Title: Ordonner le Groupe Symétrique: Pourquoi Utiliser l'Alg\`ebre de Iwahori-Hecke ?

The Bruhat order on the symmetric group is defined by means of subwords of reduced decompositions of permutations as products of simple transpositions. Ehresmann gave a different description by considering any permutation as a chain of sets and comparing componentwise the chains. A third method reduces the Bruhat order to the inclusion order on sets, by associating to any permutation a set of bigrassmannian permutations. This amounts to embed the symmetric group into a lattice which is distributive. The last manner to understand the Bruhat order is to use a distinguished linear basis of the Iwahori-Hecke algebra of the symmetric group, and this involves computing polynomials due to Kazhdan \& Lusztig; we explicit these polynomials in the case of vexillary permutations.

1991 Mathematics Subject Classification: 05E10, 20C30

Keywords and Phrases: Symmetric group, Bruhat Order, Kazhdan-Lusztig Polynomials

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