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:[email protected]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 ％ Ｆ ｓ ａ ・ `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 ＦＩ Ｉ ︱ Ｉ Ｍ 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. 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