Table Of Content

February1,2008 7:33 WSPC/TrimSize: 9inx6inforProceedings 1param2 4 0 0 2 n a J 0 2 ONE-PARAMETER GROUPS AND COMBINATORIAL 1 PHYSICS v 6 2 1 GE´RARDDUCHAMP 1 Université de Rouen 0 Laboratoire d’Informatique Fondamentale et Appliquée de Rouen 4 76821 Mont-Saint Aignan, France 0 / h p KAROLA. PENSON - Université Pierre et Marie Curie, t n Laboratoire de Physique Théorique des Liquides, CNRS UMR 7600 a Tour 16, 5ie`me étage, 4, place Jussieu, F 75252 Paris Cedex 05, France u q : ALLANI. SOLOMON v The Open University i X Physics and Astronomy Department r Milton Keynes MK7 6AA, United Kingdom a ANDREJHORZELA H. Niewodniczan´ski Institute of Nuclear Physics, Polish Academy of Sciences Department of Theoretical Physics ul. Radzikowskiego 152, PL 31-342 Kraków, Poland PAWEL BLASIAK H. Niewodniczan´ski Institute of Nuclear Physics, Polish Academy of Sciences Department of Theoretical Physics ul. Radzikowskiego 152, PL 31-342 Kraków, Poland & Université Pierre et Marie Curie, Laboratoire de Physique Théorique des Liquides, CNRS UMR 7600 Tour 16, 5ie`me étage, 4, place Jussieu, F 75252 Paris Cedex 05, France 1 February1,2008 7:33 WSPC/TrimSize: 9inx6inforProceedings 1param2 2 Inthiscommunication,weconsiderthe normal orderingofoperatorsofthetype Ω= cα,β(a+)αa(a+)β, α+β=e+1 X where a(resp. a+)is aboson annihilation (resp. creation) operator; these satisfy [a,a+]≡aa+−a+a=1, and forthe purposes of this note may bethought of as a ≡ d/dx and a+ ≡ x. We discuss the integration of the one-parameter groups eλΩ andtheircombinatorialby-products. Inparticularweshowhowthesegroups canberealizedasgroupsofsubstitutions withprefunctions. 1. Introduction This text is the continuation of a series of works on the combinatorialand analytic aspects of normal forms of boson strings[1,2,3,4,5,11,13,14,15,18,19,20]. Let w a,a+ be a wordin the letters a,a+ (i.e. a bosonstring), and ∗ ∈{ } { } define (as inBlasiak,PensonandSolomon1,2,3,4)by r,s ande,respectively w (the number of creationoperators), w (the number ofannihilation a+ a | | | | operators) and r s (the excess), then the normal form of wn is − ∞ (wn)=(a+)ne S (n,k)(a+)kak (1) w N ! k=0 X wheneispositive(i.e. therearemorecreationthanannihilationoperators). In the opposite case (i.e. there are more annihilations than creations) the normal form of wn is ∞ (wn)= S (n,k)(a+)kak (a)ne (2) w | | N ! k=0 X in each case, the coefficients S are defined by the corresponding equation w (1 and 2). Now, for any boson string u one has (u)=(a+)|u|a+a|u|a + λvv. (3) N v<u |X| | | It has been observed [12] that the numbers λ are rook numbers. v Consider, as examples, the upper-left corner of the following (doubly infinite) matrices. For w = a+a, one gets the usual matrix of Stirling numbers of the February1,2008 7:33 WSPC/TrimSize: 9inx6inforProceedings 1param2 3 second kind. 10 0 0 0 00 ··· 01 0 0 0 00  ··· 01 1 0 0 00 ··· 01 3 1 0 00   ··· (4) 01 7 6 1 00  ··· 01 152510 10  ··· 01 319065151  ···  .. .. .. .. .. .. .. ..  . . . . . . . .   For w=a+aa+, we have  1 0 0 0 0 0 0 ··· 1 1 0 0 0 0 0  ··· 2 4 1 0 0 0 0 ···  6 18 9 1 0 0 0   ··· (5)  24 96 72 16 1 0 0  ··· 120 600 600 200 25 1 0  ··· 720432054002400450361  ···  ... ... ... ... ... ... ... ...   For w=a+aaa+a+, one gets  1 0 0 0 0 0 0 0 0 ··· 2 4 1 0 0 0 0 0 0  ··· 12 60 54 14 1 0 0 0 0  ··· (6)  144 1296 2232 1296 306 30 1 0 0  ··· 2880403201094401051204500095041016521  ···  ... ... ... ... ... ... ... ... ... ...   