under the tace map of ale「
temary
led thc
Fibonacci latice l131 and that Of quasipe五 odic lattices of rank r[14].lmis class of surfaces were c」
れυαr′ αれ′s“ r/aθ ιS.It is a natural extension of the F五 cke― Klein surface of rank 2[10]rediscovered by Kohmoto,
Kadanoff,and Tang(KKT)[41 in the Fibonacci lattice problem,which played an essenda1 lole in the binary
quasiperiodic latuces[15].
The purpose of this Letter is to give an altemat市 e proof for■ e ABG result on the minimum dimension of
ve gcncralizc the ABG result to
the trace map, using the existencc of the above hypersurfaces. Moreover, 、
systems witll an arbitrary number of vectors in a highcr dilnensional space.Our results will be presented as
meorems l and 2 in the next section.
、
2.ⅣIain theorems
ｌ
Ｊ
We first summarize the main dleorems of the present paper.
‘
(1)As was stated in Seclon l,when we have a trace map of rank r,we would like to know the minimal
Dim[Im(GoX)]=:′ (r+1) fOr lく
for 3<′
=3″ -3
rく
3,
(1)
.
π
Wc
ler dimensions.It
by￢7.Deine the′ ″ん ′κaκ ε
′between two vectors ν,and tt as χ″=(ν ′
,均 )=`考 °
均=Σ l-lυ ム
μり
`′
,μ
Let X bethe mapping χ :W7→ 町 ,Such that X:均 's∈ ￢;→ χり'S∈ ト
Let G be the mapping,G:R→ R Such that G:χ り's∈ 町 → Gr■ De〈 χけ)∈ R.
R・
Let us consider me prOblem of inding out how many imer products in Eq。
R.
This is the dimension of the image of the mapping G o X:wr→
(27)are essentially independent.
,
軸岬︻ ⑤
(2)ne second meorem is a generalization of Theorem l to that for a linear system in hi」
contains Theorern l as a special case.
dimensional Euclidean space R″ 。Let us denote mis set Of r vectors
Let均 (1くプく r)be r vectors in anれ ―
which is me mapping,P:￢ 7× ￢7→
ｗ％Ｆｓａ
・
`77e f.The dimension ofthe image of the mapping G o X is given by
ｌ
ｓ
7乃 ι
O′
ｄ
problem becomes equivalent to a problem of finding out how many inner products are essentially independent.
Itis the dimension of the image of the mapping G o X:ス r→ R.
ｓ
ι
ノ
κi“αJ di鵞
04 of the trace map.
number of independent traces that represent the trace map.We call it the″
`4s′
勁 is was irst answered by ABG[5].We call it the ABG theorem and itis summarized as Theorem l.
Let a finite matrix‐ vdued alphabet be Ar={Al,A2,・ … ,Ar}where Al,A2'・・
・'Ar∈ SL(2,R).Deine dle
1ち
imer product of A′ and Aブ by χゥ=(A′ ,Aプ )=:[Tく A,Aブ )一 Tく A「
)]=ちブーちら.Let X be the mapping
X:Ar→ ら ,wch that x:Aブ s∈ A″ → χり∈ 弓 ,where tt iS the set of all inner products χ″ between the
∈ → Q=Det(χ )∈ R.ne abOve
elements of A″ .Let G be dle mapping,G:烏 → R,Such that C:χ ′
ブ弓
ゥ
K. Iguchi
Now Theorem
Theorem 2.
I
/
Physics Letters A 216 (1990 37-46
can be embedded in the following more general statement:
T\e dimension of the image of the mapping G " X is given by
DimIIm(COX)]=:r(r+1)
forl<r<n,
=:″ 一 ″(″ -1) fOrん </.
The details of the notations will be given in Sections
given in Sections 5 and 6, respectively.
