1 Combinatoire, Informatique et Physique Algèbre des Diagrammes et Physique I-III Gérard H. E. Duchamp, Université de Paris XIII, France Collaborateurs : Karol A. Penson, LPTMC, Université de Paris VI, France Allan I. Solomon, The Open University, United Kingdom Pawel Blasiak, Instit. of Nucl. Phys., Krakow, Pologne Andrzej Horzela, Instit. of Nucl. Phys., Krakow, Pologne Jean-Yves Thibon, Instit. G. Monge, Marne-la-Vallée, France Florent Hivert, Instit. G. Monge, Marne-la-Vallée, France 2ème Colloque Algéro-Français de Théorie des Nombres Alger 7-9 Mai 2005

2 Content  Preamble A simple formula giving the Hadamard product of two EGFs (Exponential Generating Fonctions )  First part : A single exponential  One-parameter groups and the Normal Ordering Problem  Substitutions and the « exponential formula » (discrete and continuous cases, Scheffer’s conditions)  Discussion of the first part  Second part : Two exponentials  Expansion with Feynman-type diagrams  Link with packed matrices  Discussion of the second part  Third part : Hopf algebra structures  Two structures of Hopf Algebras on the space of packed matrices  Classes of packed matrices and structure results  Link with Hopf algebras of rooted trees Concluding remarks

3 A simple formula giving the Hadamard product of two EGFs In a relatively recent paper Bender, Brody and Meister (*) introduce a special Field Theory described by a product formula (a kind of Hadamard product for two exponential generating functions - EGF) in the purpose of proving that any sequence of numbers could be described by a suitable set of rules applied to some type of Feynman graphs (see Second Part of this talk). These graphs label monomials and are obtained in the case of special interest when the two EGF have a contant term equal to unity. Bender, C.M, Brody, D.C. and Meister, Quantum field theory of partitions, J. Math. Phys. Vol 40 (1999)

4 If we write these functions as exponentials, we are led to witness a surprising interplay between the following aspects: algebra (of normal forms or of the exponential formula), geometry (of one-parameter groups of transformations and their conjugates) and analysis (parametric Stieltjes moment problem and convolution of kernels). This will be the first and second parts of these talks Some 5-line diagrams

5 The Hadamard product of two sequences is given by the pointwise product We can at once transfer this law on EGFs by but, here, as we get Product formula

6 Writing F and G as free exponentials we shall see that these diagrams are in fact labelling monomials. We are then in position of imposing two types of rule: On the diagrams (Selection rules) : on the outgoing, ingoing degrees, total or partial weights. On the set of diagrams (Composition and Decomposition rules) : product and coproduct of diagram(s) This leads to structures of Hopf algebras for spaces freely generated by the two sorts of diagrams (labelled and unlabelled). Labelled diagrams generate the space of Matrix QuasiSymmetric Functions, we thus obtain a new Hopf algebra structure on this space This will be the third part of these talks We conclude with some remarks…

7 A single exponential The (classical, for bosons) normal ordering problem goes as follows. Weyl (two-dimensional) algebra defined as Known to have no (faithful) representation by bounded operators in a Banach space. There are many « combinatorial » (faithful) representations by operators. The most famous one is the Bargmann-Fock representation a  d/dx ; a +  x where a has degree -1 and a + has degree 1.

8 A typical element in the Weyl algebra is of the form (normal form). When  is a single monomial, a word i.e. a product of generators a +, a we have a wonderful solution to the normal ordering problem (and thus, by linearity to the general problem).

9 Example with  = a + a a + a a + a + a a + a a +

10 a + a a + a a +

11 a + a a + a a +

12 a + a a + a a + a + aa + aa + = 1 a + a + a + aa + 3 a + a + a + 1 a +

13 This property holds for any word W: associate a path with north east steps for every a + and a south east step for every a. construct the ferrers diagram B over this path the normal form of W is where R(B,k) is the k rook number of the board B.

14 Now, we return to the general problem of a homogenous operator . As can be seen from the Bargmann-Fock representation  is homogeneous of degree e iff one has

15 Due to the symmetry of the Weyl algebra, we can suppose, with no loss of generality that e0. For homogeneous operators one has generalized Stirling numbers defined by Example:  1 = a +2 a a +4 a + a +3 a a +2 (e=4)  2 = a +2 a a + + a + a a +2 (e=2) If there is only one « a » in each monomial as in  2, one can use the integration techniques of the Frascati(*) school (even for inhomogeneous) operators of the type =q(a + )a + v(a + ) ( * ) G. Dattoli, P.L. Ottaviani, A. Torre and L. Vàsquez, Evolution operator equations: integration with algebraic and finite difference methods La Rivista del Nuovo Cimento Vol 20 1 (1997).