Remark 1.1. In each case, the matrix S is of staircase form and the  w “step”depends onthe number ofa’s in the wordw. More precisely,due to equation(3)onecanprovethateachrowendswitha’one’inthecell(n,nr), wherer = w andwenumbertheentriesfrom(0,0). Thusallthematrices a | | arerow-finiteandunitriangulariffr =1,whichcasewillbeofspecialinter- est in the following. Moreover,the first column is (1,0,0 ,0, ,0, ) ··· ··· ··· iff w ends with an a (this means that (wn) has no constant term for all N n>0). In this communication, we concentrate on boson strings and more gen- erally (homogeneous) boson operators involving only one “a”. We will see February1,2008 7:33 WSPC/TrimSize: 9inx6inforProceedings 1param2 4 that this case is closely related to one-parameter substitution groups and their conjugates. The structure of the paper is the following. In section 2 we define the framework for our transformation matrices (spaces,topologyand decomposition),then we concentrateon the Riordan subgroup (i.e. transformations which are substitutions with prefunctions) and adapt the classical theory (Sheffer condition) to the present context. Insection3we analyseunipotent transformations(Lie groupstructureand combinatorial examples). The divisibility property of the group of unipotent transformations tells us that every transformation is embedded in a one-paramater group. This will be analysed in section 4 from the formal and analytic points of view. Section 5 is devoted to some concluding re- marks and further interesting possibilities. 2. The algebra L(CN) of sequence transformations N Let C be the vector space of all complex sequences, endowed with the N Frechet product topology. It is easy to check that the algebra (C ) of N N L allcontinuousoperatorsC C is the space ofrow-finitematrices with → complex coefficients. Such a matrix M is indexed by N N and has the × property that, for every fixed row index n, the sequence (M(n,k)) has k 0 finite support. For a sequence A = (a ) , the transformed seq≥uence n n 0 ≥ B =MA is given by B =(b ) with n n 0 ≥ b = M(n,k)a (7) n k k 0 X≥ Remark that the combinatorial coefficients S defined above are indeed w row-finite matrices. Wemayassociateaunivariateserieswithagivensequence(an)n N,us- ∈ ing a sequence of prescribed (non-zero)denominators (dn)n N , as follows: ∈ zn a . (8) n d n n 0 X≥ Forexample,withd =1,wegettheordinarygeneratingfunctions(OGF), n with d =n!, we getthe exponentialgenerating functions (EGF) and with n d =(n!)2,thedoublyexponentialgeneratingfunctions(DEGF)andsoon. n Thus,oncethedenominatorshavebeenchosen,toevery(linearcontinuous) transformation of generating functions, one can associate a corresponding matrix. February1,2008 7:33 WSPC/TrimSize: 9inx6inforProceedings 1param2 5 N Thealgebra (C )possessesmanyinterestingsubalgebrasandgroups, L suchasthe algebraoflowertriangulartransformations (N,C),thegroup T (N,C) of invertible elements of the latter (which is the set of infinite inv T lowertriangularmatriceswithnon-zeroelementsonthediagonal),thesub- groupof unipotent transformations (N,C) (i.e. the set ofinfinite lower UT triangularmatriceswithelementsonthediagonalallequalto1)anditsLie algebra (N,C), the algebra of locally nilpotent transformations (with NT zeroes on the diagonal). One has the inclusions (with (N,C), the set inv D of invertible diagonal matrices) N (N,C) (N,C) (N,C) (C ) inv UT ⊂T ⊂T ⊂L N (N,C) (N,C) and (N,C) (C ). (9) inv inv D ⊂T NT ⊂L We remark that (N,C) = (N,C)⊲⊳ (N,C) because is inv inv T D UT UT normalized by and = . (every invertible transformation inv inv inv D T D UT is the product of its diagonal by a unipotent transformation). We now examine an important class of transformations of as well as T their diagonals: the substitutions with prefunctions. 