3
(2)
and 4, and the proofs of Theorems
1
and
2 will
be
3. Substitution scheme
Let us now define the substitution scheme [tZ] to obtain arbitrary deterministic potential sequences. To do so,
letusconsiderthefinitealphabet >,:{1,2,...,r}. Let
to ;,., the set of all (finite) words over 2,,
σ(た )=%1%2・ … %9た
where lく
たくr andら ,∈
o beasubstitutionon;,,thatisamappingfrom -I,
(3)
,
a fOr lく
′
We
く9た。
ex“nd
σto
a mapping iom
Σア t02ネ
by
σ (χ l χ2… ・ χ5)=σ (χ l)σ (χ 2)・ … σ (χ 5)'
n'here
(4)
x,, x2,...,x,e E,. Thus, a is an endomorphism from I,- to ;,- [9]. However, o is not
always
necessarily to be kept the same operation each time when we operate a series of substitutions. For example, see
the cases of the binary, ternary and Nary quasiperiodic lattices [S,t:,tS]. In these systems, since the mapping
(i.e., the scaling transformation) follows the generalized continued fraction expansions, the mapping of any
stage can be different from that of other stages. If we have such a sequence of different substitutions, let us
denotetheoperationatthe ath stageby o.(k)(l (ft(r,0< a<N),whichrelatesthe setof wordsatthe ath
stage to the set of words at the (a + l)st stage. Then, the sequence is defined as
恥
=Π
%,
α=0
(5)
a.(k) (l < k < r) stands for a substitution scheme with a finite word as in Eq. (:) (i.e.,
o,(k) = ckl,o6k2,o . . . oonr,o).There are some cases where the limit S: lima - -S7,, exists. The first one is the
case of co: a , where the substitution is the usual one that provides an exact self-similarity, and the second one
is that of oo*o: cd, where p is the period of a sequence that comes from the periodic continued fraction
expansions [S,t:,tS].
Irt us next define the canonical projection rr !gl. lt is defined by a mappin g r : 2,- --> Z' such that
'ir:WeE,* -'>(wr,w2,...,w,)eZ',where W is a finite word in I,- and w, are the numbers of 7's in the
u'here each
L
u'ord W. For example, for oo in Eq. (3) the canonical projection acts
as
ヽＰ ″
nlo(tc)l=(n*.r, n*.2,...,n*.,) for I <k<r,
(6)
rrhere the nr,,'s are the total numbers of j's in the word o(ft), respectively. Thus, the set of the above r
equations provides an rX r matrix D(o) (:- nlol), such that r is the dimension of the image of the mapping
; (i.e.. the dimension of the representations of the canonical projection). And in the same way for a.(k) in nq.
(-5)
r'e define the canonical projection of the substitution
olo,(f)] =(flkt.o,flkz,o,...,nt,.o),
as
for I <k<r,
and
a: 0, 1,2,...
(7)
m
K. Iguchi
/
Physics Lerters A 216 (1990 37-46
of fr's in
D(o6):1,,..,
since
rXr
matrix
by
a
unit
the initial words if we take a:0. Obviously, it is represented
initial
stage,
orc
generation
d0
as
the
with
an arbitrary
ao(k): t for I < ft < r. However, since one can start
the
X
M,.Therefore,
r
r
matrix
by
I-et
the
us denote
can regard the r X r matrix D(o,^) as the initial matrix.
(Eq.
(5))
X
products
r
r
matrices,
of
the
matrix
defines a sequence of
canonical projection of the sequence S,
which represents the development of the total numbers of ft 's in the system [9]. Let us denote by M i the set of
all matrix products of D(oo)'s on M,.
When oo is kept the same as od: o, the canonical projection induces an Anosou automorphism in the basis
vectorspaceof the rXr matricessuchthat ua+t:D(o)u,,where zo isan r-dimensionalvector eR.'.But
when oo can be different in generation by generation, the canonical projection induces pseudo-Anosou
automorphism,r [18] such that uo*r:D(o,)uo. It has been recognized [9] that a class of the series of
substitutions S, falls into the old mathematical concept of Nielsen transformations [11]. If the o" are the
generators of Nielsen transformations, then, as was recently found, their canonical projections are related to the
generalized continued fraction expansions for the set of r - I real numbers [19], where the periodic generalized
continued fraction expansions with a cycle I are related to the Anosov automorphisms and the others with an
arbitrary indefinite cycle the pseudo-Anosov automorphisms. The algorithm of the generalized continued
fraction expansions was found to be equivalent to the Jacobi-Perron algorithm [ZO] (see also Vol. 3 of Hodge
which then defines the r X
r
matrices, O(o,). This works as well to give the initial set of the numbers
and Pedoe [12]).
