# Action du groupe symétrique sur certaines fractions rationnelles suivi de Puissances paires du Vandermonde

Abstract : The main purpose of this document is the symmetric group. In particular, we study the two following problems: First, the symmetric group acts naturally on the rational function $\psi_{12\dots n} = \frac{1}{(x_1 - x_2)(x_2-x_3)\dots(x_{n-1}-x_n)},$ by permuting the variables. With the help of some operations on the graphs, we give algorithms and combinatorial formulas allowing us to compute the reduced fraction $\Psi_P = \sum_{w \in \mathcal{L}(P)} \psi_w$ where $P$ is a partially ordered set and $\mathcal{L}(P)$ is the set of the linear extensions of $P$. The author C. Greene has introduced these rational functions in the aim to generalize some identities related to the Murnaghan-Nakayama rules. We use these properties to give an original algorithm to perform partial decompositions of fractions with the help of graphs. In the second problem, we study the expansion of the even powers of the Vandermonde in several basis of symmetric functions. In this part, we give identities between symmetric functions and hyperdeterminants and we use them to obtain an hyperdeterminental expression of the coefficients in Schur's basis. We investigate also the relation between the even powers of the Vandermonde and Jack's functions. Finally, we introduce a $q$-déformation of the even powers of the Vandermonde and we relate it to some specialisations of Macdonald's polynomials.
Keywords :
Document type :
Theses
Domain :

https://tel.archives-ouvertes.fr/tel-00502471
Submitted on : Thursday, July 15, 2010 - 9:30:41 AM
Last modification on : Friday, November 9, 2018 - 1:12:55 AM
Long-term archiving on : Friday, October 22, 2010 - 11:58:42 AM

### Identifiers

• HAL Id : tel-00502471, version 1

### Citation

Adrien Boussicault. Action du groupe symétrique sur certaines fractions rationnelles suivi de Puissances paires du Vandermonde. Mathématiques [math]. Université Paris-Est, 2009. Français. ⟨tel-00502471⟩

Record views