16 The matrices of coefficients for expressions with only a single « a » turn to be matrices of substitutions with prefunction factor. This is, in fact, due to a conjugacy phenomenon. Conjugacy trick: The one-parameter groups associated with the operators of type =q(x)d/dx+v(x) are conjugate to vector fields on the line. Let u 2 =exp(  (v/q)) and u 1 =q/u 2 then u 1 u 2 =q; u 1 u’ 2 =v and the operator q(a + )a+v(a + ) reads, via the Bargmann-Fock correspondence (u 2 u 1 )d/dx+ u 1 u’ 2 =u 1 (u’ 2 + u 2 d/dx)= u 1 d/dx u 2 = 1/u 2 (u 1 u 2 d/dx ) u 2 Which is conjugate to a vector field and integrates as a substitution with prefunction factor.

17 Example: The expression  = a +2 a a + + a + a a +2 above corresponds to the operator (the line below  is in form q(x)d/dx+v(x)) Now,  is a vector field and its one-parameter group acts by a one parameter group of substitutions. We can compute the action by another conjugacy trick which amounts to straightening  to a constant field.

18 Thus set exp( )[f(x)]=f(u -1 (u(x)+)) for some u … By differentiation w.r.t. at (=0) one gets u’=1/(2x 3 ) ; u=-1/(4x 2 ) ; u -1 (y)=(-4y) -1/2

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20 In view of the conjugacy established previously we have that exp( )[f(x)] acts as which explains the prefactor. Again we can check by computation that the composition of (U  )s  amounts to simple addition of parameters !! Now suppose that exp( ) is in normal form. In view of Eq1 (slide 9) we must have

21 Hence, introducing the eigenfunctions of the derivative (a method which is equivalent to the computation with coherent states) one can recover the mixed generating series of S  (n,k) from the knowledge of the one-parameter group of transformations. Thus, one can state

22 Proposition (*): With the definitions introduced, the following conditions are equivalent (where f  U [f] is the one-parameter group exp()). Remark : Condition 1 is known as saying that S  (n,k) is of « Sheffer » type.

23 For these one-parameter groups and conjugates of vector fields G. H. E. Duchamp, K.A. Penson, A.I. Solomon, A. Horzela and P. Blasiak, One-parameter groups and combinatorial physics, Third International Workshop on Contemporary Problems in Mathematical Physics (COPROMAPH3), Porto-Novo (Benin), November arXiv : quant-ph/ For the Sheffer-type sequences and coherent states P Blasiak, A Horzela, K A Penson, G H E Duchamp and A I Solomon, Boson Normal Ordering via Substitutions and Sheffer-type Polynomials, (To be published in Physics Letters A)

24 Example : With  = a +2 a a + + a + a a +2 (Slide 11), we had e=2 and Then, applying the preceding correspondence one gets Where are the central binomial coefficients.

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27 Autre exemple : transformation idempotente. I(n,k)=nombre d’endofonctions de [1..n] idempotentes avec k points fixes. ceci est un cas particulier de la « formule exponentielle »

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29 Substitutions and the « connected graph theorem» « exponential formula» A great, powerful and celebrated result: (For certain classes of graphs) If C(x) is the EGF of CONNECTED graphs, then exp(C(x)) is the EGF of ALL (non void) graphs. (Touchard, Uhlenbeck, Mayer,…) This implies that the matrix M(n,k)=number of graphs with n vertices and having k connected components is the matrix of a substitution (like S  (n,k) previously but without prefactor).

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31 One can prove, using a Zariski-like argument, that, if M is such a matrix (with identity diagonal) then, all its powers (positive, negative and fractional) are substitution matrices and form a one-parameter group of substitutions, thus coming from a vector field on the line which could (in theory) be computed. For example, to begin with the Stirling substitution z  e z -1. We know that there is a unique one-parameter group of substitutions s (z) such that, for integer, one has the value (s 2 (z)  partition of partitions) But we have no nice description of this group nor of the vector field generating it.

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33 Two exponentials

34 The Hadamard product of two sequences is given by the pointwise product We can at once transfer this law on EGFs by but, here, as we get

35 Nice combinatorial interpretation: if the L n are (non-negative) integers, F(y) is the EGF of set- partitions for which 1-blocks can be coloured with L1 different colours. 2-blocks can be coloured with L2 different colours ……………………………………… k-blocks can be coloured with Lk different colours. As an example, let us take L 1, L 2 > 0 and L n =0 for n>2. Then the objects of size n are the set- partitions of a n-set in singletons and pairs having respectively L 1 and L 2 colours allowed When we write

36 Without colour, for n=3, we have two types of set-partition: the type (three possibilities, on the left) and the type (one possibility, on the right). With colours, we have possibilities. This agrees with the computation. A C B A C B A C B

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39 In general, we adopt the notation for the type of a (set) partition which means that there are a 1 singletons a 2 pairs a 3 3-blocks a 4 4-blocks and so on. The number of set partitions of type  as above is well known (see Comtet for example) Thus, using what has been said in the beginning, with

40 Now, one can count in another way the expression numpart()numpart(), remarking that this is the number of pair of set partitions (P1,P2) with type(P1)=, type(P2)=. But every couple of partitions (P1,P2) has an intersection matrix... one has

41 {1,5} {2} {3,4,6} {1,2} {3,4} {5,6} {1,5}{1,2} {2}{3,4} {3,4,6}{5,6} Feynman-type diagram (Bender & al.) Packed matrix see NCSF VI (GD, Hivert, and Thibon)

42 Now the product formula for EGFs reads The main interest of this new form is that we can impose rules on the counted graphs !