Let us define the finite matrix-valued alphabet A,={Ar, A2,..., A,} corresponding to the alphabet 2,,
where A,, A2,...,A, are some fixed matrices in SL(2, R) with R:R or C. According to the o (nq. (:)),
which is a mapping from -I, to 2,* , this induces a mapping o on the matrices from ,4, to Ai, the set of all
(finite) words (i.e., monomials) over A,,
σ :Aた →
4=Aσ
(た
)=Aσ 々lAσ た2… ・Aσ た
,
(8)
9人
たくr.We then extend to a mapping from A,to A'in a silllllar way as Eq.(の by
σ (A142・ … As)=σ (Al)σ (A2)… ・σ (As),
(9)
where A,, A2,...,A"eA, and n(A):Ac(k). Thus, o is an endomorphism on the transfer matrices from
A; to A: [g]. r'rom this we can define the sequences of matrices {Ar,"H:g by
for lく
Ao,o*,:
Aoor,o Aoor,o "' Aoyno,o,
(10)
for α =0,1,2,…
.,and lくたく r.In the same way we can define the sequences of matrices associated witt the
subsututiOn sequence S of Eq。 (3),just replacing ale θlブ 'sin Eq.(10)by q,;'s.
Ⅵ硫en we consider an automorphism such as Eq.(7),we need the set of the r disttnct mamces,
SL(2,R)″ .Since a matnx Aた ∈ SL(2,R)has four componcnts with a single constraint
(Al'A2'・ …
'Ar)∈
κsiaη αιo司 cct・ Therefore,the mapping σ on the transfcr mamces is
αj“ ι
Dc(Aた )=1,it is essentially a ttrι ι―
dimensional.However,as
was
mentioned
by ABG[5],tllere may exist a ixed matrix y such alat
3r―
1,satisfying Tく
4々 υ
Aた →イ
=め
)=T(じ 4たび 1)=Tく Aた )for lくたくr.nis υdOes not change the
ィ々
た
trace of any monomial in the given matrices such as Eq。 (10)。 So,it defines a kind of translational motion(i.e.,
the parallel transport)in the three― dimensional space,which provides three more constralnts.Hence,the
dimension of the mapping must be reduced to 3r-3.
4.Trace inaps
た=:T<Aそ )fOr lくたくr.■ is is a mapping
∈
′
SL(2,R)→
(i.e.a praCctioD″
た∈ R.Similarly,let us defineちノた
.,Ω
(“ )the set of al1 4 for lくたくr,the
Aた
(2),…
A,Aプ
.…
),and so foral.Let us denote by Or(1),Ω ″
″
=:Tく
Let us consider the iaces of ransfer mamces in/1r.Deine′
:S12,R)→
R such tllat II: Aた
K. Iguchi
/
Physics Letters A 216 (1990
37-46
41
of all t,, for 1<i< i<r,-.., and the set_of all ti,i,...i,, for 1( irliz< ... <
Therefore, the cardinality of A,(m) is CarIA,(m)l : (;). Fbi example , if m: r, then O,(r)
element tn...,. kt us define P, by
set
Q,= U ;:
contains only one
,a,(*)'
(11)
where U means a disjoint union.
Now the mapping o induces a mapping from J2,(1)
b Al,
the set of all traces of the monomials
in ,zli,
such that
cf : tk--+ /*:
to(t): *r.( Aoo,Aoor...
(12)
Arooo).
We know that a matrix― vducd identity[5,8]
・A:5=Σ
へ A'2・・
Σ … Σ Ql,2-ε β。 ε
′
(13)
1′ 2…
brlく
TD8篤 16鶴亀五
:ill∫ [鳳議
∞
ettdm c… Ⅲ attμO血 J
た
く
)H∝e∽ぬ
…A'7 (ぅ =0,1,and lくノくr)・
(13)in Eq。 (12),the mapping o (Eq. (tZ)) ls
ofコ ″ls giVen by
(14)
=AflA,2・
41● 2-‐ ε
′
Ъ erefore,usmg Eq。
where dle cardinality
represented
Car[コ r]=2r_1,
in terms of all traces in 4,,
(15)
which is equivalent to the result of Avishai and Berend [8]'
We then extend the mapping o to a mapping from O,*
o Al. It is an automorphism from f,),* to A,- ,
to Ai. Therefore, the automorphism o can be represented by f,), as
well. This is called the trace map l9]. The mathematical structure of the above trace map system is
induced fiom an automorphism from -zlj
schemarically shown in Fig.
1.
