Some combinatorial properties of the graded algebra of Schubert polynomials

We consider the action of the symmetric group S n on the Schubert poly-nomials of xed degree. This action induces a well known family of representations of S n. We point on Kazhdan-Lusztig structures which appear in these representations. First, a combinatorial rule for restricting these representations to Young subgroups is obtained. This rule is an exact analogue of Barbasch-Vogan's rule for restricting Kazhdan-Lusztig representations. Secondly, we compare the action of the simple reeections on the Schubert polynomials and the action of these reeections on the Kazhdan-Lusztig basis. This comparison yields a combinatorial formula for the characters of the graded algebra of the Schubert polynomials. The Major index of a permutation plays a crucial role in this work. The rst result (the restriction rule) follows from a remarkable theorem of Garsia and Gessel, which describes the distribution of the major index of shuues. We construct an explicit bijection between Young diagrams of xed size and shuues with corresponding major index. This bijection gives a purely combinatorial proof to Garsia-Gessel's theorem. The second result (the characters' formula) is equivalent to Kraskiewicz-Weyman-Stanley combinatorial rule for decomposing the Schubert polynomials graded algebra to irreducible representations. This rule involves Major indices.