43 Weight 4

44 Diagrams of (total) weight 5 Weight=number of lines

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46 Hopf algebra structures on the diagrams

47 Hopf algebra structures on the diagrams From our product formula expansion one gets the diagrams as multiplicities for monomials in the (L n ) and (V m ).

V2V2 V2V2 V2V2 L2L2 L1L1 L3L3 For example, the diagram below corresponds to the monomial (L 1 L 2 L 3 ) (V 2 ) 3 We get here a correspondence diagram  monomial in (L n ) and (V m ). Set m(d,L,V,z)=L (d) V (d) z d

49 Question Can we define a (Hopf algebra) structure on the space spanned by the diagrams which represents the operations on the monomials (multiplication and doubling of variables) ? Answer : Yes First step: Define the space Second step: Define a product Third step: Define a coproduct

50 Hopf algebra structure (H,,,1 H,,) Satisfying the following axioms  (H,,1 H ) is an associative k-algebra with unit (here k will be a – commutative - field)  (H,,) is a coassociative k-coalgebra with counit   : H -> HH is a morphism of algebras   : H -> H is an anti-automorphism which is the inverse of Id for convolution (the antipode). Convolution is defined on End(H) by =  (  )  with this law End(H) is endowed with a structure of associative algebra with unit 1 H .

51 First step: Define the spaces Diag= ddiagrams C d LDiag= dlabelled diagrams C d at this stage, we have an arrow LDiag  Diag (finite support functionals on the set of diagrams). Second step: The product on Ldiag is just the superposition of diagrams d 1  d 2 = So that m(d 1 *d 2,L,V,z)= m(d 1,L,V,z)m(d 2,L,V,z) d1d1 d2d2 Remark: Juxtaposition of diagrams amounts to do the blockdiagonal product of the corresponding matrices.

52 This product is associative with unit (the empty diagram). It is compatible with the arrow LDiag  Diag and so defines the product on Diag which, in turn is compatible with the product of monomials. LDiag x LDiag Mon x Mon LDiag Diag Diag x Diag Mon

53 Third step: For the coproduct on Ldiag, we have several possibilities : a) Split wrt to the white spots (two ways) b) Split wrt the black spots (two ways) c) Split wrt the edges Comments : (c) does not give a nice identity with the monomials (when applying d  m(d,?,?,?)) nor do (b) and (c) by intervals. (b) and (c) are essentially the same (because of the WS  BS symmetry) In fact (b) and (c) by subsets give a good representation and, moreover, they are appropriate for several physics models. Let us choose (a) by subsets, for instance…

54 The « white spots coproduct » is  ws (d)=  d I  d J the sum being taken over all the decompositions, (I,J) being a splitting of WS(d) into two subsets. For example, with the following diagrams d, d 1,d 2, one has  ws (d)=d + d + d 1 d 2 + d 2 d 1

55 This coproduct is compatible with the usual coproduct on the monomials. If  ws (d)=  d (1)  d (2) then  m(d (1),L’,1,z) m(d (2),L’’,1,z) = m(d, L’+L’’,1,z) It can be shown that, with this structure (product with unit, coproduct and the counit d   d, ), Ldiag is a Hopf algebra and that the arrow Ldiag  Diag endows Diag with a structure of Hopf algebra.

56 Concluding remarks i) We have many informations on the structures of Ldiag and Diag (multiplicative and Hopf structures). (by Cartier-Milnor-Moore theorem). ii) One can change the constants V k =1 to a condition with level (i.e. V k =1 for kN and V k = 0 for k>N). We obtain then sub-Hopf algebras of the one constructed above. These can apply to the manipulation of partition functions of physical models including Free Boson Gas, Kerr model and Superfluidity.

57 iii) The labelled diagram are in one-to-one correspondence with the packed matrices as explained above. The product defined on diagrams is the product of the functions (S P ) p packed of NCSF VI p 709 (*). Here, we have adopted a different coproduct. (*) Duchamp G., Hivert F., Thibon J.Y. : Non commutative symmetric functions VI: Free quasi-symmetric functions and related algebras, International Journal of Algebra and Computation Vol 12, No 5 (2002).

58 iv) A picture of Hopf algebras

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