%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Content of talk
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Preamble :  A simple, surprizing and dissymmetric formula giving the Hadamard product of two EGFs (Hadamard exponential product, HEP).

First part : A single exponential
One-parameter groups and the Normal Ordering Problem
Substitutions with prefunctions, Riordan group, classical Sheffer 
conditions and the « exponential formula » 
Discussion of the first part

Second part : Two exponentials 
How the  Feynman-like diagrams arise naturally 
in the expansion of a free HEP 
Hopf algebra structures
Deformations, crossing and superposing parameters.
Link with packed matrices and onther Hopf algebras
Discussion of the second part, notably combinatorial 
twisting and shifting. 

Conclusion

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Feynman-like combinatorial diagrams and the EGF Hadamard Product
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Abstract
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this talk, We consider two aspects of the product formula for formal power series applied to combinatorial field theories. 
Firstly, we remark that the case when the functions involved in the product formula have a constant term is of special interest as often these functions give rise to substitutional groups. The groups arising from the normal ordering 
problem of boson strings are naturally associated with explicit vector 
fields, or their conjugates, in the case when there is only one 
annihilation operator. These groups, which can be analytically expressed as groups of "Substitutions with prefunctions" are equivalent to classical notions like Riordan groups, classical Sheffer conditions and the « exponential formula ». 
Secondly, we discuss the Feynman-like graph representation resulting 
from the product formula. Natural deformations (counting graph parameters as crossings and superpositions) can be introduced in the product law to give a three parameter true Hopf deformation of this algebra of Feynman-like diagrams. We show that, for some values of the parameters, one recovers the algebra of Noncommutative Matrix Quasi-symmetric functions.  
In particular, we obtain a new Hopf algebra structure over the space 
of matrix quasi-symmetric functions and a natural commutative Hopf 
algebra structure on the space of diagrams themselves. 
It is shown that this Hopf algebra originates from the formal 
doubling of variables in the product formula.