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Graphes Combinatoires et Théorie Quantique des Champs
Gérard Duchamp , Université de Rouen, France Collaborateurs : Karol Penson , Université de Paris VI, France Allan Solomon , Open University, Angleterre Pawel Blasiak , Instit. of Nucl. Phys., Cracovie, Pologne Andrzej Horzela , Instit. of Nucl. Phys., Cracovie, Pologne
Congrès de l’ACFAS, 11 mai 2004
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Content of talk
A formula from QFT giving the Hadamard product of two EGFs
Development in case F(0)=G(0)=1 Expression with (Feynman-Bender and al.) diagrams
Link with packed matrices Back to physics : One parameter groups of substitutions and normal ordering of boson strings (continuous case)
Substitutions and the « exponential formula » (discrete case)
Lie-Riordan group
Conclusion
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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 3
In case we can set if, for example, the L n are (non-negative) integers, F(y) is the EGF of set-partitions (see the talk of M. Rosas yesterday) which k-blocks can be coloured with L k different colours.
As an example, let us take L allowed 1 , L 2 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 4
For n=3, we have two types : the type (three possibilities without the colours, on the left) and the type (one possibility without the colours, on the right). It turns out that, with the colours, we have which agrees with the computation.
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In general, we adopt the denotation 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 with type well known (see Comtet for example) as above is Thus, using what has been said in the beginning, with 6
one has Now, one can count in another way the expression
numpart(
type(P1)=
)numpart(
)
, type(P2)=
,
remarking that this is the number of couples of set partitions (P1,P2) with . But every couple of partitions (P1,P2) has an intersection matrix ...
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{1,5} {2} {3,4,6} {1,2} 1 1 0 {3,4} 0 0 2 {5,6} 1 0 1 Packed matrix see NCSFVI (GD, Hivert, and Thibon) Feynman-Bender (& al.) diagram {1,5} {1,2} {2} {3,4} {3,4,6} {5,6} Remark: Juxtaposition of diagrams amounts to do the blockdiagonal product of the corresponding matrices which are then indexed by the product of set partitions defined by M. Rosas yesterday.
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Now the product formula for EGFs reads The main interest of this new form is that we can impose rules on the counted graphs !
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Some Model Graphs
* 2. Lines and Vertices EX: 4 lines
V 1 *V 3
Single
(V 2 ) 2
Single and double
V
4
Quadruple
(V
1
)
4
Singles
*C.M. Bender, D.C. Brody, B.K.Meister ,
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Quantum Field Theory of Partitions, J.Math. Phys. 40, 3239 (1999)
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Back to physics:
(continuous case) One parameter groups of substitutions and normal ordering of boson strings 13
Fermion Normal Ordering Problem
*
f
1
f
2
f
3
f
4
f
5
f
6
f
7 satisfying the usual
f i f j
f j
f i
ij
In this elementary case there are 12 terms
f 3
8 terms
f +
1 term
f +2 f 4 f 5
The numbers 1,2, 12,.. are combinatorial in origin (see Navon reference) 14
*Combinatorics and Fermion Algebra, AM Navon, Il Nuovo Cimento 16B,324(1973)
Boson Normal Ordering Problem
* satisfying
bb
b b
1
(
b b
)
n
k n
1
k
S n k b b
k
The integers
S(n,k)
are the Stirling Numbers Second Kind.
of the 15
*Combinatorial Aspects of Boson Algebra, J Katriel, Lett. Nuovo Cimento 10,565(1974)
From now on, we will denote u=b+ (raising operator) and d=b (lowering op.) satisfying [d,u]=1.
With w=ud, one has the Stirling matrix 16
In this way, two parameters families of new Bell and Stirling numbers could be defined by means of the normal ordering
[(
a
)
r
a
s n
]
with r s, and
(
a
)
)
ns
S
)
k
a
k
ns ks
see for example, P Blasiak, KA Penson and AI Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, Annals of Combinatorics 7, 127 (2003)
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With w=udu, one has the following matrix 18
With w=udduu, one gets Each of these matrices are row-finite and induce a sequence transformation just by multiplication on the left and they form an algebra.
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With the sequence and the (infinite) matrix , is given by and the transformation induced over the generating series is f --> g such that 20
We can observe that, if there is only one derivative in the word w the matrix is a matrix of substitution with prefactor function i.e. a transformation of the type this is due to the fact that we can represent u,d by operators over the functions on the line (Bargmann-Fock): multiplication by substitution corresponding to x and differentiation. The resulting operator being either a vector field or the conjugate of a vector field by an automorphism. Let us compute, for example the 21
On gets the first special cases and some others 22
Substitutions and the « exponential formula » (discrete case) (For certain classes of graphs) If Well known to enumerative combinatorists:
C(x)
is the EGF of CONNECTED graphs, then exp(C(x)) is the EGF of all graphs. 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.
One can prove 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 can be computed . But no nice combinatorics seems to emerge. 23
Conclusion
• We have, following Bender and al., given a « coupled » decomposition of the product formula. This can be used to give sections of EGFs (a non-trivial problem, trisection of Hermite EGF, by Ira Gessel and al. has been obtained very recently) • Continuous and Discrete exponentials arising from physics and combinatorics have been presented. Remains some problems as, for example, a nice combinatorial description of the (existing) one-parameter groups associated to a substitution (say, to begin with, the Stirling substitution, which seems to induce what I could call a « Lambert phenomenon »).
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