Combinatoire algébrique et géométrique des nombres de Hurwitz

Abstract : This thesis is meant to be a digest, adressed to the combinatorician community, of some tools developped to tackle the problem of Hurwitz, as well as an exhibition of the thus-harvested results. The problem of Hurwitz consists of computing, in a symmetric group, the (so-called Hurwitz) number of transitive factorisations of the identity permutation whose factors have prescribed cyclic types. We first describe the topological layout of this problem through the enumeration of the ramified coverings of the sphere. We also present a natural algebraic frame, the monoid of split permutations, which allows to describe Hurwitz numbers as structure coeffcients of the algebra of this monoid, more precisely of the subalgebra spanned by the conjugacy classes, whose natural basis is indexed by multipartitions (or split partitions). The representation theory of this algebra yields an algoithm to compute one-partition Hurwitz numbers whose complexity (minimal, uniform and exponential) is far better than that of a naive edging about. This algebraic frame yields a formula describing several-partition Hurwitz series as polynomials in one-partition Hurwitz series. We secondly present the geometric frame in which are been expressed on the one hand the ELSV formula, which describes one-partition Hurwitz numbers as functions of some integrals, one the other hand a theorem of M. Kazarian expressing one-partition Hurwitz series as polynomials in some formal power series whose asymptotics is completly understood. Once the using of this integration frame has been described, we derive the asymptotics of all Hurwitz numbers
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Marc Sage. Combinatoire algébrique et géométrique des nombres de Hurwitz. Autre [cs.OH]. Université Paris-Est, 2012. Français. ⟨NNT : 2012PEST1102⟩. ⟨tel-00804228⟩



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