″
一
・
Ω
一一
Ⅵ
Ⅵ
一
﹁
］
︿
Ｋ
払
ＦＩＩ︱ＩＭ
川
︲Ｍ
Σ
一
＼嘲Ｎ
Σ
Fig. 1. The schematic diagram of the hace map system. X, stands for the alphabet and -I.* its extension. 21, stands for the set of the r
distinct matrices € SL(2, R) where R: R or C, and ,4j its extension. M, stands for the initial ,'X r matrix such as D(ooo) and M.- its
extension, the set of all matrix products of D(o.) on M,. A, ( = O, (l)) stands for the sets of tracas of the matrices in 21., and O.* its
extension. A solid arrow means a projection, where rr is a canonical projection and II a projection onto the trace space, respectively. A
bold arrow means the extension and a dashed arrow the correspondence,
K. Iguchi
/
Physics Letters A 216 (1996) 37-46
The trace maps must be carried out with many constraints as follows. Let us define the inner (scalar)
products in lhe modffied Killing form 2
*,i: (A,, A) = i[rr( A,A) - rr(a;'or)] : tij-
titj,
(16)
泄
.鵠灘
蹴
蹴
Ψ
仇
ぢ
雛装
轟
暑
ゝ
響難茸幕1樵焦FII
К
慮よ猟算鷲
昴∬庶e鴛鼎霜塊
t儀
::鴇
lmon
crA,, * c, A,rI ... l r rtrt, :
(17)
狙
%訂 e zerO cOn威趾も,Tよ hg■ e hner produ∝ sI他
bodl sides,respect市 ely,we ind tlle g linear relations Σ場_16″ (A‐ ,Aら
mutt hdd,where n∝
the Grammian Gg(Jl,′ 2'… ・,′ g)as a symmetic g× g deterlninant,
g)=
Gg(′ 1'′ 2… ・ノ
χH
χ 12
χ13
χlg
χ12
χ 22
χ23
χ2g
χ13
χ23
χ33
χ38
χlg-l
χ2g-l
χlg
χ2g
・・・
χg-lg-l
χg-lg
χgg-l
χgg
:1ル
7
:Det( x,^)
ミ :ミかミ
漁
指 &驚
(18)
,
wherea x,^:(A,,,;,,,).I-hisgivesthemapping,G: Pr--+R,suchthatG: x,.epr-G,
Det(x,.)e R.
thesetof allGrammiansforthe g transfermatrices e ,4,.Hence,thecardinalityof
. ^L.,"T.d.n9-t:by{
{
(!) for.g,> l. In addition, let us denote O.(l) by l-r. If 1 < g < 3, then each elemenr of such
is Car{{J:
{
as Gr(71, iz,...,jr) does not generally vanish due to the linear independence of the g vectors. On the other
hand, if 4 < g, then it vanishes identically due to their linear dependence. Let us simply write this situation as
1,ら ,F3≠
「
O and尋 =o(4く r),
which represent the 2″
5。
(19)
-l algebraic relations 5.
Proof of Theorem l
g革
胤
網鵠
艦聯
認T,ケ Iダ鳥新
鳳を
郷茎硯蝋
]鷲耐斐
1,き
TheК fore,■ e mdependent hner products are■
j鷲
eケ (r―
number is g市 en as tlle sum between the numbers of off― diagonal components in rows(Or columns)of me
2 ne馴 hgゎ mc年 oの
two thrcc―
お ■
iom tt h me蜘ぬ
」
『驚:羅[ヤF器禽Pi胤燎』
ォ
lw[::i漁
llml器
dlffcК
).Ollr Killhg fonll has bccn uま
″=:TKス「
dimensional Minkowskian vcctors.
and Suthcrlalld[221χ
lAプ
)r matHces rather man the fOur dimensiona1 0ne
漁)計
It,I・
l」語s惚ま
等鳳督神品1塩七
lk 4 and ie identity of ABG[5]is publishcd in Rc■
[23].
κ・セ ″C力 j/Pり SiCS Lι ″
`ぉ
determinant such that:r(r_1)=1+2+.…
:r(r+1)independent v頷あたs[9].
(B)For 4く ″,the rank of the deterlninants
A2f6″
ーイ6
996り
43
`ア
+(r-1).And we have the r traces in
witl1 3く
l-,.
So, we have totally
gく r is now 3. Let us first consider the case
of r:
4.
We then have dle six Grammians of rank 2,
G2(′
'プ )≠
0
for lく ′<ブく4,
(20)
the four GralnIIuans of rank 3,
G3(1,2,3),G3(1'2,4),G3(1'3,4),G3(2,3,4)≠
and■ e
0,
(21)
onl、 olle Grallllnian of rank 4,
G=(1,2,3,4)=0.