2.1. Substitutions with prefunctions Let (d ) bet a fixed set of denominators. We consider, for a generating n n 0 ≥ function f, the transformation Φ [f](x)=g(x)f(φ(x)). (10) g,φ The matrix of this transformation M is given by the transforms of the g,φ monomials xk hence dk xn xk φ(x)k M (n,k) =Φ =g(x) . (11) g,φ d g,φ d d n 0 n (cid:20) k(cid:21) k X≥ If g,φ=0 (otherwise the transformation is trivial), we can write 6 xl xr xm xs g(x)=a + a , φ(x)=α + α (12) l r m s d d d d l r m s r>l s>m X X with a ,α =0 and then, by (11,12) l m 6 xk xl+mk xt Φ =a (α )k + b . (13) g,φ d l m d dk d td (cid:20) k(cid:21) l m k t>l+mk t X One then has February1,2008 7:33 WSPC/TrimSize: 9inx6inforProceedings 1param2 6 M is row finite φ has no constant term (14) g,φ − ⇐⇒ and in this case it is always lower triangular. Theconverseis trueinthefollowingsense. LetT (CN)be amatrix ∈L with non-zero two first columns and suppose that the first index n such that T(n,k)= 0 is less for k = 0 than k = 1 (which is, from (11) the case 6 when T =M ). Set g,φ xn d xn g(x):=d T(n,0) ; φ(x):= 1 T(n,1) (15) 0 d g(x) d n n n 0 n 0 X≥ X≥ then T =M iff, for all k, g,φ xn φ(x)k T(n,k) =g(x) . (16) d d n k n 0 X≥ Remark 2.1. Eq. (11)iscalledtheSheffer condition(see[16,20,21,23])and, for EGF (d =n!) it amounts to stating that n xn T(n,k) yk =g(x)eyφ(x). (17) n! n,k 0 X≥ From now on, we will suppose that φ has no constant term (α =0). 0 Moreover M if and only if a ,α =0 and then the diagonal term g,φ inv 0 1 ∈T n 6 with address (n,n) is a0 α1 . We get d0 d1 (cid:16) (cid:17) a α M 0 = 1 =1. (18) g,φ ∈UT ⇐⇒ d d 0 1 In particular for the EGF and the OGF, we have the condition that g(x)=1+higher terms and φ(x)=x+higher terms. (19) Note 2.1. In classical combinatorics (for OGF and EGF), the matrices M (n,k) are known as Riordan matrices (see [16,17] for example). g,φ 3. Unipotent transformations 3.1. Lie group structure We first remark that n n truncations (i.e. taking the [0..n] [0..n] sub- × × matrix of a matrix) are algebra morphisms τ : (N,C) ([0..n] [0..n],C). (20) n T →M × February1,2008 7:33 WSPC/TrimSize: 9inx6inforProceedings 1param2 7 We can endow (N,C) with the Frechet topology given by these mor- T phisms. We will not develop this point in detail here, but this topology is metrisable and given by the following convergence criterion: a sequence (M ) of matrices in (N,C) converges iff k T for all fixed n N ∈ the sequence of truncated matrices (τ (M )) converges. n k This topology is compatible with the C-algebra structure of (N,C). T The two maps exp : (N,C) (N,C) and log : (N,C) NT → UT UT → (N,C) are continous and mutually inverse. NT 3.2. Examples 3.2.1. Provided by the exponential formula The “classicalexponential formula” [7,9,21] tells us the following: Consider aclassa offinitelabelledgraphs . Denote by c thesubclassofconnected C C graphsin . Thenthe exponentialgeneratingseriesof and c arerelated C C C as follows: EGF( )=eEGF(Cc). (21) C The followingexamplesgiveus someinsightintowhy combinatorialmatri- ces of the type: T(n,k)= Number of graphs of C on n vertices having k connected components give rise to substitution transformations. Example 3.1. Stirling numbers. Considertheclassofgraphsofequivalencerelations. Thenusingthestatis- tics x(numberofpoints)y(numberofconnectedcomponents) we get xn S(n,k) yk = n! n,k 0 X≥ x(numberofpointsofΓ) y(numberofconnectedcomponentsofΓ) = (numberofpointsofΓ)! allequivalencegraphsΓ X aClosedunderrelabelling(ofthevertices),disjointunion,andtakingconnectedcompo- nents. February1,2008 7:33 WSPC/TrimSize: 9inx6inforProceedings 1param2 8 x(numberofverticesofΓ) exp y(numberofconnectedcomponentsofΓ) = (numberofpointsofΓ)! ! Γ connected X exp yxn =ey(ex−1) (22)  n! n 1 X≥   we will see that the transformation associated with the matrix S(n,k) is f f(ex 1). → − Example 3.2. Idempotent numbers. We consider the graphs of endofunctionsb. Then, using the statis- tics x(numberofpointsoftheset)y(numberofconnectedcomponentsofthegraph) and denoting by I(n,k) the number of endofunctions of a given set with n elements having k connected components, we get xn I(n,k) yk = n! n,k 0 X≥ x(numberofverticesofΓ) y(numberofconnectedcomponentsofΓ) = (numberofverticesofΓ)! allgraphsofendofunctionsΓ X x(numberofverticesofΓ) exp y(numberofconnectedcomponentsofΓ) = (numberofverticesofΓ)! ! Γ connected X exp ynxn =eyxex (23.)  