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

Abstract : The first part (in french) recalls the previous results in the field and shows how our results fit in it. The following parts (in english) correspond to the papers written during my thesis.

In the second part, we prove a combinatorial formula for normalized character values $\hat{\chi}^\lambda(\sigma)$ (called Stanley's formula). This formula, conjectured by R.P. Stanley, gives irreducible character values in terms of the coordinates $\s{p}$ and $\s{q}$ of multirectangular diagrams. This formula is proved is in two different ways. The first one is based on the properties of Jucys-Murphy elements and shifted Schur functions. The second one is a computation of trace in the symmetric group algebra: the main tool is the description of the irreducible representation associated to $\lambda$ with Young's idempotent.

A very interesting aspect of this formula is its complexity, which only depends of the size of the support of the permutation $\sigma$ and not of the size of the permutation itself. Thus it is very useful in an asymptotic study of character values $\hat{\chi}^\lambda(\sigma)$ on a fixed permutation $\sigma$ (completed with fixed points) when the size of $\lambda$ goes to infinity. We can recover this way a combinatorial formula for the homogeneous equivalent of character value on a cycle and, also, an upper bound. This bound, optimal up to a multiplicative factor for fixed permutations, can be extended to permutations $\sigma$ whose length increases with $|\lambda|$. We improve this way the previous results in this direction.

In the third part, we focus on Kerov's polynomials. Once again, we propose two different approaches, both using Stanley's formula. The first one is based on map combinatorics. Indeed, the character value can be written as an alternate sum of functions on the set of Young diagrams indexed by maps. We obtain an explicit a relation between these functions. By iterating it, we write in a canonical way the function of a labeled map as an alternate sum of products of tree functions. This gives a combinatorial interpretation of the coefficients of Kerov's polynomials. With this method, explained in chapter 6, we prove a generalized version of Kerov's conjecture and compute some coefficients.

The second way to attack the problem is to introduce a new family of functionals of Young diagrams. Then we can deduce from Stanley's formula a new combinatorial formula for character values, in which all terms belong to the algebra $\Lambda^\star$. Using it, we can express the coefficients of Kerov's polynomials as an alternate sum of numbers of some factorizations. After a non trivial combinatorial work, we manage to simplify this expression to obtain an explicit combinatorial expression of the coefficients. This implies immediately Kerov's conjecture.

The subject of the fourth part is quite different from the others. It explains how the combinatorial structure which appear in our work on Kerov's polynomials can be used in an other domain: rational identities. We look at partial symmetrizations of the simple rational function $\prod_i (x_i-x_{i+1})^{-1}$. The main object is a sum of its image by some permutations of the variables. The sets of permutations we consider are linear extensions of posets, which can be represented by oriented graphs. Thus, we define a family of rational functions indexed by graphs.

But these rational functions happen to verify relations close to those which appear in the analysis of Kerov's polynomials. These relations give an algorithm to compute the rational functions and easy proofs (by induction) of some links between their algebraic properties and the combinatorics of the associated graphs. As the map structure is very important in the study of Kerov's polynomials, one may wonder whether it is interesting to endow our graphs with arbitrary map structure: this gives a non-inductive combinatorial formula of our rational function.

This shows the contribution of the combinatorial approach used in this thesis. In a small conclusion, we present some directions of research suggested by these results.
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https://tel.archives-ouvertes.fr/tel-00418482
Contributor : Valentin Feray <>
Submitted on : Friday, September 18, 2009 - 5:08:35 PM
Last modification on : Wednesday, February 3, 2021 - 7:54:25 AM
Long-term archiving on: : Tuesday, June 15, 2010 - 11:54:01 PM

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Valentin Féray. Fonctions sur l'ensemble des diagrammes de Young : caractères du groupe symétrique et polynômes de Kerov. Mathématiques [math]. Université Paris-Est, 2009. Français. ⟨tel-00418482v1⟩

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