(22)
Ж
lぶmftte鶴恩p訛織 neⅧ 織[∬ 器 2‰ 71黙
黒淵瑶
II.ふ瓶
:1。
this for 134,then χ34 Can bC represented as thc roots of the quadratic equation by the other nine inner products,
'府
ttデ L誼 ξ
ξ
ttldよIt酔魅籠
ち
ブちら
'We
τ箸和 1∬ 電斉
:;可 23'ち
btty have″
'器
れ
ιO.a,4+3+2=3×
4
4・
Let us condder next■ e case of r=5.In mヽ case we have the(3)(=10)Grammians of rank 2
G,(ゴ ,プ )≠ O for lく J<ノく5,tlle(3)(=10)Grammians of rank 3:G3(′
<ブ <たく5,the
)≠ O for lく ′
'ブ 'た
‐
■、e GraIIIIllllanS Of rallk 4:G4(′
,た
,′ )=O for lく ′<ノ <た <″ く 5,and the only one Grammian of rank 5:
'ブ
G5(1.2,3,4,5)=0,We then ind tllat all components in me Grammians of ranks 2,3,and 4 appear in that Of
rank 5 and he Gramnlians of rank 4 are quadratic equations for all off―
the components
G二 (ゴ ,
diagonal components,respectively.So,
χ35
and χ45 in G5(1'2,3,4,5)are not independent variables,while χ15 and χ25 in
ブ, た,′ ) are new independent variables. Thus, we have twelve independent variables,
χll'122,■ 33,χ 44,χ 55;χ 12'χ 13'χ 14,χ 15;χ 23,χ 24,χ 25,in Which we have ive traces in「 l as wen as seven
r,メ S Since χ =ちー
ι′
υι(i.e.,5+4+3=3× 5-3)independent variables.
・ThuS,We totally haκ 加ノ
り
ブちら
‐
習ヽe can extend tllis argument to the case of an arbitrary rank
′>4.The inner products χ′ with 3<′ ,′ く ′
プ
are not independent so that they are represented by the other inner products.We ind tllat the independent inner
products are the diagonal elements and the off― diagonal elements in tlle irst and second rows(or c01umns),
where the tota numbers of tlle inner prOducts in the diagonal elements and in the irst alld second rows(or
coluIIms)are r,r-l and r-2,respectively.]hese give us totally the suln″
QED
十 (r-1)+(r-2)=3′
-3.
6.Pr00f of Theorem 2
レ t us next generalize the above argument to the systenl witll tlle r vectors in an 4-dimensional linear space.
レ t竹 (1くノく r)be tlle r vectors in an 4-dimensional Euclidean space R″ .Let us call this set of r vectors Vy_
Deflne theれ 4`′ ρκグ c′ between the two vectors,ν ′and釣 ,as
“
。
′り
・ .(ν ,,釣 )=し′均=Σ1 υ
,μ
,μ
(23)
,
μ
which is me mapping,P:￢ ;× ￢;→
￢
7・
Hence the cardinality of■
Car[町
R・ Let us denote by tt the set of all inner products among the r vectors in
iS given by
]=:r(r+1),
(24)
which is equivdent to tlle result flrst given by lguchi[9].■ us we can deine the mapping,X:￢ 7→
that X:ち 's∈ ￢7→ χり'S∈ 町・
鳥,Such
grrc力 j/P″ ySJcs
κ′
Zj′
″′
rs A 276(f996'37-イ 6
“
Tlle linear independence is described as follo、
vs: Given tlle condition
Σら均=0,
(25)
ブー1
and only if the condition is satisfied for cr: 0 for all i, then the z, are linearly independent, otherwise the u,
are linearly dependent. In the n-dimensional case the statement of the linear independence is always true for
if
r
(
n, but it is not true
i
i:
r,@,,aj)
if
r > n. Multiplying Eq. (25) by u, from the left, we find
:0,
(26)
I
r
for 1 < i < r. Therefore, the above equation can be represented by an rX
matrix
as
〓
０
らの。・・ら
みちら
均わら
めらわ
ｈち為
″ ν ν
′ヽ
ν ν
(27)
Lct us cal this matix c.ne de“ minantl c l of c iS Called the Crα 777′ α
れ,Whた h is parametized byぬ e
“
)∈ ■ if the ν
:ズ ″-1)(:r(′ +1))inner products χ =(ν
,are norlnalized(not nOrlnalized).The
,ブ
Grammian is dle mapping,G:鳥 → R.We knOw''均
tllat if rくれ,then the tt can be linearly independent.
nerefOre,there exists a tti宙 d soludon of Eq.(27)such mat allら =o.■ atisl Q I≠ 0・ On the other hand,if
″>4,tllen tlle tt Can be linearly dependent such tllat no t五宙al solution e対 sts.勁 erefore,from Eq.(27)the
determinant of e must vanish so that l Q I=0.Using the samc notation as Eq.(18),we arc able to simply
M/rite the above sltuatlon as
and4:0("+l<r),
(28)
、
vhich represent ale 2″ -l algebraic relations as wel1 6.