n!  n 1 X≥   Corresponding to these numbers we get the (doubly) infinite matrix 10 0 0 0 0 0 ··· 01 0 0 0 0 0  ··· 02 1 0 0 0 0 ··· 03 6 1 0 0 0   ··· (24) 04 24 12 1 0 0  ··· 05 80 90 20 1 0  ··· 06240540240301  ···  ... ... ... ... ... ... ... ...     bFunctions fromafinitesetintoitself. February1,2008 7:33 WSPC/TrimSize: 9inx6inforProceedings 1param2 9 and we will see that the transformation associated with this matrix is f f(xex) → 3.2.2. Normal ordering powers of boson strings To get unipotent matrices,one has to consider bosonstrings with only one annihilation operator. In the introduction, we have given examples with a+a,a+aa+ (the matrix of the third string, a+aaa+a+, with two annihila- tors, is not unipotent). Such a string is then w = (a+)r pa(a+)p and we − will see shortly that if p=0, S (n,k) is the matrix of a unipotent substitution w • if p > 0, S (n,k) is the matrix of a unipotent substitution with w • prefunction To cope with the matrices coming from the normal ordering of powers of boson strings we have to make a small detour to analysis and formal groups. 4. One-parameter subgroups of UT(N,C) 4.1. Exponential of elements of NT(N,C) Let M =I+N UT(N,C) (I =IN is the indentity matrix). One has ∈ t Mt = Nk (25) k k 0(cid:18) (cid:19) X≥ t where is the generalized binomial coefficient defined by k (cid:18) (cid:19) t t(t 1) (t k+1) = − ··· − . (26) k k! (cid:18) (cid:19) One can see that, for k n, due to the local nilpotency of N, the ma- ≤ trix coefficientMt(n,k) is well defined and, in fact, a polynomialof degree n k in t (for k > n, this coefficient is 0). We have the additive property − Mt1+t2 = Mt1Mt2 and the correspondence t Mt is continuous. Con- → versely,lett M be acontinouslocalone-parametergroupinUT(N,C); t → that is, for some real ǫ>0 t and t <ǫ= M M =M (27) | 1| | 2| ⇒ t1 t2 t1+t2 February1,2008 7:33 WSPC/TrimSize: 9inx6inforProceedings 1param2 10 then there exists a unique matrix H NT(N,C) such that M = t ∈ exp(tH). (This may be proved using the projections τ and the classical n theoremabout continuousone-parametersubgroups ofLie groups,see [10], for example). When M =Mt is defined by formula (25) we have t H =log(I+N)= k 1 (−1k)k−1Nk. ≥ Themappingt MtwillbecalledaoneparametergroupofUT(N,C). P → Proposition 4.1. Let M be the matrix of a substitution with prefunction; then so is Mt for all t C. ∈ The proof will be given in a forthcoming paper and uses the fact that “tobe the matrix ofasubstitution withprefunction”is apropertyofpoly- nomial type. But, using composition, it is straightforward to see that Mt is the matrix of a substitution with prefunction for all t N. Thus, using ∈ a “Zariski-type” argument, we get the result that the property is true for all t C. ∈ 4.2. Link with local Lie groups : Straightening vector fields on the line We first treat the case p = 0 of subsection (3.2.2). The string (a+)ra cor- rresponds, in the Bargmann-Fock representation, to the vector field xr d dx defined on the whole line. Now, we can try (at least locally) to straighten this vector field by a dif- feomorphism u to get the constant vector field (this procedure has been introducedby G.Goldin inthe contextofcurrentalgebras[8]). As the one- parametergroupgeneratedbyaconstantfieldisashift, theone-parameter (local) group of transformations will be, on a suitable domain U [f](x)=f u 1(u(x)+λ) . (28) λ − Now,weknowfromsection(4.1)tha(cid:0)t,iftwoone-pa(cid:1)rametergroupshavethe same tangent vector at the origin, then they coincide (tangent paradigm). Direct computation gives this tangent vector : d 1 f u 1(u(x)+λ) = f (x) (29) − ′ dλ u(x) (cid:12)λ=0 ′ (cid:12) (cid:0) (cid:1) and so the local o(cid:12)ne-parameter group U has 1 d as tangent vector (cid:12) λ u′(x)dx field.