Fonctions sur l'ensemble des diagrammes de Young : caractères du groupe symétrique et polynômes de Kerov

Abstract : The main object of this thesis is the (normalized) irreducible character values of the symmetric group, seen as a function of the partition indexing the representation (and not of the permutation on which we compute the character value). With a good rescaling, the characters can be written as polynomials in so-called Stanley coordinates or in terms of free cumulants (the latter are observables of the diagram, which appear naturally in the asymptotics study of character values). We give a combinatorial interpretation for the coefficients of these two expressions. More precisely, the summans are indexed by maps, whose genus is linked with their asymptotic behaviour. This kind of expression is very useful to obtain asymptotic results : for example, one has given upper bounds on character values and enlarged the domain of validity of some known equivalents. Moreover, the combinatorics involved in these questions is interesting and has been applied to identities on rational functions
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Valentin Feray. Fonctions sur l'ensemble des diagrammes de Young : caractères du groupe symétrique et polynômes de Kerov. Mathématiques générales [math.GM]. Université Paris-Est, 2009. Français. ⟨NNT : 2009PEST1013⟩. ⟨tel-00418482v2⟩

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