Let us consider Gg(プ 1'ぁ ,… .,プ g)f° rlく gく ′
.(A)For lく rく
れ,thc
rank of the deterlninants is go When
e:島
幌tf鮭驚l:皐虚無
l魚ま
:鳳
念
!∬
盤
や
l亀
鳥
tL彙
:魚
電淵
;f弔
e鍛
を
きを
じ
[壼ふ
lt〔
for lく ′<“ く κ.SO,we have totally:′
(″
+1)independent vmables,where tlle number is g市 en as the sum
+1)=1+2+.… 十′[9].
れ
・ In this case we have■ e(1)
between the numbers of rows(or columns)in tlle determinant such tllat:r(′
(B)For 4<r,the rank of the deterlllinants with
″く gく
Gramnllans of rank ″2,
G″ (プ 1,あ ,…
for lく
。
,ス ″
)≠ O
<・
…<九くr,
(29)
くれ and the(1)Grammians of rank g,
“
Gg(プ 1'あ ,… .,ス
for″
With lくプ
1<ぁ
r is noW
<gく
)=0
with lくプ
1<ぁ <・ … <ノgく
r,
(30)
′
.
+1.We dlen have C″ +1(プ 1'Ji2,… .,九十1)=°・ We nOw observc tllat
an compOnents in Eq.(29)appear in G″ 十
,九 +1)=0,which is a quadratic equation for each
Kブ 1'あ ,… 。
Let us irst consider the case of r=η
off diagonal component.If we solve this for χ″″+1'then tllis is represented in terms of tlle:4(κ
6 nC matllemadcal smucture of hc Grammians hele has bcen shown tO flt he conccpt of dle deterlllinant ideals,which are
wih lcngぬ ″[18].
7 1n this casc thc diagonal clcmcnts
χ″for lくメく ′play tllc samc rolc as the r traces in Ar h dle tracc map system
+3)imcr
Ⅳοι″′
′
Jα れ
K. Iguchi
/
Physics Letters A 216 (1996)
37-46
_1″ +1・ Therefore,we have totally
, χ″
-1))independent variables.
Lt us next consider the case of an arbitr暉 ′>れ .By the same argument we ind th■ the inner products χ
り
products: χH,.… ,χ ″+1′ +1; χ12'・・・ l η+1; χ23'・ … 2′ +1;・・・; χ″_1″
'χ
'χ
dle:″ (″
+3)(=4(′ +1)一
:ん (κ
with ″<ゴ ,ブくr are not independent such that tlley can be reprcsented by the other inner productま
χll,… .,■ rr:χ 12'… ・ l″ ;χ 23'… ・ 2″ ;… ・;χ ″_1″ ,… 。,χ ″_lr。 lmiS gives us the sum,r+(r-1)+.…
'χ
'χ
一″+1)=″ ′―:κ (κ -1).nus,we tOtJly have theん r― :η (4-1)inner prOducts.QED
+(′
7.Conclusion
ln conclusion we have g市 en an dtemative proof for thc important rcSult of ABG on the minimal dimension
of■ e trace map wim arbitraly rank r as Theorem l,and generalizc it to the systems with an ttrbitrary number
of vectors in higher― dimensional linear spaccs as neorerl1 2, using the Grammians in tllc system. Since a
Vanett iS defined as the common zeros(1.e.,intcrsections)of a finite set of algebraic ninctions[12],what we
fouコ d here is that the minimd dimcnsion is ale dimension of the variety defined by tlle set of Grammians(Eq.
(19)or Eq.(28))。
■lerefOre,we call conclude that it is the dimension of the intersection of thc sct of
h)pesurfaces dnce mc Granlmian of rank r brms a hypersuttce of degree r h anケ
(′
+1)dmettOnd
space[131.
Acknowledgement
vork is
va for very helpful discussions. This 、
l‐ he autllor would like to acknowledge Toshio Yoshika、
supported in part by the Special Researcher's Basic Sciencc Program froln tte lnsdtute of Physical and
Chemical Research(RIKED.
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